Quantum Mechanical Modeling and Calculation of ...

Quantum Mechanical Modeling and Calculation of Multi-Photon Transition Probability of Some Selective Organic Molecules

THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (SCIENCE) IN CHEMISTRY BY MD MEHBOOB ALAM

DEPARTMENT OF CHEMISTRY UNIVERSITY OF CALCUTTA 2014

Acknowledgements Its thanks giving part and I swear any attempt to make it complete will be failed - it's endless. My roof of Education is balanced by so many hands - so many supports that I'll remain in debt for the whole of my life. At first, I would like to recall my PhD Mentor (Dr. Swapan Chakrabarti). He is interested in many topics including Science, Literature, Philosophy, Movies, Sports and Politics etc. A versatile personality - I have never seen before. His thirst for knowledge, his understanding and everything has a touch of sophistication. His meticulous involvement in considering every aspect of a problem and its solution is the most precious thing which we all learnt from him. He provided me every possible exposure for increasing my ability (Project with Dr. Emmanuel Fromager in France; handling the Cluster machine in CRNN; solving research problems by own; Writing the article; Writing letter to the editor of a journal; Writing response to the reviewer's comments and many more to mention). Apart from the scientific field - his relation with us is just like a friend. I clearly remember his help and support when blood-sugar level of my Father was very high and I was feeling very helpless and frightened. He arranged an appointment with a renowned Doctor for this. I also remember his help in my Passport making and in buying my first air-ticket. I have discussed my personal life too, with him and he was always there for help. Because of the research-frustration I sometimes behaved rudely with him but he understood it and always forgave me for my non-sense activities. Thanks for everything, Sir. This thesis will never be completed without your lenience. I would like to thank my lab-mates - Prasenjit Da, Sabyasachi Da, Mausumi Di, Sayantanu and Lopa. My PhD research work was almost parallel with that of Mausumi Di. We, together, have accomplished many project works and have solved many computational problems. Apart from these five, I would also like to remember our Seniors Atanu Da, Anita Di, Shankar Da, Bijan Da, Sharmistha Di, Sasanka Da, Ipsita Di and many more, my friends particularly Krishnaka, Biswajit, Sudhanshu, Soumya, Shreejita, my adolescence friends Manish, Ramesh, Kalyan, Nishant, Yashpal, Yusuf, Azad, Amit, Chandan, Rakesh, Sanjay, Late Rudal, my teachers (Particularly Late Mr. Ratnakar Pathak, Agarwal sir, R.P.Singh and Kaphil Sir). I would like to give my gratitude to the University of Calcutta, Department of Chemistry, all the professors of this Department (particularly mentioning my postgraduate mentor Prof. Dr. Kamal Bhattacharya), all the research scholars and all of my collaborators (Dr.

Acknowledgement

Emmanuel Fromager from UDS-France, Prof. Dr. Kenneth Ruud from CRNN-Norway and Prof. Dr. Antonio Rizzo from IPCF-CRN Italy) Next are my parents (Father: Md. Salauddin Ansari and Mother: Najbun Nisha) and my brothers and sister (Md. Saheb Alam). They made sacrifices of many things for nothing but to keep me up - to keep my studies going on. I cannot forget their endless endeavor right from the start of my school days and is still continue. My Father never completed his schoolings but he tells me (even now) "Education differentiates a human being from an animal. I was unfortunate, so I could not get proper education but I'll provide everything for making my children educated" and he did. I'm writing this is THE PROOF that he has succeeded. I have learnt patience from my mother. She never complained for her 24×7 hard-work for us (now I realize how hard it is to cook good food and to wash clothes). I learnt "what responsibility is?" from Saheb bhai. I must also recall the contributions from my elder sister (Sabana Khatoon) and younger brother (Sonu). I'm not a short-tempered man but many times I shouted on them for no reason and they never complained. Thanks to all of you for understanding me. I must not forget the baccha-party (kids-army) of my house - Shoaib, Shadab, Muskan, Sana, Aashiya, Reyan, Sara and Atif. They have always made my Sunday a very special day. There is a special thanks to my wife (Susmita Haldar). It was the time spent with her which realizes me that there is a world beyond the scientific research and I'm really grateful to her for this and everything else. At the end, I must thank the Council of Scientific and Industrial Research (CSIR), India for providing me the fellowship and of course for allowing me to go abroad (France) for research purpose.

Acknowledgement

Contents Preface...................................................................................................................................I-II Chapter 1 - Introduction 1.1 - Light-Matter Interaction............................................................................................2/1-6/1 1.1.1 - Linear and Non-Linear Optical Processes................................................2/1-4/1 1.1.2 - TPA and 3PA processes - Invention and applications.............................4/1-5/1 1.1.3 - Selection rules for One- and two-photon absorption processes....................6/1 1.2 - Quantum Chemical Methods..................................................................................6/1-21/1 1.2.1 - Time-Dependent Perturbation Theory...................................................9/1-16/1 1.2.2 - Response Theory................................................................... ...............16/1-20/1 1.2.3 - Few-State Model Approach.................... .............................................20/1-21/1 1.3 - Density Functional Theory (DFT).................................. .....................................22/1-25/1 1.4 - Time-Dependent Density Functional Theory (TDDFT)............... .......................25/1-28/1 1.5 - Outline of the Thesis.............................................................................. ..............28/1-29/1 1.6 - References..................................................................................... ......................29/1-31/1

Chapter 2 - A Critical Theoretical Study on the Two-Photon Absorption Properties of Some Selective Triaryl Borane-1-Naphthylphenyl Amine Based Charge-Transfer Molecules 2.1 - Abstract................................... ...........................................................................2/2 2.2 - Introduction...................................... ............................................................3/2-5/2 1.3 - Computational Details.............. ...................................................................5/2-7/2 2.4 - Results and discussion..................... ...........................................................8/2-20/2 2.5 - Conclusion.................................... ............................................................20/2-21/2 2.6 - References................................................................................................21/2-23/2 Chapter 3 - Channel Interference - The concept and its applications General Introduction.............................................................................................2/3-3/3

Contents

Part-3.1 - Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions 3.1.1 -Abstract................................................................................................5/3 3.1.2 - Computational Details........................................................................6/3 3.1.3 - Results and discussion................................................................7/3-25/3 3.1.3.1 - Geometry of the systems studied...................................7/3-9/3 3.1.3.2 - One-Photon Absorption Process...................................9/3-15/3 3.1.3.3 - Two-Photon Absorption Process...............................15/3-18/3 3.1.3.4 - Key steps of full GFSM derivation............................18/3-25/3 3.1.4 - Conclusion................................................................................25/3-26/3 Part-3.2 - High-Polarity Solvents Decreasing the Two-Photon Transition Probability of Through-Space Charge-Transfer Systems - A Surprising In Silico Observation 3.2.1 -Abstract...............................................................................................28/3 3.2.2 - Results and discussion...............................................................28/3-37/3 3.2.3 - Conclusion........................................................................................37/3 Part-3.3 - Enhancement of Twist Angle Dependent Two-Photon Activity through the Proper Alignment of Ground to Excited State and Excited State Dipole Moment Vectors 3.3.1 -Abstract...............................................................................................39/3 3.3.2 - Computational Details...............................................................39/3-41/3 3.3.3 - Results and discussion..............................................................42/3-52/3 3.3.4 - Conclusion................................................................................52/3-53/3 References……………………………………………………………………54/3-56/3 Chapter 4 - On the origin of very strong two-photon activity of squaraine dyes - A standard/damped response theory study 4.1 - Abstract...................................................................................................2/4 4.2 - Introduction......................................................................................2/4-5/4 4.3 - Computational Details......................................................................5/4-6/4

Contents

4.4 - Results and discussion...................................................................6/4-13/4 4.5 - Conclusion...................................................................................13/4-14/4 4.6 - References....................................................................................14/4-17/4 Chapter 5 - Role of Donor-Acceptor Orientation on Solvent-Dependent Three-Photon Activity in Through-Space Charge-Transfer Systems - Case Study of [2,2]Paracyclophane Derivatives 5.1 - Abstract...................................................................................................2/5 5.2 - Introduction......................................................................................2/5-4/5 5.3 - Computational Details......................................................................4/5-5/5 5.4 - Results and discussion...................................................................5/5-14/5 5.5 - Conclusion....................................................................................14/5-15/5 5.6 - References...................................................................................15/5-17/5 Chapter 6 - Summary and Conclusion Summary and Conclusion........................................................................2/6-3/6 List of Publications-I (Related to the Thesis)...........................................4/6-5/6 List of Publications-II (Not Related to the Thesis)..................................5/6-6/6 Appendix Appendix D....................................................................................I/AD-VII/AD Appendix F...................................................................................... I/AF-IV/AF Appendix T.......................................................................................I/AT-VI/AT

Contents

I

Preface The interaction of light with material systems gives useful information about the quantum mechanical part of matters and spectroscopy is one of the ways to study such interactions. The field of multi-photon absorption (MPA) processes is very important in the sense that they provide insight which is different from that obtained from the conventional one-photon spectroscopy. Apart from this, MPA processes can be utilized in different advanced fields ranging from the computer science to the modern medical fields and because of this even now (after ~80 years of its first theoretical prediction) researchers try to develop more and more MP active substances. In order to design new and efficient MP active materials it is very important to understand the underlying physics, for which, theoretical calculations and mathematical modeling plays vital role. In the present thesis, we have theoretical studied the multi-photon transition probabilities of some selective organic molecules using the well known response theory and the few-state model approaches. At first, we have shown that the two-state model approach, which has previously been established to work fine for the noncentrosymmetric molecules, failed to explain (even qualitatively) the trend of two-photon transition probability in boron-nitrogen containing donor-π-acceptor type of molecules. We have also found that higher excited states should be included in the calculation for proper explanation. In the next project we have derived a generalized few state model (GFSM) formula to explain the TP transition probability in both the centro- as well as noncentrosymmetric molecules and identified that a phenomenon called channel interference arises because of using the GFSM formula. We have applied GFSM to study the channel interference in different types of molecules and proposed a design strategy to control it in betaine type of molecules. Apart from channel interference studies we have also studies the case of double resonance in some squaraine based molecules using the recently developed

Preface

II

damped response theory approach and found that the chosen systems can exhibit very large two-photon activity because of having very small detuning energy which can further be tuned by using suitable substitutions. As the last thesis-related works, we have studied the effect of orientation of donor and acceptor groups on three-photon transition probability of some through space charge transfer systems.

Kolkata, India April, 2014

Md. Mehboob Alam

Preface

Chapter-1

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Chapter -1 Introduction

Introduction

Chapter-1

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1.1 Light-Matter Interaction "Physics would be dull and life most unfulfilling if all physical phenomena around us were linear. Fortunately, we are living in a nonlinear world." (By Y. R. Shen in - The Principles of Nonlinear Optics).1 Yes, we live in a non-linear world but the nonlinearity we are talking about is not geometrical but has the optical origin. That means, here we are going to discuss the nonlinear optical (NLO) processes which constitute a significant part of the vast arena of interaction between light and material medium. Such interactions can be studied either by two extreme methods (i.e. classical or quantum mechanical) or by the semi-classical treatment. In the two extreme cases both the light and matter are treated entirely either by the Classical mechanics or by the quantum mechanics. On the contrary, in the semi-classical approach, light is treated classically and matter quantum mechanically. It is obvious that the classical treatment is of no use in this modern era of quantum mechanics so the quantum mechanical treatment would be the best method, however, due to high complexity of this approach, it is not always affordable. Therefore, the semi-classical approach will provide a nice balance between the accuracy and the intricacy of the phenomenon of our interests. That's why, the whole description in all the works presented in this thesis is based on the semi-classical treatment, where, light is considered as a classical electromagnetic perturbation consisting of two perpendicularly oscillating electric and magnetic fields and a material medium is considered as a collection of positively charged nucleus and negatively charged rapidlymoving electron cloud described by the laws of quantum mechanics.

1.1.1 LINEAR- AND NON-LINEAR OPTICAL PROCESSES When light interacts with matter, the electric field of the former distorts the electron cloud of the latter by a forced charged separation, manifested by the induced dipole moment in the material medium. This induced dipole moment per unit volume is called polarization (P(t)) Introduction

Chapter-1

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and can be measured in terms of polarizability of the material medium. Polarization depends on the nature of the material medium as well as on the strength of the electric field vector (E(t)) of the incident radiation. When the said electric field is weak, polarization obeys a linear relationship with it and can be expressed by the equation (t) =

( )

(t)

(1.1)

Here, χ(1) is the linear susceptibility and is proportional to the well known linear polarizability of the material medium. The imaginary part of χ(1) is related to the one-photon absorption (OPA) process whereas its real part is associated with the refractive index of the medium. However, when the electric field strength is sufficiently large (as in case of a LASER source) the above linear relationship does not hold and then P(t) depends upon the higher powers of E. Under this condition equation (1.1) is modified to ( )=

Or,

( )

( )

( )+

( )=

( )

( )+

( )+ ( )

( )

( )+

( )+⋯ ( )

(2.1)

( )+⋯

Where, P(1)(t), P(2)(t) and P(3)(t) are the first, second and third order polarizations whereas

χ(2) and χ(3) are the second and third order nonlinear optical susceptibilities respectively. P(1)(t) is also called linear polarization and P(n)(t) [with n>1] are called non-linear polarizations. In the time averaging procedure the even terms vanish and only the odd terms (i.e. P(1), P(3) and P(5) or χ(1), χ(3), χ(5)... etc.) survive in equation (2.1). Considering the vector nature of P(t) and E(t) it can easily be shown that the nth order susceptibility is an (n+1)st rank tensor.2,3 Thus, χ(1), χ(2) and χ(3) are the second, third and fourth rank tensors having 9, 27 and 81 elements respectively. Although the number of components in these tensors increases enormously with the increase of their rank yet in real situation these numbers decrease drastically due to the Kleinman symmetry relation.4 For example, due to Kleinman Introduction

Chapter-1

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relation the number of components in χ(2) decreases from 27 to 10 and that in χ(3) decreases from 81 to 15. The optical processes related to the above mentioned nonlinear optical susceptibilities are in general called NLO processes which can further be classified into two categories viz. parametric and non-parametric processes. Parametric processes are those in which the system does not change its state from one real quantum mechanical state to another rather it resides in an imaginary (non-real) level for the time-period allowed by the Uncertainty principle. Second-Harmonic Generation (SHG), Sum and Difference-Frequency Generation (SFG and DFG), Optical Rectification (OR), Electro-Optical Pockel-Effect (EOPE), Third-Harmonic Generation (THG), Electro-Optical Kerr-Effect (EOKE), Intensity-Dependent Refractive Index (IDRI), Electric-Field-Induced Second-Harmonic generation (EFISHG) etc are the well known parametric processes. On the other hand non-parametric processes are those in which the system changes its state from one quantum mechanical state to another either by absorbing or releasing energy in the form of electromagnetic radiation. The well known nonparametric

processes

are

one-photon

absorption/emission

(OPA/E),

two-photon

absorption/emission (TPA/E), three-photon absorption/emission (3PA/E) processes, etc. Another important point to mention is that all parametric processes are described by the real part of susceptibility whereas the non-parametric processes are described by the imaginary part of susceptibility. For example, TPA process is associated with the imaginary part of χ(3) and 3PA is related with the imaginary part of χ(5). In general, the n-photon absorption process can be described by the imaginary part of (2n-1)st order nonlinear optical susceptibility.

1.1.2 TPA AND 3PA PROCESSES - INVENTION AND APPLICATIONS TPA was first theoretically predicted by Maria Göppert-Mayer in 1931 in her doctoral dissertation (supervised by Max Born).5 In her theory, she calculated the transition

Introduction

Chapter-1

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probability of two-quantum processes and predicted that such processes are possible in a high intensity radiation field. Under this condition, the system may absorb two-quanta of light simultaneously and thereby exciting from one electronic state to another. The intensity of the conventional light sources of her time was insufficient and hence was unable to fulfill this important condition. Owing to this technological dearth, the study of TPA processes was not feasible at that time and was achieved only after the invention of a suitable source of light i.e. LASER6 - after a long gap of ~30 years. The first experimental verification of TPA process was provided by Kaiser and Garrett7 in 1961 who observed TPA-induced frequency-up conversion fluorescence while irradiating CaF2 crystals containing Eu2+ ions with 694.3 nm radiation from a ruby-crystal laser. This experimental observation extended the boundary of TPA process from theoretical world to the real life and now-a-days multi-photon absorption processes like 3PA, 4PA and even 5PA are possible.8-10 Just one year later, in 1962 Isaac Abella11 reported the phenomenon of TPA in Cesium vapor. These experimental verifications opened a new dimension of the molecular spectroscopic world and since then several experimental as well as theoretical studies have been done to explore the various facets of multi-photon absorption process.12-15 Parthenopoulos and Rentzepis16 have shown that TPA can play significant role in three-dimensional optical data storage; Kawata et al.17 have shown its application in 3D micro-fabrication. In addition to these, multi-photon absorption processes have found immense applications in various cutting-edge technologies like upconverting lasing,18 optical power limiting,19 fluorescence imaging,20 photo-dynamic therapy,21 nano-bio-photonics applications to mention a few.22 All these applications are based mainly on two features of multi-photon absorption processes - (a) high spatial selectivity in three dimensions which in turn arises due to their non-linear dependence on intensity of the incident light and (b) excitation by longer wavelength lights.

Introduction

Chapter-1

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1.1.3 SELECTION RULES FOR ONE- AND TWO-PHOTON ABSORPTION PROCESSES OPA and TPA processes obey completely different selection rules. For example, for centrosymmetric molecules, only those OP transitions are allowed which involve states of different symmetry i.e. gerade to ungerade (g-u) and u-g transitions. All the other transitions are forbidden. However, in case of TPA g-g and u-u transitions are allowed and others are forbidden. Owing to this exclusive nature, in such molecules, the TP active states are OP inactive and vice-versa and hence these molecules are very interesting in the sense that their TP activity is not contaminated by their OP activity. Furthermore, due to this mutual exclusion, the TPA processes becomes highly important since we now can access those states which are otherwise not accessible by using the conventional OPA process. It is worth noting that in case of non-centrosymmetric molecules, a particular OP active state may be TP active as well and vice-versa. With this introduction on linear and non-linear optical processes, we now move to discuss the various quantum chemical methods available for studying such processes.

1.2 Quantum Chemical Methods A plane polarized electromagnetic wave propagating through a medium is represented as -

= Where,

and

(

.

)

=

.

(

.

)

(3.1)

are respectively the imaginary and real part of refractive index of the

medium, k is the wave vector and ω is the angular frequency of the radiation. When such a radiation is passed through a material medium, the average change of absorbed energy per unit volume is given by23,24 -

Introduction

Chapter-1

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〈

〉=〈. 〉

(4.1)

Here, j is the current density induced in the medium which can be expressed in terms of electric dipole (P), magnetic dipole (M) and electric quadrupole polarization (Q). However, for nonlinear optical phenomenon the dependence of j on M and Q are neglected and hence only the P dependence is of importance. Therefore,

=

∂ ∂t

(5.1)

Now, using equations 2.1 and 3.1 and vector nature of χ, the expression for components of first, second and third order polarizations can be written as23,24 ( )

( )

( )

= 10

( )

=3

( )

=

( )

(− ; )

+ . +⋯

∗

(− ; , − , )

∗

(− ; , − , , − , )

(6.1)

+ . +⋯

∗

+ . +⋯

(7.1)

(8.1)

Therefore, the rate of change of average absorbed energy per unit volume becomes ( )

〈

〉=〈 . 〉=〈

( )

∙ 〉+〈

( )

∙ 〉+〈

∙ 〉+⋯

(9.1)

Hence, ( )

〈

∙ 〉 = 2 Im

( )

〈

∙ 〉 = 6 Im

( )

( )

(− ; )

(− ; , − , )

Introduction

∗

(10.1)

∗

∗

(11.1)

Chapter-1

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〈

( )

∙ 〉 = 20 Im

(− ; , − , , − , )

∗

∗

∗

(12.1)

In CGS unit, the relation between the intensity of radiation and the electric field is given by2

=

(13.1)

2 ħ

Using this expression the rate of change of average absorbed energy per unit volume due to one-, two- and three-photon absorption processes become ( )

〈

∙ 〉=

( )

〈

∙ 〉=

( )

〈

∙ 〉=

4

24

160

Im

Im

( )

Im

( )

( )

(− ; )

(− ; , − , )

(− ; , − , , − , )

(14.1)

(15.1)

(16.1)

Finally the two- and three-photon absorption cross-sections (σTPA and σ3PA respectively) can be written as-

=

=

160

24

ħ

ħ

Im

Im

( )

( )

(− ; , − , )

(− ; , − , , − , )

(17.1)

(18.1)

The relationships between these cross-sections and the corresponding coefficients (β and γ respectively for TPA and 3PA) are24 -

Introduction

Chapter-1

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=

=

ħ

(19.1)

ħ

(20.1)

The quantum mechanical method available for calculating these cross-sections/coefficients is the time-dependent perturbation theory. In general any molecular property can be regarded as the response of the system to some kind of perturbation. As in case of optical processes, the perturbation is a time-dependent electromagnetic field, in the next section a brief description of the time-dependent perturbation theory is given.

1.2.1 TIME-DEPENDENT PERTURBATION THEORY In perturbation theory,25-27 it is assumed that before perturbation the system resides in an unperturbed state and the Schrödinger equation for this unperturbed state is completely solvable. It is also assumed that the stationary state eigen-functions of unperturbed system are ortho-normal to each other. If the system exists in one of its stationary state | 〉 before perturbation then we can write ( )

=

( )

( )

(21.1)

When perturbation is applied the Hamiltonian of the system becomes =

Here,

+ ( )

(22.1)

( ) is the time-dependent perturbation. In order to solve the time-dependent

Schrödinger equation for the perturbed system, its wave-function ( ) is expanded in terms of that of the unperturbed system

( )

as -

Introduction

Chapter-1

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( )| 〉

=

(23.1)

( )

=

(24.1)

ħ

( ) are arbitrary time-dependent constants. Substituting the expression for

in time-

dependent Schrödinger equation we get -

( )| 〉

ħ

=(

( )| 〉

⇒ ħ

=

( )| 〉

⇒ ħ

+ ( ))

( )

( )

( )| 〉

| 〉

+

| 〉

+ ħ

(25.1)

( ) ( )| 〉

(26.1)

( ) (27.1)

=

( )

Multiplying both sides by 〈 |

( )

( )+ ħ

( )

=

1 ħ

( ) = 〈 | ( )| 〉 and

( )

( )

| 〉

( ) ( )| 〉

+

and using the ortho-normal property of

=

( )

( )

( )〈 | ( )| 〉

=

( )〈 | ( )| 〉

+

(

−

Introduction

)

=

1 ħ

( )

( )

we get -

(28.1)

( )

(29.1)

Chapter-1

Thus, ̇ ( ) =

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ħ

∑

( )

( )

. This expression is exact but to get one coefficient

one needs all the other coefficients. One solution to this can be obtained by expanding

( )

as -

( )=

( )

( )+

( )

( )+

( )

( )+⋯

(30.1)

Then by integrating we can arrive at the recursive relation for these coefficients. t

c

( k 1) n

1 (t )    cm( k ) (t ')Vnm (t ')e iωnm t ' dt ' i m t0

After determining the coefficient

( )

(31.1)

( ) , a similar expansion for wave-function is

constructed. ( )=

( )(

)+

( )(

)+

( )(

)+

( )(

)+⋯

(32.1)

Where,

ψ ( k ) (t )   cm( k ) (t )e iωmt m

(33.1)

m

If we assume that at t=0 the system is in the unperturbed state | 〉 then,

( )

=

. For

particular transition to the final state, | 〉, the first order correction would be t

t

1 1 iω t ' iω t ' c (t )    cm(0) (t ')Vqm (t ')e qm dt '    δmpVqm (t ')e qm dt ' i m t 0 i m t0 (1) q

(34.1)

t

1 iω t ' c (t )   Vqp (t ')e qp dt ' i t 0 (1) q

Using this expression other higher order corrections and hence the correction to the wavefunction can easily be obtained. This expression is valid for any time dependent V.

Introduction

Chapter-1

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1.2.1.1 ONE-PHOTON ABSORPTION PROCESS In order to study the interaction with the radiation field, we need to consider the following perturbation ( )=− ∙

=− ̂

+ .

(35.1)

This is called dipole approximation. Here, ̂ is the dipole moment operator. infinitesimal quantity. The term

is an

is used to ensure that the perturbation is switched on

adiabatically and mathematically it eliminates the unnecessary contributions when integration limit is from −∞. With this perturbation the first order correction,

( )

( )=

1 ħ

( )

=−

( )

=−

1 ⟨ | ̂ ħ

( )=

1 ⟨ | ̂ ħ

| ⟩

| ⟩

1 ⟨ | ̂ ħ

( )

( ) becomes -

+ .

−

| ⟩

+

−

−

−

−

(36.1)

For absorption process, only the second term is of importance (this is called rotating wave approximation) and for simplicity we can also eliminate the dependent terms. Thus, we can write -

( )

Here, ⟨ | ̂

| ⟩=

1 ( )= ⟨ | ̂ ħ

cos ⟨ | ̂ | ⟩ =

cos

| ⟩

and

−

(37.1)

is the angle between the transition

dipole moment vector µ and the vector corresponding to the polarization of electric field (E)

Introduction

Chapter-1

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of the incident radiation. The probability of transition from pth state to qth state can therefore be written as -

( )

( )=

( )

(0)

=

ħ

cos

|

|

−

(38.1)

Another quantity which is frequently used to discuss the relative intensities in spectra is the oscillator strength. It is given by -

=

2

, ∈ { , , }

3

(39.1)

1.2.1.2 TWO- AND THREE-PHOTON ABSORPTION PROCESS Likewise one-photon absorption process, the transition probability for two- and three-photon absorption processes (from state p to q) depends upon the second and third order corrections to the coefficients i.e.

( )

( ) and

( )

( ). Therefore we need to derive the expression for

these correction terms. For getting these correction terms we need to use the recursive relation (equation 31.1). Thus we can write t

c

(2) q

Putting the expression for ( )

( )=

1 ħ

1 i t ' (t )    c m(1) (t ' )Vqm (t ' ) e qm dt ' i m t 0

( )

(40.1)

( ) (equation 37.1) in the above equation, we get 1 ⟨ | ̂ ħ

| ⟩

−

( )

On doing similar treatment as for one-photon case, we can write -

Introduction

(41.1)

Chapter-1 ( )

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( )

=−

1 ħ

=−

1 ħ

⟨ | ̂

1 ħ

⟨ | ̂

=− =

1 ħ

1 ⟨ | ̂ ħ

| ⟩

| ⟩

̂

−

̂ −

| ⟩

(42.1) ̂

− ⟨ | ̂

| ⟩

+

−2

+

̂ −

+

−2

−

Here, transition is taking place from the initial state | 〉 to some intermediate state | 〉 and then to the final state | 〉. Therefore,

+

=

. Removing the related terms, we

get -

( )

( )=

1 ħ

⟨ | ̂| ⟩ ̂ −

(43.1)

−2

On symmetrizing over the dummy indices and interchanging the states one can get the final expression for TP tensor elements as -

 p ˆ i m m ˆ j q p ˆ j m m ˆ i q 1  S ij    2 m   mp   mp   

   

(44.1)

For a linearly polarized single beam of monochromatic light and considering the orientational averaging procedure, the relation between the two-photon tensor elements and two-photon transition probability

is given by -

=

1 6 30

+

+

+8

+

+ (45.1)

+4

+

Introduction

+

Chapter-1

P a g e | 15/1

In a similar manner one can derive the expression for third order correction

( )

( ). From the

recursive relation (equation 31.1) we can write -

( )

( )=

( )

Putting the expression for

( )

( )=

1 ħ

( )

⟨ | ̂| ⟩

=−

1 ħ

⟨ | ̂| ⟩

( )

(46.1)

( ) in this, we get -

̂ −

1 ħ

( )

−2

̂ −

⟨ | ̂ | ⟩

−2

(47.1)

⟨ | ̂ | ⟩ ̂

−

−2

( )

=−

=

( )

⟨ | ̂| ⟩

1 ħ

=−

( )

1 ħ

⟨ | ̂| ⟩

1 ħ

⟨ | ̂ | ⟩ ̂

− ⟨ | ̂| ⟩

1 ħ

̂

−

+

Here again,

( )

( )=

After removing the

+

+

−3

+

(48.1)

⟨ | ̂ | ⟩ −2

+

1 ħ

−2

=

+

+

−3

−

, therefore,

⟨ | ̂| ⟩ −

̂

⟨ | ̂ | ⟩ −2

−3 −

(49.1)

related terms, symmetrizing over the dummy indices and re-ordering

the states, we finally get the expression for the three-photon tensor elements as -

Introduction

Chapter-1

P a g e | 16/1

Tijk 

Here,

, ,

1 2

p ˆ i u u ˆ j m m ˆ k q

 p i , j ,k 



u ,m

mp

   up  2 

(50.1)

represents the permutation with respect to the indices i, j, k. For a linearly

polarized single beam of monochromatic light and considering the orientational averaging procedure, the relation between the three-photon tensor elements and three-photon transition probability

is given by -

 3P 

1 2TiijTkkj  3Tijk2  35 i , j , k



 (51.1)

In all these derivations we have assumed that radiation consists of single frequency photons. By considering the electric field expression as,

=∑

one can easily generalize

the above expressions to radiation containing multiple frequencies photons. The approach of time-dependent perturbation theory is quite simple, but it leads to the expressions which require summation over all the states i.e. sum-over-states expression - which in principle is very difficult, if not impossible, to implement. Another approach which removes this difficulty is the Response theory approach. It is also a time-dependent perturbation theory approach but it replaces the sum-over-states expression by some algebraic equations and hence is possible to use in real practice. In the next section, a brief discussion of this Response theory approach is given.

1.2.2 RESPONSE THEORY Likewise the normal perturbation theory, response theory28-30 also starts with the separation of the Hamiltonian (H) into unperturbed time-independent Hamiltonian (H0) and the perturbation or interaction ( ) terms.

Introduction

Chapter-1

P a g e | 17/1

=

+ ( )

(52.1)

Where, the unperturbed states of the systems are well known as |0 〉 =

|0〉;

| 〉=

| 〉

(53.1)

The perturbation or interaction operator, ( ) can be expressed in terms of the frequencies as -

( )=

( )

(

)

(54.1)

As before, , is a positive infinitesimal quantity which ensure (−∞) = 0. The time development of the state |0〉 is defined by the unitary transformation as |0 〉 =

ĸ( ) |

0〉

(55.1)

The operator ĸ( ) is an anti-Hermitian operator and is defined by -

ĸ( ) =

(

( )| 〉〈0| −

∗

( )|0〉〈 |) =

( )

(56.1)

In the next step, the parameterized wave-function, |0〉 is then expanded as |0 〉 = 0 ( ) 〉 + 0 ( ) 〉 + 0 ( ) 〉 + 0 ( ) 〉 + ⋯

(57.1)

Then, the time-development of the wave-function is calculated using the Ehrenfest theorem. Since the operator O does not have any explicit time dependence, we can write -

0

0̇ + 0̇

0 = − ⟨0|[ ,

Introduction

+ ( )]|0⟩

(58.1)

Chapter-1

P a g e | 18/1

The dot over the states represent the corresponding time-derivative. If A represents an observable molecular property and the perturbation

( ) due to the external field is

sufficiently small, the time-development of the expectation value of the operator

can be

written as 〈 〉( ) = 0

= 0

+

1 2!

+

1 3!

0

0 +

〈〈 ; (

〈〈 ; (

)〉〉

), (

)〉〉

〈〈 ; (

), (

), (

), … , (

(59.1)

), (

)〉〉

+⋯ Here, 〈〈 ; (

), (

)〉〉 are the response functions which describe the

response of the property A to the external perturbation. By choosing suitable operator one can, in principle, determine the corresponding property using the above equation. As for example, if

= ̂ i.e. the dipole moment operator, the corresponding response functions

represent the polarizability and hyperpolarizabilities , , . ..etc. Thus, (−

(−

(−

;

;

;

,

,

,

) = − 〈〈 ̂ ; ̂ )〉〉

) = − 〈〈 ̂ ; ̂ , ̂ )〉〉

) = − 〈〈 ̂ ; ̂ , ̂ , ̂ )〉〉

Introduction

(60.1)

(61.1)

,

,

,

(62.1)

Chapter-1

〈〈 ; ( (

P a g e | 19/1

)〉〉 is called linear response function because it contains all the terms linear in

). Similarly, the quadratic response function 〈〈 ; (

terms 〈〈 ; (

quadratic ), (

), (

in )〉〉

perturbation ,

and

), (

cubic

)〉〉

,

response

contains the function

contains terms which are cubic in the perturbation. In

,

order to study the interaction with the electromagnetic radiation, one needs to consider perturbation of type ( ) = − ∙ . Using the Poles and residues analysis one can extract, from the response functions, the information regarding the excited states and also the transition processes in a molecular system. A brief discussion regarding the one-, two- and three-photon absorption processes are given below.

ONE-PHOTON ABSORPTION PROCESS IN RESPONSE THEORY Linear response function with dipole moment operator contains information regarding onephoton absorption process. The poles of linear response function signify the position of the excitation energy. Similarly, from residue analysis transition dipole moment between the ground, |0〉 and excited state, | 〉 can be obtained. Thus,

−

〈〈 ̂ ; ̂ )〉〉

= ⟨0| ̂ | ⟩

̂ 0

(63.1)

TWO- AND THREE-PHOTON ABSORPTION PROCESS IN RESPONSE THEORY The single residue of quadratic response function gives information about the two-photon tensor elements which in turn can be used to calculate the two-photon transition probability. Thus, the single residue of quadratic response function is given by -

Introduction

Chapter-1

P a g e | 20/1

〈〈 ̂ ; ̂ , ̂ )〉〉

−

⟨0 | ̂ | ⟩

=−

+

0 ̂

,

̂ − 0 ̂ 0 −

⟨ |( ̂ − ⟨0| ̂ |0⟩)| ⟩ ⟨ | −

(64.1)

|0 ⟩

Similarly, the double residue of the quadratic response function gives the transition dipole moment between two excited states. Thus,

(

−

−

) 〈〈 ̂ ; ̂ , ̂ )〉〉

,

(65.1) = −⟨0| ̂ | ⟩

̂ − ⟨0| ̂ |0⟩

⟨ | ̂ |0 ⟩

Similar to the two-photon absorption process, by considering the resonant condition of =

we can get the 3P tensor elements from the single residue of cubic response

function as -

=

, , ,

⟨0 | ̂ | ⟩ − ⁄3 ̂

⟨ | ̂ | ⟩ − 2 ⁄3

(66.1)

Tijk terms, in turn, can be used to evaluate the 3P transition probability.

1.2.3 FEW-STATE MODEL APPROACH So far we have seen two methods to study the interaction between light and matter - viz. sumover states approach and response theory. Both of these approaches can satisfactorily explain the said interaction, however, for larger molecules these methods are computationally very expensive. Few-state model approach is a less accurate method but it is computationally cheaper than the other two methods and hence it provides a good balance between the cost and accuracy for the quantum chemical calculations. The most widely used few-state model Introduction

Chapter-1

P a g e | 21/1

approach is perhaps the two-state model. In this case, the sum-over-states expressions are truncated to involve only two states viz. the ground state and the final excited state. By involving only two states the whole expression is drastically reduced to a smaller and easily comprehensible expression. Say in case of two-photon absorption process, by involving only the ground state |0〉, and the final state | 〉, the expression for the TP transition probability reduces to31 -

=8

Here,

∆ ⁄2

2 cos

is the transition dipole moment and

∆

(67.1)

+1

is the excitation energy for transition

between ground state, |0〉 and the final state | 〉. ∆ is the dipole moment difference between the two involved states and

∆

is the angle between

and ∆ vectors.

Similarly, for 3PA process, using 2SM the SOS expression for 3P tensor elements reduces to32 -

=

9 2

2 ∆

∆

+∆

∆

+∆

∆ (68.1)

−3

Although few-state-model is computationally cheaper, it should be used with caution. The success of any FSM depends upon the quality of the excited states involved in the calculation. By the term quality we mean "whether that particular state is an important state or not in the concerned process?" If the involved states do not contribute significantly to the process then the calculation may give erroneous results. Thus in using the FSM one must choose the states very carefully.

Introduction

Chapter-1

P a g e | 22/1

1.3 Density Functional theory (DFT) In this part, we shall discuss the basics of DFT.33-36 One of the weird facts of wave-function based quantum mechanics is that it involves the wave-function which is not an observable quantity. At the same time this wave-function (for N-particle system) depends upon 3N numbers of spatial co-ordinates and N number of spin co-ordinates, although the Hamiltonian of a system contains not more than two-body terms. Therefore, one can intuit that the wavefunction of the system contains information more than required. Apart from this, the computational cost of the wave-function based methods increases enormously with the increase in the number of atoms and hence for larger systems it is really troublesome. In DFT, the term density refers to the electron density (ρ(r)) which, for an N-electron wavefunction,

(

,

,…

) in kth state, is defined as -

( )=

… |

(

,

,…

)|

…

(69.1)

Here, { } represents collectively the spatial coordinates { } and the spin coordinates { } of the ith electron. ( ), in principle, is the probability density and characterizes the probability of finding any of the N-electrons within a small volume element

with any spin (either up

or down) while the other N-1 electrons have random spin and spatial position in the kth state of the system. ( ) is a positive quantity and is function of three spatial variables only. At the same time it is an experimental observable quantity and can be measured easily in simple experiments like X-ray diffraction. In DFT, the electron density is used as a basic variable in place of the wave-function as a result of which the dimensionality and hence the complexity of the problem is drastically reduced. The conceptual root of DFT lies in the Thomas-Fermi Model (TFM), proposed

Introduction

Chapter-1

P a g e | 23/1

independently by L. H. Thomas37 and E. Fermi38 in 1927 but because of the lack of chemical binding in it, for many years, TFM was not accepted as a valid theory for chemistry. Later on, Hohenberg and Kohn have recognized the power of this idea and proposed two theorems39 regarding this which provided a firm theoretical footing to this theory. Its miraculous accuracy and low computational cost have made DFT an option of choice for quantum chemists and its popularity can be anticipated from the number of publications - from year 2000 to 2014, a search in SciFinder search engine shows 187306 publications containing the phrase "density functional theory".

1.3.1 HOHENBERG-KOHN THEOREM First Hohenberg-Kohn Theorem - It states that "the external potential

( ) is

(within a constant) a unique functional of ( )". In other words, the ground state electron density uniquely determines the external potential of the system and hence all of its ground state properties. From this theorem, we can say that the true ground state energy is a funct ional o f the true ground state electron densit y and hence its components should also follow the same dependency. Thus, we can write – [ ]= [ ]+

[ ]+

[ ]

(70.1)

Here, E 0 is the true ground state energy, T is the kinet ic energy, E e e =electronelectron repulsion, E Ne =Nucleus-electron attraction=External potential. The first two terms on right hand side is universal in nature, as it is independent of the system under ∫

( )

study but,

the

last

term

is

system dependent.

. Therefore, we can separate the above equation as -

Introduction

[ ]=

Chapter-1

P a g e | 24/1

[ ]= [ ]+

[ ]+

( )

[ ]+

=

Where,

( )

(71.1)

[ ] is called Hohenberg-Kohn funct ional and is known for any of the

system. The funct ional form of F H K is unknown except the coulomb part which lies inside the E e e part. Second Hohenberg-Kohn Theorem - It may be stated as "The ground state energy ( [ ]) calculated using a trial electron densit y ( ≠ greater than the

[ ]. However, when =

, [ ] =

) will always be

[ ]." This, in principle,

is nothing but the Variat ional principle. Therefore, we can write

= [ ]+

The

[ ]+

( )

= [ ]≥

[ ]=

is an unknown quant it y. Therefore, to find the value of

(72.1)

[ ], one needs

to minimize the energy funct ional over all the densit ies which correspond to the true wave-funct ion. According to Levi constrained formalism, this minimizat ion can be expressed as -

=

→

[ ]+

( )

(73.1)

Here, the universal funct ional is given by [ ]=

→

+

(74.1)

Thus, we see that in principle using the two HK theorems the whole of the DFT formalism can be constructed. However, its pract ical implementat ion cannot be done just by using these two theorems. The real applicat ion of DFT as a

Introduction

Chapter-1

P a g e | 25/1

computational tool became possible after the formalism by W. Kohn and L. J. Sham proposed in 1965. 4 0 In their approach they introduced a non-interact ing reference system built from one-electron funct ions (orbitals) to calculate the major part of the kinet ic energy to good accuracy and all t he remaining parts were combined wit h the non-classical part of the electron-electron repulsion. Therefore, by this approach they have succeeded in calculat ing much of the informat ion accurately leaving a small part to be determined by an approximate funct ional called exchange-correlat ion funct ional. Therefore, using a proper exchange-correlat ion funct ional we can calculate the ground state energy and other propert ies accurately. Some of the well-known exchange-correlat ion funct ionals are – VWN 4 1 (named after the developers Vosko, Wilk and Nusair. It is a LDA t ype of funct inoal), LSDA (local spin densit y approximat ion), B3P86 (is similar to VWN but contains non-local correlat ion provided by Perdew 86), 4 2 B3LYP (Becke 3-parameters Lee, Yang, Parr), 4 3 B3PW91 (similar to B3P86 but contains non-local correlat ion from Perdew/Wang 91), 4 4 long-range corrected funct ionals (LC-wPBE, 4 5 CAM-B3LYP, 4 6 wB97XD 4 7 ) etc. There are several other funct ionals available in different quantum chemistry packages.

1.4 Time-Dependent Density Functional Theory (TD-DFT) Till now, we have discussed the version of DFT which is strict ly applicable to the ground state and excit ed state properties cannot be studied wit h this formulat ion. For the excited state studies t ime-dependent (TD) version of DFT is used which is based on the Runge-Gross theorem. 4 8 Runge-Gross theorem is also

Introduction

Chapter-1

P a g e | 26/1

called the t ime-dependent version of the Hohenberg-Kohn theorem. In this sect ion, a very brief discussion on TD-DFT is given. 4 9 -5 1 Runge-Gross Theorem: It can be stated as - The densities ρ(r,t) and ρ'(r,t) evolving from a common initial state ψ 0 = ψ (t=0) under the influence of two external potentials

( , ) and

( , ) are al ways different provided that the

potentials differ by more than a purely time-dependent function. Here the external potent ials are always expandable through the Taylor series about the 

init ial t ime t = 0

Vk (r ) k  t  t0  . Thus, m 0 k !

i.e. V  r , t   

∆

( , )= ( , )−

( , )≠ ( )

(75.1)

If this is the case, then there exists a unique one-to-one correspondence between the densities and the external potentials. Thus similar to ground-state DFT, in this case also the timedependent

potentials

can

be

written

as

a

functional

of

the

density

i.e.

V ( r , t )  V [ n, ψinitial ]( r , t ) from which the time-dependent Hamiltonian of the system and the

wave-function and also other observable properties can be expressed as a density functional. The dependence on initial state ψ0 is removed if the system starts from the ground state. In analogous to ground-state DFT, here also a non-interacting system is assumed which can produce the time-dependent density equal to that for the interacting system

( , )=

( , )

This satisfies the time-dependent Kohn-Sham equation.

Introduction

(76.1)

Chapter-1

P a g e | 27/1

( , )

Here,

[ ;

= −

∇ + 2

[ ;

]( , )

( , )

(77.1)

]( , ) is the external potential of the non-interacting system which produces

the density of the real system. This term is further decomposed into three terms - the external time-dependent field, time-dependent Hartree potential and the exchange-correlation potential.

[ ;

]( , ) =

[ ;

]( , ) +

′

( , ) + | − ′|

[ ;

,

]( , )

(78.1)

The last term is approximated by some standard functionals of density, the Kohn-Sham initial state and the true initial state. The knowledge of this term is equivalent to the exact solution of all the time-dependent Coulomb interacting problems. In actual practice, time-dependent Kohn-Sham calculation works as follows. At first, an initial set of N ortho-normal Kohn-Sham orbitals is taken which reproduce the correct density of the true initial state ψ0 and its first time-derivative. Then, equation (78.1) changes these orbitals under the influence of external potential (depending upon the situation/problem), the Hartree potential and the chosen exchange-correlation functional. The accuracy of this approach depends completely on the chosen exchange-correlation functional. In principle, if we use an exact exchange-correlation functional then we get the exact data for the properties under investigation. However, the above formalism has its own limitations. Some of those are as follows. (1) Similar to ground-state DFT, TDDFT also suffers from the question of interacting and non-interacting v-representability which was largely resolved by the van Leeuwen theorem.52 (2) van Leeuwen theorem is not applicable to non-Taylor expandable densities. (3) The above formulation of TDDFT is applicable for systems in time-dependent scalar potentials but not in vector potentials which basically exclude discussions on the

Introduction

Chapter-1

P a g e | 28/1

interaction between light and matter. Ghosh and Dhara53 were provided a significant contribution to resolve this matter and their formulation (also called time-dependent currentDFT) can be applied to such cases. Several other modifications and improvement have made the TDDFT approach one of the most reliable computational tool for studying the excited state properties particularly in the field of molecular spectroscopy.

1.5 Outline of the Thesis In Chapter 2, we have shown that the previous notion of better performance of two-state model for non-centrosymmetric molecules is no longer valid to capture the essence of TPA process in some selected boron-nitrogen containing donor-π-acceptor type of organic molecules. The response theory results can be reproduced only when we include higher excited states in the calculations but this agreement is not more than qualitative in nature indicating that higher excited states play important role in controlling the TP activity of the molecules under consideration. Chapter 3 consists of three parts. In Part 3.1, we have developed a generalized few-state model (GFSM) formula for studying the TP activity in three-dimensional molecules and applied the same to two real systems (o-betaine and p-betaine). We have studied how the phenomenon of channel interference arises because of using the GFSM. In Part 3.2, we have studied the solvent phase behavior of channel interference in two through-space chargetransfer (TSCT) systems. Finally in Part 3.3, we have proposed an experimental strategy to control the channel interference in o-betaine molecule. In Chapter 4, we have studied the phenomenon of double resonance in case of some squaraine dyes using the newly formulated damped response theory approach and found that

Introduction

Chapter-1

P a g e | 29/1

because of having very small detuning energy these systems possess very high TP activity which can further be increased by suitable substitutions. Finally, in Chapter 5, the effect of donor-acceptor orientation on the solvent dependent 3P activity of some selected TSCT systems have been studied and we found that for a particular orientation of donor and acceptor groups their 3P activity can further be enhanced to a large extent.

1.6 References 1. Y. R. Shen, The Principles of Nonlinear Optics, Wiley, New York, 1984, pp. 23–25. 2. R. W. Boyd, Nonlinear Optics, Academic Press, London 2003. 3. P. N. Prasad and D. J. Williams, Introduction to nonlinear optical effects in molecules and polymers, Wiley, New York, 1990. 4. D. A. Kleinman, Phys. Rev. 1962, 126, 1977. 5. M. Göppert-Mayer, Ann. Phys. (Leipzig), 1931, 9, 237. 6. T. H. Maiman, Nature, 1960, 187, 493. 7. W. Kaiser and C. G. B. Garrett, Phys. Rev. Lett., 1961, 7, 229. 8. S. S. Mitra, N. H. K. Judell, A. Vaidyanathan and A. H. Guenther, Opt. Lett. 1982, 7, 307. 9. I. M. Catalano, A. Cingolani and A. Minafara, Solid State Commun. 1975, 16, 417. 10. Q. Zheng, H. Zhu, S.-C. Chen, C. Tang, E. Ma and X. Chen, Nature Photonics, 2013, 7, 234. 11. I. D. Abella, Phys. Rev. Lett., 1962, 9, 453. 12. L. Frediani, Z. Rinkevicius and H. Ågren, J. Chem. Phys. 2005, 122, 244104. 13. D. H. Friese, C. Hättig and K. Ruud, Phys. Chem. Chem. Phys. 2012, 14, 1175. 14. K. Liu, Y. Wang, Y. Tu, H. Ågren and Y. Luo, J. Phys. Chem. B. 2008, 112, 4387. 15. M. M. Alam, M. Chattopadhyaya and S. Chakrabarti, Phys. Chem. Chem. Phys. 2012, 14, 1156. 16. D. A. Parthenopoulos and P. M. Rentzepis, Science, 1989, 245, 843. 17. S. Kawata, H. B. Sun, T. Tanaka and K. Takada, Nature, 2001, 412, 697. 18. L. W. Tutt and T. F. Boggess, Prog. Quantum Electron., 1993, 17, 299. 19. J. E. Ehrlich, X. L. Wu, L. Y. S. Lee, Z. Y. Hu, H. Rockel, S. R. Marder and J. W. Perry, Opt. Lett., 1997, 22, 1843. 20. B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikel, S. M. Kuebler, L. Y. S. Lee and D. McCord-Maughon, et al., Nature, 1999, 398, 51.

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21. J. Arnbjerg, A. Jiménez-Banzo, M. J. Paterson, S. Nonell, J. I. Borrell, O. Christiansen and P. R. Ogilby, J. Am. Chem. Soc., 2007, 129, 5188. 22. G. S. He, L.-S. Tan, Q. Zheng and P. N. Prasad, Chem. Rev. 2008, 108, 1245. 23. H. Mahr, in Quantum Electronics, Vol. IA:225, H. Rabin and C. L. Tang, Eds. Academic Press, New York, 1975. 24. R. L. Sutherland, Handbook of Nonlinear Optics, Marcel Dekker, Inc., New York, 1996. 25. P. W. Atkins and R. S. Friedman, Molecular Quantum Chemistry, Oxford University Press, Oxford, 1997. 26. J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Reading, 1995. 27. B. H. Bransden and C. J. Joachain. Introduction to Quantum Mechanics, Longman Scientific&Technical, 1994. 28. J. Olsen and P. Jørgensen, J. Chem. Phys. 1985, 82, 3235. 29. O. Christiansen, P. Jørgensen and C. Hättig, Int. J. Quant. Chem. 1998, 68, 1. 30. H. Hettema, H. J. Aa, Jensen, P. Jørgensen and J. Olsen, J. Chem. Phys. 1992, 97, 1174. 31. M. Albota, D. Beljonne, J.-L. Brédas, J. E. Ehrlich, J.-Y. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W. W. Webb, X.-L. Wu and C. Xu. Science, 1998, 281, 1653. 32. M. M. Alam, M. Chattopadhyaya, S. Chakrabarti and K. Ruud, Phys. Chem. Chem. Phys. 2012, 3, 961. 33. W. Koch and M. C. Holthausen, A Chemist's Guide to Density Functional Theory, WileyVCH, Weinheim, 2002 34. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989. 35. C. Fiolhais, F. Nogueira and M. Marques, eds. A Primer in Density Functional Theory, Springer-Verlag, New York, 2003. 36. R. M. Dreizler and E. K. U. Gross, Density Functional Theory: An Approach to the Quantum Many-Body Problem, Springer-Verlag, 1990. 37. L. H. Thomas, Prceedings of the Cambridge Philosophical Society, 1927, 23, 542. 38. E. Fermi, Rend. Accad. Lincei, 1927, 6, 602. 39. P. Hohenberg and W. Kohn, Phys. Rev. B. 1964, 136, 864. 40. W. Kohn and L. J. Sham, Phys. Rev. A. 1965, 140, 1133. 41. S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 1980, 58, 1200. 42. J. P. Perdew, Phys. Rev. B, 1986, 33, 8822. 43. A. D. Becke, J. Chem. Phys. 1993, 98, 1372. 44. J. P. Perdew, in Electronic Structure of Solids ‘91, Ed. P. Ziesche and H. Eschrig (Akademie Verlag, Berlin, 1991) 11.

Introduction

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45. Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, and K. Hirao, J. Chem. Phys., 2004, 120, 8425.; O. A. Vydrov and G. E. Scuseria, J. Chem. Phys., 2006, 125, 234109.; O. A. Vydrov, J. Heyd, A. Krukau, and G. E. Scuseria, J. Chem. Phys., 2006, 125, 074106.; O. A. Vydrov, G. E. Scuseria, and J. P. Perdew, J. Chem. Phys., 2007, 126, 154109. 46. T. Yanai, D. Tew, and N. Handy, Chem. Phys. Lett., 2004, 393, 51. 47. J.-D. Chai and M. Head-Gordon, Phys. Chem. Chem. Phys., 2008, 10, 6615. 48. E. Runge and E. K. U. Gross, Phys. Rev. Lett. 1984, 52, 997. 49. M. A. L. Marques, N. T. Maitra, F. M. S. Nogueira, E. K. U. Gross and A, Rubio Eds. Fundamentals of Time-Dependent Density Functional Theory, Lecture Notes in Physics, Vol. 837. Springer-Verlag, Berlin, 2012. 50. M. A. L. Marques and E.K.U. Gross, Annu. Rev. Phys. Chem. 2004, 55, 427. 51. C. A. Ullrich, Time-Dependent Density-Functional Theory: Concepts and Applications, Oxford University Press, Oxford, 2012. 52. R. van Leeuwen, Phys. Rev. Lett. 1998, 80, 1280; Int. J. Mod. Phys. B. 2001, 15, 1969. 53. S. K. Ghosh and A. K. Dhara, Phys. Rev. A. 1988, 38, 1149.

Introduction

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Chapter - 2 A critical theoretical study on the twophoton absorption properties of some selective triaryl borane-1naphthylphenyl amine based chargetransfer molecules (Phys. Chem. Chem. Phys. 2011, vol. 13, pp. 9285-9292)

A critical theoretical study on the two-photon absorption properties of some selective triaryl borane-1-naphthylphenyl amine based charge-transfer molecules

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2.1 Abstract In the present work, we have studied the two-photon absorption (TPA) properties of some selective molecules containing triarylborane and 1-naphthylphenylamine as the acceptor and donor moiety, respectively. The calculations are performed by using the state-of-the-art linear and quadratic response theory in the framework of the time dependent density functional theoretical method. The TPA parameters are calculated with CAMB3LYP functional and the cc-pVDZ basis set. The one-photon results indicate that both the electronic transitions (S0–S1 and S0–S2) are associated with the charge transfer interaction between the donor and acceptor moieties along with the reorganization of the π-electron density. All these chromophores are found to have very strong TP active modes. In order to find out the origin of large TP transition probability of these molecules, we have performed two-state model (TSM) and sum-over-states (SOS) calculations. We have found that the TSM failed to reproduce the correct trend of the TP transition probability of the molecules obtained from the response theory, while SOS is quite successful in doing so. The whole study indicates that the transition moments between the excited states play a pivotal role in controlling the TP transition probabilities of these molecules. The role of solvent in their TPA process has meticulously been scrutinized within the polarized continuum model (PCM). Furthermore, we have benchmarked our theoretical findings by calculating the TPA cross-section of a boron and nitrogen containing a charge transfer molecule for which the experimental result is available and we found that our theoretical result is in good agreement with the experimental one which definitely demonstrates the potential of all these light-emitting diode molecules as TP active materials too.


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2.2 Introduction The interaction of light with matter results in the polarization of the latter. This polarization of the material medium depends upon the strength of the applied optical field, particularly on its electric field vector. Based on the nature of this dependence, two types of optical phenomena are observed, namely linear and non-linear. The linear optical processes are characterized by the fact that the polarization is proportional to the first power of the electric field vector and for the non-linear processes the polarization depends on the higher power of the electric field vector. Like many others, TPA is a non-linear optical (NLO) process. The possibility of TPA was first theoretically predicted by Maria Gӧppert-Mayer in 1931. In her paper,1 she showed that the electronic transition(s) in a molecule or atom can take place by the simultaneous absorption of two photons. From theoretical perspective, it is now well established that TPA is related to the imaginary part of third order non-linear optical susceptibility and its strength depends upon the square of the intensity of the incident light. Therefore, a high local intensity of the incident light is required to have noticeable TP activity and as a consequence the first experimental verification2,3 of the TPA process became possible only after the invention of a laser source in 1961. The recent surge of interest for synthesizing and characterizing two photon active materials stems from their potential use in diverse fields, like 3D optical storage,4 3D micro-fabrication,5 up-converting lasing,6 optical power limiting,7 fluorescence imaging,8 photo-dynamic cancer therapy,9 etc. The TPA activity of a material is quantitatively measured in terms of its cross-section value. The TPA cross-section of a molecule depends upon many factors. Few of those10,11 are the dimensionality of charge transfer network, push-pull strength and length of conjugation of organic molecules. Sometimes, solvent also plays an important role in deciding the net crosssection value. Apart from these well known factors, Lin et al.12 highlighted the crucial role of


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vibronic coupling in TP transition probability. Chakrabarti and Ruud showed that through space long range charge transfer interaction13 and weak interaction14 may control the TPA activity of host–guest type molecules. The effect of branching of dipole and quadrupole has been examined by Terenziani et al.15 and they found that depending upon the nature of interbranch coupling, cooperative enhancement and weakening of the intensity of the TPA signal may occur. The formation of Frenkel excitons in multibranched organic chromophores and their role in the TP activity have also been reported. The nature of the biradicals in the organic open-shell molecules and their impact on the TPA intensity is well described by Jha et al.16 All these earlier theoretical and experimental findings suggest that compounds like porphyrins and metalloporphyrins, aromatic push-pull molecules, multi-branched dendrimers, quantum dots, molecular tweezer complexes have large TPA cross-section at a desirable wavelength. It is also very important to mention here that the measurement of the TPA crosssection at the experimental level is always a difficult task and the net cross-section value depends on many factors.17 These include the nature of the experimental technique used, the chosen excitation wavelength, spatial and temporal fluctuation of the laser beam, multiple reflection effect and time duration of the pulsed laser source. A slight variation of any of these factors can change the TPA cross-section value by an order of magnitude. Therefore, it is always instructive to perform sophisticated ab initio calculation to evaluate the TPA crosssection of the target molecule prior to the experimental verification. This will not only help us to develop a new design strategy but also help to understand the underlying mechanism of the TPA activity of a molecule. In the present work, we have studied the TPA properties of a series of molecules having triarylborane as the acceptor and 1-naphthylphenylamine as the donor. A number of experimental and theoretical studies18 have already been done on absorption and emission properties of these compounds which eventually establish their worth as non-linear optical


CHAPTER 2

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materials,19,20 chemical sensors,21 organic light emitting diodes (OLEDs)22,23 etc. We have chosen these systems for the TPA study because these are all D–π–A type of systems (D = Donor, A = Acceptor) and such molecules are known to have strong TP activity. At first we have calculated the TPA parameters of the chosen systems using the quadratic response theory in the framework of TDDFT with CAMB3LYP functional and the cc-pVDZ basis set and found that all three systems have large TP transition probability. Since all these molecules are non-centrosymmetric, we have performed two-state model (TSM) calculation as suggested in the earlier theoretical works.24,25 Unfortunately in the present study, TSM results fail to rationalize the response theory results correctly. To find out the origin of the strong TP activity of these molecules more rigorously, we then adopt a sum-over states (SOS) approach for re-evaluating the TP transition probability. Considering the large size of the molecules with increasing computational cost, we have restricted ourselves to involve up to four states in the SOS calculation and we found that the SOS approach can predict the correct trend of the TP transition probability for the first two excitations of all the systems. We have also studied the effect of solvents on the TPA activity of these molecules and found that solvents have very little impact on the TPA parameters. Furthermore, we have benchmarked our findings by calculating the TPA cross-section of a molecule containing boron and nitrogen for which the experimental TPA cross-section value is available in the literature26 and interestingly our theoretical TPA cross-section value corroborates the experimental finding quite well which decisively points out the reliability of our present theoretical prediction.

2.3 Computational Details The gas phase geometry optimization of all the molecules has been done with the 6-311G (d, p) basis set and B3LYP functional at the DFT level of theory and we have used the Gaussian


CHAPTER 2

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03 suite of program27 for this purpose. The optimized geometries of the three target molecules for this purpose are depicted in Figure 1/2 and are designated as BN1, BN2 and BN3, respectively. After geometry optimization, we have checked the frequency of the optimized geometry, no imaginary frequencies was obtained. In the next step, the OPA and TPA parameters are calculated from the poles and residues of linear and quadratic response functions,28 respectively. These calculations are performed within the framework of the TDDFT as implemented in the DALTON code.29 The TDDFT calculations are carried out using the Coulomb attenuating method B3LYP (CAMB3LYP),30 a functional which has already proved its worth in reproducing various experimental excited-state properties of charge transfer systems and Rydberg excitations of small molecules.31,32 In the long range corrected CAMB3LYP functional, the B3LYP functional33a,b has been modified as 1

=

1 − [ + erf (

)]

+

+ erf (

)

(1.2)

The first term in RHS accounts for the short-range interactions and is evaluated by DFT whereas the second term represents the long-range interaction and is calculated by using Hartree–Fock exchange. Here the following relations i.e. 0 ≤ + ≤ 1, 0 ≤ ≤ 1 and 0 ≤ ≤ 1 are satisfied. The parameter α incorporates the HF exchange contribution over the whole range by a factor of α and the parameter β includes the DFT counterpart over the whole range by a factor of 1 − ( + ). These parameters i.e. α, β and μ are obtained by the optimization of a set of parameters, namely ionization potential, atomic energies and atomization energies, for some benchmark molecules. It has been found30 that the CAMB3LYP functional contains just 19% exact exchange at the short range (like a conventional hybrid) and 65% at the long range. To justify the use of CAMB3LYP functional, the OP and TP parameters for the BN1 molecule are also evaluated with B3LYP functional and the TDHF method as well.


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The expression for the OPA transition probability is given by34 =

2 3

|⟨0| ̂ | ⟩|

(2.2)

Where α = {x, y, z}, 〈0| and 〈 |are the ground and final excited states, respectively. corresponding OP excitation frequency, ̂

is the

is the αth component of the electric dipole

moment operator and the sum runs over the Cartesian components. The reference quantity for the TP calculations is the TP transition moment tensor elements, Sab which can be expressed by the formula35 =

⟨0| ̂ | ⟩⟨ | ̂ | ⟩ ⟨0| ̂ | ⟩⟨ | ̂ | ⟩ + − ⁄2 − ⁄2

Where, the sum runs over all the excited states | 〉 .

and

(3.2)

are the frequencies

corresponding to the excitation energies from the ground to the intermediate state | 〉 and final TP state | 〉, respectively. ̂ and ̂ are the cartesian components of the electric dipole moment operator, ̂ . The Sab can be extracted from the single residue of the quadratic response function which is implicitly a SOS approach. For linearly polarized light, the relation between the TP transition probability and TPA tensor elements is given by36 . .

=6

+

+

+8

+

+

+4

+

+

(4.2)


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2.4 Results and Discussion

Figure 1/2. Optimized geometry of (a) BN1, (b) BN2, (c) BN3 and (d) benchmark molecules (These pictures are drawn using MOLEKEL software, version 5.4.0.8).

The optimized geometries of the target molecules, namely BN1, BN2 and BN3, belong to the C1 point group and may be considered as an asymmetric D–π–A aromatic push–pull system. In all these molecules, the B- and N-atoms are lying along the z-axis and is evident from the molecular Cartesian axes as depicted in Figure 1/2. Between B- and N-atoms, the benzene ring is acting as a spacer group. The intermediate benzene rings are not co-planar in BN2 and BN3. In BN2, the two benzene rings in between B and N are at a dihedral angle of 144.1221°. In BN3, the benzene ring attached with B makes a dihedral angle of 142.4911° with the middle one while the corresponding angle between the middle and N connected benzene rings is 143.1261°. The non-planarity among the intermediate benzene rings may facilitate intramolecular charge-transfer processes as observed in well known twisted intra-molecular charge transfer systems.


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Table 1/2. Gas phase OPA parameters for BN1, BN2 and Bn3 molecules, H ≡ HOMO and L ≡ LUMO

Molecules

Ex. State

Ex. energy/eV

Λ

δOPA

1

3.741

0.507

0.606

2

4.038

0.532

0.137

1

3.777

0.455

0.776

2

4.013

0.494

0.404

1

3.811

0.505

0.785

2

4.040

0.493

1.017

BN1

BN2

BN3

Orbitals H-L H-L+1 H-L+1 H-L H-L+1 H-L H-L+1 H-L H-L+1 H-L H-L+2 H-L H-1-L

The calculated gas phase OPA parameters for the first two excited states of all the target molecules are presented in Table 1/2. We found that excitation energies corresponding to S0– S1 and S0–S2 transitions are not changing significantly on going from BN1 to BN3. Table 1/2 also suggests that the intensity of S0–S1 OP transition is relatively higher than that of S0–S2 and this is consistent for BN1 and BN2. However in BN3, the S0–S2 OP transition is more intense than that of the S0–S1 and is evident from the oscillator strengths associated with the transitions. Table 1/2 also furnishes the orbital contributions involved in the transition processes of S0–S1 and S0–S2. In the case of BN1, the major contribution for S0–S1 comes from HOMO–LUMO (0.60) but HOMO– LUMO+1 (0.26) also has some contribution. The S0–S2 transition mainly has HOMO–LUMO+1 (+0.62) character and a little contribution come from HOMO–LUMO (-0.21) as well. Thus, for BN1, HOMO–LUMO contribution appears in opposite sign in S0–S1 and S0–S2 transitions. For the BN2 molecule, both HOMO– LUMO+1 (0.47) and HOMO– LUMO (-0.43) are contributing equally in the S0–S1 transition process, while the S0–S2 transition is a mixture of HOMO–LUMO+1 (0.47), HOMO–LUMO (0.34) and HOMO–LUMO+2 (0.27). The major orbital contribution for S0–S1 transition of the BN3 molecule comes from HOMO–LUMO+1 (-0.60) along with a relatively small HOMO–LUMO (+0.25) character. On the other hand, S0–S2 associated with BN3 may be


CHAPTER 2

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considered as a linear combination of HOMO–LUMO+2 (0.38), HOMO–LUMO (0.33) and HOMO-1–LUMO (0.28). The values in the parentheses represent the contribution of each pair of molecular orbitals in a given excitation process and all the relevant molecular orbitals are depicted in Figures 2/2 and 3/2. These figures clearly demonstrate that HOMO–LUMO has dominant charge transfer character from the N to B moiety of the molecules while all other orbital pairs have significant π-electron density reorganization character. Based on the orbital pictures and their relative contributions in the excitation process we may surmise that both S0–S1 and S0–S2 have charge transfer character with π-electron reorganization. The components of transition moments (µ0f) for both S0–S1 and S0–S2 transitions are given in Table 2/2. For all the three studied molecules the Z-component appears with the highest contribution to the net transition moment and this is valid for both the excitations. In principle, a pure charge transfer excitation should have zero transition dipole moment and so does the oscillator strength associated with the transition process. Therefore, the large Zcomponent of the transition moment is a sign of coupling of the CT state with local excitation in the Z direction. For BN1 and BN2, it is also worth noting that the magnitude of the Zcomponent of transition moment associated with S0–S1 transition is higher than that of S0–S2 transition while this trend is just opposite for the BN3 molecule, indicating stronger coupling between the CT state and local excitations in S0–S2 transition compared to that of S0–S1 transition of BN3. Table 2/2. Gas phase OPA transition moments (obtained from response theory calculations)

Molecules BN1 BN2 BN3

Ex. State

OPA Transition moments (in a.u.)

1 2 1 2 1

X 0.586 0.398 -0.743 -0.712 -0.805

Y -0.393 -0.801 -0.135 -0.381 -0.242

Z -2.570 1.104 2.893 -2.018 2.896

2

-0.670

-0.005

-3.20


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P a g e | 11/2

In order to gain further insight about the OPA process i.e. the nature of S0–S1 and S0–S2 transitions, we have calculated a parameter, popularly known as Λ,32 which is a measure of the degree of spatial overlap between the moduli of all pairs of occupied and virtual orbitals involved in a particular excitation process and can be expressed as

= Where,

=

+

and

and

∑, ∑,

(5.2)

are the solution vectors for the basic TDDFT

equation in the Born–Oppenheimer approximation, i and a are the indices of occupied and virtual orbitals respectively.

is the inner product between moduli of each pair of occupied

and virtual orbitals. The value of Λ lies between 0 and 1. The larger the value of Λ the more significant the spatial overlap between the occupied and virtual orbitals will be, and this is an indication of short-range or local excitations. Similarly, a small value of Λ characterizes a long range excitation. Earlier, Chakrabarti and Ruud13 found an appreciably small value of Λ (0.16) for the tweezer-TNF complex while Peach et al.32 reported the Λ value of a well known charge transfer molecule, DMABN (4-(N,N-dimethylamino) benzonitrile), as 0.72. In the present investigation, we have found that the values of Λ for the S0–S1 and S0–S2 transitions of all the three studied molecules are lying in between 0.45 to 0.53, indicating that the electronic transitions are essentially local in nature and corroborate the findings obtained from the MO pictures which demonstrates that both S0–S1 and S0–S2 transitions are a mixture of CT and π-electron reorganization states.


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Figure 2/2. Contributing molecular orbitals for BN1 [(a) HOMO, (b) LUMO, (c) LUMO+1] and BN2 [(d) HOMO, (e) LUMO, (f) LUMO+1] molecules

Figure 3/2. Contributing molecular orbitals for BN3 molecule [(a) HOMO-1, (b) HOMO, (c) LUMO, (d) LUMO+1, (e) LUMO+2]


CHAPTER 2

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The TP transition probability for the first two excited states in the gas and solvent phases are shown in Table 3/2, which clearly illustrates that all these molecules are highly TP active with respect to standard and reference dye molecules.17 In the present investigation, we have considered the non-polar solvents, namely THF and toluene. We found that the chosen solvents have little impact on the net TP transition probability of these molecules. Table 3/2 also suggests that the gas phase TP transition probability for S0–S2 is always higher than that of S0–S1, however this trend is changed a bit in the solvent phase, where the BN1 molecule has higher TP transition probability for S0–S1 in comparison to S0–S2 transition. As mentioned earlier the results in Table 3/2 are obtained with the CAMB3LYP/cc-pVDZ level of theory. Earlier Poulsen et al.38 showed that incorporation of extra basis had little impact on the net TP transition probability. To check the role of the basis set in our case, we have calculated the TP transition moment tensor elements using two additional basis sets namely, aug-cc-pVDZ and cc-pVTZ. The use of these additional basis sets would definitely help us to check the role of diffusion and polarization functions in the magnitude of TP transition moment tensor components. The components of TP transition moment and the net TP transition probability values of BN1 obtained from the three basis sets are shown in Table T1 in appendix T. With the cc-pVDZ basis set, the TP transition probabilities are 2.369 × 105 a.u. and 2.455 × 105 a.u., respectively, for S1 and S2 states of the BN1 molecule whereas with the aug-cc-pVDZ basis set the corresponding values are 1.98 × 105 a.u. and 2.52 × 105 a.u. respectively. The effect of the additional polarization function in the basis set is also found to have small impact on the net transition probabilities and is evident from the corresponding values for S1 (2.084 × 105 a.u.) and S2 (2.403 × 105 a.u.) states of the BN1 molecule with the CAMB3LYP/cc-pVTZ basis set level of theory. So we may summarize that polarization and diffusion functions have an insignificant effect on the net TP transition probabilities of our systems.


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Table 3/2. Quadratic response results for two-photon transition probabilities (in 105 order) of BN1, BN2 and BN3 molecules (in a.u.)

Molecule BN1

BN2 BN3

Excited state

TP transition probability Gas phase

THF

Toluene

1

2.369

3.930

3.840

2

2.455

2.718

2.931

1

3.242

5.760

5.280

2

6.508

6.960

7.470

1

1.665

3.090

2.793

2

7.940

8.940

9.210

We have also made a comparative study on the ∆ ,

and

for the first molecule (BN1)

using TDHF, B3LYP, and CAMB3LYP methods and the results are presented in Table 4/2. We found that B3LYP gives higher systematically underestimates

values than CAMB3LYP, while TDHF

in comparison to CAMB3LYP and B3LYP which is

evident from Table 4/2. The reason behind such discrepancy is many fold. First of all, the experimental39 OP excitation in BN1 appears around 370 nm in the n-hexane solvent and the corresponding theoretical values in the gas phase with CAMB3LYP, B3LYP and TDHF are 331, 405 and 273 nm, respectively. This clearly suggests that HF results are not at all reliable. It is worth commenting that our target molecule (BN1) is little different from that of the molecule chosen for the experimental investigation, where the phenyl rings attached with B are replaced by the mesityl groups. If we consider the full BN1 molecule with the mesityl group, the S0–S1 excitation in the gas phase appears at 360 nm with CAMB3LYP functional which is remarkably close to the experimental39 absorption maximum in the n-hexane solvent. For reducing the computational cost, we have just considered the target molecules without these six methyl groups in the acceptor moiety.


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Table 4/2. Gas phase TDHF, B3LYP and CAMB3LYP results for OPA excitation energy, TPA tensor elements and TP transition probability (in 105 orders) of the BN1 molecule

Method

Ex. St.

ω0f /eV

1

TP transition tensor elements (in a.u.)

δTP (a.u.)

Sxx

Syy

Szz

Sxy

Sxz

Syz

4.53

-10.60

-7.11

75.70

6.00

-8.30

-2.10

0.3118

2

4.77

8.60

1.10

-94.70

0.40

19.80

-13.00

0.5511

1

3.06

-15.40

-17.60

335.10

15.80

-32.80

-5.50

6.4470

2

3.31

35.40

32.70

255.90

-35.80

76.40

58.20

5.6520

1

3.74

20.50

18.20

-207.30

-15.70

18.90

6.50

2.3690

2

4.04

-15.20

-19.70

-170.20

22.30

53.50

-44.90

2.4550

TDHF

B3LYP

CAMB3LYP

Another important issue associated with the mismatch in

values among these methods

comes from the difference in the ground and excited state dipole moments (∆ ) and the transition moment (

) between the ground and excited states. Table 5/2 illustrates that

B3LYP gives a very large difference in the ground and excited state dipole moments, which is again a source of relatively higher

value of all the molecules with B3LYP functional.

On the other hand, TDHF underestimates ∆ which makes lower

value in comparison to

the other two methods. Earlier Day et al.40 explored the TPA spectra of a well studied 4,4'dimethylaminonitrostilbene (DANS) that includes theoretical calculation of ∆ ,

and

both in gas and solvent phases. Day et al.40 also found that CAMB3LYP performs better than B3LYP with respect to the reproducibility of the experimental ∆ and

of DANS in

cyclohexane and DMSO solvents. This in turn establishes the supremacy of CAMB3LYP over B3LYP and TDHF in predicting linear and non-linear response properties of chargetransfer systems. In order to check the reliability of our present theoretical prediction, in particular the TP activity of the target molecules, we performed linear and quadratic response calculations on a benchmark molecule26 (Figure 1(d)/2) for which both OP and TP results are available. The calculations on the benchmark molecule have been carried out with the


CHAPTER 2

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CAMB3LYP/cc-pVDZ level of theory using the THF solvent within PCM. The OPA results of the benchmark molecule are presented in Table T2 in appendix T. With the present methodology, the first OP excitation (S0–S1) appears at 376 nm and the result is fairly close to the experimental one (403 nm). We have also found that S0–S1 transition is an admixture of HOMO–LUMO, HOMO–LUMO+1 and HOMO-1–LUMO, however the dominant contribution (-0.64) comes from HOMO–LUMO. The relevant molecular orbitals are given in Figure F1 of appendix F. The Λ value (0.56) associated with S0–S1 transition of the benchmark molecule suggests that this transition is local in nature like our model systems. The details of the TP transition moment tensor components and transition probabilities are presented in Table T3 in appendix T. Table T3 clearly indicates that Szz is the largest TP tensor component and is consistent with that of our model systems. The theoretical TPA cross-section value for S0–S1 transition with 0.1 eV line-width is 3.7 × 102 GM while the corresponding experimental value is 1.8 × 102 GM which clearly indicates that the response theory can give us the correct order of the cross-section although the absolute value is different. As mentioned earlier, the experimental TPA cross-section value is very sensitive and depends upon the nature of the experimental setup and many other factors and this may be attributed to the small disagreement between the theoretical and experimental absolute values of the TPA cross-section. Apart from looking at the net TP transition probability value, we have also analyzed the components of TP transition moment tensors explicitly. Table 6/2 depicts the relevant data of all the molecules in the gas phase. It is quite evident from Table 6/2 that Szz is the highest tensor component for both the S0–S1 and S0–S2 transitions and is consistent for all the three molecules although the contribution from other tensor components is also very significant. As mentioned earlier, a similar feature has been found for the components of OP transition moment, where also the Z-component appears on the top irrespective of the nature of


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molecules and excitations. This clearly demonstrates that charge-transfer from N to B along the Z-direction plays an important role in determining the net TP transition probability of these molecules. Table 5/2. Gas phase TDHF, B3LYP and CAMB3LYP results for ∆ and

Method

Ex. St.

of BN1

μ0f (in a.u.)

Δμ (in a.u.) X

Y

Z

Total

X

Y

Z

Total

1

-0.104

0.192

-0.670

0.705

0.690

-0.708

-1.950

2.185

2

0.180

-0.141

-0.986

1.012

-0.198

0.653

-1.679

1.812

1

0.041

0.348

-4.161

4.176

-0.508

0.300

2.411

2.482

2

-1.923

1.520

0.845

2.593

0.349

-0.605

0.698

0.987

1

-0.260

0.493

-2.450

2.513

0.589

-0.393

-2.570

2.665

2

-1.222

0.980

0.565

1.665

0.398

-0.801

1.104

1.421

TDHF

B3LYP

CAMB3LYP

Table 6/2. Gas phase TP transition tensor elements of BN1, BN2 and BN3 molecules in atomic units obtained by using the CAMB3LYP/ccpVDZ level of theory

TP transition tensor elements (in a.u.) Molecule

Ex. St. Sxx

Syy

Szz

Sxy

Sxz

Syz

1

20.5

18.2

-207.3

-15.7

18.9

6.5

2

-15.2

-19.7

-170.2

22.3

53.5

-44.9

1

-43.1

-2.2

242.3

-9.9

15.1

14.5

2

21.7

0.1

312.3

8.4

65.4

-9.7

1

-45.3

-2.2

170.2

-11.4

37.2

4.3

2

9.2

0.1

355.2

5.7

-50.0

-19.3

BN1

BN2

BN3

To find out the origin of strong TPA activity of these molecules we have performed two state model calculations for the S0–S1 and S0–S2 transitions in the gas phase and re-evaluated the TP transition probabilities of these molecules. Within TSM, the TP transition probability may be expressed by the following expression41


CHAPTER 2

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=8

∆ × ∆

2 cos

∆

(6.2)

+1

Here, ∆ ⃗ is the difference in dipole moment between excited and ground states i.e. ( ⃗ ⃗ ) and one can write, ∆

= ∆⃗ .

is the module of vector ⃗

,

∆

−

is the angle

between the vectors ∆ ⃗ and ⃗ . ∆ is half of the OP excitation energy from ground (0) to excited state ( f ). One of the important parameters for TSM calculation is the difference in dipole moments between the ground and excited states (∆ ) and the excited state dipole moment can be extracted from the double residue of the quadratic response function. All the relevant TDDFT calculations are performed with CAMB3LYP functional and the cc-pVDZ basis set as implemented in the DALTON code. All the TSM parameters including the angle between ∆ ⃗ and ⃗

are presented in Table 7/2. Earlier theoretical results24,25 at the TSM level on a non-

symmetric charge transfer molecule gave fairly good results comparable with that of response results. It is also found that the angle between ∆ ⃗ and ⃗

has profound impact on the net TP

transition probability value.


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Table 7/2. Parameters and results of TSM calculation in the gas phase, TPA transition probabilities are given in 105 order (a.u.)

System

Ex. State

Components of ∆ ⃗

|∆ |

cos

TP transition probability ∆

X

Y

Z

1

-0.260

0.493

-2.450

2.513

2.665

0.889

2

-1.222

0.980

0.565

1.665

1.421

1

-1.206

-0.266

-1.758

2.148

2

-0.973

-0.122

-1.408

1

-1.701

-0.662

2

-0.447

-0.193

∆

TSM

Response

27.252

1.956

2.369

-0.274

105.902

0.093

2.455

2.990

-0.647

130.316

1.258

3.242

1.716

2.174

0.960

16.260

0.582

6.508

0.144

1.831

3.015

0.353

69.329

0.620

1.665

-2.537

2.583

3.273

0.997

4.439

3.093

7.940

BN1

BN2

BN3

In the present case, we also noticed that the highest TP transition probability comes from the S0–S2 transition of the BN3 molecule, where ∆ ⃗ and ⃗

are aligned parallel. However, the

major discrepancy appears in the trend of TP transition probability for BN1 and BN2 molecules. For these two molecules (BN1 and BN2) the response results suggest higher TP transition probability for the S0–S2 transition in comparison to S0–S1, which is in direct conflict with the results obtained from TSM calculation and surely put a question mark on the reliability of the TSM even in a situation where the target molecules are all noncentrosymmetric. Earlier Brédas et al.42 also found such a limitation of the TSM. Brédas et al.42 argued that all the dipole allowed OP modes are equally TP active and as a consequence TP process may gain significant strength from the OP process. In this situation, simple TSM may fail to provide reliable results and hence use of SOS is more acceptable. Interestingly, we found that SOS results with three and four states can correctly predict the trend of the response results for all the target molecules and the results are presented in Table 8/2. However, the SOS results are still either underestimating or overestimating the results of the response theory. Brédas et al.42 also showed that the TP transition probability calculated by A critical theoretical study on the two-photon absorption properties of some selective triaryl borane-1-naphthylphenyl amine based charge-transfer molecules

CHAPTER 2

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the SOS approach could be converged only if one involves twenty or more intermediate states which is definitely computationally bit expensive at the TDDFT level of theory. Nevertheless, the failure of TSM and the success of the SOS approach have helped us to arrive at a conclusion that neither the difference in the dipole moments between the ground and excited states nor the OP transition moment between ground and excited states is alone responsible for the observed in silico trend in the TP transition probability of the S0–S1 and S0–S2 transitions, rather the transition moment between the excited states plays a dominant role in the fine tuning of the absolute TP activity of these molecules. Table 8/2. Comparison of TP transition probability (in a.u.) as obtained from the quadratic response theory, TSM and SOS approach Molecule

Ex. State

Response

TSM

3-States

4States

BN1

1

2.369

1.956

3.816

3.860

2

2.455

0.093

4.728

4.739

1

3.242

1.258

0.352

0.363

2

6.508

0.582

13.396

13.265

1

1.655

0.620

6.552

6.431

2

7.940

3.093

18.074

18.655

BN2

BN3

2.5 Conclusion In summary, we have studied the TPA properties of a series of triarylboranes associated with the 1-naphthylphenylamine group as the electron donor. All the studied molecules are found to have strong TP active modes. The comparison of HF, B3LYP and CAMB3LYP results using the same basis set (cc-pVDZ) indicates that HF highly underestimates while B3LYP overestimates the net TP transition probability as compared to CAMB3LYP. In all these systems, we have found that the TP transition probability is larger for S0–S2 compared to S0– S1 state in the gas phase. To explain this relative trend in the TP transition probability values


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of S0–S1 and S0–S2, we have performed the TSM calculation involving the effect of dipole orientation of transition moments and found that TSM fails to reproduce the correct order of TP transition probability as obtained from the response theory calculation and this is in direct conflict with the earlier theoretical studies. The reason of this failure is attributed to the fact that a non-centrosymmetric molecule may gain TPA strength from OP transition since all the dipole allowed OP states are equally TP active. In such a situation an explicit SOS calculation is more reliable and we have done the SOS calculation considering up to four excited states. Interestingly, the SOS results can successfully explain the trend of TP transition probability that is obtained from the quadratic response theory. The success of the SOS approach clearly dictates that the transition moment between the excited states is very crucial in determining the net TP transition probability of these molecules. Finally, to give a proper judgment on our theoretical findings, we have calculated the TPA cross-section of a similar molecule for which the experimental value is already reported. We have found that our theoretical result is fairly accurate and from this result we can conclude that the three target molecules chosen in the present investigation have the potential to act as strong TPA active material in addition to their other material properties.

2.6 References 1.

M. Göppert-Mayer, Ann. Phys. (Leipzig), 1931, 9, 237.

2.

W. Kaiser and C. G. B. Garrett, Phys. Rev. Lett., 1961, 7, 229.

3.

I. D. Abella, Phys. Rev. Lett., 1962, 9, 453.

4.

D. A. Parthenopoulos and P. M. Rentzepis, Science, 1989, 245, 843.

5.

S. Kawata, H. B. Sun, T. Tanaka and K. Takada, Nature, 2001, 412, 697.

6.

L. W. Tutt and T. F. Boggess, Prog. Quantum Electron., 1993, 17, 299.

7.

J. E. Ehrlich, X. L. Wu, L. Y. S. Lee, Z. Y. Hu, H. Rockel, S. R. Marder and J. W. Perry, Opt. Lett., 1997, 22, 1843.


CHAPTER 2

8.

P a g e | 22/2

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikel, S. M. Kuebler, L. Y. S. Lee and D. McCord-Maughon, et al., Nature, 1999, 398, 51.

9.

J. Arnbjerg, A. Jiménez-Banzo, M. J. Paterson, S. Nonell, J. I. Borrell, O. Christiansen and P. R. Ogilby, J. Am. Chem. Soc., 2007, 129, 5188.

10. K. Liu, Y. Wang, Y. Tu, H. Ågren and Y. Luo, J. Phys. Chem. B, 2008, 112, 4387. 11. L. Ferrighi, L. Frediani, E. Fossgaard and K. Ruud, Chem. Phys., 2007, 127, 244103. 12. N. Lin, X. Zhao, A. Rizzo and Y. Luo, J. Chem. Phys., 2007, 126, 244509. 13. S. Chakrabarti and K. Ruud, Phys. Chem. Chem. Phys., 2009, 11, 2592. 14. S. Chakrabarti and K. Ruud, J. Phys. Chem. A, 2009, 113, 5485. 15. F. Terenziani, C. Katan, E. Badaeva, S. Tretiak and H. Ågren, Adv. Mater., 2008, 20, 4641. 16. P. C. Jha, Z. Rinkevicius and H. Ågren, J. Chem. Phys., 2009, 130, 014103. 17. P. C. Jha, Y. Wang and H. Ågren, ChemPhysChem., 2008, 9, 111. 18. Z. M. Hudson and S. Wang, Acc. Chem. Res., 2009, 42, 1584. 19. C. D. Entwistle and T. B. Marder, Angew. Chem., Int. Ed., 2002, 41, 2927. 20. Z. Yuan, J. C. Collings, N. J. Taylor, T. B. Marder, C. Jardin and J.-F. Halet, J. Solid State Chem., 2000, 154, 5. 21. Y. Kim and F. P. Gabbai, J. Am. Chem. Soc., 2009, 131, 3363. 22. T. Noda and Y. Shirota, J. Am. Chem. Soc., 1998, 120, 9714. 23. M. Elbing and G. C. Bazan, Angew. Chem., Int. Ed., 2008, 47, 834. 24. P. Cronstrand, Y. Luo and H. Ågren, J. Chem. Phys., 2002, 117, 11102. 25. J.-D. Guo, C.-K. Wang, Y. Luo and H. Ågren, Phys. Chem. Chem. Phys., 2003, 5, 3869. 26. Z.-Q. Liu, Q. Fang, D. Wang, D.-X. Cao, G. Xue, W.-T. Yu and H. Lei, Chem.-Eur. J., 2003, 9, 5074. 27. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. A critical theoretical study on the two-photon absorption properties of some selective triaryl borane-1-naphthylphenyl amine based charge-transfer molecules

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B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. G. Johnson, W. Chen, M. W. Wong, C. Gonzalez and J. A. Pople, GAUSSIAN 03 (Revision B.03), Gaussian, Inc., Wallingford, CT, 2004. 28. J. Olsen and P. Jørgensen, J. Chem. Phys., 1985, 82, 3235. 29. T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. Ågren, A. A. Auer, K. L. Bak, V. Bakken, O. Christiansen, S. Coriani, P. Dahle, E. K. Dalskov, T. Enevoldsen, B. Fernandez, C. Hattig, K. Hald, A. Halkier, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman, M. J. Packer, T. B. Pedersen, T. A. Ruden, A. Sanchez, T. Saue, S. P. A. Sauer, B. Schimmelpfennig, K. O. Sylvester-Hvid, P. R. Taylor and O. Vahtras, DALTON, a molecular electronic structure program, Release 2.0, 2005, see. http://www.kjemi.uio.no/software/dalton/dalton.html. 30. T. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett., 2004, 393, 51. 31. M. J. G. Peach, A. J. Cohen and D. J. Tozer, Phys. Chem. Chem. Phys., 2006, 8, 4543. 32. M. J. G. Peach, P. Benfield, T. Helgaker and D. J. Tozer, J. Chem. Phys., 2008, 128, 044118. 33. (a) A. D. Becke, J. Chem. Phys., 1993, 98, 1372; (b) C. Lee, W. Yang and R. G. Parr, Phys. Rev. B, 1988, 37, 785. 34. P. Macak, Y. Luo, P. Norman and H. Ågren, J. Chem. Phys., 2000,113, 7055. 35. Y. R. Shen, The Principles of Nonlinear Optics, Wiley, New York, 1984. 36. W. M. McClain, J. Chem. Phys., 1971, 55, 2789. 37. U. Varetto, MOLEKEL Version 5.4.0.8, Swiss National Supercomputing Centre, Manno, Switzerland. 38. T. D. Poulsen, P. K. Frederiksen, M. Jørgensen, K. V. Mikkelsen and P. R. Ogilby, J. Phys. Chem. A, 2001, 105, 11488. 39. D. -R. Bai, X.-Y. Liu and S. Wang, Chem.-Eur. J., 2007, 13, 5713. 40. P. N. Day, K. A. Nguyen and R. Pachter, J. Chem. Phys., 2006, 125, 094103. 41. P. Cronstrand, Y. Luo and H. Ågren, Chem. Phys. Lett., 2002, 352, 262. 42. E. Zojer, W. Wenseleers, P. Pacher, S. Barlow, M. Halik, C. Grasso, J. W. Perry, S. R. Marder and J. L. Brédas, J. Phys. Chem. B, 2004, 108, 8641.


CHAPTER 3

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Chapter - 3 Channel interference and its applications

Channel interference and its applications

CHAPTER 3

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General Introduction Both the theoretical as well as experimental researches of past few decades have confirmed that two-photon absorption (TPA) process can be tuned by several factors including the dimensionality of the charge-transfer network and length of conjugation1-5 and the strength and position of the donor-acceptor groups.6,7 It is also affected by the vibronic coupling,8-10 nature of the solvents,11-15 possibility of H-bonding14,15 and aggregation16 as well as the micelle effects.17 Other factors, particularly those which are not related to chemistry but which have significant impact on the TPA of a system are the time duration of the pulsed laser source,18 spatial and temporal fluctuation of the laser beam and also the wavelength used for the excitation process.19 In addition to all these factors, a new analysis tool to understand the physics of the TPA process at the molecular level was introduced by Cronstrand, Luo and Ågren.20,21 This new factor can easily be comprehended if one looks at the basic mechanism of TPA process. From the nature of the TPA process we know, it is a simultaneous absorption of two photons of same or different frequencies by a molecular system which then excited from one electronic state to the other even if either of the photons, individually, don’t have sufficient energy for the said transition. In this process, the system first absorbs one photon and gets excited to some virtual state, then, before relaxing back to its initial state it absorbs the second photon almost instantaneously and ultimately gets excited to the final state. A pictorial representation of the TPA process is shown in Figure 1/3. Thus, in a TPA process at least three states are involved – two real and one virtual which in turn indicates the involvement of at least three transition moment (TM) vectors – µgv, µvf and µgf (where g, v and f represent the ground, virtual and final states respectively and µij represents the TM vector for i ↔ j transition). Each of these TM vectors is also called an optical channel because each of them refers to some transition pathway/channel. Therefore


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the TPA process should depend upon the relative orientations of these TM vectors. This new mechanism was termed as channel interference. Obviously it is an inherent property of any multi-photon absorption process and can hence be used to optimize the TP activity of different classes of molecules. It is important to emphasize that the term "interference" has nothing to do with the interference between different light waves; instead it refers to the interaction between different optical channels through their transition moment vectors. Unlike light-wave interference, channel interference is a microscopic property.

Figure 1/3 Pictorial representation of Two-Photon Absorption process

For convenience, this chapter has been divided into three parts - at first the full theoretical development of generalized few-state model (GFSM) formula for the study of channel interference in 3D molecules (henceforth called 3D-GFSM) is given along with its application to a real molecular system (in Part 3.1) then in parts 3.2 and 3.3 further applications of 3D-GFSM and strategy to control the channel interference in a particular class of molecules are discussed.


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Part - 3.1 Solvent induced channel interference in the two-photon absorption processa theoretical study with a generalized few-state-model in three dimensions (Phys. Chem. Chem. Phys. 2012, vol. 14, pp. 1156-1165)

Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions

Part – 3.1

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3.1.1 Abstract In this part, we, for the first time, report the effect of interference between different optical channels on the TPA process in 3D molecules. We have employed response theory as well as a sum-over-states (SOS) approach involving few intermediate states to calculate the TPA parameters like transition probabilities (δTP) and TPA tensor elements. In order to use the limited SOS approach, we have derived a new formula for a GFSM in 3D. Owing to the presence of additional terms related to the angle between different transition moment vectors, the channel interference associated with the TPA process in 3D is significantly different and much more complicated than that in 1D and 2D cases. The entire study has been carried out on two simplest Reichardt’s dyes, namely 2- and 4-(pyridinium-1-yl)-phenolate (ortho- and para-betaine) in gas phase, THF, CH3CN and water solvents. We have meticulously inspected the effect of the additional angle related terms on the overall TPA transition probabilities of the two 3D isomeric molecules studied and found that the interfering terms involved in the δTP expression contribute both constructively and destructively as well to the overall δTP value. Moreover, the interfering term has a more conspicuous role in determining the net δTP associated with charge transfer transition in comparison to that of π-π* transition of the studied systems. Interestingly, our model calculations suggest that, for o- and pbetaine, the quenching of destructive interference associated with a particular TP process can be done with high polarity solvents while the enhancement of constructive interference will be achieved in solvents having relatively small polarity. All the one- and two-photon parameters are evaluated using a range separated CAMB3LYP functional.


Part – 3.1

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3.1.2 Computational Details The systems considered in the present work are optimized in the gas phase, THF, water and acetonitrile solvents at the CAMB3LYP/6-311++G(d,p) level of theory using DALTON22 suite of program. Due to charge separation in the ground state geometry of both o- and pbetaine, we have used a range separated CAMB3LYP23 functional. The polarizable continuum model (PCM)24,25 has been used for the solvent phase geometry optimization. On the optimized geometries, we have performed frequency calculations and no imaginary frequency was found. After geometry optimization and frequency checking, the one-photon absorption (OPA) and TPA parameters are evaluated with linear and quadratic response theory26 using Dunning’s aug-cc-pVDZ basis set and CAMB3LYP functional and the calculations are performed within the framework of time-dependent density functional theory (TDDFT) as implemented in DALTON31 program package. Here also PCM is used for the solvent phase time dependent calculations. The response calculations using PCM rely upon non-equilibrium formulation27,28 as implemented in DALTON code. For PCM based response calculations, a spherical cavity model has been used where the solvent cavities are consisted of interlocking spheres. The cavity radii used for different atoms are as follows: R(C) = 2.28 Å , R(N) = 2.04 Å and R(O) = 1.8 Å. The static and optical dielectric constants of different solvents are chosen as supported by the DALTON code. After the completion of all the time dependent calculations, we have employed the GFSM approach to re-evaluate the TPA transition probabilities of o- and p-betaine. In order to understand the channel interference effect associated with the TPA process of the studied systems, we have calculated the different angles between the transition moments and various other terms appearing in the expression of the overall δTP. The theoretical description including GFSM in 3D is incorporated in the Results and Discussion section.


Part – 3.1

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3.1.3 Results and Discussion 3.1.3.1 Geometry of the systems studied

Figure 1/3.1. Optimized structure of (a) ortho- and (b) para-betain in ground state (the planes of the two rings and the co-ordinate axes are also shown to visualize the 3D structure of the molecules).

The representative geometries of o- and p-betaine are shown in Figure 1/3.1. The optimized geometries of both the isomers manifestly deviated from the planar structure and both of them belonged to the C1 point group. The important geometric parameters of o- and p-betaine are presented in Tables 1/3.1 and 2/3.1, respectively. For o-betaine, the solvent has strong impact on the twisted angle, with the highest value of 56° in water and the corresponding highest dihedral angle (49°) between the two rings of p-betaine is obtained with the acetonitrile solvent. It is also noticeable that the relevant twisted angle in the gas phase is much smaller than it is in the solvent phase and this fact is true for both the isomers. Although the dihedral angle between the two rings of o- and p-betaine has strong solvent dependence with respect to the gas phase result, the other geometric parameters namely bond angles and various bond lengths of both the isomers do not change appreciably while moving from the gas to solvent phase, which is evident from Tables 1/3.1 and 2/3.1. Apart from using the CAMB3LYP functional, we have optimized the geometries of the isomers with the Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions

Part – 3.1

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B3LYP functional as well. However, the experimentally29 available geometric parameters of p-betaine are more close to the CAMB3LYP results and small disagreement may be attributed to the failure of PCM in capturing the specific interactions, like hydrogen bonding.30,31 It is worth mentioning that the experimental structure of o-betaine is not known to us and hence we are not in a position to make any extreme comment on the validity of the theoretically predicted geometry of o-betaine. Nonetheless, due to strong resemblance between the experimental and CAMB3LYP/H2O geometry of p-betaine, there is no reason to presuppose that the geometric parameters of o-betaine, if experimentally available, would not agree with the present theoretically predicted geometry, and therefore it is fair enough to perform all the time dependent calculations on o- and p-betaine considering CAMB3LYP optimized geometries. Table 1/3.1. CAM-B3LYP/6-311++G(d,p) optimized geometry parameters of ortho-betain in gas, THF, CH3CN and H2O solvents. Bond lengths are given in Å, bond angles and dihedral angles are given in degrees

Geometrical Parameters

Gas phase

THF

CH3CN

H2O

C1-N6-C12-C14

-35.80

-48.39

-55.54

-56.54

C1-N6-C12-C13

146.82

133.32

125.45

124.71

O22-C14-C12

123.49

123.02

122.80

122.83

N6-C12

1.428

1.443

1.447

1.447

C14-O22

1.255

1.269

1.277

1.278

C1-N6

1.355

1.348

1.347

1.347

C5-N6

1.357

1.350

1.348

1.348

C12-C14

1.444

1.434

1.429

1.429

C12-C13

1.399

1.390

1.388

1.388


Part – 3.1

P a g e | 9/3

Table 2/3.1. CAM-B3LYP/6-311++G(d,p) optimized geometry parameters of para-betain in gas, THF, CH3CN and H2O solvents. Bond lengths are given in Å, bond angles and dihedral angles are given in degrees. The values given in parentheses are the relative deviation of the experimental and the theoretical values in %. (Crystallographic data correspond to the crystallization in water medium.)

Geometrical Parameters

Gas phase

THF

CH3CN

H2O

Crystallographic data

C1-N6-C12-C14

147.68

135.26

130.37

133.06

133.00 (0.047)

C1-N6-C12-C13

-32.33

-44.75

-49.94

-46.95

-47.00 (0.119)

O22-C19-C15

122.64

122.52

122.31

122.45

121.66 (0.652)

N6-C12

1.404

1.438

1.444

1.442

1.453 (0.750)

C19-O22

1.234

1.264

1.278

1.272

1.320 (3.658)

C1-N6

1.366

1.352

1.351

1.350

1.358 (0.552)

C5-N6

1.366

1.352

1.351

1.350

1.358 (0.552)

C12-C14

1.414

1.397

1.393

1.395

1.392 (0.216)

C12-C13

1.414

1.397

1.393

1.395

1.392 (0.216)

3.1.3.2 One-Photon Absorption Process At first, the OPA parameters for the first two excited states (S0–S1 and S0–S2) of the CAMB3LYP/6-311++G(d,p) optimized geometries of o- and p-betaine in the gas phase, THF, water and acetonitrile solvents are calculated using cc-pVDZ basis set and CAMB3LYP functional. In the next step, we have calculated the said parameters using the same functional in combination with aug-cc-pVDZ as the basis set. For the first two excited states, we have found that the maximum difference in the excitation energy of the two results is ~0.201 eV and for oscillator strength, the maximum difference is 0.0191 a.u. which may be significant for evaluating the TP transition moment tensor elements in the SOS approach. To check the reliability of the basis set, we need to compare the theoretical findings with the available experimental results. In an earlier experimental32 study, it was found that the electronic excitation of o- and p-betaine, in CH3CN solvent at the one-photon level, appeared at energies 2.600 and 2.674 eV, respectively, and the corresponding values in water are 3.264 and 3.400 eV. Our calculations at the CAMB3LYP/aug-cc-pVDZ level of theory have yielded 2.584 and 2.93 eV as excitation energies of S0–S1 transitions of o-betaine in CH3CN Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions

Part – 3.1

P a g e | 10/3

and water solvents, respectively. For p-betaine, the corresponding theoretical data are 2.693 and 3.05 eV, respectively. Although the first excitation (S0–S1) energies of both the isomers in CH3CN solvent are well in agreement with the experimental data, an underestimation has been observed for the same in the case of water. To check whether or not it has happened due to the CAMB3LYP functional, we have used a LC-PBE33 functional for OPA calculation too and in this case we have noticed an overestimation of the first excitation energies for both the isomers and the results are given in Table T4 of appendix T. Furthermore, due to charge separation in the ground state geometries of o- and p-betaine, the possibility of having diradical characters in the ground state geometries of the systems can’t be ruled out. For this purpose, we have carried out NEVPT234 calculations on CAMB3LYP optimized geometries of both the isomers considering 12 active electrons in 10 orbitals and the selection of the active space was done by a prior MP2 calculation. In this case, we have used 6-31+g(d) as the basis set. From NEVPT2 orbital occupations, we have estimated the diradical character of the ground state o- and p-betaine. We have found only 0.3 and 0.1% diradical character of o- and p-betaine, respectively, and it indicates that both the systems have essentially closed shell electronic configurations in their ground state. Earlier, Paley et al.35 also argued that the diradical character in betaine systems could exist only in the excited state. Therefore, we may surmise that the difference between the experimental and theoretical excitation energies of the two isomers in water medium is arising out of the inability of the PCM model to describe the possible hydrogen bonding interaction between water and o-/p-betaine correctly.30,31 Comparing the experimentally observed OPA data with that of the present theoretical findings including closed shell nature of the systems and non-availability of other long range corrected functionals for evaluating excited–excited state transition moments in Dalton code,22 we have decided to go ahead with the rest of the investigations with the CAMB3LYP/aug-cc-pVDZ level of theory. Furthermore, Alam and co-workers36 have shown


Part – 3.1

P a g e | 11/3

that additional polarization functions in the basis set do not have significant effect on the OPA and TPA parameters. The excitation energy, oscillator strengths, transition moments, and other relevant OPA data for the first two excited states of the target molecules are presented in Table 3/3.1. The results clearly indicate that the excitation energy increases slightly on moving from gas phase to water phase. It has happened due to large charge separation in the ground state compared to the excited state of the two isomers and hence polar solvents stabilize the ground state more than the excited state. This is supported by the observation that the excitation energy in CH3CN is less than that in water medium. It is also clear from the results in Table 3/3.1 that the overall excitation in both the isomers is dominated by the first transition (S0–S1) which has the highest oscillator strength than the other transition, namely S0–S2. It should also be noted that p-betaine has larger oscillator strength (0.525) than the o-form (0.110) for S0–S1 transition. This difference can be explained by considering the origin of oscillator strength. It is well known that the oscillator strength is directly related to OPA transition probability which in turn is proportional to the square of the ground to excited state transition moment (

) and the excitation energy (

).


Part – 3.1

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Table 3/3.1. Excitation energy (in eV), oscillator strength (in a.u.), components and total value of m0i, L-value (the first value in parentheses in the last column) and contributing orbitals involved in the first two transitions in ortho- and para-betain in gas, THF, CH3CN and H2O solvents. Calculations have been done with CAM-B3LYP/6-311++G(d,p) optimized geometries and at the CAM-B3LYP/augcc-pVDZ level of theory.H = HOMO and L = LUMO

System

Ex. State

Ex. Energy (eV)

Os. Str. (a.u.)

X

Y

Z

Total

Gas

2.39

0.110

-1.343

0.082

0.738

1.535

(0.49) H-L [-0.69]

THF

2.56

0.070

-1.027

0.023

0.607

1.194

(0.36) H-L [-0.69]

CH3CN

2.58

0.044

-0.806

0.007

0.409

0.947

(0.30) H-L [-0.70]

H2O

2.93

0.042

0.785

-0.005

-0.488

0.925

(0.29) H-L [-0.70]

Gas

3.32

0.040

0.647

-0.073

-0.506

0.825

(0.33) H-L [0.66]

THF

3.64

0.003

0.192

0.012

-0.073

0.206

(0.24) H-L [-0.63]

CH3CN

3.70

0.002

-0.140

-0.016

0.034

0.145

(0.22) H-L [-0.66]

H2O

4.07

0.002

0.136

0.017

-0.031

0.141

(0.22) H-L [0.67]

Gas

2.48

0.525

0.0

0.0

-2.934

2.934

(0.66) H-L [0.70]

THF

2.56

0.362

0.0

0.0

2.396

2.396

(0.47) H-L [0.70]

CH3CN

2.69

0.283

0.0

0.0

2.073

2.073

(0.41) H-L [-0.70]

H2O

3.05

0.313

0.0

0.0

2.166

2.166

(0.43) H-L [0.70]

Gas

2.96

0.004

-0.074

0.220

-0.001

0.232

(0.37) H-L [-0.70]

THF

3.43

0.003

-0.095

0.187

0.0

0.210

(0.27) H-L [-0.70]

CH3CN

3.62

0.005

-0.236

-0.002

0.0

0.236

(0.26) H-L [0.69]

H2O

4.03

0.004

-0.086

0.196

0.0

0.214

(0.26) H-L [0.69]

Solvent

(a.u.)

(Λ values) & orbitals

o-betaine

1

2

p-betaine

1

2

The expression for OPA transition probability is given by:37 =

2 3

|⟨0| ̂ | ⟩|

Here, ̂ is the ath component of the dipole moment operator. From 3/3.1), it is clear that both

and

(1/3.1)

and

data (Table

are larger for the first excitation in p-betaine than for


Part – 3.1

the ortho isomer and hence the former has higher

P a g e | 13/3

than the latter one. The variation of

for other excitations can similarly be explained by considering the relative magnitudes of

and

, however, it is worth noting that

has larger effect on the

values than

the excitation energy, which is evident from the present study. A closer inspection of the components indicates that, for the para-isomer, the Z-component of

is most contributing

in S0–S1 transition but for S0–S2, the Y-component has the largest value. For o-betaine, the X and Z components have significant contributions in both S0–S1 and S0–S2 transitions. The analyses of orbital contributions clearly indicate that, for both the isomers, the major contribution for S0–S1 transition comes from HOMO-LUMO. However, this is not the case for S0 –S2 transition. For o-betaine in the gas phase, HOMO-1-LUMO (0.66) has major contribution but in THF, acetonitrile and water solvents, the major contribution for S0 –S2 transition comes from HOMO–LUMO+1 and the respective contributions are also presented in Table 3/3.1. The orbital pictures demonstrate that HOMO-1-LUMO transition in o-betaine is more likely to be of charge transfer type and HOMO-LUMO transition is mostly associated with π electron reorganization. For p-betaine, the major contribution in S0 –S2 transition comes from HOMO-LUMO+1 both in the gas and solvent phases. Like o-betaine, the HOMO-LUMO transition in the para isomer has more π-π* character while the HOMOLUMO+1 transition is of charge transfer type. The relevant orbital pictures of both the isomers are shown in Figure 2/3.1.


Part – 3.1

P a g e | 14/3

Figure 2/3.1. (a) HOMO-1, (b) HOMO, (c) LUMO, (d) LUMO+1 of ortho-betaine and (e) HOMO, (f) LUMO, (g) LUMO+1 of para-betaine in gas phase.

To quantify the nature of the OPA process, we have calculated the well accepted Λparameter,38 which is an estimation of the degree of spatial overlap between the moduli of all pairs of occupied and virtual orbitals associated with a specific electronic excitation. Mathematically, the Λ-parameter can be expressed as

∑, = ∑, Where,

=

+

and

and

(2/3.1)

are the solution vectors for the basic TDDFT

equation under the clamped nucleus approximation, i and a are the indices of the occupied and virtual orbitals, respectively. The inner product between the moduli of each pair of virtual and occupied orbitals is taken care by

. The value of Λ lies in between 0 and 1. The more

the value of Λ, the more significant the spatial overlap between the occupied and virtual orbitals will be, and this is a sign of short-range or local excitation. On the other hand, a small value of Λ describes long range nature of the excitation. Earlier Chakrabarti and Ruud39 predicted a significantly small value of Λ (0.16) for the tweezer–TNF complex while Peach


Part – 3.1

P a g e | 15/3

et al.38 reported the Λ value as high as 0.72 for a well known charge transfer molecule, DMABN (4-(N,N-dimethyl-amino)-benzonitrile). Recently Alam et al.36 have found a value of Λ ~0.5 for a group of boron-nitrogen containing charge transfer molecules and they have argued that the relatively large Λ value involved with the electronic transitions in this class of molecules is dominated by a mixture of charge transfer and π electron reorganization for both the S0–S1 and S0–S2 transitions studied. For o-betaine, the values of Λ for S0–S1 transition in gas phase, THF, acetonitrile and water medium are 0.49 0.36, 0.30 and 0.29, respectively, whereas the corresponding values for S0 – S2 transition are 0.33, 0.24, 0.22 and 0.22 respectively. Similarly, for p-betaine, the Λ values for S0–S1 and S0–S2 transitions are 0.66, 0.47, 0.41, 0.43 and 0.37 0.27, 0.26, 0.26, respectively, in gas phase, THF, acetonitrile and water medium. These data give us a clear revelation of the fact that S0-S2 has more long range character than S0–S1 and this is valid for both the isomers. Moreover, we have found that for both the isomers the values of Λ decrease sharply on moving from gas to solvent phase. Irrespective of the nature of the isomers, the values of Λ combined with orbital pictures are in favor of the assignment of S0–S1 transition as π-π* and S0–S2 as charge transfer type. 3.1.3.3 Two-Photon Absorption process As stated earlier, the main focus of the present study is to investigate the role of different optical channels in the overall TPA transition probability of three dimensional molecules. For this, we have calculated the TPA parameters, such as TPA tensor elements (Sab) and twophoton transition probabilities (

) of o-betaine and p-betaine in gas phase and three

different solvents with the help of quadratic response theory at the CAMB3LYP/aug-ccpVDZ level. The Sab can be extracted from the single residue of response function and is given by the formula40


Part – 3.1

P a g e | 16/3

⟨0| ̂ | ⟩⟨ | ̂ |0⟩ ⟨0| ̂ | ⟩⟨ | ̂ |0⟩ + ∆ − ⁄2

=

Here, , { , , } , and ∆

=

−

⁄2 .

and

(3/3.1)

are the excitation energies from

ground |0〉 to the intermediate state | 〉 and final state | 〉, respectively. For an excitation by a linearly polarized single beam of light, . .

The

=6

+

+

+8

is evaluated by the following expression41 +

+

+4

+

+

(4/3.1)

for CAMB3LYP/6-311++G(d,p) optimized geometries for the first two excited states

are shown in Table 4/3.1. The results clearly show that both the isomers have large TPA activity. The

for the first excited state is about hundred times larger than that of the

second excited state and this is true for both the isomers. It is also worth noting that

for

the first excited state of o-betaine and for the second excited state of p-betaine alters slowly on shifting from gas to solvent phase. However, for the second excited state of the ortho isomer, gas phase

value is about eight times larger than that in solvent phase. Almost

similar trend is found for the 1st excited state of the para-isomer. The analysis of TPA tensor elements reveals that the

component has major contribution in the TPA process for the

first excited state of the ortho isomer. The contributions (with

>

role in determining the net

and

components also have some moderate

) but the rest of the tensor elements do not have any significant values associated with S0–S1 transitions of o-betaine and

situation remains the same for the S0–S2 transition in gas phase and THF. On the contrary, the pattern of the relative contribution of the various TP tensor components associated with S0–S2 transition is quite different in water and acetonitrile solvents and except

and

, all the

tensor elements have significant contributions in these solvents. In the case of p-betaine, is the most contributing tensor component for S0–S1 transition and this finding holds good irrespective of the nature of the solvent considered while for S0–S2 transition, all the elements except

and

have zero contribution.


Part – 3.1

P a g e | 17/3

Table 4/3.1. TPA tensor elements (Sij) (in a.u.) of ortho- and para-betain in gas, THF, CH3CN and H2O solvents System

Ex. St.

o-betaine

1

2

p-betaine

1

2

Solvent

TPA tensor elements (in a.u.) Sxx

Syy

Szz

Sxy

Sxz

Syz

Gas

-117.0

-0.6

-28.4

8.3

60.8

-3.9

THF

137.2

-0.6

35.1

-4.5

-72.4

1.6

CH3CN

-115.3

0.6

-30.7

2.7

61.8

-0.5

H2O

112.3

-0.6

30.3

-2.3

-60.6

0.3

Gas

-23.5

-0.6

-14.1

4.3

19.0

-2.3

THF

-12.1

1.0

4.0

-11.6

-0.2

7.1

CH3CN

1.1

-0.6

-4.2

10.2

3.7

-6.2

H2O

0.0

-0.5

-4.3

10.0

Gas

-0.7

4.0

-138.6

1.7

4.1 0.0

-6.1 0.0

THF

0.5

5.6

-376.0

0.6

0.0

0.0

CH3CN

-4.2

-1.4

360.3

1.8

0.0

0.0

H2O

-0.6 0.0

-5.3 0.0

354.8 0.0

-0.5 0.0

0.0

0.0

8.1

-33.1

0.0

0.0

0.0

0.0

-23.3

29.6

0.0

0.0

0.0

0.0

29.9

12.4

0.0

0.0

0.0

0.0

21.4

-24.8

Gas THF CH3CN H2O

To make an extensive analysis on the microscopic origin of the TPA process of these molecules, we have employed the SOS approach to evaluate the

values involving up to

eight intermediate states in the gas phase. For comparison, the SOS and response theory results on

are presented in Table T5 of appendix T. Table T5 of appendix T indicates that

results obtained using the SOS approach are fluctuating with the number of intermediate states involved and even with eight states the results did not fully converge to the response theory values. Earlier it has been found that the full convergence of the SOS result can be achieved if one incorporates a large number of intermediate states in such calculations.20,49,50 However, the computational cost of such calculations will be prohibitively expensive. Nevertheless, looking at the data of Table T5 of appendix T, it is quite evident that the values with the SOS approach involving two intermediate states are close to the response theory results and hence it is fair enough to go forward with the channel interference study Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions

Part – 3.1

P a g e | 18/3

within the framework of a four-state model in 3D. As mentioned earlier, Cronstrand et al.21,20 addressed the issue of channel interference involved in the TPA process, however, their theoretical investigations were restricted to the two dimensional molecules only. To decipher the interfering effect of different optical channels and the alignment of different transition moment vectors in the TPA process of three dimensional systems, we need to derive a new GFSM formula that can capture all the essential features of 3D molecules. A detailed derivation is given in the appendix D and the key steps of the full derivation are presented in the next section. 3.1.3.4 Key steps of full GFSM derivation By putting equation (3/3.1) in equation (4/3.1) with two intermediate states i and j, we get the in 3D as-

expression for overall

=6

2

+

∆

2

2

+

∆

∆

+ ∆

+8

+

+ ∆

+

+ ∆

+

+ ∆

+

+ ∆

2

+

+

+

∆ 2 ∆ 2 ∆

+

+

∆

∆

∆ 2

∆

+

∆ 2

2

∆

2

2

2

+

2 ∆

+

2 ∆

+ ∆

+

+

+4

2

∆

+

+

2 ∆

2 ∆ 2 ∆

This expression can be reduced to


Part – 3.1

P a g e | 19/3

=

=

+

,

Where,

(5/3.1) , (

's are the same as three-state model terms and

The cross-term in the derivation is represented as interference in a TPA process. The

and is the source of channel

+

+ ∆ ∆

+ 16

+ ∆

×

+ ∆

+ 16

+ ∆

×

+ ∆

+ 16

+ ∆

×

+ ∆

+

+ 16

+ ∆ ∆

+

+ 16

16 ∆ ∆

(with j ≠ i) is the cross-term.

in 3D can be explicitly written as -

= 48

=

)

+ ∆ ∆

3

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions

Part – 3.1

=

P a g e | 20/3

16 ∆ ∆

3 +

+

−2

Here,

+

+

+ +

⃗ ∙⃗ ⃗ ∙⃗ ⃗ ∙ ⃗ ⃗ ∙ ⃗ × + × | || | | || | | || | | || |

⃗ ∙ ⃗ ⃗ ∙ ⃗ ⃗ ∙ ⃗ ⃗ ∙⃗ × + × | || | | || | | || | | || |

cos

∆ ∆ ⇒

=

+

+

+

3

16

=

+

+

+

∆ ∆

+

+

+

16

=

+

+

−2

=

+

=

8 ∆ ∆

+

cos

=2

cos

cos

+ cos

cos

+ cos

cos

. Therefore, we can write

+ cos

cos

+ cos

cos

(6/3.1)

Earlier Cronstrand et al.20,21 derived an apparently similar expression which was valid for only planar molecules and the few state model formula derived by them reads

=

8 ∆ ∆

3cos

cos

+ sin

sin

− 2 sin

sin

(7/3.1)

For deriving this formula, Cronstrand et al.20,21 used only those components of the TPA tensor elements which correspond to the plane of the molecule. For example, for a planar molecule oriented in the xy plane, the contributing TPA tensor elements are Sxx, Syy and Sxy. However, we have used all the components of TPA tensor elements in deriving equation (6/3.1). Therefore, in principle, equation (6/3.1) is valid for all the one, two and three dimensional molecules. For two dimensional molecules, the tensor elements in the direction perpendicular to the molecular plane will be very small and the expression will produce the Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions

Part – 3.1

P a g e | 21/3

same result as that obtained from equation (7/3.1). Similarly, for 1D molecule, the only contributing TPA tensor element will be that corresponding to the orientation of the linear molecule and the other tensor elements will have no impact on the net cross term and the cosine terms become 1 which ultimately reduces equation (6/3.1) to

=

24

(8/3.1)

∆ ∆

If the excitation process is contributed mainly by two different excitations viz., ground state, |0〉, to some intermediate state, | 〉, and then from | 〉 to the final state | 〉 then the above expression is reduced to 3SM.

=

It is evident that

=8

∆

2 cos

(9/3.1)

+1

will be higher if the magnitudes of the transition moment vectors are

sufficiently large and the excitation energies are relatively small. Moreover, the have maximum value when the involved transition moment vectors,

and

will

, are aligned

either parallel or anti-parallel to each other. Similarly, if the excitation process involves two intermediate states, | 〉 and | 〉 , then equation (6/3.1) reduces to a four-state-model (4SM) ,

formula and now has contribution from four different terms viz.

,

and

, where

the first two are the 3SM terms and the last two are known as the interfering terms. The present expression of the interfering term (equation (6/3.1)) for 3D molecules is significantly different from that of equation (7/3.1) which is valid for 2D molecules only. Equation (6/3.1) contains some extra angle terms namely

and

which are absent in

equation (7/3.1). Moreover, their sine and cosine dependencies are also quite dissimilar. The extra angle terms render the TPA process with channel interference in 3D much more complicated than 2D and thus controlling TPA activity of a 3D system is a challenging task. Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions

Part – 3.1

P a g e | 22/3

Apart from the larger transition moment values and low excitation energy, the TP transition probabilities also depend on the relative orientation of the transition moment vectors i.e.

,

and

. For relative enhancement of the

,

value, the alignment of the transition

moment vectors must be in a direction so that the overall value of the angle terms in the parentheses of equation (6/3.1) becomes positive and this ensures that interfering term ( contributes constructively. To maximize the positive

)

, all the -vectors are needed to be

parallel with each other and under this condition; the highest value of the term in the parentheses of equation (6/3.1) will be +3. In contrast, the lowest possible value of the angle term in equation (6/3.1) will be -3 when the transition moment vectors are aligned antiparallel and in this case the interfering term plays a completely destructive role in regulating the net

value. It is worth recalling that the value of

in 2D also lies in the range of +3

and -3. However, in the intermediate angle range, the behavior of the interfering term in 2D and 3D will be drastically different. This will happen because only cosine functions are involved in equation (6/3.1) (3D case) while both sine and cosine terms appear in equation (7/3.1). Table 5/3.1 furnishes the relevant information about the relative orientation of the

-

vectors and its effect in controlling the overall sign of the angle term in 3D. The product of cosine terms will be positive when both the angles involved in each term in parentheses of equation (6/3.1) are either greater than or less than 90°. When some of the cosine products are positive and some are negative, the overall sign depends on their relative magnitudes. Since the alignment of all the

-vectors in a particular direction may not be feasible in 3D,

alternatively one can follow the arguments of Table 5/3.1 to maximize the interfering term as much as possible in the intermediate angle range.


Part – 3.1

P a g e | 23/3

Table 5/3.1. Sign of different cosine terms and their effect on the overall sign of cross-term

Relative Orientation of the following pair of vectors

Sign of

,

,

,

,

,

,

+ve ≤ 90°

+ve ≤ 90°

+ve ≤ 90°

+ve ≤ 90°

+ve ≤ 90°

+ve ≤ 90°

+ve

+ve ≤ 90°

-ve ≥ 90°

+ve ≤ 90°

-ve ≥ 90°

+ve ≤ 90°

-ve ≥ 90°

-ve

-ve ≥ 90°

-ve ≥ 90°

-ve ≥ 90°

-ve ≥ 90°

-ve ≥ 90°

-ve ≥ 90°

+ve

To scrutinize the role of different angles between the transition moment vectors in the interfering term (

) and net

thoroughly, we have calculated the pertinent angles and

interfering term both in the gas and solvent phases. Since the four-state-model gives us results close to that of the response theory, the angles and the components of ,

and

, that is,

for o- and p-betaine in the gas phase, THF, water and acetonitrile solvents

are evaluated within the framework of a four-state model in 3D and the results presented in Table 6/3.1. It is manifestly clear from Table 6/3.1 that both the cross terms and alignment of different transition moment vectors relative to each other have profound impact on the overall of the systems studied. For S0–S1 transition of o-betaine in the gas phase, the substantially larger in comparison to the interfering term, interference has little impact on the net

is

, and as a consequence, channel

. It is also noticeable that the constructive channel

interference associated with S0–S1 transition of o-betaine, which is more likely to be a π-π* one, decreases sharply on going from the gas to the solvent phase and is evident from Table 6/3.1. In p-betaine also, S0–S1 transition has sufficient π-π* character and in this case too,

is much higher than the interfering term

and the interfering term is destructive in

nature both in gas and solvent phases. The S0–S2 transition of o- and p-betaine is more akin to charge transfer type. In the gas phase of o-betaine, the interfering term

associated with

S0–S2 transition has destructive contribution while it is constructive for p-betaine. The magnitude of the interfering term involved with S0–S2 transition of both the systems is quite significant. For S0–S2 transition in the gas phase of the o-isomer, the cross term is in fact Solvent induced channel interference in the two-photon absorption process-a theoretical study with a generalized few-state-model in three dimensions

Part – 3.1

larger than appears as 2

. .

P a g e | 24/3

It should be kept in mind that, in the overall

expression, the cross term

In general, the dominance of the cross term in determining the net

value

is evident for the S0–S2 transition of the two isomers in the gas and solvent phases. A meticulous examination of Table 6/3.1 also reflects another important aspect of channel interference. In the presence of solvent, the strength of destructive interference involved with S0–S2 transition of o-betaine is quenched considerably. The effect of solvent is also pronounced in the case of p-betaine where the strength of constructive interference associated with S0–S2 transition increases on moving from the gas to solvent phase. More interestingly, we have noticed that the use of a solvent with relatively smaller polarity will be more effective to increase the magnitude of the constructive interference and thereby increasing the net

value

while the opposite is true if the nature of the interference is destructive, that

means, to reduce the strength of the destructive interfering term, the use of a solvent having high polarity would be more wise. The whole study indicates that channel interference in 3D is strikingly different from that of 2D and the solvent plays a decisive role in determining the strength of the interfering term.


Part – 3.1

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Table 6/3.1. Four-states-model results, for (in 104 a.u.) and different angles (in degrees), of ortho- and para-betaine in gas, THF, CH3CN and H2O solvents (Data for the 2nd excited state are given in bold font) System

Solvent 17.05

0.01

0.43

17.92

180.00

148.76

170.75

139.64

38.92

10.89

0.39

2.96

-1.04

1.26

141.13

5.80

170.59

25.41

165.35

31.16

24.48

0.0

0.04

24.56

175.62

97.92

169.18

104.68

73.00

7.09

0.16

0.40

-0.10

0.36

107.00

8.40

169.18

74.42

177.21

82.08

17.89

0.0

0.03

17.95

174.27

98.57

19.54

70.49

111.04

165.47

0.11

0.23

-0.07

0.20

68.96

164.80

19.44

109.88

174.87

81.43

13.47

0.0

0.02

13.52

5.64

81.29

0.00

69.95

68.46

15.37

0.07

0.18

-0.05

0.15

111.63

15.96

0.00

110.51

5.09

98.79

9.70

0.0

-0.05

9.6

0.00

179.43

89.84

89.96

89.96

89.84

0.36

0.20

0.11

0.87

90.00

89.83

89.84

90.00

0.00

0.57

93.06

0.0

-0.12

92.82

180.00

22.02

90.04

90.00

90.00

89.96

0.41

0.24

0.29

1.23

90.00

89.96

90.04

90.00

180.00

157.98

88.91

0.0

-0.10

88.71

180.00

38.46

90.00

90.00

90.00

90.00

0.28

0.31

0.23

1.05

0.00

90.00

90.00

90.00

180.00

141.54

68.55

0.0

-0.08

68.39

180.00

30.82

0.00

90.06

90.02

90.02

0.23

0.20

0.18

0.79

90.06

90.02

0.00

90.03

180.00

149.27

o-betaine

Gas

THF

CH3CN

H2O

p-betaine

Gas

THF

CH3CN

H2O

3.1.4 Conclusion In summary, we have derived a generalized few states model (GFSM) formula for calculating the TPA transition probability (

) of 3D molecules and applied it to the simple unit of

Reichardt’s dyes namely 2- and 4-(pyridinium-1-yl)phenolate in gas phase, THF, acetonitrile and water solvents. We have found that the GFSM for 3D molecules is notably different from that already available for 2D molecules and contains some additional terms related to the angles between different transition moment vectors. The presence of these terms makes the


Part – 3.1

P a g e | 26/3

TPA process of 3D molecules much more complicated and difficult to control as compared to that in 1 and 2D molecules. We have meticulously studied the effect of the interference term on the net

of the systems studied. We have noticed that this interference term plays a

more prominent role in determining the overall

of the charge transfer transitions than

that of π-π* transition. Finally, the present study reveals that, for o- and p-betaine, the use of a solvent having moderate polarity is more suitable for increasing the magnitude of the constructive interfering term vis-à-vis the overall

value while a high polarity solvent is

more effective in quenching the destructive interfering term. To check whether or not this interesting observation holds good for the other 3D systems, we need to explore the field both at the QM/MM and PCM levels of theory further.


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Part - 3.2 High-polarity solvents decreasing the two-photon transition probability of through-space charge-transfer systems - A surprising in silico observation (J. Phys. Chem. Lett. 2012, vol. 3, pp. 961-966)

High-polarity solvents decreasing the two-photon transition probability of through-space charge-transfer systems - A surprising in silico observation

Part – 3.2

P a g e | 28/3

3.2.1 Abstract In the part, we have applied the 3D-GFSM formula to address the question as to why larger TPA cross sections are observed in non-polar than in polar solvents for through-space chargetransfer (TSCT) systems such as [2,2]-paracyclophane (henceforth called PCP) derivatives. In order to answer this question, we have performed ab initio calculations on two well-known TSCT systems, namely, a PCP derivative and a molecular Tweezer-trinitrofluorinone (TNF) complex and found that the TP transition probability values of these systems decreases with increasing solvent polarity. To rationalize this result, we have analyzed the role of different optical channels associated with the two-photon process and noticed that, in TSCTs, the interference between the optical channels is mostly destructive and that its

magnitude

increases with increasing solvent polarity. Moreover, it is also found that a destructive interference may sometimes even become a constructive one in a non-polar solvent, making the TP activity of TSCTs in polar solvents less than that in non-polar solvents.

3.2.2 Results and Discussion

Figure 1/3.2. Gas-phase-optimized geometry of (a) the doubly positively charged [2.2]paracyclophane (PCP) derivative and (b) the tweezer-TNF complex

The systems studied in this contribution are shown in Figure 1/3.2. In the donor-acceptorsubstituted doubly positively charged PCP, the donor moieties -NMe2 are placed at the para


Part – 3.2

P a g e | 29/3

position of one benzene ring, while the acceptor moieties -(+)NMe3 are located at the para position of the other benzene ring of the PCP. This configuration ensures the TSCT character of the substituted PCP. Figure 1b/3.2 shows a host-guest complex, made up of a tweezer molecule, 4,6-bis-(6-acrid-9-yl)-pyridin-2-yl)-pyrimidine and TNF. The geometry of PCP has been optimized at the B3LYP/6-311G(d,p) level of theory using the Gaussian 03 suite of programs,44 while the optimized coordinates of the tweezer-TNF complex have been taken from an earlier work by Chakrabarti and Ruud,45 where optimization was carried out at the MPW1B95/6-311G(d,p) level of theory. After geometry optimizations, we calculated the OPA and TPA parameters of the first excited state of PCP and the first two excited states of the tweezer-TNF complex, which can be classified as being of TSCT nature from an analysis of the poles and residues of the linear and quadratic response functions in the framework of time dependent density functional theory (TDDFT)26 as implemented in the DALTON22 code. All OPA and TPA parameters are evaluated using the CAMB3LYP46 functional in combination with Dunning’s cc-pVDZ basis set, which have been demonstrated to be an accurate model for such intramolecular TPA-active charge-transfer states.47 For solvent-phase calculations, the non-equilibrium formulation of quadratic response theory14 in the polarizable continuum model (PCM)24 has been used, and the relevant calculations were performed using the DALTON code. To shed light on the microscopic origin of the unusual solvent dependency of the TP transition probability (δTP) of the TSCTs, we have re-evaluated the δTP values following the recently developed 3D generalized model by Alam et al.,48 and these calculations have been performed with the aid of different OPA parameters, namely, the excitation energies, ground to excited state (μ0i) and excited-excited state (μij) transition moment vectors. It is worth recalling that geometrical relaxation in the presence of a solvent sometimes makes important indirect contributions to the TPA and other NLO properties49 of a material. However, in our cases, we have found that the change in the geometrical


Part – 3.2

P a g e | 30/3

parameters in the different solvents at the PCM level of theory is insignificant, and as a consequence, indirect contributions to the TPA of these TSCTs are not expected. Nonetheless, we have performed PCM-response calculations on the corresponding solventoptimized geometries of PCP and have noticed that the trend in the δTP is quite similar to the PCM-response results obtained from the gas-phase-optimized geometry. The gas- and solvent-phase OPA data of the target systems are presented in Table 1/3.2. From the data in Table 1/3.2, it is evident that with increasing solvent polarity, the excitation energy (ω0i) gradually increases for PCP, whereas the reverse trend is observed for the tweezer-TNF complex. At the same time, it is also important to note that the increasing trend in ω0i for PCP is much more rapid than the corresponding decreasing trend for the tweezerTNF complex. It is also clear from this table that the oscillator strength (δOPA) for the S0-S1 transition is very small and does not change significantly when going from one solvent to another. The very small values for δOPA in different solvents are attributed to the small magnitude of ground- to excited-state transition moment vectors (μ0i).

Figure 2/3.2. Gas-phase orbital pictures of PCP [(a) HOMO, (b) LUMO] and the tweezer-TNF [(c) HOMO−1, (d) HOMO, (e) LUMO] complex


Part – 3.2

P a g e | 31/3 Table 1/3.2. OPA parameters for PCP and tweezer-TNF complex

Solvent (ε)

MeCN (37.5)

THF (7.52)

CHCl3 (4.81)

C6H5CH3 (2.38)

Ex. St. (i)

PCP 0i

Tweezer-TNF μ0i (au)

ω0i (eV)

Λ

μ (au)

δOPA (au)

Orbitals

ω0i (eV)

1

3.052

0.228

0.230

0.003

H-L (-0.70)

3.114

0.195 0.423 0.009

2

-

-

-

-

-

3.218

0.180 0.343 0.007 H-1-L (-0.67)

1

2.956

0.223

0.228

0.003

H-L (-0.70)

3.122

0.196 0.433 0.009

H-L(-0.67)

2

-

-

-

-

-

3.234

0.175 0.348 0.007

H-1-L(0.68)

1

2.900

0.220

0.227

0.003

H-L (0.70)

3.128

0.197 0.439 0.010

H-L(0.67)

2

-

-

-

-

-

3.243

0.173 0.350 0.007

H-1-L(-0.67)

1

2.759

0.214

0.224

0.003

H-L (-0.70)

3.146

0.200 0.452 0.010

H-L(0.67)

2

-

-

-

-

-

3.272

0.168 0.353 0.007 H-1-L (-0.67)

1

2.511

0.204

0.193

0.002

H-L (0.70)

3.189

0.209 0.411 0.009

2

-

-

-

-

-

3.338

0.164 0.313 0.007 H-1-L (-0.67)

Λ

δOPA (au)

Orbitals H-L(0.67)

H-L(0.67)

Gas (0.0)

In order to understand the nature of the transition in the two systems, we have calculated the contributions of different orbitals involved in the transition process. The orbitals involved and the corresponding contributions in different solvent media are specified in Table 1/3.2. For both the systems, the dominating orbitals associated with the S0-S1 transition is of HOMOLUMO type, and this is true both in the gas and the solvent phases. On the other hand, the S0S2 transition of the tweezer-TNF complex is mostly governed by the transition of an electron from HOMO-1 to LUMO. The respective orbital pictures of the PCP and tweezer-TNF complex in the gas phase are depicted in Figure 2/3.2. Considering the nature of the molecular orbitals, one can unambiguously assign the S0-S1 transition of both the systems and S0-S2 transition of the tweezer-TNF complex as TSCT type, or in other words, these transitions are essentially long-range in nature. The present assignment of the transitions is in agreement with the earlier finding of Chakrabarti and Ruud45 and was recently verified by CC2 calculations.47 It is important to note that no significant change in the orbital pictures is


Part – 3.2

P a g e | 32/3

observed in the solvent calculations, and the solvent-phase orbital pictures are supplied in Figure F2 and F3 in appendix F. To quantify the long-range nature of the transitions, we have evaluated the well-known Λ parameter,38 which is a measure of the degree of spatial overlap between the moduli of all of the pairs of occupied and virtual orbitals involved in a particular excitation process and which can be expressed as

=

∑, ∑,

(1/3.2)

where κia = Xia + Yia and Xia and Yia are the excitation and de-excitation components of the solution vectors for the basic TDDFT equation in the Born-Oppenheimer approximation and i and a are the indices of the occupied and virtual orbitals, respectively. Oia is the inner product between moduli of each pair of occupied and virtual orbitals. Λ always lies between 0 and 1. A large value of Λ implies that the spatial overlap between the occupied and virtual orbitals is significant and, hence, the corresponding transition is of short-range nature. Similarly, a small value of Λ is a manifestation of the long-range nature of the transition. The Λ parameter has been recently evaluated for a number of charge-transfer systems,36,38,45,50,51 and it has been found that the Λ value is quite consistent with the qualitative description of the nature of the charge-transfer transition. In the present work, we have obtained a reasonably small value of Λ for both the PCP and tweezer-TNF complexes. For PCP, the Λ value associated with the S0-S1 transition is 0.204, whereas the corresponding values of the S0-S1 and S0-S2 transitions in the tweezer-TNF complex are 0.195 and 0.209, respectively, which certainly defines the dominant character of the transition as being of TSCT type.


Part – 3.2

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Table 2/3.2. TPA Transition Probability (all δ terms are given in 103 a.u) of PCP, As Calculated from Response Theory (δResp) and the Four-State Model (δ4SM), and the Different Terms appeared in the 4SM Calculation ( is the angle between the transition moment vectors µij and µkl) Solvent (ε) MeCN (37.5) THF (7.52) CHCl3 (4.81) C6H5CH3 (2.38) Gas (0.0)

2×

2.335

0.003

−0.160

2.179

1.716

90.1

89.9

156.3

0.1

90.1

90.0

2.514

0.001

−0.094

2.421

1.932

90.1

90.1

160.3

0.2

90.1

90.1

2.643

0.001

−0.082

2.562

2.079

90.1

89.7

161.9

0.4

89.9

90.1

2.907

0.001

−0.078

2.828

2.424

90.0

90.2

165.0

0.2

89.9

90.1

2.229

0.001

−0.070

2.159

2.055

90.0

90.0

168.5

0.0

90.0

90.0

With this OPA background, we now discuss the peculiar solvent dependence of the TPA process of the chosen TSCT systems. The δTP for the S0-S1 transition of PCP and S0-S1 and S0-S2 transitions of the tweezer-TNF complex are presented in Tables 2/3.2 and 3/3.2, respectively. From Table 2/3.2, it is evident that the δTP obtained from quadratic response theory, δResp, of PCP gradually decreases with increasing solvent polarity, consistent with the experimental findings of Woo et al.10 It is worth recalling that although quadratic response theory gives us fairly accurate TPA cross sections, it cannot easily shed light on the underlying microscopic origin of the TP activity of a molecule. To decipher the microscopic mechanism of the TP process in a system, few-state models are often beneficial.


Part – 3.2

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Table 3/3.2. TPA Transition Probability (all δ terms are given in 103 a.u) of Tweezer−TNF, As Calculated from Response Theory (δResp) and the Four-State Model (δ4SM), and the Different Terms appeared in the 4SM Calculation ( is the angle between the transition moment vectors µij and µkl) Solvent (ε)

Ex. St. (f)

2×

MeCN (37.5)

1

25.695

0.188

-3.196

22.687

22.650

154.2

22.0

56.6

161.8

41.3

141.1

2

0.272

25.151

-3.719

21.703

29.160

138.8

13.1

56.6

163.9

57.0

158.1

THF (7.52)

1

26.633

0.165

-3.003

23.795

23.580

153.1

22.2

57.9

161.6

42.3

140.4

2

0.243

25.978

-3.506

22.715

29.730

137.7

12.9

57.9

163.6

58.2

157.8

CHCl3 (4.81)

1

27.230

0.156

2.924

30.311

24.150

27.8

22.4

58.6

18.5

137.0

140.0

2

0.233

26.481

3.420

30.134

30.300

43.1

12.7

58.6

16.5

121.1

157.7

C6H5CH3

1

27.725

0.125

2.578

30.429

24.810

30.0

22.9

60.9

18.6

135.3

139.2

(2.38)

2

0.194

26.816

3.040

30.050

30.000

44.6

12.2

60.9

16.8

119.4

157.2

Gas (0.0)

1

12.680

0.062

-1.172

11.570

16.980

127.8

15.7

62.9

163.4

58.0

147.9

2

0.077

17.503

-1.506

16.074

19.740

122.0

1.1

62.9

163.1

63.0

164.2

In this context, we have re-evaluated the δTP of the systems using a four-state model (4SM) (hereafter, represented as δ4SM) within the newly derived generalized few-state model (GFSM) formula for 3D molecules. The expression of δ4SM is given by48 =

=8

Where,

+

2 cos

∆

=

= cos

cos

In eq 2, ΔEn = ωn-(ωf/2), and

+2

+ cos

∆

(2a/3.2)

+1 ;

(i=1,2)

(2c/3.2)

∆

cos

(2b/3.2)

+ cos

cos

(2d/3.2)

is the angle between the transition moment vectors μab and

μcd. The values for the different δ and θ terms that appear in eq 2 are presented in Tables 2/3.2 and 3/3.2, respectively, for the PCP and the tweezer-TNF complexes in both the gas and solvent phases. From Table 2/3.2, one can easily find that δ11 is contributing the most, while the contribution of δ22 is insignificant compared to the overall δ4SM.


Part – 3.2

P a g e | 35/3

Figure 3/3.2. Plot of δ12 for the first excited state of PCP in the gas and solvent phases

Similarly, for the tweezer-TNF complex, δ11 is contributing the most for S0-S2, and δ22 contributes the most for the S0-S2 transition. In comparison to the δ11 and δ22 terms, the δ12 has the special feature that it can have both positive and negative values because it depends on the product of several cosine functions associated with the angles between the various transition moment vectors (μ0i and μij). As a consequence, the interference term δ12 plays a crucial role in controlling the net δ4SM of the model systems. We have observed that the δ12 contribution is always destructive (i.e., it has negative values) for PCP, both in the gas and the solvent phases. More interestingly, the magnitude of the destructive δ12 term of PCP increases with increasing solvent polarity thereby explaining the net lowering of its δ4SM in more polar solvents. The solvent dependence of δ12 of the tweezer-TNF complex is even more surprising. In non-polar solvents, the δ12 of the tweezer-TNF complex changes its sign and becomes positive, ultimately increasing the net δ4SM for both TSCT transitions of the tweezer-TNF complex. The solvent-dependent variations of δ12 for PCP and the tweezer-TNF complex are presented in Figures 3/3.2 4/3.2.


Part – 3.2

P a g e | 36/3

(a)

(b)

Figure 4/3.2. Plot of δ12 for the (a) first and (b) second excited state of the tweezer-TNF complex in the gas and solvent phases

The contributions and also the variations of the different δ terms in different solvents can be explained by considering the magnitude of contributing terms appearing in eq 2b. The values of the different transition moment vectors and angle terms (Xij) for PCP and the tweezer-TNF complex are supplied, respectively, in Tables T6 to T11 in appendix T. It is obvious from these Tables that for PCP, the ordering of the magnitude of the four transition moment vectors is μ11 >> μ02 > μ01 > μ12. The large magnitude of μ11 is the main reason for the largest contribution to δ11, and the small values of μ02 and μ12 make δ22 the smallest contributing term. Similarly, for the tweezer-TNF complex, the largest contribution to δ11 for the first excited state and δ22 for the second excited state is due to the large values of μ11 and μ22, respectively. The results (Table T11 in appendix T) also reveal that for PCP, all of the angle terms (Xij) have a magnitude of ~1 or -1, and among the six different θ terms, only

and

make significant contributions. All other angles are 90° (or very close to 90°), causing the corresponding cosine terms to vanish. This observation is important in the sense that for PCP, the sign of the interference term arises from these two angles only, and the orientations of μ01 and μ02 with respect to each other play a pivotal role in making the interference term in


Part – 3.2

P a g e | 37/3

PCP destructive; this is similar for μ1f and μ2f. In contrast, all angles contribute in the tweezer-TNF complex. With the increase in polarity of the solvent, the value of μ11 decreases, whereas that of μ01, μ02, μ12, ΔE1, and ΔE2 increases slowly for PCP. As a consequence, the value of δ11 of PCP decreases and the magnitude of the destructive interference increases with increasing polarity of the solvent. The overall decrease in δ4SM with the polarity of the solvent can thus be attributed to the corresponding increase in the destructive interference. In case of tweezer-TNF complex, except for μ12, all the other transition moment vectors decrease with increasing solvent polarity. It is worth noting that the 4SM results are found to be in excellent agreement with the response theory results.

3.2.3 Conclusion In summary, we have studied the unusual solvent dependence of the TPA process in two well-known through-space charge-transfer systems, PCP and the tweezer-TNF complex, using quadratic response theory and the newly derived GFSM formula for 3D systems. We have found that the δResp of donor-acceptor-substituted PCP gradually decreases with increasing solvent polarity, which is in agreement with experimental observations. Our theoretical analysis suggests that this at first surprising result arises from the destructive interference between different optical channels in the TPA process, the strength of which increases with increasing solvent polarity. This peculiar solvent dependence of the TP activity is also found for the other TSCT system, the tweezer-TNF complex, where the interference term is found to change its sign when going from the gas phase to low-polarity solvents such as toluene/chloroform.


Part – 3.3

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Part - 3.3 Enhancement of twist angle dependent two-photon activity through the proper alignment of ground to excited state and excited state dipole moment vectors (J. Phys. Chem. A 2012, vol. 116, pp. 8067-8073)

Enhancement of twist angle dependent two-photon activity through the proper alignment of ground to excited state and excited state dipole moment vectors

Part – 3.3

P a g e | 39/3

3.3.1 Abstract In this part we are intended to provide a technique to control the channel interference in a real molecular system. For this purpose we have selected o-betaine (studied in part 3.1) and found that δTP will reach its maximum value at a twist angle around 65°. However, the potential energy scan with respect to the twist angle between its two rings indicates that the molecule in its ground state is quite unstable at this angle. Out of the different possibilities, the one having a single methyl group at the ortho position of the pyridinium ring is found to attain the optimum twist angle between the two rings, and interestingly, this particular substituted obetaine has larger δTP value than any other substituted or pristine o-betaine. The twist angle dependent variation of δTP has been explained by employing the generalized-few-state-model formula for 3D molecules. The results clearly reveal that the magnitude of ground to excited state and excited state dipole moment vectors as well as the angle between them are strongly in favor of maximizing the overall δTP values at the optimum twist angle. The constructive interference between the optical channels at the optimum twist angle also plays an important role to achieve the maximum δTP value. Furthermore, to give proper judgment on our findings, we have also performed solvent phase calculations on all the model systems in nonpolar solvents, namely, cyclohexane and n-hexane, and the results are quite consistent with the gas phase findings. The present study will definitely offer a new way to synthesize novel TP active material based on o-betaine.

3.3.2 Computational Details At first, we have optimized the ground state structures of the unsubstituted and substituted obetaine molecules in the gas phase and two solvents namely, cyclohexane and n-hexane, at the CAMB3LYP/6-311++G(d,p) level of theory, and the calculations are performed in the


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DALTON22 program package. The gas phase optimized structures are shown in Figure 1/3.3. It is worth mentioning that, in an earlier work,48 it was shown that the optimized geometry of a similar system (p-betaine) obtained from the CAMB3LYP/6-311++G(d,p) level of theory is in nice agreement with the available X-ray crystallographic data. We have also carried out the frequency calculation on the optimized geometries, and no imaginary frequency has been found. After geometry optimizations and frequency check, we have evaluated the various one- and two-photon absorption parameters of o-betaine by changing the twist angle from 0° to 180° between the pyridinium and phenolate rings, at CAMB3LYP/cc-pVDZ level of theory.

Figure 1/3.3. Optimized ground state structures of unsubstituted (a) and substituted (b−f) o-betaine in the gas phase.

The justification of using CAMB3LYP functional in this study is that this functional has already proved its worth in reproducing excitation energies in a number of previous works.36,38 In this case too, the CAMB3LYP functional along with the aug-cc-pVDZ basis set satisfactorily reproduces48 the experimental52 excitation energy of o-betaine in CH3CN


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solvent. We must also admit here that, in case of water, there is a difference between experimental32 and theoretical48 values, but this value is mainly related to the well-known limitation of PCM model, which is not able to account for the specific interactions like hydrogen bonding. For this reason also, we have used non-polar solvents in this work. In the present study, using cc-pVDZ basis set we have calculated the one-photon excitation energy of o-betaine and found that the calculated excitation energy in CH3CN solvent is 2.55 eV, which is very close to the experimental32 value of 2.6 eV. Such a nice agreement between theoretical and experimental results justifies the use of the CAMB3LYP/cc-pVDZ level of theory in this study. After the calculation of OPA and TPA parameters for different twist angles, the δTP of all the optimized substituted and unsubstituted o-betaine molecules have been computed in both the gas and solvent phases at the same level of theory. It is worth mentioning that o-betaine has sufficient zwitterionic character in the ground state, and in the presence of a polar solvent, indirect contribution to the net TP activity will be significant,53 which ultimately will complicate the comparison of TP activity of the model systems in different solvents. To minimize the indirect contribution effect, we have selected non-polar solvents like cyclohexane and n-hexane for the solvent phase response calculations. For solvent phase calculations, we have employed the non-equilibrium formulation of response theory27,28 within the polarizable continuum model (PCM) as implemented in the DALTON22 suite of programs. However, we must also mention here that the electrostatic PCM model we have employed does not include the dispersion term for solute−solvent interaction and that this may become important in the case of non-polar solvents.


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3.3.3 Results and Discussion The TP activity of a molecule is measured in terms of δTP which in turn is related to the TPA tensor elements (Sαβ). For an excitation by a single beam of linearly polarized light, the relationship between δTP and Sαβ is given by41 . .

=6

+

+

+8

+

+

+4

+

+

(1/3.3)

TPA tensor elements are related to the various transition dipole moment vectors and the excitation energies by the equation40 + −

=

(2/3.3)

⁄2

Where, terms have their usual meanings. The OP and TP parameters can be extracted from the residues of linear and quadratic response functions, respectively. The other alternative for evaluating the δTP and Sαβ is the few-states model (FSM) in which a limited number of intermediate states are involved in a calculation. Although FSM is computationally cheaper, one must keep in mind that the reliability of this approach is limited to the involvement of effective states in the calculations. The beauty of this FSM approach lies in the fact that it can be reduced to the expressions containing angles between different transition moment vectors, which provide a better understanding of the TPA process. According to Alam et al.,48 the overall δTP of a 3D system within GFSM can be written as48

= ,

Where,

= cos

8

=

cos

(3/3.3)

∆ ∆

,

+ cos

cos

+ cos

cos


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In the present work, we have studied the variation of δTP and the interference term with the twist angle between the pyridinium and phenolate rings of o-betaine. The twist angle is varied from 0° to 180° at an interval of 5°, and for each twist angle, all the OP and TP parameters for the first excited state have been calculated and then plotted against these angles.

Figure 2/3.3. Variation of μ01 and μ11 (in au) with the twist angle (in deg) of unsubstituted o-betaine

The transition probability for OPA is measured by the oscillator strength (δOPA), which is related to the excitation energy (ω0f) and the transition dipole moment (μ0f) by the following relationship37

=

2 3

|⟨0| ̂ | ⟩|

(4/3.3)

Figure F4 in appendix F shows the variation of δOPA associated with the S0−S1 transition of obetaine. The data containing different OPA parameters are supplied in Table T12 in appendix


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T. It is clear from the plot that δOPA gradually decreases with the increase in the twist angle and shows a minimum at 90°. In order to explain this behavior, we have plotted μ01 and ω01 against the twist angle. The relevant plots are shown, respectively, in Figure 2/3.3 and Figure F5 in appendix F. These plots demonstrate that, as we increase the twist angle, both μ01 and ω01 decreases continuously, which in turn is responsible for the net decrease of δOPA with the twist angle. Similar variations of δOPA for the p-betaine molecule, with minima at 90°, were obtained by Zaleśny et al.54 using the GRINDOL method and also by Fabien et al.55 at the ab initio CIS level.

Figure 3/3.3. Variation of δresp, δ3SM, and δ11 with twist angle


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Figure 4/3.3. Plot of SCF energy of unsubstituted o-betaine as a function of twist angle

After analyzing the variation of δOPA, we have investigated the variation of TP transition probability as extracted from the quadratic response theory, δresp, against the twist angle. The corresponding plot is shown in Figure 3/3.3 and different TPA parameters are supplied in Table T13 in appendix T. It is evident from Figure 3/3.3 that the value of δresp at first increases with the increase in twist angle and reaches its maximum value at 70°, then, with further increase in the angle, a sharp decrease is observed. It is worth commenting that Zaleśny et al.54 also found a similar curve with a peak at 80° for p-betaine and that Pati et al.52 found a maximum at 76° for the quinopyran system. At this stage, it is very crucial to check whether the geometry of o-betaine at the twist angles at which δresp reaches its maximum value is stable. For this purpose, we have plotted (Figure 4/3.3) the SCF energy of unsubstituted o-betaine (in gas phase) against the twist angle. The plot dictates that the geometry of o-betaine around the twist angle 70° is energetically unstable. Earlier, Niewodniczański et al.56 have also studied the potential energy curves (PEC) of o-betaine at


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the HF and MP2/6-31G(d) levels of theory, and they obtained a similar feature, although the PEC was relatively flat in comparison to the present one. The twist angle of the solvent phase optimized geometries of unsubstituted o-betaine are also far from the required value, which indicates that it is not possible for unsubstituted o-betaine to achieve the optimum twist angle to maximize the δTP either in the gas phase or in solvents. To check the role of the basis set on TP activity, we have also evaluated the δresp of o-betaine at the CAMB3LYP/aug-cc-pVDZ level of theory and noticed that the nature of the variation of δresp with twist angle is quite similar to that obtained from the CAMB3LYP/cc-pVDZ level of theory and that both results give the same range of twist angle for maximum δresp values. Since in this work we are mainly concerned with the range of twist angle at which δresp becomes the maximum and as both the cc-pVDZ and aug-cc-pVDZ basis sets give us the same range of twist angle, we have performed all the calculations with the smaller basis set (cc-pVDZ). It provides a good compromise between the computational cost and the accuracy of the results. The δresp result with aug-cc-pVDZ basis set is depicted in Figure F6 in appendix F. For explaining the variation of δresp in our case, we have applied the three-level model within GFSM48 and reevaluated all of the δ terms appearing in eq 3a/3/3. Before going into details about our results, it is important to explain the mathematical model used for this work. In this work, we have considered the first excited state as the final state, and both the first and second excited states are taken as the intermediates. A pictorial representation of the model used is shown in Figure 5/3.3. In this case, within GFSM,48 the overall δTP (we represent it as δ3SM), as suggested by eq 3a/3/3, is given by =

+

+2

(5/3.3)

Here δ11 and δ22 terms are given by the expression48


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=8

−

⁄2

2 cos

(6/3.3)

+1

Equation 6/3.3 clearly indicates that the δ11 and δ22 terms will always have positive values because these involve only the square terms and hence are always constructive in the sense that they always tend to increase the overall value of δ3SM.

i 2

 02

 21

i 1

Final State

 01 0 Figure 5/3.3. Pictorial representation of the few-state model used in this work

The δ12 term, which, as suggested by eqs 3/3/3, is given by48

=

8|

|| ∆

|| ∆

||

|

(cos

cos

+ cos

cos

+ cos

cos

) (7/3.3)

This term depends on the orientations (given by the different angles, θ) of different transition moment vectors with respect to each other. Depending on these angles, δ12 may have either a positive or a negative value, which means it can either increase or decrease the overall δ3SM value. This dual nature δ12 (or in general δij with i ≠ j) is termed as the interference term.21,48 Therefore, in our case, i.e., the 0-1 transition, the only interference term that arises is δ12.


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Figure 6/3.3. Variation of (a) δ22 and δ12 (all in au) (b) X11 and X12 (all in au) with the twist angle

The values of different δ terms as calculated using our model are plotted in Figures 3/3.3 and 6a/3.3. Figure 3/3.3 suggests that the variations of δ11, δ3SM, and δresp with twist angle are similar in nature. The plots for the three δs (Figure 3/3.3) show a crest at a twist angle of 70° and a trough at 90°. However, Figure 6a/3.3 indicates that the variation of δ22 is a bit different. The δ22 term shows a sharp increase with the twist angle and reaches a peak at around 30−35°, and then, it decreases continuously until we arrive at the twist angle 90°. However, δ12 reaches its maximum value at and around 65−70°, and again the minimum value of δ12 is found at 90°. It is interesting to note that δ11 and the interference terms show maximums and minimums at almost the same point where the overall δ3SM and δresp have their respective maximum and minimum values. We must also admit here that the value of δ12 is not very large as compared to the overall δ3SM or δ11. The variation of these δ-terms can be explained by considering the variation of other terms involved in their expressions (i.e., μ, ω, and Xij). From eq 3a/3/3, it is obvious that the δ12 term (in three-level model) depends on the four transition moment vectors, two energy terms and six angle terms, whereas δ11 depends only on two μ terms (μ01 and μ11), the angle between them, and an energy term. We have calculated all these terms by changing the twist angle, and the results are depicted in Figures 2/3.3 and 6b/3.3. We have noticed that the two Enhancement of twist angle dependent two-photon activity through the proper alignment of ground to excited state and excited state dipole moment vectors

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energy terms (ω01 and ω02) are decreasing continuously with the increase in twist angle and reaches a minimum at 90°. A closer inspection of all the μ plots reveals that, out of the four transition moment vectors, the contribution of μ11 is much higher than the other three. Moreover, Figure 2/3.3 clearly indicates that, while the peak in the variation pattern of μ11 has appeared at the twist angle 90°, μ01 shows just the opposite trend, that is, it has the minimum value at 90° twist angle. As a consequence, this particular twist angle (90°) should not be a good choice to maximize the δ3SM of the system. However, the two μ curves, namely, the twist angle dependent variations of μ11 and μ01, meet at an angle 65° with moderate values, and therefore, the product of μ11 and μ01 will have a maximum value at this point. The value of ω01 is also sufficiently small at 65° angle. This explains the highest contribution of the δ11 term. The contribution of the other two μ terms, namely, μ12 and μ02, are rather very small (Figure F7 in appendix F) in the entire range of the angle variation, and hence, the corresponding contribution from δ22 is also very small. Apart from the magnitude of the various transition moment vectors, the contribution, Xij, that comes from the angles between these vectors and hence the channel interference term may play an important role in determining the net δ3SM of the system. Therefore, it is highly instructive to check the variation of Xij with the twist angle. Within 3SM, the Xij should have three different components, namely, X11, X12 and X22 of which X11 and X22 are associated with δ11 and δ22 and always have positive values. On the contrary, the cross-term, X12 may have both positive and negative values in the range {−3 to +3}. The sign of this cross-term will ultimately dictate the nature of the channel interference, and it is always a difficult task to find out the means of controlling its sign. As stated earlier, we have found constructive channel interference for o-betaine, or in the other words, the value of X12 is positive. To examine the effect of twist angle on X11, X12 and X22, we have calculated these parameters in the entire range of the twist angle, and the relevant results are depicted in Figures 6b/3.3 and


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F8 in appendix F. Interestingly, X12 attains significantly large positive values in the angle range 65−80°, and a similar kind of variation is also observed for X11, which in fact is associated with the highest contributing term, δ11. It is very interesting to note that the value of X22 is close to the maximum value (+3) for the entire range of angles (except for 90°). However, this large value of X22 is eclipsed by the small magnitudes of related transition moment vectors, and hence, the overall contribution of δ22 is very small. The above discussion clearly reveals that the large value of δ3SM around 65−70° twist angle arises from the very large contribution of δ11 and δ12, which ultimately comes from a combined effect of the magnitudes of different transition moment vectors (mainly μ01 and μ11) and their alignment with respect to each other. From the above analyses, one can easily anticipate that, if somehow the twist angle between the two rings of o-betaine can be restricted around 65−70°, the system will show very large TP activity. As mentioned earlier, this target angle cannot be achieved with o-betaine because the system at such a twist angle is energetically unstable, and the angle between the two rings of optimized geometry is 36° only. However, the accomplishment of this target angle is not a very difficult task. One possible way to increase the twist angle between the rings of obetaine is to exploit the method of substitution(s) at the remaining three ortho positions of obetaine. On the basis of this strategy, we have considered five model systems (Figures 1bf/3.3) in which the hydrogen atoms at the ortho positions of o-betaine were substituted by one, two, and three methyl groups. It is worth commenting that, in the case of substitutions by one methyl or two methyl groups, simultaneously, there are two distinct possibilities in each case. For one methyl group substitution, CH3 can be placed on the ortho position of either pyridinium ring (Figure 1b/3.3) or the phenoxide ring (Figure 1c/3.3). Similarly, for two methyl groups substitution, the two possibilities are one in which both the methyl groups are attached to the pyridinium ring (Figure 1d/3.3) and in the other case; each ring of o-betaine


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contains one methyl group (Figure 1e/3.3). The only possibility for trimethyl substituted obetaine is shown in Figure 1f/3.3.

Figure 7/3.3. Plot of TPA cross-section (in GM unit) against wavelength, λ (in nm), for unsubstituted and substituted o-betaine in (a) gas phase (b) n-hexane solvent

After optimizing the geometry of all these model systems, we have evaluated their δresp at CAMB3LYP/cc-pVDZ level of theory. The twist angle between the rings of the model systems including o-betaine and the δresp of all the systems are presented in Table T14 in appendix T. It is evident from this table that the highest δresp (1.49 × 105 au) is obtained with the system as depicted in Figure 1e/3.3, where the twist angle between the rings is around 65°. It is also important to note that almost the same value for δresp (1.47 × 105 au) is obtained for the monosubstituted o-betaine (where the CH3 group is on the ortho position of the pyridinium ring). For this system, the twist angle is 60.5°. However, trimethyl substituted obetaine (Figure 1f/3.3) has a twist angle around 80°, and it has more than 20 times lower δresp with respect to that of the model systems shown in Figure 1b,e/3.3. All the above-mentioned calculations have been performed with the gas phase optimized geometries. To verify the robustness and reliability of the gas phase results, it is always instructive to repeat these calculations in solvent phases too. For this purpose, we have also optimized all the substituted


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and unsubstituted o-betaine molecules in two different solvents, namely, cyclohexane and nhexane, and calculated their δresp with the corresponding solvent phase optimized geometries. The solvent phase δresp values are supplied in Table T14 in appendix T, and the corresponding TPA cross-sections (in Göppert-Mayer unit) are depicted in Figures 7a/3.3, 7b/3.3 and F9 in appendix F. The TPA cross-sections are evaluated considering 0.1 eV width of the spectra. These two figures clearly elucidate that the value of δresp in both solvents is maximum for monosubstituted o-betaine having a −CH3 group on the ortho position of the pyridinium ring where the twist angle is very close to 65°. For other substituted o-betaines, in particular, for the trisubstituted system, the twist angle between the rings is very close to 90°, and as a consequence, the δresp of this system is sufficiently small. We have also performed similar calculations with the substituted p-betaine (in the gas phase), and the results are presented in Table T15 in appendix T. In this case, the effect of the twist angle is even more prominent. The substituted p-betaine with three methyl groups at the ortho positions (Figure F10 in appendix F) has more than 105 times higher δresp than that of its tetra-substituted analogue. Finally, we believe that the present investigation will attract the attention of the experimentalist to synthesize new TPA active material based on substituted o-betaine.

3.3.4 Conclusion In conclusion, we have studied the variation of TP transition probability with the twist angle between the two rings of o-betaine using both quadratic response theory and recently developed GFSM. The GFSM analysis clearly explains that the term δ11 together with the channel interference (δ12) have dominant contribution to the overall TP transition probability of the system and that these terms become the maximum in the twist angle range of 65−70°. In this range, the magnitude of ground to excited state transition moment (μ01), excited state dipole moment (μ11), and the angle between these vectors are in favor of maximum δ3SM. We


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have also shown that the required optimum twist angle can be achieved by introducing one methyl group at the ortho position of the pyridinium ring of o-betaine. To be more realistic, we have also calculated the TP transition probability of all possible ortho substituted obetaine model systems in two solvents, namely, cyclohexane and n-hexane. We have found that the monosubstituted o-betaine (where one CH3 group is on the ortho position of pyridinium ring) has a twist angle very close to the optimum value and also has larger TP transition probability than other model systems. To the best of our knowledge, this is the first successful attempt to control the alignment between different transition moment vectors with respect to twist angle and hence to control the TPA process of a 3D system.


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Chapter - 4 On the origin of very strong twophoton activity of squaraine dyes - A standard/damped response theory study (Phys. Chem. Chem. Phys. 2014, vol. 16, pp. 8030-8035)

On the origin of very strong two-photon activity of squaraine dyes - A standard/damped response theory study

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4.1 Abstract In the present work, we report the mechanism of a very large increase in the two-photon (TP) activity of squaraine based molecules upon changing the substituents. The replacement of a specific fused ring by ethylene or ethyne moieties enhances the TP transition strength of these molecules up to the order of 1013 au (~1010 GM), both in the gas phase as well as in dichloromethane solvent. Our calculations decisively establish that the reason for this large enhancement in the TP activity of the studied systems is the severe decrease in the corresponding detuning energies. We explain this fact using the damped response theory calculations and provide a novel design strategy to control the detuning energy of such molecules. The results are benchmarked against the available experimental findings.

4.2 Introduction Even after 80 years since the first theoretical prediction1 and more than 50 years since the first experimental verification,2,3 two-photon absorption (TPA) is still a central theme of research in the field of non-linear optics. The quest for more and more two-photon (TP) active materials is triggered by the long list of applications in which they can be employed, in various advanced technological domains such as, among others,4-6 3D data storage,7 photodynamic therapy8 and optical limiting.9,10 The inner mechanism and the computational aspect of the TPA process became more transparent and definitely more easily accessible after the development almost thirty years ago of modern analytical response theory (RT)11-15 and, more recently, of time-dependent density functional theory (TDDFT).16-18 Using the latter, one can easily calculate the TPA strength of large molecules.19-21 Our groups have been quite active in the field; for recent studies see for instance ref. 22 and 23. The TP strength

for a

transition induced by a linearly polarized beam between the initial |0〉 and final | 〉 states is


Chapter 4

P a g e | 3/4

connected to the TP transition tensor

through the quantity

, given by the

expression:21,24-27

=2

+4 ,

The relationship between

(1.4) ,

and the TP transition strength, which in turn is directly related

to experiment and is usually measured in GM, is discussed for example in Ref. 21. In the equation above,

represents the TP transition tensor component and it depends on

the transition dipole moments (TDMs) and on the corresponding excitation energies. In a sum-over-states formalism, for the degenerate case of two photons with equal circular frequency

can be written as21,24-27

=

Here and above, , | 〉 and | 〉 and ∆

( )

+ ∆

(2.4)

= { , , },

is the α component of the TDM vector involving states

=ħ

⁄2 is the detuning energy. In Figure 1/4 the latter is

−

depicted for the specific case of a centrosymmetric molecule, for a TP transition between the ground XAg and the first excited state of the same symmetry (1Ag), with a nearly resonant intermediate 1Bu state.


Chapter 4

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Figure 1/4. Schematic diagram showing three important states, their symmetry and detuning energy involved in TPA process in centrosymmetric molecules

From Eq. (2) it is evident that the smaller the values of ∆ consequence the larger will be

( )

, the larger will be

. This suggests that, a careful control of the ∆

and as a ( )

may

lead to a sizeable increase in the TP activity. However, in the limits of double resonant TPA (RTPA), when ∆

( )

becomes zero, we encounter a singularity problem in Eq. (2) and the

standard approach to TPA, tailored to the case of single (ground and final states) resonance, fails. The results obtained using standard RT, for the nearly double resonant TPA processes, are in principle not reliable. Recently, damped response theory (DRT)28,29 has been employed to develop a protocol for the calculation of TPA.30 Since its development, DRT has proven to be an indispensable tool for studying several absorption and dichroism processes in large molecules also.31-35 The DRT approach to TPA (DTPA from now on) has advantages with respect to the standard approach (STPA below) also beyond the cases of double resonance. With DTPA, the observable is determined directly, at a given frequency, resorting to a modified damped cubic response function. Therefore the TPA spectrum may be evaluated for selected frequency ranges, something which is especially convenient for large molecules with a high density of states, where STPA encounters problems, being targeted at individual


Chapter 4

P a g e | 5/4

excited states. Moreover, in the implementation of ref. 30 an atomic-orbital based density matrix formulation, particularly suitable for studies on large systems, was used. Herein we intend to propose a new design strategy for materials having large TP transition strengths, by controlling the detuning energy, ∆

( )

. To this end, we have considered the

D-π-A-π-D structures (D = electron donor groups; A = electron poor 1,3-disubstituted C4O2 unit) of squaraine dyes, see Figure 2/4, which are well known for their large TP activity in combination with low energy, sharp and intense one photon (OP) absorption/emission bands in the visible to near infrared region.36-45

4.3 Computational Details The ground state geometries of all these molecules, in the gas phase and in CH2Cl2 solvent, were optimized at density functional level of theory using the B3LYP functional46-48 and two sets of basis sets: (a) 6-311G(d,p) for C, H and N atoms and (b) 6-311+G(d,p) for O atoms.49,50 An extra diffuse function on the O atoms was included to account for their excess negative charge. No constraint was applied during geometry optimization. Solvent phase geometry optimizations were carried out using the polarizable continuum model (PCM) of Tomasi et al.51,52 Frequency calculations on the optimized geometries produced all real frequencies for these systems, which ensure that the geometries belong to minima on their respective potential energy surfaces. All the geometry optimizations and frequency calculations have been performed using the Gaussian 09 suite of programs.53 In the next step, we have employed the linear and quadratic response theories within the framework of timedependent density functional theory (TDDFT), as implemented in DALTON program package,54,55 to compute the OP and TP absorption parameters of these molecules in both the gas phase as well as in CH2Cl2 solvent. The calculations were performed aiming at the lowest ten excited states for the molecules in C2h symmetry and at the lowest five for the


Chapter 4

P a g e | 6/4

system in a C2 arrangement. In this case also, PCM51,52 was used for the solvent phase calculations. In the final stage of our study, we have employed the DTPA30 implemented in LSDalton.56 A damping factor Γ of 0.004 au (0.109 eV, intended as half width at half maximum, HWHM) was employed in the DTPA calculations. All the response theory calculations employed the CAMB3LYP functional57-60 and the aug-cc-pVDZ basis set.61,62 CAMB3LYP has proven recently to be particularly suited for TPA studies involving in particular excited states of diffuse, charge transfer and Rydberg character.22,63-67

4.4 Results and Discussions The ground state optimized structure of all the systems in gas phase are given in Figure 2/4. The first four SQ1-based molecular structures (SQ1, SQ1-py, SQ1-C2H3 and SQ1-C2H) have a C2h point group symmetry arrangement, whereas the last one, SQ2, belongs to the C2 point group. It is important to mention that the SQ1 and SQ2 molecules are studied here since they can be seen as model systems for two experimentally studied molecules.68,69 The first one68 has n-Octyl groups replaced by -CH3 groups in SQ1. The n-Hexyl groups present in the pyrrole rings of the squaraine studied in Ref. 69 are replaced by –CH3 groups, and the -NBu2 substituents by -NMe2 groups, in SQ2, see Figure 2/4. The replacements were made in order to reduce the computational cost.


Chapter 4

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Figure 2/4. Pictorial representation of all the molecules considered in this work

The OP absorption (OPA) data for the lowest active state of the five squaraine dyes of Figure 2/4, both in gas phase and dissolved in CH2Cl2, are presented in Table 1/4 in the form of oscillator strengths,

=2

⁄3 ħ. The first OP active state of the SQ1-based

molecules belongs to the Bu symmetry representation (| 〉 = |1〉 = 1

), whereas that of SQ2

can be labeled as a B state (| 〉 = |1〉 = 1 ). From the results in Table 1/4 it is evident that the oscillator strength

for this state is the largest (2.903 in gas phase and 3.286 in

CH2Cl2) for SQ2, whereas it is the lowest (0.787 in gas phase and 1.092 in the solvated phase) for SQ1-py. This is mainly due to the difference in the extent of π-conjugation between the two molecules. That the extent of π-conjugation is maximized in SQ2 and at its minimum in SQ1-py is somewhat expected, considering the structures shown in Figure 1/4, is confirmed by the values of the ground to excited state transition dipole moment (

), which,

in the gas phase, are 7.758 au for SQ2 and 3.390 au for SQ1-py, respectively. The corresponding values in CH2Cl2 are 8.708 au and 4.052 au, respectively. Indeed the strong correlation between extension of the π-conjugation and size of the TDM is evident


Chapter 4

P a g e | 8/4

throughout the series of molecules shown in Figure 1/4, see Table 1/4. It is also important to note that on moving from the gas to the CH2Cl2 solvent phase, in spite of a decrease in values,

for all the systems increases. We have noticed that it happens due to the large

increase in the values of

and the quadratic dependence of

on

. The decrease in

indicates that in these molecules the excited state is more stabilized by the moderately polar CH2Cl2 solvent than the ground state. This, in turn, increases the extent of charge transfer leading to an increase of

and, therefore, of the intensity of the OPA peaks for all

the systems in the solvent phase. To get a better insight into these XAg→1Bu (or XA → 1B) transitions, we have extracted information on the most important contribution to the excitation vector in the response calculation, following a procedure employed for example in Ref. 66. The results are listed in Table 1/4. It is apparent that for all the SQ1-based molecules the transitions have a HOMO (Au) to LUMO (Bu) character and HOMO (A) to LUMO (B) character for SQ2. The corresponding gas phase orbital images are shown in Figure 3/4. Whereas for all the SQ1based molecules the HOMOs and LUMOs are spread throughout the whole molecule, for SQ2, the LUMO is mainly localized in the central region while the HOMO electron density is again delocalized over the entire molecule. Therefore, in SQ1-based molecules the ground to excited state transition is predominantly local, whereas in SQ2 it involves a charge transfer from the long π-conjugated arms to the central region. In all these cases the lowest OP transition is primarily of π-π* character.


Chapter 4

P a g e | 9/4

Table 1/4 - OPA parameters of all the systems in gas phase and solvated in CH2Cl2. Solvent phase data are given in bold-italics. [H ≡ Highest Occupied Molecular Orbital (HOMO) and L ≡ Lowest Unoccupied Molecular Orbital (LUMO)] The labels attached to H or L give the symmetry representation of the molecule these orbitals belong to (e.g. HBg ≡ HOMO belonging to the Bg symmetry representation of the C2h point group). The numbers in parenthesis give the amplitude (absolute value) of the dominant excitation in the TDDFT excitation vector.66 Systems

ħ

(eV)

Orbitals involved

(au)

SQ1

2.593 (2.487) 4.039 (4.654)

1.036 (1.319)

HAu-LBg (0.70) HAu-LBg (0.70)

SQ1-py

2.796 (2.748) 3.390 (4.052)

0.787 (1.092)


SQ1-C2H3

2.707 (2.620) 3.996 (4.687)

1.059 (1.406)


SQ1-C2H

2.712 (2.641) 3.890 (4.564)

1.006 (1.346)


SQ2

2.001 (1.791) 7.758 (8.708)

2.903 (3.286)

HA-LB (0.69) HA-LB (0.68)

Figure 3/4. Orbitals involved in the transition from ground to the lowest OP active state of the systems considered in this work

The TP molecular parameter

and detuning energies ∆

( )

corresponding to the largest TP

transition strengths for each of the systems studied here, computed both in gas phase and in CH2Cl2 solvent and obtained using STPA are collected in Table 2/4. We note that the value of

for SQ2 is larger than that for SQ1 in CH2Cl2, which is consistent with the


Chapter 4

P a g e | 10/4

experimental results for their analogue molecules. The reverse apparently holds for the gas phase.68,69 The ratio of

values for these two systems in gas phase is 9.50 × 107 : 1.01 ×

107 (≈9.4:1) whereas the same in a CH2Cl2 solution is 1.29 × 107 : 4.67 × 107 (≈1:3.6). The corresponding experimental ratio68,69 for analogue molecules in CH2Cl2 is 1:60. It is important to note that on moving from gas to solvent phase, the value of

(summed over

TP all allowed transitions considered, 0f 0f  ) decreases by more than six times for SQ1, f

whereas it increases by a factor of about four for SQ2. Note also that the most intense TP active state for SQ2 changes from 5A (in gas phase, ∆ solvent, ∆

= 3.43

= 4.27

) to 2A (in CH2Cl2

) whereas it remains the same (4Ag, with a change in excitation

energy from 5.11 eV to 5.12 eV) for SQ1. The opposite trend for the TPA of these two molecules may be correlated with the corresponding change in detuning energy on moving from gas to the solvated medium. Upon addition of substituents as the vinylinic or acetylenic groups to SQ1-py, δTP increases instead to a large extent, especially when taking the effect of the solvent into account. This increase is related to the change in detuning energy ∆ substitution. The decrease of ∆

( )

( )

upon

afforded by the solvent is much larger than that in the gas

phase, particularly for the SQ1-C2H3 and SQ1-C2H molecules. Moving from the gas phase to CH2Cl2, ∆

( )

for SQ1-C2H3 changes from 0.050 eV to 0.004 eV, the same value is

obtained for SQ1-C2H down from 0.028 eV. Such a small value of ∆ large value for

( )

leads to a very

of these two systems, reaching as much as 1.15 × 1013 au for SQ1-C2H3

(δTP ≈ 1010 GM).


Chapter 4

P a g e | 11/4

TP f  Table 2/4 - TPA molecular parameter  0 f (in au) and detuning energy,  E i (in eV) corresponding to the largest TP transition strengths (among the lowest ten excited states of each TP allowed symmetry for SQ1 and its derivatives, the lowest five in both A and B symmetries for SQ2) for each of the systems studied here, both in gas phase and solvated in CH2Cl2. STPA results. The label of the final excited state considered for this analysis is also given. The intermediate nearly resonant state is in all cases the 1Bu state (1B for SQ2) state. See also Figure 4/4.

In gas phase System SQ1 SQ1-py SQ1-C2H3 SQ1-C2H SQ2

f 

Sym 4Ag 3Ag 4Ag 3Ag 5A

E i

(eV) 0.038 0.087 0.050 0.028 0.133

In CH2Cl2

 0TfP (au)

Sym

f E i  (eV)

 0TPf (au)

9.50 × 107 2.21 × 107 2.60 × 108 1.26 × 109 1.01 × 107

4Ag 3Ag 3Ag 3Ag 2A

0.072 0.064 0.004 0.004 0.079

1.29 × 107 2.91 × 108 1.15 × 1013 8.92 × 1011 4.67 × 107

As mentioned above, in the cases where the conditions of double resonance are approached in a TPA process, the limitations of STPA make its analysis particularly critical. In the case of molecules of large size, and in regions of the excitation spectrum where the density of excited states becomes notable, one way to circumvent the shortcomings of STPA can be found in DTPA, currently implemented for isolated molecules.30 Out of the five molecules discussed in this paper, SQ1 and its derivatives were found to be manageable with current computational resources in our laboratories. In order to verify the predictions of gas-phase STPA, we have analyzed all these four SQ1-based molecules using DTPA. In DTPA, an empirical complex damping parameter iΓ is introduced into the response function equations, thereby extending their realm to the entire range of frequencies, including the complex ones. The first formulation of damped response theory, not yet extended (to our knowledge) to the study of TPA, and where the empirical damping term is introduced in the standard Ehrenfest equation, was presented by Norman et al. in 2005.28 An equivalent formulation was proposed more recently by Kristensen et al.29 In ref. 29 the authors introduced empirical excited state lifetimes directly into the cubic response functions, using therefore complex excitation energies. In ref. 30 this approach to damped response theory was extended to the treatment of TPA. DTPA, unlike STPA, yields the whole absorption profile in the desired frequency range


Chapter 4

P a g e | 12/4

directly, without any reference to the individual states. Differences between STPA and DTPA arise in particular when the absorption TP wavelength approaches the excitation energy of the intermediate states. This case of double resonance (the first resonance corresponding to the conservation of energy condition in TPA, 2 which together amount therefore to ∆

( )

≈

, the second to the condition

≈

,

≈ 0 ) is problematic not only for theoretical

calculations but also for experiment, as TPA and OPA spectra tend to overlap. In the present case, we have employed DTPA to calculate the gas-phase δTP of all the four SQ1-based molecules in a range of frequencies from 0.030 to 0.095 au with a step of 0.01 au. This range covers for all these four molecules the area of the spectrum approaching the double resonance condition, placed roughly between 2.7 and 2.8 eV (the range of excitation energies of the lowest excited – nearly resonant – Bu excited state). The DTPA spectra are shown in Figure 4/4, where the DTPA results are shown together with the corresponding stick STPA data, identifying in red final states of Ag symmetry, in blue those of Bg symmetry. The final states of both symmetries with TP strengths weaker than needed to appear as sticks in the spectra are identified with dotted vertical lines with the appropriate color code. The vertical dotted lines in dark green indicate the position of the lowest OP active (Bu) state. The detuning energies discussed above are explicitly indicated. All four molecules display very large values for δTP, particularly strong in the 2.7-2.8 eV photon frequency range. The remarkable absorption is due to the particular TP efficiency of the total symmetric excited states lying in the region, and it is boosted by a near resonance with the first OP active state of Bu symmetry.


Chapter 4

P a g e | 13/4

Figure 4/4 Gas-phase TPA spectra of four out of the five molecules studies here. DTPA spectra (   ) reported together with the corresponding stick STPA results ( TP 0f  ). For the latter, a color TP

code distinguishes Ag (red) and Bg (blue) TP active states. Dotted vertical lines indicate TP active states whose TP strength is not showing in the scale of the figure. The vertical dotted line in dark green shows the location of the lowest OP active (TP transparent) 1Bustate. Note that the frequency in abscissa is that of the TP process, and therefore the sticks show at half the excitation energy of the final state, whereas the dark green dotted lines are at the excitation energy of the 1Bu state. That allows to display on each panel the detuning energy, defined here as  XA ,nA 2   XA ,1B , where g

g

g

u

the final TP active state nAg yields the most intense TP strength in the panel.

4.5 Conclusion In the first systematic ab initio TPA study carried out within the framework of the rather recently developed damped response theory approach,30 we have studied the TPA process in squaraine based D-π-A-π-D type of molecules, whose two photon activity can largely be improved by simple geometrical substitutions. Indeed our calculations clearly show that removing the fused ring and placing the ethylene or ethyne moieties with the symmetry point group unaltered, decreases the detuning energies (∆

( )

) of these molecules to a great extent,


Chapter 4

P a g e | 14/4

which in turn causes an enormous rise in their TP activity. The use of a damped TPA approach proves to be of major relevance in order to validate our results and to avoid the breakdown of standard response theory under the double resonance conditions applying in the small ∆

( )

tune the ∆

limit. In short, the present study decisively establishes that it is possible to fine ( )

values and hence the TP activity of squaraine based systems by simple

chemical modification.

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38. S.-H. Kim, J.-H. Kim, J.-Z. Cui, Y.-S. Gal, S.-H. Jin, and K. Koh, Dyes Pigm., 2002, 55, 1. 39. S. Yagi, S. Murayama, Y. Hyodo, Y. Fujie, M. Hirose and H. Nakazumi, J. Chem. Soc., Perkin Trans. 1,2002, 6, 1417. 40. S. A. Odom, S. Webster, L. A. Padilha, D. Peceli, H. Hu, G. Nootz, S.-J. Chung, S. Ohira, J. D. Matichak, O. V. Przhonska, A. D. Kachkovski, S. Barlow, J.-L. Brédas, H. L. Anderson, D. J. Hagan, E. W. V. Stryland, and S. R. Marder, J. Am. Chem. Soc., 2009, 131, 7510. 41. E. Collini, S. Carlotto, C. Ferrante, R. Bozio, A. Polimeno, J. Bloino, V. Barone, E. Ronchi, L. Beverina and G. A. Pagani, Phys. Chem. Chem. Phys., 2011, 13, 12087. 42. C.-T. Chen, S. R. Marder, and L.-T. Cheng, J. Am. Chem. Soc., 1994, 116, 3117. 43. F. Terenziani, A. Painelli, C. Katan, M. Charlot and M. Blanchard-Desce, J. Am. Chem. Soc., 2006, 128, 15742. 44. L. Beverina, M. Crippa, M. Landenna, R. Ruffo, P. Salice, F. Silvestri, S. Versari, A. Villa, L. Ciaffoni, E. Collini, C. Ferrante, S. Bradamante, C. M. Mari, R. Bozio and G. A. Pagani, J. Am. Chem. Soc., 2008, 130, 1894. 45. K. D. Belfield, M. V. Bondar, H. S. Haniff, I. A. Mikhailov, G. Luchita, and O. V. Przhonska, ChemPhysChem, 2013, 14, DOI: 10.1002/cphc.201300447, First published online 10 Sept. 2013. 46. A. D. Becke, J. Chem. Phys., 1993, 98, 5648. 47. A. D. Becke, Phys. Rev A, 1988, 38, 3098. 48. C. Lee, W. Yang and R. G. Parr, Phys. Rev. B, 1988, 37, 785. 49. A. D. McLean and G. S. Chandler, J. Chem. Phys., 1980, 72, 5639. 50. K. Raghavachari, J. S. Binkley, R. Seeger, and J. A. Pople, J. Chem. Phys.,1980, 72, 650. 51. S. Miertus, E. Scrocco and J. Tomasi, Chem. Phys., 1981, 55, 117. 52. J. Tomasi, B. Mennucci and R. Cammi, Chem. Rev., 2005, 105, 2999. 53. Gaussian 09, Revision C.01, Gaussian, Inc., Wallingford CT, 2010. 54. DALTON, a molecular electronic structure program; release Dalton2011 (Rev. 0, July 2011) http://www.kjemi.uio.no/software/dalton/dalton.html (2011). 55. WIREs Comput Mol Sci 2013. doi: 10.1002/wcms.1172 56. LSDalton, a molecular electronic structure program, Release 2013, written by V. Bakken, S. Coriani, P. Ettenhuber, S. Hoest, I.-M. Hoeyvik, B. Jansik, J. Kauczor, T. Kjaergaard, A. Krapp, K. Kristensen, P. Merlot, S. Reine, V. Rybkin, P. Sałek, A. J. Thorvaldsen, L. Thoegersen, M. Watson, M. Ziolkowski, T. Helgaker, P. Jørgensen, J. Olsen.


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57. Y. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett., 2004, 393, 51. 58. M. J. G. Peach, T. Helgaker, P. Sałek, T. W. Keal, O. B. Lutnæs, D. J. Tozer and N. C. Handy, Phys. Chem. Chem. Phys., 2006, 8, 558. 59. M. J. Paterson, O. Christiansen, F. Pawłowski, P. Jørgensen, C. Hättig, T. Helgaker, and P. Sałek, J. Chem. Phys., 2006, 124, 054322. 60. D. Shcherbin and K. Ruud, K., Chem. Phys., 2008, 349, 234. 61. T. H. Dunning, Jr., J.Chem. Phys., 1989, 90, 1007. 62. R. A. Kendall, T. H. Dunning, Jr. and R. J. Harrison, J. Chem. Phys., 1992, 96, 6796. 63. B. Jansík, A. Rizzo and H. Ågren, J. Phys. Chem. B, 2007, 111, 446;Erratum:J. Phys. Chem. B, 2007, 111, 2409. 64. N. Lin, X. Zhao,A. Rizzo and Yi Luo,J. Chem. Phys, 2007, 126, 244509. 65. B. Jansík, A. Rizzo, H. Ågren and B. Champagne, J. Chem. Theory Comput., 2008, 4, 457. 66. A. Rizzo, N. Lin and K. Ruud, J. Chem. Phys., 2008, 128, 164312. 67. M. Guillaume, K. Ruud, A. Rizzo, S. Monti, Z. Lin and X. Xu, J. Phys. Chem. B, 2010, 114, 6500. 68. L. Beverina, M. Crippa, P. Salice, R. Ruffo, C. Ferrante, I. Fortunati, R. Signorini, C. M. Mari, R. Bozio, A. Facchetti, and G. A. Pagani, Chem. Mater., 2008, 20, 3242. 69. S.-J. Chung, S. Zheng, T. Odani, L. Beverina, J. Fu, L. A. Padilha, A. Biesso, J. M. Hales, X. Zhan, K. Schmidt, A. Ye, E. Zojer, S. Barlow, D. J. Hagan, E. W. V. Stryland, Y. Yi, Z. Shuai, G. A. Pagani, J.-L. Brédas, J. W. Perry and S. R. Marder, J. Am. Chem. Soc., 2006, 128, 14444.


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Chapter - 5 Role of donor-acceptor orientation on solvent-dependent three-photon activity in through-space chargetransfer systems - Case study of [2,2]paracyclophane derivatives (Phys. Chem. Chem. Phys. 2013, vol. 15, pp. 17570-17576)

Role of donor-acceptor orientation on solvent-dependent three-photon activity in through-space charge-transfer systems - Case study of [2,2]-paracyclophane derivatives

Chapter 5

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5.1 Abstract We study the effect of donor-acceptor orientation on solvent-dependent three-photon transition probabilities (δ3PA) of representative through-space charge-transfer (TSCT) systems, namely, doubly positively charged [2,2]-paracyclophane derivatives. Our cubic response calculations reveal that the value of δ3PA may be as high as106 a.u., which can further be increased by a specific orientation of the donor-acceptor moieties. To explain the origin of the solvent cum orientation dependency of δ3PA, we have calculated different threephoton tensor components using a two-state model, noting that only a few tensor elements contribute significantly to the overall δ3PA value. We show that this dependence is due to the large dipole-moment difference between the ground and excited states of the systems. The dominance of a few tensor elements indicates a synergistic involvement of π-conjugation and TSCT for the large δ3PA of these systems.

5.2 Introduction The need for highly sensitive experimental set-up and the computational costs has made the study of multi-photon absorption (MPA) processes a challenging task. In addition to the experimental and computational obstacles, one obvious problem of MPA processes is that their transition probabilities are found to decrease with increasing order of the MPA processes. Because of these limitations, very few processes beyond the lowest-order MPA process, two-photon absorption (2PA),1-13 have been extensively studied, theoretically as well as experimentally. MPA processes are characterized by the high spatial confinement of the excitations and the use of long wavelength radiation, relying on the non-linear relation between the transition probabilities and the intensity of the incident radiation and arising because of the involvement of intermediate virtual states in the excitation process. The higher


Chapter 5

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the order of the MPA process, the more tangible these qualities of the MPA processes are expected to be.14-17 The improved sensitivity of the higher-order MPA processes over 2PA have raised expectations in technologically advanced fields. In the recent past, some theoretical and experimental works have been devoted to study the higher-order MPA processes, mainly the three-photon absorption (3PA) process. He et al.14 have shown that 3PA can be used in frequency up-conversion lasing, short-pulse optical communications and the measured15 3PA cross-section for a thiophene derivative in THF solvent is 8.8 × 10-76 cm6 s2. Maiti et al.16 in 1997, have applied the 3PA technique in the field of bio-imaging and imaged tryptophan and serotonin molecules. Similarly the applications18-21 of 3PA in optical limiting, short-pulse fiber communication and light-activated therapy have been demonstrated. From theoretical point of view, efficient theoretical/computational studies of the 3PA process of realistic systems has become possible with the implementation of cubic response (CR)29 theory in

the framework of time-dependent density functional theory

(TDDFT).30,31 Using this approach, Cronstrand et al.22-24 and Sałek et al.25 have theoretically studied some larger molecular systems. Lin et al.26 studied the effect of a solvent on the 3PA of symmetric charge-transfer molecules. As the full sum-over-states calculations are computationally very expensive, the alternative tool i.e. the few-state model has also been used recently for studying the 3PA process but unlike the case of 2PA, a non-systematic and noticeable divergence has been reported in this case. With this introductory background, it is safe to say that till now the study of real life 3PA applications using the theoretical model is limited to the calculation of 3P transition probabilities/cross-sections and is still in its infancy. In order to have a proper control of the 3PA activity of a system, several facets of this field need proper exploration. As seen for the TPA process,32-34the orientation of donor-acceptor groups in a system could have profound impact on the 3PA process. In the present work, our aim is to unravel the


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mechanisms of the 3PA process by studying the effect of donor-acceptor orientation and that of the solvent, on a hitherto unexplored class of molecules, namely a through-space chargetransfer (TSCT) system. For this purpose, we have chosen three doubly positively charged [2,2]-paracyclophane derivatives having different orientations of the donor-acceptor moieties. The three-photon (3P) transition probabilities and other required parameters of the first two excited states of all these systems in gas phase and in two different solvents (MeCN and tetrahydrofuran (THF)) are calculated using CR theory as implemented within TDDFT.29 Solvent-phase CR calculations have been performed using the Polarizable Continuum Model (PCM).35 Furthermore, the origin of this solvent cum orientation dependency have been explained using two-state model (2SM) calculations in both the gas as well as in different solvent phases.

5.3 Computational Details Geometries of the PCP1molecule (see Figure 1/5) in gas phase and in two different solvents (MeCN and THF) have been taken from our previous work4 where the optimization was done at the B3LYP/6-311G (d,p) level of theory. The other two molecules were optimized at the same level of theory using the Gaussian 03 suite of programs,36 the solvent effects in all cases described by the PCM.37With these optimized geometries we have calculated, from the residues of TDDFT-based CR functions, the 3PA parameters for the transition from the ground to the first excited state of all systems both in the gas as well as in the two solvent phases. These calculations were performed using the CAMB3LYP38 functional and Dunning’s cc-pVDZ basis set.39 This combination of exchange-correlation functional and basis set has been shown to work accurately in case of the TPA of intramolecular chargetransfer molecules6,40 and we expect it to perform well in the study of 3PA as well. For the solvent-phase calculations, the non-equilibrium formulation35 of CR theory within PCM has


Chapter 5

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been used and all the response calculations have been performed using the DALTON program package.41 Considering the huge computational cost for calculating the 3PA transition probabilities of the systems in the solvent, we restrict this study to two solvents only, one polar (MeCN) and another of intermediate polarity (THF). After the response theory calculations, we have reevaluated the 3PA parameters using a two-state model approach.

5.4 Results and discussion 1 1´ 6 5

1

1´

6´

2

2´

4

2´

6

5´ 3

6´

X

3´

3

3´

5

X 4

5´

4´

2

4´ Z

Z

PCP1

PCP2 1

2

3 1´ 2´ 6 4 3´ 6´ 5 5´ 4´

X

Z

PCP3 Figure 1/5. A schematic representation of the three derivatives of the doubly positively charged [2,2]-paracyclophane (PCP).The red-colored atom-labeling is for the C-atoms of the above-plane benzene ring which contains the acceptor -NMe3+ moieties and the black ones for the below the plane benzene ring containing the donors -NMe2.


Chapter 5

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Table 1/5. OPA parameters for S0-S1 transition in PCP1, PCP2 and PCP3 molecules in gas and solvent phases, calculated at CAMB3LYP/cc-pVDZ level of theory System

PCP1

PCP2

PCP3

Solvent

x

Δμff (a.u.) y z

μ0f (a.u.) Total

x

y

z

Total

ωf (eV)

Λ

δOPA

Gas

0.0

3.991

-0.0

3.991

0.136

0.0

-0.137

0.193

2.511

0.204

0.002

THF

0.001

4.342

-0.002

4.342

0.155

0.0

-0.155

0.219

2.951

0.216

0.002

MeCN

-0.002

4.296

0.001

4.296

0.159

0.0

-0.156

0.223

3.039

0.220

0.003

Gas

-0.353

3.976

0.001

3.992

0.010

0.046

-0.033

0.057

2.761

0.222

0.0

THF

-0.349

4.277

-0.007

4.291

0.114

0.031

-0.036

0.124

3.084

0.234

0.0

MeCN

-0.325

4.238

-0.005

4.250

0.112

0.034

-0.034

0.122

3.163

0.237

0.0

Gas

0.913

3.674

1.408

4.039

-0.072

-0.243

-0.057

0.260

2.738

0.172

0.004

THF

1.345

4.447

0.110

4.647

0.043

-0.050

-0.024

0.071

3.262

0.200

0.0

MeCN

4.207

0.643

-1.615

4.552

-0.012

0.046

-0.032

0.058

3.352

0.204

0.0

The systems studied in this contribution are shown in Figure 1/5. We will refer to these molecules as PCP1, PCP2 and PCP3, respectively. In all molecules, -NMe2 and -NMe3+ are respectively the donor and acceptor groups. The two donor groups are attached to one of the benzene rings and the two acceptors are attached to the second benzene ring of the systems. This particular configuration favors the TSCT nature of the system. In PCP1, the donor moieties are placed at the (2, 5) positions whereas the acceptor moieties are located at the (3´, 6´) positions of the two rings. Similarly, in PCP2 and PCP3 the donor groups are placed at (3, 6) and (2, 6) positions and acceptors at (3´, 6´) and (3´, 5´) positions respectively. For clarity, the atom-labeling is also shown in Figure 1/5. Furthermore, to make the discussion easier, the orientation of the donor and acceptor groups in PCP1 will be marked by “×”. In PCP2, the orientation of the donors and acceptors can be represented by “=”, indicating that both donor moieties are placed on one benzene ring, while the acceptors are attached to the other ring. Finally, PCP3 can be symbolized by “X” and here the upper part of “X” represents the donors and lower one represents the acceptors. We first consider the one-photon absorption (OPA) process in the three systems. The OPA data as obtained using linear response theory for the first excited state of all three systems in


Orbitals H-L (0.70) H-L (0.70) H-L (-0.70) H-L (-0.70) H-L (0.70) H-L (-0.70) H-L (-0.66) H-L (-0.69) H-L (-0.69)

Chapter 5

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both the gas as well as in different solvent phases are reported in Table 1/5. From the table, we note that the first excited state of all three molecules is weakly OPA active in both of the two solvents as well as in gas phase. The oscillator strength (δOPA), which determines the strength of the one-photon transition in a system, is directly proportional to the product of the ground- to excited-state transition energy and the square of the corresponding transition dipole moment vector (µ0f). It is obvious from Table 1/5 that both of these quantities (in a.u.) have very small values for all the three molecules in both the gas as well as in the different solvent phases, making their δOPA very small and thus the excitation is only weakly OPA active. These results are also consistent with our previous work4 on the one- and two-photon absorption of PCP1. We have already rationalized the long-range nature of the S0-S1 transition for PCP1 in our previous work. Similar to that study, we have here computed the contributions of different orbitals involved in the S0-S1 transition of the three molecules and the results are given in Table 1/5. The results clearly show that irrespective of the nature of the solvent, the dominant contribution to the S0-S1 transition comes from the HOMO-LUMO orbital pairs for all molecules. These orbitals are depicted in Figures 2/5, 3/5 and 4/5 for the three molecules, and clearly show that the HOMO is mainly located on the donor side of the molecules whereas the LUMO is localized on the acceptor side. This reveals the TSCT nature of the S0-S1 transition in all these molecules in both the gas as well as in different solvent phases.


Chapter 5

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Figure 2/5. HOMO, LUMO and differential electron density mapping (HOMO-LUMO) of PCP1 (a, b, c)



Chapter 5

P a g e | 9/5


In order to understand the nature of this transition in more detail, we have plotted the differential electron density plot between the HOMO-LUMO orbital pair for all three molecules. These plots are generated by subtracting the Gaussian cube files containing the HOMO and LUMO electron densities (isovalue = 0.06) and are collected in Figures 2/5, 3/5 and 4/5. It is obvious from the plots that, for both PCP1 and PCP2, no changes occur on moving from either the gas phase to the solvent phase or from one solvent to another. In contrast, PCP3 shows significant changes in the differential electron density mapping in the gas and solvent phases. These changes are observed only in the electron density of the donor group containing the benzene ring, where one of the -NMe2 is not participating in the TSCT process in the solvent phases. To further clarify the nature of the S0-S1 transition, we have computed the Λ parameter,42,43 which can quantify the long-/short-range nature of a transition. It may have values between 0 and 1, and the long-range nature of a transition is identified by a small value of Λ, indicating small orbital overlap. The results are given in Table 1/5. From these data, we see that irrespective of the nature of solvent the value of Λ for all three molecules is very small (around 0.2), which clearly indicates that the S0-S1 transition Role of donor-acceptor orientation on solvent-dependent three-photon activity in through-space charge-transfer systems - Case study of [2,2]-paracyclophane derivatives

Chapter 5

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in all these molecules is truly of long-range nature. We note that although the validity of DFT for studying TSCT transitions may be questioned, the reliability of the CAMB3LYP functional for TSCT transitions has been verified against CC2 calculations for transitions of the same kind as those studied here.44 After the above OPA discussion, we now move to the 3PA process in the three molecules. However, before discussing our results, let us briefly outline the theoretical basis for our calculations. Being related to the fifth-order susceptibility, the three-photon transition probability (δ3PA) is a challenge for ab initio calculations. The δ3PA of a system can be expressed in a simple manner in terms of 3P transition tensor elements (Tijk), and for linearly polarized light this relation can be written as45 =

1 35

2

+3

(1.5)

, ,

Where, the factor (1/35) arises because of orientational averaging. The Tijk terms appearing in Eq.1.5 can be obtained from the single residue of the CR function. Alternatively, for a resonant absorption, Tijk can be expressed in terms of different components of third-order transition dipole moment vectors and the corresponding transition energies and these can easily be extracted from the residues and poles of linear () or quadratic () response functions. The relationship between Tijk, different transition moments and excitation energies is given by ⟨0| | ⟩

= ,

⁄3

−

⟨ |

| ⟩

−2

⁄3 (2.5)

= ,

−

⁄3

−2

⁄3

where, i   i  is the ith component of the transition dipole moment vector for a transition from the αth state to the βth state, ωα is the transition energy from the ground to the


Chapter 5

αth state, and

P a g e | 11/5

p

ijk

represents the summation over all the permutations of indices i, j and k

which runs over the Cartesian coordinates x, y and z. Table 2/5. δ3PA (in 106 a.u.) and 3P tensor elements (Tijk) of the first excited state of PCP1, PCP2 and PCP3 molecules in gas and different solvent phases, calculated using response theory as well as 2SM at the CAMB3LYP/cc-pVDZ level of theory. The first number in the last row represents the response theory results, whereas the second one is from two-state model calculations System

PCP1

PCP2

PCP3

Solvent

Gas

THF

MeCN

Gas

THF

MeCN

Gas

THF

MeCN

Txxx

249.66

248.0

230.20

-582.25

535.80

495.26

-42.49

-139.11

562.93

Tyyy

-29.53

-45.66

-43.79

-2140.42

1350.81

1355.02

-7651.19

-1223.07

-266.04

Tzzz

-352.22

-461.86

-441.18

-28.60

-102.36

-96.42

-1203.78

-48.82

-82.75

Txxy

-0.06

0.01

1.87

119.23

-146.12

-115.29

-129.07

-141.56

-24.78

Txxz

-234.25

-259.95

-247.36

69.64

-84.11

-76.18

-33.58

47.66

-72.42

Tyyx

2358.76

1979.14

1801.20

-1867.03

1881.99

1703.36

-1313.69

-447.25

-115.70

Tyyz

-2379.52

-2079.93

-1904.57

659.98

-737.07

-683.65

-3310.65

-182.78

88.91

Tzzx

217.47

244.10

235.42

18.92

-47.06

-46.84

-302.56

-40.95

-77.88

Tzzy

-5.22

-7.16

-8.42

7.85

-36.20

-36.70

-1852.35

-36.76

-77.59

Txyz

0.01

-0.63

1.29

-0.06

2.46

3.40

-529.38

-95.83

60.69

3.39

2.65

2.23

1.85

1.50

1.28

16.30

0.34

0.05

2.71

2.56

2.25

0.93

0.74

0.65

17.91

0.48

0.14

With this brief theoretical background, we now turn to the results of our 3PA study. The values of δ3PA for the first excited state of all the three systems in gas as well as in different solvent phases, along with the different three-photon tensor elements Tijk are reported in Table 2/5. The data in Table 2/5 clearly indicate that irrespective of the nature of the solvent, all three systems have large δ3PA values (≥ 106 a.u.). At the same time, all these molecules are consistently more 3PA active in gas phase as compared to the solvent phase. The δ3PA in the less polar solvent, THF, are always larger than that in the more polar MeCN solvent. Exactly the same solvent dependence for the two-photon transition probability of PCP1 was observed


Chapter 5

P a g e | 12/5

in our previous study.4 This is not surprising since the basic mechanisms of both the 2PA and 3PA processes are very similar and hence can be expected to be affected in a similar manner by different solvents. The results also show that in the gas phase, PCP3 have much larger δ3PA values than the other two molecules. The value of gas phase δ3PA for PCP3 is indeed the largest value obtained in this work. When going from gas phase to solvent phase, PCP3 becomes much less 3PA active than both PCP1 and PCP2. To understand the origin of this orientation cum solvent-dependent 3PA activity of the three systems, we can inspect the relative contributions of the different 3PA tensor elements (Tijk). These are supplied in Table 2/5, and shows that the largest contributions in the “×” orientation (PCP1) are Tyyx and Tyyz both in the gas and solvent phases. Similarly, in the “=” orientation (PCP2), the largest contributions come from Txxx, Tyyy, Tyyx and Tyyz. However, in case of the “X” orientation (PCP3), unlike the other two molecules, most of the Tijk terms contribute significantly to the overall δ3PA values. It is important to note that if we divide the molecules in two parts (by cutting perpendicular to the C2-C3 and C6-C5 bonds, see Figure 1/5) then in both the “×” and “=” orientation the two parts will have both donors and acceptors, but in the “X” orientation, one part will have only the two donor groups whereas the other part will have two acceptor groups. This characteristic distinguishes PCP3 from the other two systems and is probably the origin of the large solvent dependency of its 3PA activity. In order to understand the dominance of the selective Tijk terms and the effect of donoracceptor orientation, we have considered the sum-over-states expression of the 3PA transition probability in Eq. 2.5. Although Eq.2.5 contains easily understandable quantities, the analysis of the expression is made difficult by the large number of intermediate states that appear in the summations. To get insight into the qualitative origin of the 3PA process of the systems, we have simplified Eq. 2.5 by using a two-state model (2SM) approach involving only the


Chapter 5

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ground (S0) and the first excited states (S1). Unlike 2PA, the expression for the tensor elements involved in the 3PA process is still complicated. Within the 2SM, after expanding Eq. (2.5), each Tijk will have 19 terms, given by

Tijk 

9 0f 00 00 0 f 00 00 0 f 00 00 0 f 0f i00  00   k00 i00  0j f   k00  00 j  k  i  k  j   j i  k   j  k i j i 2 2f





9 00 0 f ff 0f ff 00 0 f ff 00 0 f ff 00 0 f ff i  j k  i00  k0 f  jff   00 j i  k   j  k i   k  i  j   k  j i 2 f



9 i0 f  jff  kff  i0 f  kff  jff   0j f i ff  kff   0j f  kff i ff   k0 f i ff  jff  k0 f  jff i ff 2 2 f









 

27 i0 f  0j f  k0 f 2 2f (3a.5)

This equation can be reduced to a simpler form using i ff to indicate the difference in dipole moment between the excited and ground states of the molecule

Tijk

9  2 2f

 i ff  jff  k0 f  i ff  kff  0j f   jff  i ff  k0 f   jff  kff  i0 f  ff ff 0f ff ff 0f 0f 0f 0f    k  i  j   k  j  i  3 i  j  k

  

(3b.5)

And finally, Tijk 

9 2  iff  jff  k0 f   i ff  kff  0j f   jff  kff  i0 f  3 i0 f  0j f  k0 f 2 2f







(3c.5)

Eq.3c.5 shows that the values of the different Tijk within the 2SM depend on ωf and the different components of Δµff and µ0f. It is also apparent from this equation that large positive values of different components of Δµff and a concomitant small value of ωf will ensure a large value of the corresponding Tijk. It must also be noted that large positive values of different components of µ0f decrease the overall Tijk because of the last term in equation 3c.5. However, in the calculations one must keep in mind that the overall values of different Tijk strongly depend on both the magnitude as well as on the sign of the three components of Δµff and µ0f. From Table 1/5, it can be noted that the magnitude of Δµff is much larger (more than


Chapter 5

P a g e | 14/5

75 times, e.g. for PCP3 in MeCN solvent) than that of µ0f for all the systems in both the gas and solvent phases. For PCP1 and PCP2, the x and z components of Δµff are very small compared to its y component. In PCP1, the y component of µ0f is zero, and for this reason, all terms except Tyyx and Tyyz are very small for this system in the gas phase. The large contribution of Tyyx and Tyyz in PCP2 can be explained in a similar way. In PCP3, all the components of Δµff and µ0f are much higher than in PCP1 and PCP2, which explains the significantly larger 3P tensor elements of the PCP3 system. The large magnitude of all the components of Δµff and µ0f is due to the orientation of the donor-acceptor groups in PCP3. However, on moving from the gas phase to the solvent phases, a noticeable decrease in the value of the y component of µ0f in PCP3 is observed, causing a significant decrease in a large number of the 3P tensor elements which, in turn, is responsible for the overall reduction in the δ3PA values of PCP3 in THF and MeCN as compared to that of the gas phase. The solvent-dependent quenching of the charge-transfer strength in the y direction of the PCP3 molecule is evident from the differential electron density plot in Figure 4/5, which explains the origin of this anomalous solvatochromic 3PA activity of PCP3.

5.5 Conclusion In conclusion, using TDDFT/CR theory and a two-state model, we have studied the effect of donor-acceptor orientation on the solvent-dependent 3PA of an unexplored class of compounds, i.e. TSCT type of molecules, namely the doubly positively charged [2, 2]paracyclophane derivatives. The results show that the gas phase 3P transition probabilities are as high as 107 a.u. for a particular orientation of the donor-acceptor groups and for this orientation it decreases dramatically on going from gas to solvent phase. This orientation cum solvent effect has been analyzed by inspecting the magnitude of the dominant 3P tensor components. In the gas phase, for this specific orientation of the donor-acceptor groups, the


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difference in the dipole moment between the ground and excited states becomes very large, thereby allowing most of the tensor elements to contribute to the overall δ3PA, making the “X” orientation the most favorable in terms of boosting the 3PA. However, in the solvent phases, the value of the ground- to excited-state transition moment decreases significantly for this particular orientation, resulting in a reduction in the corresponding δ3PA values. The differential electron density plot clearly suggests that the lowering of the charge-transfer strength in the y direction in presence of the solvent can probably be attributed to this orientation cum solvent-dependent 3PA activity for the PCP3 molecule. Although, this study is restricted to [2,2]-paracyclophane type of molecules, in future other class of compounds including other TSCT systems may be explored for such orientational dependency of 3P activity.

5.6 References 1

M. G. Mayer, Ann. Phys.1931, 9, 273.

2

M. M. Alam, M. Chattopadhyaya and S. Chakrabarti, Phys. Chem. Chem. Phys. 2012, 14, 1156.

3

M. M. Alam, M. Chattopadhyaya and S. Chakrabarti, Phys. Chem. Chem. Phys.2011, 13, 9285.

4

M. M. Alam, M. Chattopadhyaya, S. Chakrabarti and K. Ruud, J. Phys. Chem. Lett.2012, 3, 961.

5

M. M. Alam, M. Chattopadhyaya and S. Chakrabarti, J. Phys. Chem. A. 2011, 115, 2607.

6

S. Chakrabarti and K. Ruud, Phys. Chem. Chem. Phys. 2009, 11, 2592.

7

L. Ferrighi, L. Frediani, E. Fossgaard and K. Ruud, J Chem. Phys. 2007, 127, 244103.

8

P. Cronstrand, Y. Luo and H. Ågren, J. Chem. Phys. 2002, 117, 11102.

9

P. Sałek, O. Vahtras, T. Helgaker and H. Ågren, J. Chem. Phys, 2002, 117, 9630.

10 W. Kaiser and C. G. B. Garrett, Phys. Rev. Lett.1961, 7, 229. 11 M. Albota, D. Beljonne, J-L. Brédas, J. E. Ehrlich, J-Y. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Mardar, D. McCord-Maughon, J. W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W. W. Webb, X-L. Wu and C. Xu, Science.1998, 281, 1653.


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12 D. A. Parthenopoulos and P. M. Rentzepis, Science. 1990, 245, 843. 13 H. Y. Woo, J. W. Hong, B. Liu, A. Mikhailovsky, D. Korystov and G. C. Bazan, J. Am. Chem. Soc.2005, 127, 820. 14 G. S. He, P. P. Makowicz, T. Lin and P. N. Prasad, Nature (London). 2002, 415, 767. 15 G. S. He, J. D. Bhawalkar, P. N. Prasad and B. A. Reinhardt, Opt. Lett.1995, 14, 1524. 16 S. Maiti, J. B. Shear, R. M. Williams, W. R. Zipfel and W. W. Webb, Science.1997, 275, 530. 17 J. R. Lakowicz, I. Grycznski, H. Malak, M. Schrader, P. Engelhardt, H. Knao and S. W. Hell, Biophys. J.1997, 72, 567. 18 T-C. Lin, G. S. He, Q. Zheng and P. N. Prasad, J. Mater. Chem.2006, 16, 2490. 19 A. Gandman, L. Chuntonov, L. Rybak and Z. Amitay, Phys. Rev. A. 2007, 76, 053419. 20 I. Cohanoschi, L. Echeverría and E. Hernández, Chem. Phys. Lett.2006, 419, 33. 21 S. Singh and L. T. Bradley, Phys. Rev. Lett.1964, 12, 612. 22 P. Cronstrand, Y. Luo, P. Norman and H. Ågren, Chem. Phys. Lett.2003, 375, 233. 23 P. Cronstrand, B. Jansik, D. Jonsson, Y. Luo and H. Ågren, J. Chem. Phys.2004, 121, 9239. 24 P. Cronstrand, P. Norman, Y. Luo and H. Ågren, J. Chem. Phys.2004, 121, 2020. 25 P. Sałek, H. Ågren, A. Baev and P. N. Prasad, J. Phys. Chem. A. 2005, 109, 11037. 26 N. Lin, L. Ferrighi, X. Zhao, K. Ruud, A. Rizzo and Y. Luo, J. Phys. Chem. B. 2008, 112, 4703. 27 P. C. Jha, Y. Luo, I. Polyzos, P. Persephonis and H. Ågren, J. Chem. Phys. 2009, 130,174312. 28 V. Galasso, J. Chem. Phys.1990, 92, 2495. 29 J. Olsen and P. Jørgensen, J. Chem. Phys.1985, 82, 3235. 30 B. Jansik, P. Sałek, D. Jonsson, O. Vahtras and H. Ågren, J. Chem. Phys.2005, 122, 054107. 31 U. Ekström, L. Visscher, R. Bast, A.J. Thorvaldsen and K. Ruud, J. Chem. Theory Comput. 2010, 6, 1971. 32 W. J. Yang, D. Y. Kim, M-Y. J. H. M. Kim, Y. K. Lee, X. Fang, S-J. Jeon, and B. R. Cho, Chemistry-A European Journal. 2005, 11, 4191. 33 E. A. Badaeva, and T. V. Timofeeva, J. Phys. Chem. A. 2005, 109, 7276. 34 Z. C. Wei, H. H. Fan, N. Li, H. Z. Wang, and Z. P. Zhong, J. Mol. Stru. 2005, 748, 1. 35 L. Ferrighi, L. Frediani and K. Ruud, J. Phys. Chem. B 2007, 111, 8965. 36 GAUSSIAN 03 (Revision B.03), Gaussian, Inc., Wallingford, CT, 2004. Role of donor-acceptor orientation on solvent-dependent three-photon activity in through-space charge-transfer systems - Case study of [2,2]-paracyclophane derivatives

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37 S. Miertus, E. Scrocco and J. Tomasi, Chem. Phys. 1981, 55, 117. 38 T. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett. 2004, 393, 51. 39 T. H. Dunning, J. Chem. Phys. 1989, 90, 1007. 40 D. H. Friese, C. Hättig and K. Ruud, Phys. Chem. Chem. Phys. 2012, 14, 1175. 41 DALTON, a molecular electronic structure program, Release Dalton2011 (Rev. 0, July2011), see http://www.kjemi.uio.no/software/dalton/dalton.html. 42 M. J. G. Peach, P. Benfield, T. Helgaker and D. J. Tozer, J. Chem. Phys. 2008, 128, 044118. 43 M. J. G. Peach, C. R. Le Sueur, K. Ruud, M. Guillaume and D. J. Tozer, Phys. Chem. Chem. Phys. 2009, 11, 4465. 44 D. H. Friese, C. Hättig and K. Ruud, Phys. Chem. Chem. Phys. 2012, 14, 1175. 45 W. M. McClain, J. Chem. Phys. 1971, 55, 2789.


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Chapter - 6 Summary and Conclusion

Summary and Conclusion

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Summary and Conclusion Finally, I end this exposition with the summary and conclusion of all the works presented in this thesis. In Chapter 1, I have presented an introduction to the field of linear and nonlinear optical phenomena - mainly the one-, two- and three-photon (OP, TP and 3P) absorption processes. Along with this I've also discussed the various quantum chemical methods, like the timedependent perturbation theory, Response theory and few-states model approaches, which are available for the study of such processes in molecular systems. Chapter 2 represents my first research work, in which, the TP activity of some boronnitrogen containing donor-π-acceptor type of organic molecules have been studied. We have reported that the previous notion of the better performance of two-state model (2SM) for noncentrosymmetric molecules is no longer valid for the studied systems and one need to include higher excited states to reproduce the response theory results - at least qualitatively. This work indicates that higher excited states play pivotal role in controlling the TP activity of these systems. In Chapter 3, we have elaborately discussed how one can included higher excited states in the calculation of TP transition probability of multi-dimensional systems and what new consequences can arise because of this. In this context, we have derived a generalized fewstate model (GFSM) formula (in Part 3.1) which can be applied equally to any dimensional molecules and identify the phenomenon of channel interference. We have studied the channel interference mechanism in different types of molecules and found that solvents play a very important role in controlling this phenomenon. In case of o-betaine and p-betaine molecules the magnitude of channel interference gradually decreases with the increase in polarity of the


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solvent. In Part 3.2, we have studied the same for two through-space charge-transfer systems (a paracyclophane derivative and a tweezer-TNF complex) and found that in case of tweezerTNF complex the nature of channel interference changes from destructive to constructive on moving from highly polar solvents to less polar solvents because of which this system possess larger TP activity in less polar solvents as compared to that in highly polar solvents. Finally, in Part 3.3, we have studied the dependence of channel interference on twist angle between the two rings of o-betaine molecule and proposed a design strategy to control it. In Chapter 4 we have reported the mechanism of very large TP activity of some selective squaraine dyes and noticed that the replacement of a specific fused ring by ethylene and ethyne moieties increases their activity up to the order of 1013 a.u. We have applied both the standard as well as damped response theory to unfold the mystery behind this and found that the severe decrease in detuning energies (in the substituted systems) is the main reason of their unprecedented TP activity. Finally, in Chapter 5, we have studied the effect of orientation of donor-acceptor groups on the 3P activity of some selective TSCT systems. We found that for a particular orientation of donor-acceptor groups most of the 3P tensor elements contribute significantly and hence the corresponding system also possesses very large 3P activity, particularly in the gas phase. We have explained this observation by using 2SM formula for 3PA process and found that the large dipole-moment difference between the ground and excited states of the systems is one of the reasons behind this in-silico observation.


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List of Publications-I (Related to the Thesis) 1. M. M. Alam, M. Chattopadhyaya and S. Chakrabarti, A Critical Theoretical Study on the Two-Photon Absorption Properties of Some Selective triaryl borane-1naphthylphenyl amine Based Charge Transfer Molecules. Phys. Chem. Chem. Phys. 2011, 13, 9285. 2. M. M. Alam, M. Chattopadhyaya and S. Chakrabarti, Solvent induced channel interference in the two-photon absorption process – a theoretical study with a generalized few-state-model in three dimensions. Phys. Chem. Chem. Phys. 2012, 14, 1156. 3. M. M. Alam, M. Chattopadhyaya, S. Chakrabarti and K. Ruud, High-Polarity Solvents Decreasing the Two-Photon Transition Probability of Through-Space Charge-Transfer Systems – A Surprising In Silico Observation. J. Phys. Chem. Lett. 2012, 3, 961. 4. M. M. Alam, M. Chattopadhyaya and S. Chakrabarti, Enhancement of Twist Angle Dependent Two-Photon Activity through the Proper Alignment of Ground to Excited State and Excited State Dipole Moment Vectors. J. Phys. Chem. A 2012, 116, 8067. 5. M. M. Alam, M. Chattopadhyaya, S. Chakrabarti and K. Ruud, Role of DonorAcceptor Orientation on Solvent-Dependent Three-Photon Activity in Through-Space Charge-Transfer Systems - Case Study of [2,2]-Paracyclophane Derivatives. Phys. Chem. Chem. Phys. 2013, 15, 17570. 6. M. M. Alam, M. Chattopadhyaya, S. Chakrabarti and A. Rizzo, A. On the origin of abnormally high two-photon activity of squaraine dyes - A damped response theory study. Phys. Chem. Chem. Phys. 2014, 16, 8030-8035.


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7. M. M. Alam, M. Chattopadhyaya, S. Chakrabarti and K. Ruud, The Chemical Control of Channel Interference in Two-Photon Absorption Process (Acc. Chem. Res. dx.doi.org/ 10.1021/ar500083f).

List of Publications-II (Not Related to Thesis) 8. M. M. Alam, M. Chattopadhyaya and S. Chakrabarti, On the Origin of Large TwoPhoton Activity of DANS Molecule. J. Phys. Chem. A 2012, 116, 11034. 9. M. M. Aam and E. Fromager, Metallophilic interactions in A-frame molecules [S(MPH3)2] (M = Cu, Ag, Au) from range separated density-functional perturbation theory. Chem. Phys. Letts. 2012, 554, 37. 10. M. M. Alam and M. Chattopadhyaya, Solvent Dependent One-, Two- and ThreePhoton Absorption Properties of PRODAN based Chemo-Sensors. (J. Chem. Sci. Accepted 2014) 11. M. M. Alam, Donors Contribute More than Acceptors to increase the Two-Photon Activity - A Case study with Cyclopenta[b]naphthalene Based Molecules. (Communicated) 12. M. Chattopadhyaya, M. M. Alam and S. Chakrabarti, New Design Strategy for the Two-Photon Active Material Based on Push-Pull Substituted Bisanthene Molecule. J. Phys. Chem. A 2011, 115, 2607. 13. M. Chattopadhyaya, S. Sen, M. M. Alam and S. Chakrabarti, The role of relativity and dispersion controlled inter-chain interaction on the band gap of thiophene, selenophene, and tellurophene oligomers. J. Chem. Phys. 2012, 136, 094904. 14. M. Chattopadhyaya, N. A. Murugan, M. M. Alam and S. Chakrabarti, Spatial spincharge separation in neutral endohedral metallofullerene: A combined restricted open-


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shell MP2 and Car–Parrinello molecular dynamics study. Chem. Phys. Letts. 2013, 557, 71. 15. M. Chattopadhyaya, M. M. Alam and S. Chakrabarti, On the microscopic origin of bending of graphene nanoribbons in the presence of a perpendicular electric field. Phys. Chem. Chem. Phys. (Comm). 2012, 14, 9439. 16. M. Chattopadhyaya, M. M. Alam, S. Sen and S. Chakrabarti, Electrostatic Spin Crossover and Concomitant Electrically Operated Spin Switch Action in a Ti-Based Endohedral Metallofullerene Polymer. Phys. Rev. Letts. 2012, 109, 257204. 17. M. Chattopadhyaya, S. Sen, M. M. Alam and S. Chakrabarti, On site Coulomb repulsion dominates over the non-local Hartree-Fock exchange in determining the band gap of polymers. (J. Phys. Chem. Sol, 2013 Accepted) 18. M. Chattopadhyaya, M. M. Alam and S. Chakrabarti, Chemical control of a molecular spin switch in presence of gate. RSC Adv. (Communication) 2013, 3, 19894. 19. M. Chattopadhyaya, M. M. Alam and S. Chakrabarti, All electrically controlled electronic spin based molecular quantum bit originating from spin state dependent quantum interference. (ChemPhysChem, Accepted 2014). 20. M. Chattopadhyaya, M. M. Alam and S. Chakrabarti, Enhancement of the twophoton transition probability through the quenching of destructive interference in polar solvents: A case study with D-Luciferin. (Manuscript under preparation)


Appendix D

APPENDIX-F

P a g e | I/AF

Appendix F

Figure F1 – Orbital pictures of benchmark molecule

Figure F2 - Orbital pictures of PCP in different solvent phases

APPENDIX-F

P a g e | II/AF

Figure F3 - Orbital pictures of tweezer-TNF complex in different solvent phases

APPENDIX-F

P a g e | III/AF

Figure F4 - Variation of oscillator strength (δOPA in a.u.) with twist angle in unsubstituted o-betaine molecule

Figure F5- Variation of excitation energies (ω01 and ω02 in eV) with twist angle in unsubstituted o-betaine molecule

Figure F6 - Variation of δresp with the twist angle of unsubstituted obetaine molecule (calculated at CAMB3LYP/aug-cc-pVDZ level of theory)

Figure F7 - Variation of μ02 and μ12 (both in a.u.) with the twist angle in unsubstituted o-betaine molecule

APPENDIX-F

Figure F8 - Variation of different Xij terms with twist angle of unsubstituted o-betaine molecule

P a g e | IV/AF

Figure F9 - Plot of TPA cross-section of different substituted and unsubstituted obetaine molecules against the twist angle, in cyclo-hexane solvent

Figure F10 - Optimized geometries of unsubstituted and substituted p-betaine molecules.

APPENDIX-T

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Appendix T Table T1 – Table for TPA tensor elements and two photon transition probability (in a.u) of BN1 molecule using cc-pVDZ, aug-cc-pVDZ and cc-pVTZ basis sets and CAMB3LYP functional (Gas Phase results) Basis set

Ex. State

cc-pVDZ Aug-cc-pVDZ cc-pVTZ

1 2 1 2

Sxx 20.5 -15.2 -21.1 -13.2

Syy 18.2 -19.7 -20.4 -17.2

1 2

-20.3 13.6

-18.8 17.8

TP tensor elements Szz Sxy -207.3 -15.7 -170.2 22.3 190.4 17.1 -175.6 19.5 194.9 170.6

16.2 -20.4

Sxz 18.9 53.5 -15.8 52.8

Syz 6.5 -44.9 -9.1 -45.5

-16.9 -52.0

-7.7 43.7

δTP (in 105 a.u.) 2.369 2.455 1.980 2.520 2.084 2.403

Table T2 – Table for One-photon absorption data – Excitation energy (in eV), Transition moment (in a.u), Λ parameter, kappa values and Orbitals involved in the transition ( H ≡ HOMO, L ≡ LUMO) for S1 state of benchmark molecule in THF solvent Ex. State

Ex. Energy/eV

1

3.30

Transition moment in a.u. X Y Z -0.1249

0.0401

-4.6139

Λ

κ

0.5673

-0.6442 -0.1840 -0.1545

Orbitals involved H-L H-L+1 H-1-L

Table T3 – TPA tensor elements and TP transition probability (in 105 a.u order) of the benchmark molecule using CAM-B3LYP/cc-pVDZ basis set level of theory in THF solvent Ex. State

Sxx

Syy

Szz

Sxy

Sxz

Syz

δTP in 105 a.u.

1

12.5

2.2

-573.1

5.3

-8.3

4.2

19.39

Table T4 - Theoretical & experimental excitation energy (in eV) of o- and p-betain in water System o-betaine p-betaine

Ex. State 1 2 1 2

(eV) calculated using aug-cc-pVDZ basis set CAMB3LYP LC-PBE 2.934 3.551 4.072 4.612 3.048 3.651 4.030 4.432

Experimental (eV) 3.280 3..397

APPENDIX-T

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Table T5 - SOS results (in 104 a.u. order) involving different number of intermediate states Test of convergence behavior No. of intermediate states 1 2 3 4 5 6 7 8 Response theory

TP transition probability (in 104 a.u.) o-betaine p-betaine 1st ex. 2nd ex. 1st ex. 2nd ex. 17.05 2.96 9.70 0.11 17.93 1.26 9.60 0.87 17.95 1.34 9.89 0.87 18.72 1.52 9.88 0.87 18.72 1.52 9.90 0.84 17.66 1.59 10.09 0.81 13.54 1.84 10.06 0.81 13.22 1.84 9.99 0.82 13.10 0.90 11.40 0.93

Table T6 - µ01 and µ02 (in a.u.) of PCP µ01

solvents MeCN THF CHCl3 Toluene Gas

x 0.130 -0.128 0.128 -0.126 0.108

y -0.005 0.005 -0.004 0.003 -0.001

µ02 z -0.190 0.188 -0.188 0.185 -0.159

Total 0.230 0.228 0.227 0.224 0.193

x -0.35 0.325 0.308 0.279 0.213

y -0.258 0.192 0.168 0.126 0.075

z 0.514 -0.470 -0.451 -0.408 -0.311

Total 0.674 0.601 0.571 0.510 0.384

z 0.046 0.028 -0.025 0.024 -0.031

Total 0.082 0.050 0.044 0.043 0.055

z

Total 0.344 0.348 0.313 0.353 0.313

Table T7 - µ11 and µ21 (in a.u.) of PCP μ11

Solvents MeCN THF CHCl3 Toluene Gas

x -3.442 -3.490 -3.519 -3.562 -3.295

y 0.000 0.000 0.000 0.000 0.000

μ21 z -2.350 -2.384 -2.403 -2.434 -2.252

Total 4.168 4.226 4.261 4.314 3.991

x 0.068 0.041 -0.036 0.036 -0.045

y 0.000 0.000 0.000 0.000 0.000

Table T8 - µ01 and µ02 (in a.u.) of tweezer-TNF complex

MeCN THF CHCl3 Toluene Gas

μ02

μ01

Solvents x 0.172 0.181 0.287 -0.206 0.287

y 0.327 0.328 -0.273 -0.327 -0.273

z -0.208 -0.217 0.109 0.236 0.109

Total 0.424 0.433 0.411 0.452 0.411

x 0.096 0.097 0.028 -0.094 0.028

y 0.292 0.295 0.290 -0.297 0.290

0.154 0.157 0.115 -0.165 0.115

APPENDIX-T

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Table T9 - µ11 and µ21 (in a.u.) of tweezer-TNF complex μ11


x -0.644 -0.631 -0.622 -0.600 -0.516

y -4.593 -4.619 -4.630 -4.622 -4.182

μ21 z 0.918 0.948 0.961 0.967 0.792

Total 4.728 4.757 4.769 4.760 4.287

x 0.067 0.064 0.063 -0.058 -0.047

y 0.513 0.476 0.465 -0.419 -0.333

z 0.064 0.059 0.058 -0.051 -0.035

Total 0.522 0.487 0.473 0.426 0.338

Table T10 - µ22 (in a.u.) of tweezer-TNF complex μ22


x 0.392 0.412 0.424 0.449 0.430

y 5.199 5.239 5.263 5.289 4.866

z 2.210 2.253 2.274 2.299 2.030

Total 5.663 5.718 5.749 5.785 5.290

Table T11 - Different angle terms (Xij) of PCP and Tweezer-TNF complex in gas and different solvent phases Solvents MeCN THF CHCl3 Toluene Gas

Excited states 1 2 1 2 1 2 1 2 1 2

PCP -0.92 -0.94 -0.95 -0.97 -0.98

Tweezer-TNF -1.94 -1.77 -1.90 -1.72 1.87 1.69 1.80 1.61 -1.48 -1.40

APPENDIX-T

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Table T12 - Data for the plot of Oscillator strength, ω01, ω02, μ01, μ11, μ02 and μ12 against the twist angle of unsubstituted o-betaine molecule Twist angle 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180

Osc. Str. (a.u.) 0.2201 0.2027 0.1811 0.1617 0.1452 0.1307 0.1176 0.1052 0.0933 0.0817 0.0702 0.0587 0.0472 0.0359 0.0249 0.015 0.0068 0.0016 0.0 0.0016 0.0068 0.015 0.0249 0.0359 0.0472 0.0587 0.0702 0.0817 0.0933 0.1052 0.1176 0.1307 0.1452 0.1617 0.1811 0.2027 0.2201

ω01 (eV)

ω02 (eV)

µ01 (a.u.)

µ11 (a.u.)

µ02 (a.u.)

µ12 (a.u.)

2.83757 2.79001 2.7164 2.62939 2.53686 2.44287 2.3489 2.25483 2.15971 2.06238 1.96178 1.8572 1.7485 1.63667 1.52445 1.41764 1.32631 1.26502 1.24855 1.26502 1.32631 1.41764 1.52445 1.63667 1.7485 1.8572 1.96178 2.06238 2.15971 2.25483 2.3489 2.44287 2.53686 2.62939 2.7164 2.79001 2.83757

3.052 3.05 3.043 3.031 3.014 2.992 2.963 2.927 2.88 2.823 2.755 2.675 2.583 2.48 2.365 2.239 2.106 1.973 1.953 1.973 2.106 2.239 2.365 2.48 2.583 2.675 2.755 2.823 2.88 2.927 2.963 2.992 3.014 3.031 3.043 3.05 3.052

2.0048 1.9367 1.8511 1.7746 1.7091 1.6505 1.5949 1.5386 1.4796 1.4156 1.3445 1.2628 1.1661 1.049 0.9047 0.7264 0.5074 0.2489 0.0347 0.2489 0.5074 0.7264 0.9047 1.049 1.1661 1.2628 1.3445 1.4156 1.4796 1.5386 1.5949 1.6505 1.7091 1.7746 1.8511 1.9367 2.0048

1.61499 1.6347 1.67955 1.73631 1.80187 1.8773 1.9643 2.06418 2.17829 2.30856 2.45847 2.63467 2.84715 3.10932 3.43467 3.8272 4.25378 4.60652 4.72318 4.60652 4.25378 3.8272 3.43467 3.10932 2.84715 2.63467 2.45847 2.30856 2.17829 2.06418 1.9643 1.8773 1.80187 1.73631 1.67955 1.6344 1.61499

0.19543 0.51584 0.7114 0.81417 0.86087 0.87337 0.86202 0.83078 0.7844 0.73232 0.686 0.64851 0.61859 0.59401 0.5736 0.55765 0.54708 0.54319 0.54732 0.54319 0.54708 0.55765 0.5736 0.59401 0.61859 0.64851 0.686 0.73232 0.7844 0.83078 0.86202 0.87337 0.86087 0.81417 0.7114 0.51584 0.19542

0.07771 0.17711 0.23747 0.26746 0.28215 0.29046 0.29638 0.30075 0.30224 0.3002 0.2967 0.29278 0.28837 0.28202 0.27114 0.25107 0.21155 0.12725 0.04598 0.12725 0.21155 0.25107 0.27114 0.28202 0.28837 0.29278 0.2967 0.3002 0.30224 0.30075 0.29638 0.29046 0.28218 0.26746 0.23747 0.17711 0.07771

APPENDIX-T

P a g e | V/AT

Table T13 - Data for plot of different δ-terms (in a.u.) against the twist angle (in degree) of unsubstituted o-betaine molecule Twist angle (degree) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180

δresp (a.u.)

δ4SM (a.u.)

δ11 (a.u.)

δ22 (a.u.)

δ12 (a.u.)

74400 72600 72600 75000 80100 87600 97500 110100 125400 144300 167100 193500 222600 250500 266100 248700 171900 53700 1029 53700 171900 248700 266100 250500 222600 193500 167100 144300 125400 110100 97500 87600 80100 75000 72600 72600 74400

92473 93268 97193 103482 111922 122513 135371 150499 168116 188533 212183 239015 267771 294179 307060 283898 195590 60787 1320 60787 195590 283898 307060 294179 267771 239015 212183 188533 168116 150499 135371 122513 111922 103482 97193 93268 92473

92459 91466 93095 97594 104741 114352 126430 140968 158187 178303 201532 227798 255950 281947 294984 273135 188076 58306 1214 58306 188076 273135 294984 281947 255950 227798 201532 178303 158187 140968 126430 114352 104741 97594 93095 91466 92459

0 22 83 144 185 208 218 215 202 184 168 158 150 141 129 108 76 27 3 27 76 108 129 141 150 158 168 184 202 215 218 208 185 144 83 22 0

7 889 2007 2872 3498 3977 4362 4658 4863 5023 5241 5529 5836 6046 5974 5327 3719 1227 51 1227 3719 5327 5974 6046 5836 5529 5241 5023 4863 4658 4362 3977 3498 2872 2007 889 7

APPENDIX-T

P a g e | VI/AT

Table T14 - Table containing twist angle and the corresponding TP transition probability of different substituted and unsubstituted o-betaine molecules in gas and solvent phases System Unsubstituted One CH3 on pyridinium ring One CH3 on phenoxide ring One CH3 on each ring Two CH3 on pyridinium ring Trimethyl substituted

In cyclohexane Angle δTP 41.6 2.05 × 105 64.2 2.14 × 105 52.6 1.90 × 105 69.9 1.77 × 105 76.6 9.18 × 104 87.7 7.59 × 101

In n-hexane Angle δTP 41.2 1.96 × 105 63.9 2.09 × 105 52.1 1.84 × 105 69.4 1.78 × 105 76.0 9.72 × 104 87.0 7.50 × 101

In gas Angle δTP 35.8 1.13 × 105 60.5 1.47 × 105 47.5 1.17 × 105 65.12 1.49 × 105 71.09 1.22 × 105 80.0 3.60 × 104

Table T15 - Table containing twist angle and the corresponding TP transition probability of different substituted and unsubstituted p-betaine molecules in gas phase Systems (a) Unsubstituted (b) Tetra-substituted (c) Tri-substituted (d) Two CH3 on pyridinium ring (e) Di-substituted (both methyl groups on phenolate ring) (f) Di-substituted (one methyl group on each ring)

Twist angle (in degree) 32.3 90.0 75.34 67.25 52.96 60.23

δresp (in a.u.) 1.0 × 105 6.45 × 100 5.67 × 105 4.05 × 105 1.38 × 105 2.33 × 105