Valence bond wavefunctions

We consider in the following the optimization of modem valence bond wavefunctions for states that are second or higher within a particular symmetry. If CASSCF solutions... 

We focus in this Section on particular aspects relating to the direct interpretation of valence bond wavefunctions. Important features of a description in terms of modem valence bond concepts include the orbital shapes (including their overlap integrals) and estimates of the relative importance of the different stmctures (and modes of spin coupling) in the VB wavefunction. We address here the particular question of defining nonorthogonal weights, as well as certain aspects of spin correlation analysis. 

A. F. Voter, W. A. Goddard, III, Chem. Phys. 57, 253 (1981). A Method for Describing Resonance between Generalized Valence Bond Wavefunctions. 

The valence bond wavefunction for H2 is simply a linear combination of in... 

On the other hand, the (un-normalized) ground and excited (if-) state valence bond wavefunctions are... 

Wavefunction Vq refers to the situation where both electrons are on nucleus a or nucleus b, i.e., ionic structures. Now it is obvious that the valence bond wavefunction covalent structure, while the molecular orbital wavefunction t has an equal mixture of covalent and ionic contributions. Similarly, expanding the wavefunction in eq. (3.2.27) yields... 

So, put another way, the valence bond wavefunctions for the ground state of H2 has equal contributions from configurations and ax 2, and the combination coefficients have different signs. Such a wavefunction, which employs a mixture of configurations to describe the electronic state of an atom or molecule, is called a configuration interaction (Cl) wavefunction. 

The optimal c2/ci ratio is -0.73. More importantly, the improved valence bond wavefunction p + and the improved molecular orbital wavefunction t/fj are one and the same, thus showing these two approaches can lead to identical quantitative results. 

The Ab Initio Valence Bond program TURTLE has been under development for about 12 years and is now becoming useful for the non-specialist computational chemist as is exemplified by its incorporation in the GAMESS-UK program. We describe here the principles of the matrix evaluation and orbital optimisation algorithms and the extensions required to use the Valence Bond wavefunctions in analytical (nuclear) gradient calculations. For the applications, the emphasis is on the selective use of restrictions on the orbitals in the Valence Bond wavefunctions, to investigate chemical concepts, in particular resonance in aromatic systems. 

In the development of the TURTLE program [3], we started by considering a multi-structure Valence Bond wavefunction and added the capability to optimise the orbitals. We tried to avoid putting restrictions on the way the wavefunction is built and to allow great flexibility in the choice of orbitals. For... 

The SuperCI itself is usually quite stable, but involves solving a non-orthogonal Cl of a considerable dimension, with each Brillouin state containing the same number of determinants as the Valence Bond wavefunction, which is rather time consuming. The SuperCI matrix can be approximated by its first row (the Brillouin theorem elements) and the diagonal at a considerable time saving. Then the Brillouin state coefficients by are estimated following... 

J.H. van Lenthe and G.G. Balint-Kurti, VBSCF The optimisation of non-orthogonal orbitals in a general (Valence Bond) wavefunction, in 5th seminar on Computational Methods in Quantum Chemistry (Groningen, 1981). 

It can be seen that the first two terms are the same as the valence bond wavefunction, and there are an additional two terms. The first two are commonly called the covalent terms because each has the electrons associated with both centers. The final two are known as ionic terms because each places two electrons at one center (i.e. H+ H and H H ). Valence bond theory generally neglects ionic terms of this type, whereas in MO theory the covalent and ionic terms are treated equally. 

With a suitably chosen IVb, more than 99% of a ground-state CASSCF wavefunction may typically be brought to VB-like form. Provided CASSCF solutions are available, it is straightforward to extend this strategy to obtain modern valence bond wavefunctions for states that are second or higher within a particular symmetry [46,47]. 

The final spin-coupled valence bond wavefunction is of the form (17), but where the number of structures is of course now finite. The coefficients Cgkiii, 2 > > iv) are determined by constructing the matrix of the Hamiltonian (2) over the chosen set of structures and diagonalizing. 

The first two terms are exactly the wavefunction for the valence bond description while the last two terms are the type of ionic components that were described earlier. In the MO wavefunction, these ionic terms are given much more weight (50%) than in the valence bond wavefunction. This is exactly the improvement that the configuration interaction procedure makes. If the Cl wavefunction described above is expanded as in (18)... 

The relative success of the valence bond wavefunction comes about because it keeps the two electrons apart. When electron 1 is on atom A, electron 2 is on atom B, and vice versa. This can be compared with the MO wavefunction, which assumes that the electrons move independently of one another so that the probability of finding one electron at a particular point in space is independent of the position of the other electron. In reality, of course, electrons tend to avoid one another because they are negatively charged. Thus, the motion of one electron at a particular instant is dependent upon the position of the other electron, and their motions are correlated. 

Figure 2.9. The qualitative behavior of the ground state energy of titanium oxide and nickel oxide calculated with a VB model that includes some ionic contribution (amount X). A pure valence bond wavefunction has = 0 and a molecular orbital state has A = 1. For titanium oxide the MO model predicts an energy closer to the real value than a VB calculation. For nickel oxide the valence bond model is more realistic.

Hunt WJ, Hay PJ, Goddard WA (1972) Self-consistent procedures for generalised valence bond wavefunctions. Applications H3, BH, H2O, C2H6 and O3. J Chem Phys 57 738-748... 

Table 4. Huckel Valence Bond Wavefunctions for the v(3,l) and V(3,l) States.

First, let s try a qualitative picture of what s going on. We find in Section 5.2 that the simplest MO wavefunction for the H2 molecule is really very poor because it predicts that half the time the molecule dissociates into ions (Eq. 5.19). We fix this inadequacy in Section 5.2 by constructing a valence bond wavefunction (Eq. 5.24), iAvB(ground) = [1sa(1)1sb(2) + 1sb(1)1sa(2)]. ... 

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