The VBCI Method

As discussed in Chapter 9, the VBCI method provides results that are at par with the BOVB method, the difference being that the electrons of the spectator orbitals are correlated too in the VBCI method. The wave function starts from a VBSCF wave function and augments it with subsequent local configuration interaction that can be restricted to single excitations (VBCIS level), or single and double excitations (VBCISD), or higher excitations. Here, we will consider only the VBCISD level, which is a good compromise between accuracy and cost efficiency. 

The VBCI method is implemented in the XMVB program and is fairly straightforward. This is done in exactly the same way as in a VBSCF or BOVB calculation. Thus, for a VBCISD calculation in which all the orbitals are requested to be localized on their respective fragment, the input will simply be Input 10.2 in which the keyword vbscf is replaced by vbcisd . This calculation will be referred to as L-VBCISD. Of course, it is also possible to delocalize the TT-spectator orbitals as has been done above in the BOVB framework, which is accomplished by replacing bovb by vbcisd in Input 10.6. This latter level is referred to as tt-D-VBCISD in Table 10.2. 

In the above examples, the VBCI calculations include 549 and 1089 configurations, respectively, at the L-VBCISD and tt-D-VBCISD levels. However, note that the VBCI outputs also provide the VB information in condensed form, in terms of the three fundamental VB structures. This is shown in Output 10.2 for the L-VBCISD calculation, which displays a 3 x 3 Hamiltonian matrix, the corresponding overlap matrix between the fundamental VB structures (which are defined according to Eq. 9.14) and the weights of these structures. 

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All the calculations of F2 are carried out with a simple basis set of double-zeta polarization type, the standard 6-31G(d) basis set, and are performed at a fixed interatomic distance of 1.44 A, which is approximately the optimized distance for a full Cl calculation in this basis set. Only the corresponding orbitals are referred to as the active orbitals , while the orbitals representing the lone pairs, so-called spectator orbitals , remain doubly occupied in all calculations. A common point to the various VB methods we use, except the VBCI method, is that at the dissociation limit, the methods converge to two F fragments at the restricted-open-shell Hartree Fock (ROHF) level. 

The VBCI method can be viewed as a MRCI extension of the VBSCF approach. This method, which has been developed as a spin-free approach, starts with the calculation of a VBSCF wavefunction. The orbitals used to construct the initial wavefunction are formed as linear combinations of AOs from different subsets (or blocks ) as in eqn (3.6). The virtual orbitals needed for the additional VBCI configurations come from the orthogonal complements to the occupied orbitals for each subset from the original VBSCF wavefunction. The most convenient way of finding these virtual orbitals to diagonalise the representation of the projection onto the occupied space operator for each subset. 

The VBCI method, recently developed by Wu et al. is a post-VBSCF calculation that uses configuration interaction to supplement the VBSCF energy with dynamic correlation. At the same time, the method preserves the interpretability of the final wave function in terms of a minimal number of VB structures, each having a clear chemical meaning. The VB structures that are used in the VBSCF calculations are referred to as fundamental structures, denoted as T , and the orbitals that appear in the VBSCF calculation are referred to as occupied orbitals. Depending on the problem at hand, the VBSCF calculation may use semidelocalized CF orbitals, or orbitals that are each localized on a single atom or fragment in the latter case the fundamental structures will explicitly involve the covalent and ionic components of the bonds. 

The VBPCM procedure is not, in principle, restricted to the VBSCF method it has the potential ability to be implemented to more sophisticated methods like BOVB, VBCI, or other methods. The method is implemented in XMVB. 

VB (BOVB) method, which also utilizes covalent and ionic structures, but in addition allows them to have their own unique set of orbitals. This method is now incorporated into the programs TURTLE and XIAMEN-99. Very recently, Wu et al. developed a VBCI method that is akin to BOVB, but which can be applied to larger systems. The recent biorthogonal VB method (bio-VB) of McDouall has the potential to carry out VB calculations on systems with up to 60 electrons outside the closed shell. And finally, Truhlar and co-workers " ° developed the VB-based multiconfiguration molecular mechanics method (MCMM) to treat dynamical aspects of chemical reactions, while Landis and co-workers " introduced the VAL-BOND method that predicts the structures of transition metal complexes using Pauling s ideas of orbital hybridization. In the section dedicated to VB methods, we mention the main software and methods that we used, and outline their features, capabilities, and limitations. 

VBCI Valence bond configuration interaction. A VB computational method that starts with a VBSCF wave function, which is further improved by CI. The Cl involves virtual orbitals that are localized on exactly the same regions as the respective active orbitals. There are a few VBCI levels that are denoted by the rank of excitation into the virtual orbitals, for example, VBCISD involves single and double excitations. 

VBPCM Valence bond polarized continuum model. A VB computational method that incorporates solvent effect by using the PCM solvation model. The method can be coupled with VBSCF, BOVB, and VBCI. 

Xiamen-99 is a pure ab initio valence bond program. One can use the package to do any types of VB calculations with any forms of VB orbitals. This means that VBSCF, BOVB, and VBCI calculations may be carried out with the package, and it is also feasible to combine the valence bond method with some advanced molecular orbital methods, like VB-DFT [49],... 

Fig. 23.15. Ab initio computations of VBSCDs for the exchange processes, X + X-X — X-X + X, for X = H and Li (adapted with permission from Ref. [22], 1990, American Chemical Society). The reaction coordinate is 0.5( i — 2 + 1) where Ui and 2 are the X-X bond orders in X-X-X. The computations were based on a VBCI type method with single excitations, using the 6-3IG basis set for H3 and 6-3IG for Li3.

In summary, the VBSCF, ° VBCI, and BOVB methods are ideal tools for studying bonding and for generating VBSCDs for chemical reactions. However, while VBSCF will provide a qualitatively correct picture, both VBCI and BOVB methods will give quantitatively good results in addition to a lucid chemical picture. 

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