Modern Ab-Initio Valence Bond Methods

Chapter 8

Modern Ab-Initio Valence Bond Methods Philippe C. Hiberty and Sason Shaik

Quantum mechanics has provided chemistry with two general theories of bonding: valence bond (VB) theory and molecular orbital (MO) theory. VB theory is essentially a quantum mechanical formulation of the classical concept of the chemical bond wherein the molecule is regarded as a set of atoms held together by local bonds. This is a very appealing model as it represents the quantum mechanical translation of the classical basic concepts that are deeply rooted in chemistry, such as Lewis' structural formulas, chemical valency, hybrid orbitals, and resonance. MO theory, on the other hand, uses a more physics-related language and has sprung as a means to interpret the electronic spectra of molecules and deal with excited states. However, with its canonical MOs delocalized over the entire molecule, this theory bears little relationship to the familiar language of chemists in terms of localized bonds and this is probably the reason why it was initially eclipsed by VB theory, up to the mid-1950s. Then the situation reversed and MO theory took over, among other reasons, because of the efficient implementations, which provided the chemical community with computational software of ever increasing speed and capabilities. Nowadays, with the advent of modern computational ab initio VB methods and the progress in computer and coding technologies, VB theory is coming of age. Indeed, starting with the 1980s onward, several methodological advances in VB theory have been made, and allowed new and more accurate applications. Thus, dynamic correlation has been incorporated into VB calculations, so that at present, sophisticated VB methods combine the accuracy of post-HF methods with the specific advantages of VB theory such as extreme compactness of the wave 3

4

Modern Ab Initio Valence Bond Methods

functions that are readily interpreted in terms of Lewis structures, ability to calculate diabatic states, resonance energies and so on. Moreover, VB theory has been recently extended to handle species and reactions in solution, and is also capable of treating transition metal complexes. These newly developed tools have been used to verify and quantify fundamental concepts such as aromaticity, resonance energies, hybridization and so on, and to develop new ideas and models in chemical reactivity, that were not foreseen from the empirical VB model. The combination of the lucid insight inherent to VB theory and its new computational capabilities is discussed in this chapter. We hope that this chapter makes a case also for using these modern ab initio VB methods as routine tools in the service of chemistry. 8.1 Basic Principles and Survey of Modern Methods 8.1.1 VB vs MO wave functions in the 2-electron/2-center case The description of the chemical bond in VB theory can be illustrated with the example of the H2 molecule. Let φa and φb be two atomic orbitals (AOs) localized on the left and right hydrogen atom, respectively (Scheme 8.1). To a first approximation, one may consider the fully covalent description of the bond, as in the pioneering article of Heitler and London.1 Dropping normalization constants hereafter, the corresponding wave function, ΦHL in eq. 1, displays two Slater determinants (see the definition of a Slater determinant in next box), each representing a situation where both atoms are neutral and bear electrons of opposite spins: ΦHL(H–H) = "a"b # "a"b

(1)

The bar over an orbital indicates spin-down (β), while the absence of a bar indicates spin-up (α). ! By itself, each Slater determinant in eq. 1 is not much lower in energy than the two separate atoms, and is therefore practically nonbonding.2 It is the superposition of the two determinants, or “resonance”


5

between the two spin arrangements, that creates bonding, as represented in 1 in Scheme 8.1. This early description was remarkably successful Definition of Slater determinants The Pauli principle requires the electronic wave function to be antisymmetric with respect to the exchange of two electrons. A simple orbital-product wave function, e.g. φ1(1)φ2(2) for a two-electron case, does not satisfy this requirement. The product can however, be antisymmetrized, as in D(2-e) below: D(2-e) =

1 2

[φ1(1)φ2(2) – φ2(1)φ1(2)]

(B1)

Eq. B1 is then expressible as a determinant, called “Slater determinant”: !

1 #"1(1)"1(2) & % ( D(2-e) = 2 $"2 (1)"2 (2)'

(B2)

Generalizing, a Slater determinant for a state involving N electrons is an antisymmetrized product of N orbitals and therefore contains N! terms, corresponding to the N! possible orbital permutations applied to the ! diagonal term, e.g., φ1(1)φ2(2) in (B2). This can be written in the form of an N×N-dimensional determinant, multiplied by a factor (N!)-1/2. This determinant is usually represented with a simplified notation, using the diagonal term: D(N-e) = "1"2 ..."N

(B3)

This notation contains all the information needed to generate the N×Ndimensional Slater determinant. The factor (N!)-1/2 is implicitly assumed in (B3). ! since it accounted for ~75% of the total bond energy of H2. For a complete description, it is necessary to include two ionic terms, as is done in eq. 2 that represents the full VB wave function for a general single bond between two atoms A and B:

(

)

ΨVB(A–B) = C1 "a"b # "a"b + C2 "a"a + C3 "b"b

!

(2)

6


where the two last Slater determinants represent A– B+ and A+B– ionic situations, as in 2 (Scheme 8.1). Thus, the full VB wave function is a linear combination of three VB functions (generally referred to as “VB structures”), each representing a particular bonding scheme. The AOs that are used to construct the VB structures in eq. 2, are defined as linear combinations of the basis functions, χµ, centered on a single atom, eq. 3: (3)

#i = ! Tìi " µ µ

and taken from standard basis sets. The AOs are 1s types for hydrogen atoms, and hybrids of ns and np basis functions for heavier atoms, giving rise to the concept of hybridization. For example, the so resulting hybrid atomic orbitals (HAOs) of carbon in a C-H bond in different molecules will resemble the sp3, sp2 or sp types, well known from the important hybridization concept. The VB description of the A–B bond can be compared with the simple MO description, in which a unique delocalized MO is doubly occupied, 3 in Scheme 8.1, eq. 4: ΦMO(A–B) = " g" g ;

" g # ($a + $b )

(4)

where the coefficients of φa and φb in σg are assumed to be equal for simplicity. ΦMO wave function is insufficient however, and a ! ! This simple better description of the bond requires configuration interaction (CI)

MO- Wavefunction

Covalent-Ionic Superposition in an A–B Bond

HL- Wavefunction

!u H

H

H

1

H

A

B

A

B

2

A

B

!g

3

Scheme 8.1. VB and MO wave functions for the 2-electron/2-center case.


7

between ΦMO and some “excited” configurations, as in eq. 5, for an accurate description of the A–B bond:

(

)

ΨMO-CI(A–B) = " # g# g + µ # g# u $ # g# u + % # u# u ; " u # ($a % $b )

(5)

Clearly, ! ΨVB and ΨMO-CI may seem to speak different languages. However simple algebra, consisting of expanding the MO determinants ! of eq. 5 into AO determinants like those of eq. 2, shows that these two wave functions are exactly equivalent and therefore provide the same bonding energy, and any other property of the A–B molecule.3,4 This equivalence between MO-CI and VB descriptions is general, so that any MO-CI wave function can be transformed into a VB wave function and vice versa.5 It is therefore clear that both MO-CI and VB methods, if pushed to a sufficient level of sophistication, are accurate theories. For example, despite some still existing erroneous views, VB methods at any level of sophistication, perfectly account for the triplet state of dioxygen, the valence ionization spectrum of methane and water, the aromaticity/antiaromaticity dichotomy and so on3,4 (see Appendix-1). 8.1.2 Writing VB functions beyond the 2-electron/2-center case. Writing VB wave functions for polyelectronic cases are simple extensions of the 2-electron/2-center example discussed above. Let us consider butadiene and restrict the description to the π system. Denoting the π AOs of the C1–C4 carbons by φa, φb, φc, and φd , respectively, the fully covalent VB structure Φcov for the π system of butadiene displays two covalent bonds: one between φa and φb, and one between φc and φd, as represented in 4 in Scheme 8.2. It follows that the covalent VB structure can be expressed in the form of eq. 6 as a product of the bond wave functions. Φcov(4) = ("a"b # "a"b )("c"d # "c"d )

(6)

Upon expansion of the product, one gets a sum of four determinants as in eq. 7:

!

8


Φcov(4) = "a"b"c"d # "a"b"c"d # "a"b"c"d + "a"b"c"d

4

2

!

(7)

3

1

4 Scheme 8.2. Covalent structure in the π-bonding system of butadiene

The product of bond functions in eq. 6 involves so-called perfect pairing, whereby the covalent structures of the molecule involves a product of all the bond-pair wave functions of the individual bonds in the corresponding Lewis structure. As a rule, such a perfect-pairing polyelectronic VB structure having n bond pairs will be described by 2n determinants, displaying all the possible spin permutations between the orbitals that are singlet-coupled. Inclusion of the ionic components of the bonds results in the final VB wave function expressed in eq. 8 as a linear combination of VB structures ΦK:

"VB = # CK $K

(8)

K

!

One drawback of this description of a molecular system, as in eq. 8, in the early VB theory was the exponential growth in the number of VB structures with the number of bonds in the molecule. This had in turn two undesired consequences: (i) the wave function lost lucidity; and (ii) the calculation became time-consuming. Thus, since the Hamiltonian VB matrix elements are not as easily calculated as in MO-CI theory, owing to the so-called “N! problem” (see box), the growth in the number of VB configurations posed a formidable computational challenge. Additionally, the early VB theory used the same AOs as in the free atoms, and consequently the calculations suffered from the lack of efficient methods for orbital optimization and for incorporation of dynamic correlation. As will be seen below, these difficulties have been overcome by the new generation of VB methods.


9

The non-orthogonality problem or “N! problem” In MO theory, two different Slater determinants have a zero-overlap, owing to the orthogonality of the orbitals. In VB theory, the orbitals are generally non orthogonal, and the overlap between two N-dimensional determinants is calculated as a sum of N! products of orbital overlaps, a complication that has been termed “the N! problem”. Similarly, the Hamiltonian matrix elements between Slater determinants are calculated by means of the very simple Slater-Condon rules when the orbitals are orthogonal, whereas for non-orthogonal orbitals these simple rules are not applicable, and there are many more terms multiplied by overlaps. It follows that the evaluation of these matrix elements is much more timeconsuming in the VB framework than in configuration interaction in the MO framework. However, the term “the N! problem” is actually a misnomer, since the difficulty due to non-orthogonality does not imply that the computational effort required to perform a non-orthogonal configuration interaction scales as N!. Modern ab initio VB methods generally scale as N4. 8.1.3 Some landmark improvements of the early VB method To keep the advantage of VB theory while being at the same time efficient, the new VB methods had to meet several challenges : (i) to provide compact VB wave functions that would be clearly interpretable in terms of Lewis structure, (ii) to be as little time-consuming as possible, and (iii) to provide energetics as accurate as sophisticated MOCI methods. All three challenges have been achieved by major modernization of the early VB method. Some of these landmark improvements are briefly described below. 8.1.3.1 VB wave functions with semi-localized atomic orbitals. A great step for obtaining compact wave functions was made by Coulson and Fischer6 who proposed to describe the two-electron bond as a formally covalent singlet-coupling between two orbitals ϕa and ϕb, the

10


latter being optimized with freedom to delocalize over the two centers. This is exemplified in eq. 9 for the A–B bond: ΨCF(A–B) = " a " b # " b" b

!

(9a)

ϕa = φa + εφb

(9b)

ϕb = φb + ε’φa

(9c)

Here φa and φb are purely localized AOs (or HAOs), while ϕa and ϕb are delocalized orbitals. In fact, experience shows that the Coulson-Fischer orbitals ϕa and ϕb, which result from the energy minimization, are generally not extensively delocalized (ε, ε’ < 1), and as such they can be viewed as "distorted atomic orbitals". However, minor as this may look, this slight delocalization renders the Coulson-Fischer wave function equivalent to the ΨVB(A–B) wave function (eq. 2) with the three classical VB structures. A straightforward expansion of the Coulson-Fischer wave function leads to the linear combination of the classical structures, in eq. 10.

(

)

"CF (A# B) = (1 + $$ %) &a&b # &a&b + 2$ &a&a + 2$ % &b&b

(10)

Thus, the Coulson-Fischer representation keeps the simplicity of the covalent picture while treating the covalent/ionic balance by embedding the effect of the ionic terms in a variational way, through the delocalization tails of the VB orbitals. The Coulson-Fischer idea has later been generalized to polyatomic molecules and gave rise to the Generalized Valence Bond (GVB)7 and Spin-Coupled (SC)8 methods. The advantage of using wave functions of Coulson-Fischer type becomes obvious when one wishes to treat all the bonds of a molecular system in a VB way. For example, the GVB wave function representing methane with its four C-H bonds needs a single formally covalent structure (eq. 11),

!

ΨGVB = ("1h1 # "1h1)(" 2 h2 # " 2 h2 )(" 3h3 # " 3h3 )(" 4 h4 # " 4 h4 )

!

(11)


11

where the ϕI’s are the four HAOs of the carbon atom, which are singletcoupled to the orbitals hi of the hydrogen atoms; both ϕi and hi are localized on their respective centers while bearing small delocalization tails to the other centers. This is a great simplification compared with the mixed covalent/ionic wave function that possesses 81 mixed structures based on localized orbitals. h1

!1 !4 h4

C

!2 h2

!3 h3

Scheme 8.3. The schematic GVB wave function for methane

Letting all the orbitals in eq. 11 be determined variationally leads to four sp3 type HAOs ϕi pointing in tetrahedral directions toward the corresponding hydrogen atoms, as shown in Scheme 8.3. These HAOs, which come out from a variational calculation without the input of any qualitative preconception, clearly demonstrate the validity of the universally used hybridization model. Incidentally, this GVB wave function is much lower in energy than the simple MO wave function with its delocalized canonical MOs.9 8.1.3.2 Efficient orbital optimization by Self-Consistent-Field VB. Despite their remarkable usefulness, the VB methods which utilize Coulson-Fischer orbitals are not able to distinguish ionic from covalent structures, both types of structures being implicitly embedded in a formally covalent wave function. For example, in the transition state of the SN2 reaction (12), X– + H3 C–Y → [X…CH3 …Y]– → X–CH3 + Y–

(12)

12


the ionic structure 7 (Scheme 8.3) plays a fundamental role, and it is important to be able to calculate its relative energy and relative weight with respect to the covalent structures 5 and 6. To account for the ionic structures, one has to go back to the classical VB representation in terms of VB structures built with pure HAOs. The compactness of the VB wave function is maintained by treating only the active orbitals and electrons (those that are implied in bond-breaking/bond-forming in the reaction, red in Scheme 8.4) in a VB fashion, while the other orbitals (called “inactive” or “spectator” orbitals, black in Scheme 8.4) are treated as doubly occupied bonding orbitals throughout the reaction, i.e. treated in a MO way. This way, the transition state of an SN2 reaction is represented by a total of six VB structures, three of which being represented in Scheme 8.4 while the three remaining ones are practically negligible. H

H C

X

C

X

Y

HH

HH 5

Y

6

X– H3C–Y

X–CH3 Y–

H C

X

Y

HH 7

X– H3C+ Y–

Scheme 8.4. The main VB structures necessary to describe an SN2 transition state.

The calculation of such a multi-structure VB wave function, in the spirit of eq. 8, can then be made by means of the Valence Bond Self Consistent field method (VBSCF),10 which optimizes the coefficients CK and the orbitals of the VB structures ΦK simultaneously. The VBSCF method has been implemented in ab initio codes by van Lenthe et al.,11 with efficient algorithms of orbital optimization that get rid of the N!


13

problem. The algorithm has been further improved just recently by Wu et al.,12 and even faster versions are currently in progress. 8.1.3.3 Improving the accuracy by including dynamic electron correlation. The accuracy of the GVB, SCVB and VBSCF methods is comparable to that of valence-CASSCF, which is sufficient for many applications. However, none of these VB or MO methods are capable of yielding accurate reaction barriers or dissociation energies. The reason for that becomes clear if one examines the pictorial representation of the A–B bond as described at the VBSCF level in Scheme 8.5a (recall that the GVB, SC or MO-CASSCF wave functions are practically equivalent to the VBSCF ones). (a) !VBSCF C1

A

B

+ C2

A

8

B

+ C3

A

9

B

10

(b) !BOVB C'1

A

B

8

+ C'2

A

B

9

+ C'3

A

B

10

Scheme 8.5. Representation of the A–B bond by the VBSCF (a) and BOVB (b) methods. The spectator orbitals in black may be lone pairs or bond orbitals of bonds between A and/or B to substituents on A and B.

In ΨVBSCF, the necessary VB structures (8-10) are present and their coefficients and orbitals are optimized simultaneously, thus covering the main part of electron correlation, so-called “static correlation”. Now the set of HAOs is common to the three VB structures 8-10, with the consequence that the active orbitals (red in Scheme 8.5a) have the same size and shape whether they are singly occupied as in 8 or doubly occupied as in 9 or 10. Similarly, the inactive orbitals (black in Scheme 8.5a) are optimized for an average mean-field situation while they

14


experience the fields of neutral atoms in 8, versus ions in 9 and 10. Clearly, a better wave function would be allowed to have different orbitals for different VB structures. Such a wave function is represented in Scheme 8.5b, where it is seen that the orbitals surrounding, e.g., A– in 9 or B– in 10, are drawn bigger than those surrounding A and B in 8. This is the essence of the “breathing-orbital valence bond” method (BOVB),13 and this improvement, that keeps the wave function as compact as in VBSCF, brings some dynamic correlation that is necessary for getting accurate dissociation energies. Another recent VB method that takes care of dynamic correlation is the VBCI method.14 This is a post-VBSCF approach, where the VBSCF wave function serves as a reference wave function for the CI procedure. Thus, excited VB structures are generated from the reference wave function by replacing occupied (optimized VBSCF) orbitals with virtual orbitals, and CI is performed between the reference VB structures and the excited ones. To generate physically meaningful excited structures, the virtual orbitals are constructed to be strictly localized, like the occupied VB orbitals. After the CI has been done, the reference and all the excited VB structures that represent the same bonding scheme are condensed into a single structure. In this manner, the extensive VBCI wave function is condensed to a minimal set of fundamental structures, which ensures that VBCI keeps the VB advantage of compactness. A much faster variant than VBCI is the VBPT2 method,15 in which the excited VB structures are treated by perturbation theory to second order rather than by CI. According to our experience and to benchmark calculations,15 the BOVB, VBCI and VBPT2 methods all provide reasonably accurate reaction barriers and dissociation energies. This latter feature is crucially important for VB applications to chemical reactivity, as will be exemplified below. 8.1.3.4 Inclusion of solvent effects. Coupling the VB method with the polarizable continuum model (PCM) generates the VBPCM method, which was developed for exploring the solute-solvent interactions at the ab initio VB level.16 To incorporate solvent effects into a VB scheme, the state wave function is expressed in the usual form as a linear combination of VB structures, but now, these VB structures interact with


15

one another in the presence of the polarizing field of a solvent. VBPCM enables one to study the energy curve of the full VB state as well as that of individual VB structures throughout the path of a chemical reaction, and then reveal the solvent effect on the different VB structures as well as on the total VB wave function. Another solvation model, the SMx model developed by Truhlar and coworkers,17 has also been incorporated into VB theory, leading to the VBSM method.18 For large systems such as biological systems, VB theory has been combined with molecular mechanics (MM). The resulting VB/MM method19 utilizes the ab initio VB approach for the reactive fragments and MM for the environment and has been applied to an SN2 reaction in solution.19 8.2. Strengths of the Valence Bond Approach 8.2.1 Interpretability combined with accuracy of the wave functions As mentioned above, modern ab initio VB methods like BOVB, VBCI or VBPT2 can combine extreme compactness and clear physical meaning of the wave functions with accuracy of the calculated energies. This combination of compactness and accuracy can be appreciated with the example of the dissociation energy of difluorine, a well-known difficult test case. At the MO (Hartree-Fock) level, even with very large basis sets, this molecule is unbound,20 in contrast to an experimental dissociation energy of 38.3 kcal/mol.21 At the GVB, SC, VBSCF and valence-CASSCF levels, the calculated dissociation energy is about half the experimental value.13 Finally, a simple BOVB/cc-pVTZ calculation provides a dissociation energy of 37.9 kcal/mol, very close to the experimental value, with only three VB structures (8-10 in Scheme 8.5).22 By comparison, in the MO framework, an MCSCF treatment must go far beyond the valence CASSCF level, and the resulting dissociation energy oscillates between too small to too large values until as many as 968 configurations are included and the dissociation energies converges.23 VB theory provides a clear picture of the very important concept of electron correlation. For a single bond, the static electron correlation is

16


accounted for if the three VB structures are given optimized coefficients, as in Scheme 8.5a, whereas the weights of the ionic structures are systematically overestimated at the Hartree-Fock level. This equilibration of ionic vs covalent coefficients is also called “left-right” correlation. On the other hand, the subtle “breathing-orbital” effect, by which the orbitals rearrange in size and shape to follow the charge fluctuation in the bond (as shown in Scheme 8.5b) is associated with dynamic correlation, more precisely, with the change in dynamic correlation that attends bondbreaking/bond-formation of the A–B bond (also called differential dynamic correlation). This latter term is the dominant correlation term in three-electron/two-center (3e/2c) bonds, in which there is no left-right correlation. To illustrate this point, consider the AB– anion that possesses a three-electron bond, noted [A∴Β]–. At the Hartree-Fock level, the bonding and antibonding orbitals σg and σu (see Scheme 8.1) are doubly and singly occupied, respectively. It turns out that the Hartree-Fock wave function is equivalent to the VBSCF description, i.e. a resonance between two Lewis structures, as an expansion of the Hartree-Fock determinant would show (eq. 13): ΦMO[A∴B]– = " g" g" u = #b#b#a $ #a#a#b ;

" g # ($a + $b ) , " u # ($a % $b )

(13)

Since the Hartree-Fock description involves the two Lewis ! structures that can be drawn for this system with the right coefficients ! equivalent!electronegativities), no left-right (50:50 if A and B have correlation is necessary to re-equilibrate these coefficients by CI, so that the only electron correlation that has to be accounted for is dynamic.24 In accord, a simple BOVB description of the [A∴Β]– bond, in terms of two VB structures 11 and 12 each having their own set of optimized orbitals, contains all the physics of 3e/2c interactions and provides accurate bonding energies.25 Besides the very simple picture provided by VB theory for dynamic correlation, it should be noted that the contribution of this term to the 3e/2c bonding energy is very large in all cases.26,27 For example, the bonding energy of F2–, as calculated at the Hartree-Fock level, is close to zero, compared to 28.0 kcal/mol with a 2-structure BOVB calculation and 30.2 kcal/mol experimentally.21


17

The BOVB method not only provides dissociation energies, but also accurate dissociation curves at any interatomic distance,13b and accurate relative energies of transition states relative to reactants and products.28-31

A

B

A

11

B

12

Scheme 8.6. The 2-structure BOVB description of the [A∴B]– bond

The BOVB method not only provides dissociation energies, but also accurate dissociation curves at any interatomic distance,13b and accurate relative energies of transition states relative to reactants and products.28-31 The root cause of this combination of accuracy and compactness (e.g. 6 structures for an SN2 reaction,29 and 8 for a radical abstraction reaction31) probably lies in the intuitive principle that if an electronic state is described by all relevant VB structures and if each of these VB structures has its optimal specific set of orbitals, then the relative energy of this electronic state should be balanced throughout a potential energy surface. Another point that favors compactness is that one does not take care of all the electron correlation in a molecular system (as full CI would do), but only the part of electron correlation that varies throughout a reaction coordinate or potential surface is taken into account. It is in this spirit that spectator orbitals can be held doubly occupied as in a simple MO treatment, and that only the differential part of dynamic correlation is taken into account. 8.2.2 A simple solution to the symmetry dilemma The so-called “symmetry dilemma” has been first reported by Löwdin32 and it refers to an artefactual symmetry-breaking of the wave function, which occurs in some MO-based methods, even sophisticated ones. The problem is encountered each time a molecular system qualitatively corresponds to a resonance between two (or more) Lewis

18


structures of equal or quasi-equal weights, as for example in a threeelectron bond [A∴A]–. In such cases, an unphysical symmetry-broken solution may happen to be lower in energy compared with the symmetryadapted one, resulting in poor relative energies, inaccurate molecular properties and so on. The problem is general and exists in a variety of open-shell electronic states involving radicals of allylic types, coreionized diatoms, charged clusters, etc. The symmetry dilemma is clearly illustrated with the example of the [A∴A]– species (eq. 13 and Scheme 8.6 with A = B). As has been said, the symmetry-adapted Hartree-Fock wave function for [A∴A]– corresponds to two identical VB structures, like in Scheme 8.6 but with the same set of orbitals for the two structures. In such a case, the wave function is stabilized by resonance, but the orbitals are not optimal for each individual VB structure. Another solution, upon which a HartreeFock calculation may possibly converge, optimizes the orbitals for one VB structure (e.g. 11), to the detriment of the other (12). Then the VB structures have different energies and 11 ends up having a larger coefficient than 12. In this symmetry-broken solution, the resonance energy is diminished relative to the symmetry-adapted solution. Here is the dilemma: at the Hartree-Fock level, one cannot have simultaneously good orbitals and full resonance energy.32 A classical remedy consists of imposing the symmetry and doing CI. However, in many cases there is only quasi-symmetry (e.g. in [A∴B]– with A ≠ B), and in such a case there is no way to avoid the artificial favoring of one structure over the other. As a consequence, it is then very difficult to correct the initial deficiency by subsequent CI. While the problem is currently solved in the MO framework with elaborate methods such as Coupled-Cluster calculations using Brueckner orbitals,33 the symmetry-breaking artefact vanishes at the BOVB level. Indeed, as this method provides a superposition of two VB structures each having its optimal set of orbitals, the BOVB wave function involves at the same time both optimal orbitals and full resonance effect at any molecular geometry, and the root cause for the symmetry-breaking disappears. It follows that the BOVB method is, by nature, free from the symmetry-breaking artefact. Historically, the first calculation of that kind was done by Jackels and Davidson in 1976 for the NO2• radical.34 A


19

standard BOVB calculation was later performed for the potential surface of the HOOH– anion.35 8.2.3 Calculations of diabatic energy curves along a reaction coordinate 8.2.3.1 General model. There are two fundamental questions that any model of chemical reactivity would have to answer: What are the origins of the barriers? And what are the factors that determine reaction mechanisms? Since chemical reactivity involves bond breaking and making, VB theory with its focus on the bond as the key constituent of the wave function, is able to provide a lucid model that answers these two questions in a unified manner. The centerpiece of the VB model is the VB state correlation diagram (VBSCD), displayed in Figure 8.1,36 which traces the energy of the VB configurations along the reaction coordinate, and provides a mechanism for the barrier formation and generation of a transition state in an elementary reaction. R*

P*

Gr

Gp !r

R

B !p

!E! !Erp

P

Reaction Coordinate Figure 8.1: VBSCD for a general reaction R → P. R and P are ground states of reactants and products, R* and P* are promoted excited states.

This diagram applies to elementary reactions wherein the barrier can be described as the interplay of two major VB states, that of the reactants

20


and that of the products. It displays the ground state energy profile for the reacting system (bold curve), as well as the energy profiles for individual VB states (thinner curves); these latter curves are also called sometimes “diabatic” curves, while the full state energy curve (in bold) is called “adiabatic”. Thus, starting from the reactant geometry on the left, the VB structure Ψr that represents the reactant's electronic state, R, has the lowest energy and it merges with the ground state. Then, as one deforms the reacting molecules towards the product geometry, Ψr gradually rises and finally reaches an excited state P* that represents the VB structure of the reactants in the product geometry. A similar diabatic curve can be traced from P, the VB structure of the products in its optimal geometry, to R*, the same VB structure in the reactant geometry. Consequently, the two curves cross somewhere in the middle of the diagram. At each point of the diagram, the adiabatic ground state of the system (bold curve) is generated by mixing of the individual VB states. This mixing is stabilizing, so that the ground state is stabilized by a resonance energy term, labeled as B, in the region of the crossing point of the diabatic curves, which corresponds to the transition state. The barrier is thus interpreted as arising from avoided crossing between two diabatic curves, which represent the energy profiles of the VB state curves of the reactants and products. The nature of the R* and P* promoted states depends on the reaction type and will be specified below using a few examples. In all cases, the promoted state R* is the electronic image of P in the geometry or R, while P* is the image of R at the geometry of P. The G terms are the corresponding promotion energy gaps, B is the resonance energy of the transition state (TS), ΔE≠ is the energy barrier, and ΔErp is the reaction energy. The simplest expression for the barrier is given by eq. 14:

"E # = fGr ! B

(14)

Here, the term ƒGr is the height of the crossing point, expressed as some fraction (ƒ) of the promotion gap at the reactant side (Gr). A more explicit expression is eq. 15:


21

"E $ # f 0G0 + 0.5"Erp ! B ; G0 = 0.5 (Gr + Gp), ƒ0 = (ƒr + ƒp)

(15)

which shows the effects of the two promotion gaps and ƒ factors through their average quantities, G0 and f0. Eq. 15 expresses the barrier as a balance of the contributions of an intrinsic term, f0 G0 -B and a “driving force” term, 0.5∆Erp. The model is general and has been described in details before,3,4,37 and applied to a large number of reactions of different types. Here we will briefly summarize some VB computational applications on hydrogen abstraction reactions and various SN2 reactions. 8.2.3.2 Application to hydrogen-abstraction reactions. Consider a general hydrogen abstraction reaction that involves cleavage of a bond H–Y by a radical X•↑ (X, Y = a univalent atom or a molecular fragment): X•↑ + H–Y → X–H + Y•↑

(16)

Practically, Ψr is a linear combination of covalent and ionic forms that contribute to the Lewis structure “X•↑ + H–Y”, as shown below: Ψr = C1(X•↑ + H•–•Y) + C2(X•↑ + H+ :Y–) + C3(X•↑ + H:– Y+)

(17)

This combination is maintained in Ψr from R to P* throughout the reaction coordinate, while the coefficients of the contributing structures change and adapt themselves to the geometric change (e.g., at infinite H--Y distance C1=1). The curve Ψp, which stretches between P and R* is defined in an analogous manner. Since the promoted state R* is the VB structure of P in the geometry of R, its electronic state is illustrated by eq. 18: R* = (X•↑H•)---------•Y

(18)

where the H–Y bond is infinitely long, while the X•↑ radical (spin-up) experiences some Pauli repulsion with the electron of H which is 50-50% spin-up and spin-down. Thus, the XH fragment has a wave function which is 75% triplet and 25% singlet, and hence the promotion gap, from

22


the ground state to R*, is ¾ of the corresponding singlet-to-triplet excitation of the X-H bond. In the case of identity reactions (X = Y), it has been demonstrated that the promotion energy Gr required to go from R to R* is indeed proportional to the singlet-triplet gap of the X-H bond,31,37 which in turn is proportional to the X-H bond energy D(X-H). Actually, systematic VB ab initio calculations by the VBCI method have shown that, to a good approximation, Gr can be expressed as follows: 31 Gr " 2 D(X ! H )

(19)

In the general case of non-identity reaction, the X-H and Y-H bond strengths are different, one being the weakest (DW) and the other the strongest (DS). VB calculations for a panel of 14 reactions showed that B is approximately half of the weakest bonding energy, DW, or in other words, of the bond energy of the bond that is broken in the reactants of the exothermic direction of the reaction, while G0 in eq. 15 is given by the sum of both bonding energies:31c B = 0.5DW ; G0 = DW + DS

(20)

Moreover, the f factor appears to be relatively constant in both identity and non-identity reactions. Thus, by taking f0 ~ 1/3 as in identity reactions (accurate VB calculations yield f0 = 0.32-0.36),31c one gets the very simple eq. 21:

!E # = K (DS " 0.5 DW )+ 0.5!Erp ; K ≈ 1/3

(21)

Eq. 21 is valid for non-identity as well as identity reactions, in which case DS = DW. The so-calculated barriers were shown to match fairly well the corresponding CCSD(T) barriers for a series of identity abstraction reactions (X = Y = H, CH3 , SiH3, GeH3, SnH3, PbH3),31a with an average deviation of 2.1 kcal/mol and a maximum deviation of 4.8 kcal/mol. While the limitations of this expression have been discussed in detail, e.g., in case where the TS is not co-linear,3 still eq. 21 yields good orders of magnitudes and correctly reproduces the trends in the series.


23

For non-identity reactions (X ≠ Y), Figure 8.2 displays a good correlation of the barriers calculated through eq. 21 plotted against the VBCISD calculations. Thus, it appears that the VBSCD model is able to assess semi-quantitatively barriers for H-abstraction reactions in terms of the dissociation energies of the bonds of reactants and products.

45

!E! (eq. 21)

40 35 30

R2 = 0.974

25 20 15 10 5

10

15

20

25

30

35

!E! (VB ab initio)

40

45

Figure 8.2. VBSCD-derived barriers plotted against ab initio VB calculated barriers. Energies in kcal/mol.

Recent applications of eq. 21 to the reactivity of the enzyme cytochrome P450 in alkane hydroxylation shows that a good correlation with DFT-computed barriers is achieved with eq. 21 using a constant B value, which is very close to 0.5DW.38 8.2.4 Quantitative evaluation of common chemical paradigms 8.2.4.1 Direct calculation of resonance energies and hyperconjugation energies. Estimating resonance energies (REs) is a simple matter in the framework of VB theory. The RE is a measure of the magnitude of contributions from resonance structures other than the principal Lewis structure to the ground state of a conjugated molecule.

24


Starting from the description of an electronic system in terms of interacting VB structures (eq. 8), if we assume that a given Lewis structure L is the most stable (i.e., has the greatest weight) among all resonance contributors, the RE would be RE = E(ΦL) – E(ΨVB)

(22)

in which ΨVB is the fully delocalized ground state, and ΦL is the reference Lewis structure (which, depending on the reference ‘state’, may be represented by a single VB structure or by a group of VB structures). Eq. 22 has been used with ab initio VB calculations to calculate the resonance energies of benzene,39 butadiene,40 allyl radical and ions,41,42 transition states of organic reactions,3,29,31, 43-46 and so on, and to quantify the σ-aromaticity of cyclopropane.47 Eq. 22 has also been used to calculate the resonance energy arising from the mixing of covalent and ionic VB structures in a bond, leading to the discovery of a new type of chemical bonding (see next subsection).22,48,49 Another technique for calculating resonance energies or delocalization energies consists of defining the reference Lewis structure, by a so-called block-localized wave function (BLW), in which the orbitals are doubly occupied but optimized with some localization constraints. Thus, the orbitals that represent e.g., a π-bond in a conjugated system can be optimized while being constrained to be strictly localized on the two bonded atoms, and the orbitals that represent a lone pair are localized on a single atom. The orbital optimization can be carried out at the Hartree-Fock level,50 but a recent version using orbital optimization at the DFT level also exists.51 The resonance energy is then calculated as the difference between the BLW wave function representing the reference Lewis structure and the fully delocalized wave function of the ground state. More generally, the BLW method can be used to calculate delocalization energies by defining a diabatic state in which delocalization is “turned off”. In his latter state, the molecule or interacting system is partitioned into subgroups, and each localized MO is expanded in terms of basis functions belonging to only one subgroup. As the BLW method involves optimization of non-orthogonal orbitals, and since the BLW wave function represents a Lewis structure, the BLW


25

technique can be considered as belonging to the VB family, actually the simplest VB-variant. The above BLW method has been used to calculate the resonance energies of many organic molecules. For example, it has been used to quantify the role of resonance in the rotational barriers of amides,52 and in the acidities of carboxylic acids and enols as compared to alcohols.53 It was also used to provide accurate estimations of the vertical and adiabatic resonance energies of benzene,54 allyl radical and anions,55 and so on. Calculations of delocalization energies by VBSCF or BLW methods have also been used to get accurate estimates of the magnitudes of hyperconjugation. This has been applied to trace the origin of Saytzeff’s rule,56 the role of hyperconjugation in the rotational barrier of ethane57 or in the exceptional short bond length of tetrahedranyltetrahedrane,58 and so on. 8.2.4.2 Characterization of a novel type of bonding: Charge-shift bonds. The resonance energy arising from the mixing of covalent and ionic VB structures in a bond A–B can be calculated by means of Eq. 8, in which Φcov is the pure covalent structure (first term in Eq. 2) while ΨVB is the full VB wave function involving both covalent and ionic terms (all three terms in eq. 2). Such calculations were done at the BOVB and VBCI levels,22,48 and the resonance energy, referred to as covalent-ionic resonance energy (RECI), was determined. Thus, RECI was found to be a minor component of the bonding energy in the two classical families of covalent and ionic bonds, which are mainly stabilized by the purely covalent interactions in the first case, and by purely electrostatic interactions in the second. However, alongside these two classical types, a third category of bonds appeared, in which RECI is the major component of the bonding energy, even in some homonuclear bonds. This type of bonding was called “charge-shift bonding”, because the electron-pair fluctuation plays the dominant role in the bonding mechanism.22,48 An extreme case of charge-shift bonding is the F2 molecule, in which the covalent repulsion is repulsive at all distances. More generally, the charge-shift bonding territory involves, homopolar bonds of compact electronegative and/or lone-pair-rich elements,

26


heteropolar bonds of these elements among themselves and with other atoms (for example, the metalloids, such as silicon and germanium), hypercoordinated molecules, and bonds whose covalent components are weakened by exchange-repulsion strain (as in [1.1.1]propellane),49 nodensity bonds,22,48d and so on. Charge-shift bonding has experimental consequences, such has the barriers for halogen-transfer reactions having much larger barriers than the corresponding hydrogen-transfer processes,48b the rarity of silicenium ions in condensed phases,59 the surprising strength of inverted bonds in the series of propellanes molecules.49 A related bonding mechanism was discovered in maximum spin clusters devoid of any electron pairs between the constituent atoms, and nevertheless quite strongly bound. Thus, for example, the 3Σu+ state of 3 Cu2 is bonded despite of the fact that its σ and σ* orbitals are singly occupied and hence the bond order of the two atoms is zero.60 As the cluster grows, this no-pair bonding increases and reaches 18-19 kcal/mol per atom. The bonding arises from covalent-ionic mixing of triplet configurations and constitutes a mechanism of bound triplet pairs. 8.2.4.3 σ vs π driving force for the D6h geometry of benzene. The regular hexagonal structure of benzene can be considered as a stable intermediate in a reaction that interchanges two distorted Kekulé type isomers, each displaying alternating C-C bond lengths as shown in Figure 8.3. It is well known that the D6h geometry of benzene is stable against a Kekulean distortion (of b2u symmetry), but one may still wonder which one of the two sets of bonds, σ or π, is responsible for this resistance to a b2u distortion. The σ frame, which is just a set of identical single bonds, prefers by nature a regular geometry with equal C-C bond lengths. It is not obvious whether the π electronic component has, by itself, the same tendency and hence contributes to the D6h geometry of benzene. Could the π electronic component be actually distortive, but overwhelmed by the propensity of the σ frame to maintain a D6h geometry? To answer this question, consider in Figure 8.3 the VBSCD that represents the interchange of Kekulé structures along the b2u reaction coordinate; the middle of the b2u coordinate corresponds to the D6h


27

structure, while its two extremes correspond to the bond-alternated mirror image Kekulé geometries. (b)

(a)

K2*

K1*

K2*

K1*

1B (!) 2u 1B 2u

1A (!) 1g

K1

K1 K2

K2 1A

1g

RC RC

Figure 8.3. VBSCDs showing the crossing and avoided crossing of the Kekulé structures of benzene along the bond alternating mode, b2u for: (a) π-only curves, (b) full σ+π curves.

Part (a) of the figure considers π energies only. Starting from the lefthand side, Kekulé structure K1 correlates to the excited state K2* in which the π bonds of the initial K1 structure are elongated, while the repulsive non-bonding interactions between the π bonds are reinforced. The same argument applies if we start from the right-hand side, with structure K2 and follow it along the b2u coordinate; K2 will then rise and correlate to K1*. To get an estimate for the gap, we can extrapolate the Kekulé geometries to a complete distortion, in which the π bonds of K1 and K2 would be completely separated (which in practice is prevented by the σ frame that limits the distortion). At this asymptote the promotion energy, Ki → Ki* (i = 1,2), is due to the unpairing of three π bonds in the ground state, Ki, and replacing these bonds by three non-bonding interactions, in Ki*. According to qualitative VB theory,3 the latter are repulsive by a quantity that amounts to half the size of a triplet repulsion. The fact that such a distortion can never be reached is of no concern.

28


What matters is that this constitutes an asymptotic estimate of the energy gap G that correlates the two Kekulé structures, and that eventually determines if their mixing results in a barrier or in a stable situation, in the style of the VBSCD above. According to the VB rules,3 G is given by eq. 23: G(K→K*) = 3[0.75Δ EST(C=C)] = 9/4 Δ EST(C=C)

(23)

Since the Δ EST value for an isolated π bond is well over 100 kcal/mol, eq. 23 places the π electronic system in the region of large gaps. Consequently, the π-component of benzene is predicted by the VBSCD model to be an unstable transition state, 1A1g(π), as illustrated in Figure 8.3a. This “π-transition state” prefers a distorted Kekulean geometry with bond alternation, but is forced by the σ frame, with its strong symmetrizing driving force, to adopt the regular D6h geometry. This prediction, which was derived at the time based on qualitative considerations of G in the VBSCD of isoelectronic series,61 was later confirmed by a variety of rigorous ab initio σ-π separation methods.62 The prediction was further linked63 to experimental data associated with the vibrational frequencies of the excited states of benzene. The spectroscopic experiments63 show a peculiar phenomenon. This phenomenon is both state specific, to the 1B2u excited state, as well as vibrational mode specific, to the bond-alternating mode, i.e. the Kekulé mode b2u. Thus, upon excitation from the 1 A1g ground state to the 1 B2u excited state, with exception of b2u all other vibrational modes behave "normally" and undergo frequency lowering in the excited state, as expected from the decrease in π-bonding and disruption of aromaticity following a π→π* excitation. By contrast, the Kekulé b2u mode undergoes an upward shift of 257-261 cm–1. As explained below, this phenomenon is predictable from the VBSCD model and constitutes a critical test of π distortivity in the ground state of benzene. A simple vibronic coupling mechanism cannot account for this mode specificity and state specificity. Indeed, the VBSCD model is able to make predictions not only for the ground state of an electronic system, but also for a selected excited state. Thus, the mixing of the two Kekulé structures K1 and K2 in


29

Figure 8.3a leads to a pair of resonant and antiresonant states K1±K2; the1A1g ground state K1+K2 is the resonance-stabilized combination, and the 1B2u excited state K1-K2 is the antiresonant mixture (this is the first excited state of benzene64). In fact, the VBSCD in Figure 8.3a predicts that the curvature of the 1 A1g(π) ground state (restricted to the π electronic system) is negative, whereas by contrast, that of the 1 B2u(π) state is positive. Of course, when the energy of the σ frame is added as shown in Figure 8.3b, the net total driving force for the ground state becomes symmetrizing, with a small positive curvature. By comparison, the 1B2u excited state displays now a steeper curve and is much more symmetrizing than the ground state, having more positive curvature. As such, the VBSCD model predicts that the 1 A1g→1 B2u excitation of benzene should result in the reinforcement of the symmetrizing driving force, which will be manifested as a frequency increase of the Kekulean b2u mode. In order to show how delicate the balance is between the σ and π opposing tendencies, we recently65 derived an empirical equation, eq. 24, for 4n+2 annulenes:

ΔEπ+σ = 5.0 (2n+1) – 5.4 (2n), kcal/mol

(24)

Here Δ Eπ+σ stands for the total (π and σ) distortion energies, the terms 5.0(2n+1) represent the resisting σ effect, which is 5.0 kcal/mol for an adjacent pair of σ-bond, whereas the negative term, – 5.4(2n), accounts for the π-distortivity. This expression predicts that for n = 7, namely the C30H30 annulene, the ΔEπ+σ becomes negative and the annulene undergoes bond localization. If we increase the π-distortivity coefficient but just a tiny bit, namely to eq. 25,

ΔEπ+σ = 5.0 (2n+1) – 6.0 (2n), kcal/mol

(25)

the equation would predict now that already the annulene with n = 3, namely C14H14, will undergo bond localization. This extreme sensitivity, which is predicted to manifest in computations and experimental data of annulenes, is a simple outcome of the VBSCD prediction that the πcomponent of these species behaves as a transition state with a propensity towards bond-localization.

30


8.3 Present Capabilities and Expected Improvements 8.3.1 Evaluation of Hamiltonian matrix elements Since the early days of VB theory, much progress has been made in improving the algorithms and speeding up the evaluation of the Hamiltonian matrix elements between VB structures. Efficient algorithms that deal with the non-orthogonality problem (see box #2) have been implemented,11 such that the calculation of Hamiltonian matrix elements between non-orthogonal determinants scales as N4. Among the methodological contributions that brought VB to the modern era, one may cite the Prosser and Hagstrom method of evaluating matrix elements,66 the generalized Slater-Condon rules,67 the spin free VB method,68 the left coset decomposition algorithm,69 the algebrant algorithm of Li et al,70 and the paired-permanent-determinant approach.71 8.3.2 Direct VBSCF/BOVB algorithm In the conventional VBSCF or BOVB calculations, the Hamiltonian matrix elements are expressed explicitly in form of orbital integrals, and thus, an integral transformation from basis functions to VB orbitals is required for each iteration, resulting thereby in costly demands for both CPU time and storage space. Furthermore, as discussed above, the conventional VBSCF/BOVB method applies the super-CI method to optimize orbitals or uses approximate Newton-Raphson type methods,11b where the energy gradients are approximately obtained. Both of the two algorithms require costly computational efforts. To reduce the cost in the VBSCF/BOVB calculations, Wu et al. developed an algorithm for evaluating analytical energy gradients.72 Most importantly, the formulas in this algorithm are orbital-free, and thus may be performed in integral direct mode easily, without integral transformation procedure. A further advantage of this algorithm is that the computational cost mainly depends on the number of basis functions, rather than the number of orbitals. This latter feature is fundamental for speeding up calculations of BOVB type, which usually deal with many more orbitals than VBSCF calculations, since BOVB wave functions allow each VB structure to


31

have its own orbital set. Thus, with the direct algorithm, the cost of one BOVB iteration is almost the same as that of a VBSCF iteration.72 8.3.3 Current calculations of medium-sized molecular systems With the advances in VB methods, made in the last two decades, ab initio VB theory is now capable not only of accurate calculations of small molecules, by using BOVB and VBCI, but also of VB calculations of medium-size molecules, or even transition metal complexes. Just recently, Wu et al.73 have applied the VBSCF method to a study of the Diels-Alder reaction, and to the rotational barrier for the organometallic complex (CO)4Fe(CH2=CH2). In these studies, a 6-31G* basis set was used for atoms of the first and second rows of the periodic table, and a standard Lanl2dz basis set was used for the Fe atom. All electrons except for the ECP electrons of Fe were included in the VB calculation, summing up to 88 electrons in 46 VB orbitals in the case of the organometallic complexes. In both Diels-Alder and organometallic complex studies, the VBSCF accuracy was found to be equivalent to the corresponding CASSCF calculations. Better accuracy can be achieved with higher levels of VB methods, especially VBPT2, which is a very cheap postVBSCF method. A new version of VBPT2 that applies contraction technique of active space is in progress, and will hopefully provide the long sought for powerful tool for VB studies of many problems involving medium-sized molecules. 8.3.4 Mixed Valence Bond - Quantum Monte Carlo methods Quantum Monte Carlo (QMC) methods74 offer interesting alternatives to basis set ab initio methods. In QMC, the trial wave function is typically constructed as the product of a determinantal expansion and a correlation function, the latter containing explicit interparticle distance terms. The Diffusion Monte Carlo (DMC) method belongs to the QMC family and is able to provide extremely accurate BDEs, provided the trial function already includes a good deal of electron correlation. Thus, truncated

32


MCSCF wave functions were used as trial wave functions but proved disappointing.75 To get better DMC trial functions, several workers had the idea to use trial wave functions related to VB theory,76 ,77 expecting two advantages : (i) for the same amount of electron correlation being taken into account, VB expansions can be more compact than MCSCF ones ; (ii) because VB orbitals are generally localized on one or two centers, a VB-based trial wave-function could be cheaper than a trial wave function of the same expansion length based on MOs delocalized over the entire molecule. Lester et al.77 calculated the BDE for the acetylenic C-H bond by performing a DMC calculation using a trial BOVB wave function with a polarized triple-ζ basis set of Slater orbitals. The accuracy is excellent, with a C-H BDE of 132.4 ± 0.9 kcal/mol, practically equivalent to the recommended experimental value of 132.8 ± 0.7 kcal/mol. These values are to be compared with DMC results obtained with single determinant trial wave functions, using Hartree-Fock orbitals (137.5 ± 0.5 kcal/mol) and local spin density (LDA) Kohn-Sham orbitals (135.6 ± 0.5 kcal/mol). Very recently, Goddard et al.78 used simple GVB wave functions as guess functions for DMC calculations, and applied this approach to the adiabatic singlet-triplet splitting in methylene, the vertical and adiabatic singlet-triplet splitting in ethylene, 2 + 2 cycloaddition, and Be2 bond breaking. In all these cases, this approach was accurate within a few tenths of a kcal/mol. Less accurate results, however, were found for the very difficult test case of the N→V transition energy of ethylene, for which dynamic correlation is crucially important. With the very recent (and actually on-going) progress of QMC algorithms, trial wave functions of VBSCF quality yield results as accurate as former BOVB trial functions. This improvement should allow quite large systems to be treated by VB-QMC methods, with possibly up to 100 VB structures, with an accuracy close to experimental error bars. Moreover, work is in progress to use QMC methods to perform calculation of VB type, allowing calculations of the weights of VB structures, and calculations of individual VB structures, i.e. diabatic states.79


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Last but not least, a further advantage of QMC methods, and soon of VB-QMC methods, is that QMC algorithms scale as N3 (N being the number of electron), and possibly as N2 in the near future.79 8.3.5 Prospective The question that may be posed at this point is whether VB theory will ever return as a mainstream method that will be used by chemists as a computational tool and/or as a conceptual framework? Let us try to discuss these prospects by separating the two aspects; the future computational capabilities, and the conceptual impact. As shown above, VB methodology is constantly developing, and surely in the not too far future we may expect to see computations of ‘real’ reactions like the Diels Alder reaction, or of organometallic species becoming more and more common. In this respect the VBPT2 method shows a great promise to be a future standard method, with the cost of VBSCF, but with accuracy competing with CASPT2. One of the present shortcomings of VBPT2 is the treatment of spinstate of transition metal complexes in e.g., bioinorganic chemistry. At present, the calculations of the nonheme oxo-iron complex, (NH3)5FeO2+, indicates that the method overestimates the stability of high-spin states.80 However, spin-state ordering seems to be one of the most difficult hurdles for computational chemistry, and even a large CASSCF(20,13) followed by spectroscopy-oriented configuration interaction (SORCI) calculations overestimate the stability of the high-spin quintet state of this reagent.81 CASPT2 performs relatively well82 by using zero order Hamiltonians with energy-shifted orbital energies, a very large basis set, and a double-shell for the occupied 3d orbitals of iron. This latter effect, which correlates the nonbonding doubly occupied 3d orbitals, is analogous to the orbital splitting used in BOVB, and can be utilized in the same manner in VBPT2, thus making this method applicable to bioinorganic complexes. When this is achieved, VB as a computational tool will be as versatile as the MO-based methods, albeit never as fast as MP2. Another future development is VBDFT, which combines the

34


dynamic correlation facility of DFT, with the static correlation inherent in VB. An early version which has been tried already more than a decade ago,83 showed some promise. However, it was already then clear that an efficient way of implementing VBDFT method could be constructed by using densities to get the energies of the VB structures, and then mixing them either with these frozen densities or by iterating on the density during the mixing procedure. Such a VBDFT method may combine speed and accuracy, and may enable the calculations of reasonably large systems. The constantly developing BLW method is now capable of incorporating charge resonance effects within the MO-VB framework.84 Once this development is generalized it will add to the growing arsenal of VB-type methods. This method will certainly be competitive with MP2. Conceptually, VB theory is extremely versatile and applicable to various domains in chemistry. There is no real hurdle to use VB theory as a conceptual framework for chemistry, other than the saddening fact that for historical reasons VB is not taught anymore in quantum chemistry courses as a mainstream method. However, the recent monograph written on VB theory,3 may ease the way for those who are willing to try and teach or study elements of VB theory. Once this wall is broken, chemists will find a beautiful theory, which can easily be incorporated into a thought process. 8.4 Concluding Remarks This short chapter has intended to show that ab initio VB theory has enjoyed impressive progress during the past two or three decades. As a result, ab initio VB algorithms are today much faster than they used to be, and by several orders of magnitude. Moreover, modern VB has also reasonably accurate computational methods, which can provide bonding energies and reaction barriers with accuracies comparable to sophisticated MO-CI methods. This has been achieved by incorporating dynamic correlation effects in the VB calculations, and this without complicating the wave functions that remain compact and easily


35

interpretable. In addition, the modern VB methods can be combined with a solvent model, and provide thereby a method that can handle molecules and reactions in solution and in proteins. Thus, from a quantitative point of view, VB theory enables today the calculations of ‘real’ chemical problems for organic molecules, as well as molecules that contain transition metals, and all these can be done in the gas phase or in solution. Further improvements in speed and capabilities of VB methods are currently in progress. Especially promising is the VBPT2 method that emerges as a fast and accurate method, and the combination of VB and QMC methods that may enable, in the future, to handle much larger systems than those presented here. Another aspect of VB theory that is emphasized in this chapter is insight. Thus, despite the sophistication and accuracy of the above VB methods, all of them rely on a compact wave function that includes a minimal number of structures in the VB-structure set. The insight of this compact wave function is projected by a set of applications including bonding in main group elements, quantitative evaluation of common paradigms such as resonance energies, hyperconjugation, aromaticity and antiaromaticity in conjugated systems, distortivity of the π system of benzene and related molecules,65 and general models of chemical reactivity. Many other applications e.g., to photochemistry, excited states, polyradicals, etc have appeared in a recent monograph.3 VB theory provides also a great deal of insights into bonding in odd-electron systems, specifically on one-electron and three-electron bonds.4,24,25,85,86 In fact, VB theory gives rise to new bonding paradigms, like ‘chargeshift bonding’, which concerns two-electron bonds which are neither covalent nor ionic but with bonding energy that is dominated by the covalent-ionic resonance interaction.22,48,49 Another paradigm is the ‘ferromagnetic bonding’ that occurs in high-spin clusters, e.g., n+1Lin, n+1 Cun, etc. that are devoid of electron pairs but still have significant bond energies.60 It is to be expected that this combination of chemical insight with ever increasing speed and accuracy will place modern VB methods in an important position among the ab initio methods that will be used in the future.

36


8.5 Appendix-1: The Myth of “VB failures” One of the major alleged “VB failures” is associated with the dioxygen molecule, O2.3 As we repeatedly stated, it is true that a naïve application of hybridization and perfect pairing approach (simple Lewis pairing) without consideration of the important effect of 4-electron repulsion would put two electrons in one π plane and four electrons in the other, leading to the erroneous prediction of a 1Δg ground state. However, proper use of elementary VB principles, even at a simple qualitative level, shows that a better bonding energy is obtained by forming two three-electron π-bonds, leading to the 3Σg– paramagnetic ground state in agreement with experiment. In agreement with the qualitative analysis, early ab initio VB calculations7d correctly provide the ordering of the ground state and low-lying excited states of O2. Another alleged VB failure deals with the photoelectron spectroscopy (PES) of CH4:3 Starting from a naïve application of the VB picture of CH4, it follows that since methane has four equivalent localized bond orbitals (LBOs), ergo the molecule should exhibit only one ionization peak in PES. However, since the PES of methane shows two peaks, ergo VB theory “fails”! This naïve argument is obviously false, since a physically correct representation of the CH4+ cation must be a linear combination of the four VB configurations that correspond to one bond ionization, and not only one configuration. Elementary application of symmetry point group theory then shows that correct linear combinations are 2 A1 and 2T2; the later being a triply degenerate VB state.3 Accordingly, VB theory predicts two ionization peaks, in agreement with experiment. 8.6. Appendix-2: Some available VB software packages Other than the GVB method that is implemented in many packages by now, here are brief descriptions of the main VB software packages we are aware of and with which we had some experience to varying degrees.


37

8.6.1 The XMVB program The XMVB software87,88 is a general program that is designed to perform multistructure VB calculations. It can execute either nonorthogonal CI, or non-orthogonal MCSCF calculations with simultaneous optimization of orbitals and coefficients of VB structures. Complete freedom is given to the user to deal with HAOs, or orbitals of CoulsonFischer type, so that calculations of VBSCF, SCVB, BLW, BOVB, VBCI and VBPT2 types can be performed. The parallel version of XMVB, based on the Message Passing Interface, is also available.87 XMVB can be used as a stand-alone program that is freely available from the author (website: http://ctc.xmu.edu.cn/xmvb/). It has also been incorporated also into GAMESS-US,89 and can be interfaced to GAUSSIAN.90 8.6.2 The TURTLE software TURTLE91,92 is also designed to perform multistructure VB calculations and can execute calculations of the VBSCF, SCVB, BLW or BOVB types. Currently, TURTLE involves analytical gradients to optimize the energies of individual VB structures or multistructure electronic states with respect to the nuclear coordinates.93 A parallel version has been developed and implemented using the message-passing interface (MPI), for the sake of making the software portable.94 TURTLE is now implemented in the GAMESS-UK program.95 8.6.3 The VB2000 software VB200096,97 is an ab initio VB package that can be used for performing non-orthogonal CI, multi-structure VB with optimized orbitals, as well as SCVB, GVB, VBSCF and BOVB. VB2000 can be used as a plug-in module for GAMESS(US)89 and Gaussian98/0390 so that some of the functionalities of GAMESS and Gaussian can be used for calculating VB wave functions. GAMESS also provides interface (option) for the access of VB2000 module.

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8.6.4 The CRUNCH software The CRUNCH (Computational Resource for Understanding Chemistry) has been written originally in Fortran by Gallup, and recently translated into C.98 This program can perform multiconfiguration VB calculations with fixed orbitals, plus a number of MO-based calculations like RHF, ROHF, UHF (followed by MP2), Orthogonal CI and MCSCF.


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