Theoretical study of rotational fine structure in

Theoretical study of rotational fine structure in radiation less transitions W. E. Henke, H. L. Selzle, T. R. Hays, and E. W. Schlag Institute 0/ Physical and Theoretical Chemistry, Technical University West Germany

0/ Munich, 8046 Garching,

S.H.Lin Department o/Chemistry, Arizona State University, Tempe, Arizona 85281 (Received 21 August 1981; accepted 29 September 1981) In the previous paper it is shown that, at least for the case of formaldehyde, individually resolved rovibronic levels show a systematic dependence on J and K. Here a general theory is evolved which includes the effects of rotational energy gap, vibration-rotation coupling, and Coriolis effects on radiationless transitions. This general theory in contrast to existing pessimism, predicts a clear systematic variation of the radiationless rate for formaldehyde, in reasonable agreement with the experiment. This shows that these variations are the results of a general effect and not due to particular unsystematic resonances in this molecule. Hence, it is felt that this constitutes an interesting tiepoint between high resolution jet measurements and a successful quantitative theory of a radiationless transition.

I. INTRODUCTION

Rotational effects have not been included in any of the widely applied, conventional theories of radiative l - 3 and nonradiative 4- 11 processes. Recently, however, there has been great interest in the theoretical study of the photophysical processes of the single rovibronic levels (SRVL) of molecules. 12-15 The purposes of this paper are to report the general calculation of SRVL nonradiative (NR) rates of polyatomic molecules, especially for internal conversion (IC), and to compare these calculations to the experimental results in the companion paper preceding this one, in which lifetimes of single, well-defined rotations of formaldehyde are measured in a hypersonic jet. The first theoretical examination of the rotational effect on the nonradiative decay of polyatomics was carried out by Howard and Schlag. 12.13 They discussed the selection rules for nonradiative processes for internal conversion within singlet and triplet states and intersystem crossing. In particular, they showed that, because of the sum rule over the Wigner 3 -J symbols which constrains the angular momentum, no rotational effect is predicted in the case of constant FranckCondon factors between initial and final states. Further on, Novak and Rice 14 •15 have also theoretically modeled the influence of rotational motion on intersystem crossing (ISC) in isolated molecules and of CorioUs coupling on internal conversion. However, detailed comparisons with experimental data were not carried out by them. They have looked at effects on the rate of intersystem crossing, of rotational state dependence of spin-orbit coupling, of differences in moments of inertia between the coupled electronic states, of Coriolis coupling, and of vibration-rotation interaction in the initial state. They demonstrate again that the sum rules, which represent the conservation of angular momentum and which constrain rotational transformations, act so as to greatly reduce the influence of rotational motion on the relaxation process. They do not treat the resonant or nearresonant interactions between the two electronic maniJ. Chem. Phys. 76(3), 1 Feb. 1982

folds involved in ISC, use the Fermi golden rule, and consider only the symmetric rotor case; they have not considered the geometry change between the two electronic states (e. g., H2CO). In this paper, we go further and consider further effects in rotational NR transitions and directly predict experimental lifetimes. In this way a direct, detailed comparison with experimental data is possible. Here we choose to study the following effects: (1) The effect of rotational energy gap on the SRVL NR rate constant; (2) the effect of the vibration-rotation coupling through the change of moment of inertia; (3) the effect of the vibration-rotation coupling through the Corio lis interaction; (4) the dynamical effect of intramolecular vibrational dephasing (i. e., the damping effect) (that is, the small or intermediate molecule case); and (5) the electric field effect on the SRVL NR rate. These effects in contrast to previous work l4 ,15 are shown here to be considerable. Both symmetric rotor and asymmetric rotor calculations will be considered and numerical calculations will be performed to demonstrate the above-mentioned effects, wherever feasible and necessary. The theoretical results will be applied to interpret the new experimental data of formaldehyde.

II. GENERAL THEORY

The rate constant of a SRVL(bvJKM) can in general be expressed as WbvJKM

2rr = -;;It

"" L...J

1(av'J'K'M'IH' 1bvJKM) 12 A

v'JI K'M'

(2.1) where

+ (Eav'J'K'M'

-EaVJKM)2] ,

(2.2)

where iI' represents the perturbation that causes the radiationless transition and M represents the magnetic

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Henke et al.: Fine structure in radiationless transitions

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quantum number along the space-fixed Z -axis. Depending on whether we are concerned with the symmetric rotor or asymmetric rotor, the rotational quantum number K will be different. In general, one can only observe W bvJ K in the absence of an external field, i. e., WbVJK =

2J1+1~WbVJKM'

(2.3)

Notice that r"v'J'K'M',bvJKM represents the so-called dephasing constant (or the damping constant). In a collision-free molecule, r"v'J'K'M',bvJKM is mainly due to vibrational redistribution. In a large molecule, one can usually make the approximation r"v'J'K'M'.bvJKM- O. In this case we have WbvJKM =

In other words, the rotational dependence of the SRVL NR rate constant is through the energy gap E'''JK -E bvJK (or E av' JT -E bvJT ), which in turn depends on the difference in moments of inertia between the two electronic states. This effect was first pointed out by Howard and Schlag. 12,13 Using the relation

_lA, I >~_ 2 ..... ; I~·~) Hao 41b6bv -. Il "T' \41.6. v' BQ BQ =

Here C I (ab)=v'(w l /21l)R 1 (ab). For an asymmetric rotor, to calculate Wbv/T, we simply replace w. J K,bJ K by W.JT,bJT' To evaluate the integral in Eq. (2.13) the saddle-point method can be used. In this case, we obtain I

where .)

WbOOO (t

(41.€l.v·lii~o I 41 b6 bv >(Y J' K' M' 1 Y J KM> X

(2.8)

That is, we obtain the selection rules ~J = 0, M = 0, Substituting Eq. (2.8) into Eq. (2.1) yields

~M=O.

Wb."K= :1TL

" .,'

=

Lt

ICI(ab)12 ( ..... 1. 2) 1£2 exp 2~j

j'" dtexprL:t~~e/tWj+it(W'b+WI)] t

,

(2. 17)

fbvJ K(i) = fbV(t;) ,

and

t;,

(2.16)

j

_«)

1\2 If L.J 1 2 H BO 1bll!lcI>b)=-F'l!bb(Q) == - F' I!bb(O) - F'

LJ (aI!Qbb) Q a J

j

+ ... , (2.85)

WbvJKM(F) == WbvJKM(F)1

0

+ WbVJKM(Flz + WbVJKM(F)3 ,

where I'bb(Q) == (cI>b II! I cI>b)' which is a function of normal coordinates Qj.

WbvJ KM(F)1 == !7T

~

I

(4?a 0 av'

. (2.88)

(2.89)

where WbVJKM(F)1 is due to the processes J' ==J,

1.iI~o l4>b0 bV) - FlJ.z(av', bv) J(~~ 1) /2 D[E(F)av'J KM -

E(F)bVJ KM]

+ 27T • F 2M 2(J-K)(J+K+1)", 1 (,t 12 [ n 4J2(J + 1)2 IJ.. av ,bv) D E(F)av'/K+1M - E(F)bvJKMl

7'

2 2

+ 27T • F M (J+K)(J-K-1)", 1 ' 12 n 4J2(J + 1)2 7'lJ.jav ,bv) D[E(F)av'JK_1M - E(F)bVJ KM] •

(2.90)

Here we have 1J..(av', bv) == (cI>a 0 av.1 J1x + iP- y 1 cI>b 0 bV>

(2.91)

and (2.92)

Notice that Jl. 1I(av', bv), Jl..(av', bv), and IJ.-Cav', bv) represent the matrix elements of the transition dipole moment for bv- av'. Similarly W bvJ KM(F)2 is due to the processes J' =J -1,

+ 27T

t;'

F2(J 2 _M2)(J -K)(J -K -1}", 1 12 4J2(4J2 -1) 1J..(av, bv} D[E(F)av'J_1K+1M -E(F)bvJKM]

7'

A

,

2 2 2 27T F (J _M )(J+K_1)", 1 ' 12 +t;. 4J2{4J2 _1) ~ I!jav,bv) D[E(F)av'J_1K_1M-E(F)bVJKM]

(2.93)

and W bvJ KM(Fh is due to the processes J' =J + 1, J. Chern. Phys., Vol. 76, No.3, 1 February 1982


1344


_ 271 • F2[(J + 1)2 _K2][(J + 1)2 _M2] '2 WbVJKM(F)3 - Ii (J + 1)2(2 J + 1)(2J + 3) ~ 1I-Le(av , bv) 1 D[E(F)avl J+1KM - E(F)bvJKM] 2 271. F [(J+1)2- M 2](J+K+1)(J+K+2)" 1 ' 2 + Ii 4(J + 1)2(2 J + 1)(2 J + 3) I-L+(av, bv) 1 D[E(F)avl J+1K+1M - E(F)bVJ KM]

7'

2 +271. F [(J+l)2- M 2](J-K+1)(J-K+2)" 1 ' 12 ;;; 4(J + 1)2(2 J + 1)(2 J + 3) ,.dav, bv) D[E(F)av l J+1K_1M - E(F)bvJKMl .

7'

Notice that

electric field effect to the SRVL rate is an order of magnitude smaller than those discussed above.

II-L+(av', bv)j2= II-Ljav', bv)j2 = 1I-L/av', bv)j2 + 1I-Ly(av', bv)j2. (2.95) It should be noted that those terms in WbvJKM(F) that involve the matrix elements of the transition moment represent the so-called electric field-induced NR transitions.

Next we consider the electric field effect on SRVL NR rates through its effect on rotational wave functions. For the case in which the rotational energy gap is much larger than the Stark energy, we can use the perturbation method; for example, for the b electronic state we have Y

_ yO JKM- JKM +

+

For the case in which the rotational energy gap is comparable to the Stark energy (i. e., the resonant interaction), one can find the electric field effect on rotational wave functions by taking the linear combination of the rotational wave functions that are in the resonant interaction. For example, if only two states are involved, then we have (2.101)

and C1(H 11 -W)+C 2H 12 =0. C1H 21 +C 2(H 22 -W)=O,

(J - 1 K M IH~ IJ K M) yO E E J-1KM JK - J-1K

(J+1KMIH~IJKM)yO E

JK -

E

J+1K

+ ...

J+1KM

(2.104)

(J- 1 K M 1fi~ 1J K M) f(J2 - K 2)(J2 - M 2)Jl/2 = - I-Lbb F [ J 2(4J2 -1) ,

and (2.97)

+ 1KM lfi~ IJK M)

H21 = - I-Lbb F {

[(J + 1)2 _K2][(J + 1)2 _M2]}1/2 (J + 1)2(2J + 1)(2J+3)

(2.98)

W. J KM =%(H 11 + H 22 ) ±t[(H11 -H 22 )2 + 4H~2]1I2 , iJiJKM(W- J KM) = - bJKM Y JKM

and I-L bb = (cI> b1 ~ z 1 cI> b) •

(2.99)

YJKM=YJKM-

iJiJKM(W+JKM)=aJKMYJKM+bJKMYJ+1KM'

_

~ (J2 _K2)(J2 _M2) 112

2B

J4(4J2-1)

(J+1)4(2J+1)(2J+3)

Y J - 1KM

(2.107a) (2.107b)

+ !71

H12 -

W+JKM )2+H21112 12

(2.108)

and 0

Y J+1KM + " ' , (2.100)

and one can see that the contribution from this type of WbvJK(W+J KM) = !71 a~KM ~ 1(cI>a6av1

aJKM-[(H11

0

~{[(J + 1)2 _K2][(J + 1)2 _M2]}1I2 2B

+ aJ KM Y J+1KM ,

(2.106)

where

It follows that

o

(2.105)

It follows that

_ _ {[(J + 1)2 _K2][(J + 1)2 _M2]}1!2 I-Lbb F (J+1)2(2J+1)(2J+3) ,

+

(2.102)

(2.103)

(2.96)

where

(J

(2.94)

(2.109)

and the corresponding SRVL NR rates are given by

IH~o 1cl>b6 bV) 12D (Eav'JKM -

E bv - W+ JKM )

b~KM ~ 1(cI>a6av1 IH~o 1cl>b6 bv) 12D(EavlJ+1KM -

E bv - W+JKM )

(2.110)

and

(2.111)

respectively. J. Chern. Phys., Vol. 76, No.3, 1 February 1982



As can be seen from the above discussion the electric field can induce NR transitions and can affect NR rates through its effect on wave functions and the energy gap involved in the Lorentzian; i. e., the electric field can shift the energy matching out of resonance, which may be quite important for small or intermediate-size molecules.

1

FORMALDEHYDE 224 SRVL NR DECAY RATES EXPERIMENTAL J'

K.'1 K,.

°00

ICl 1'0 ~ 10,

It should be noted that, for the electric field-induced

NR process, the matrix elements of the transition moment for the electronic transition b- a under consideration plays an important role; the electronic transition b - a which is dipole allowed is favorable for the electric field-induced process and the direction of the transition moment will determine which of IJ.x(bv, av'), IJ./bv, av'), and 1J..(bv, av') will participate in the fieldinduced process.

1345

13

ICl 220 2 8 " 1Cl2n

[!] [!]

202

1"

3 12 1Cl4 n 31' 4 " 30/3'2 4()I. 3"

8

5,s

4] 134,

S3

5 0S

0

THEORETIC AL

U;

J'K'

~

CD ~ :-'-4 ",0

ell, °0

'0

31 §3, 30

22

8 2,

20

0 0 42

52

5, 50

40

~

CDo

III. DISCUSSION

~ln

In the previous sections the theoretical treatment of the SRVL NR rate dependence on rotation was given. Here we will discuss the results in view of recent experimental results. In the 4~ and 4~ bands of formaldehyde it was found that the lifetimes generally decrease with increasing rotation, as shown in the preceding paper. From the theoretical treatment of the SRVL NR rates (i. e., ignoring the vibration-rotation interaction), it was shown that the SRVL NR rates should depend only on the rotational energy gap, w!Jo::bJK [see Eq. (2.9) or Eq. (2.13)]. As can be seen from Fig. l(a), this effect is Significant. The decay rate for the maximum value of the rotational quantum number in the calculation is ap-

FORMALDEHYDE 43 SRVL NR DECAY RATES EXPERIMENTAL

o ~ 3.0

~ ;:0

»

2.0

-i rT1

1.0

jIooo

-

J' K'

2D

CD