Linear combination of VB structures

Another method from the same PCM family of solvation methods, namely the IEF-PCM [24] (see also the contribution by Cances), has recently been used to develop an ab initio VB solvation method [25], According to this approach, in order to incorporate solvent effect into the VB scheme, the state wavefunction is expressed in the usual terms as a linear combination of VB structures, but now these VB structures are optimized and interact with one another in the presence of a polarizing field of the solvent. The Schrodinger equation for the VB structures is then solved directly by a self-consistent procedure. 

Once a wave function is available and is written as a linear combination of VB structures as in Equation 3.54, the major VB structures can be distinguished from the minor ones by consideration of the magnitudes of their respective coefficients. 

The Valence Bond Self-Consistent Field (VBSCF) method has been devised by Balint-Kurti and van Lenthe (32), and was further modified by Verbeek (6,33) who also developed an efficient implementation in a package called TURTLE (11). Basically, the VBSCF method is a multiconfiguration SCF procedure that allows the use of nonorthogonal orbitals of any type. The wave function is given as a linear combination of VB structures, (Eq. 9.7). 

BOVB Breathing orbital valence bond. A VB computational method. The BOVB wave function is a linear combination of VB structures that simultaneously optimizes the structural coefficients and the orbitals of the structures and allows different orbitals for different structures. The BOVB method must be used with strictly localized active orbitals (see HAOs). When all the orbitals are localized, the method is referred to as L-BOVB. There are other BOVB levels, which use delocalized MO-type inactive orbitals, if the latter have different symmetry than the active orbitals. (See Chapters 9 and 10.)... 

The modest a-jt overlap in open-shell molecular solids suggests an approach based on separated molecular fragments Structural and spectroscopic evidence supports the occurrence of essentially unperturbed molecules or molecular ions in the solid state. Since valence-bond (VB) treatments of molecules become exact in the dissociated-atom limit, a diagrammatic VB approach has been developed for open-shell molecular solids. The resulting correlated crystal states are simply weighted linear combinations of VB structures for the entire solid. 

This output displays part of the information that is given by the XMVB program at the end of an L-VBCISD calculation on F2. Each fundamental structure is a linear combination of VB functions that possess the same nature in terms of spin-pairing and charge distributions. The coefficients of these VB functions are extracted from the multistructure ground-state wave function, and are renormalized (see Eqs. 9.13-9.14). 

In the case of the ground state 2E +, the final VB wave function is given by the following linear combination of those structures shown in Fig.6 ... 

Here a and b are purely localized AOs, while cp and cp are delocalized AOs. In fact, experience shows that the Coulson-Fischer orbitals cp and cp, which result from the energy minimization, are generally not very delocalized (s < 1). As such they can be viewed as distorted orbitals that remain atomic-like in nature. However minor this may look, the slight delocalization renders the Coulson-Fischer wave function equivalent to the VB-full (Eq. [5]) wave function with the three classical structures. A straightforward expansion of the Coulson-Fischer wave function leads to a linear combination of classical structures in Eq. [7]. 

The task is now to calculate the structure and energy of the system in the transition state between A and B. Its wave function is assumed to be constmcted from a linear combination of the two. It is convenient to use VB terminology for this purpose. Let the wave function of A be denoted by a VB function A) and that of B by B). 

The final description, either in terms of a Cl wave function written as a linear combination of two determinants built from delocalized MOs (eq. (7.4)), or as a VB wave function written in terms of two VB-HL structures composed of AOs (eq. (7.7)), is identical. 

The wave function for this system can be written as a linear combination of two VB states, which represent the ionic Bu+Cl- and the covalent Bu-Cl resonance structures, namely... 

The MO theory differs greatly from the VB approach and the basic MO theory is an extension of the atomic structure theory to molecular regime. MOs are delocalized over the nuclear framework and have led to equations, which are computationally tractable. At the heart of the MO approach lies the linear combination of atomic orbitals (LCAO) formahsm... 

The results of a valence bond treatment of the rotational barrier in ethane lie between the extremes of the NBO and EDA analyses and seem to reconcile this dispute by suggesting that both Pauli repulsion and hyperconjugation are important. This is probably closest to the truth (remember that Pauli repulsion dominates in the higher alkanes) but the VB approach is still imperfect and also is mostly a very powerful expert method [43]. VB methods construct the total wave function from linear combinations of covalent resonance and an array of ionic structures as the covalent structure is typically much lower in energy, the ionic contributions are included by using highly delocalised (and polarisable) so-called Coulson-Fischer orbitals. Needless to say, this is not error free and the brief description of this rather old but valuable approach indicates the expert nature of this type of analysis. 

The quantum mechanical methods described in this book are all molecular orbital (MO) methods, or oriented toward the molecular orbital approach ab initio and semiempirical methods use the MO method, and density functional methods are oriented toward the MO approach. There is another approach to applying the Schrodinger equation to chemistry, namely the valence bond method. Basically the MO method allows atomic orbitals to interact to create the molecular orbitals of a molecule, and does not focus on individual bonds as shown in conventional structural formulas. The VB method, on the other hand, takes the molecule, mathematically, as a sum (linear combination) of structures each of which corresponds to a structural formula with a certain pairing of electrons [16]. The MO method explains in a relatively simple way phenomena that can be understood only with difficulty using the VB method, like the triplet nature of dioxygen or the fact that benzene is aromatic but cyclobutadiene is not [17]. With the application of computers to quantum chemistry the MO method almost eclipsed the VB approach, but the latter has in recent years made a limited comeback [18],... 

The resulting LBOs are fairly localized, one on Ci — C2 the other on C3—C4. Elowever, the delocalization tails are significant even though we used Hiickel orbitals. These large localization tails reflect the fact that butadiene has some conjugation between the n-bonds and in terms of VB theory is describable by a linear combination of the major Kekule structure and the minor long bond structure. 

Having defined a diabatic state as a unique VB structure, or more generally as a linear combination of a subset of VB structures leading to a specific bonding scheme, the question is now How do we calculate such a state in a meaningful way ... 

More recently Hiberty et ol[26] proposed the breathing orbital valence bond (BOVB) method, which can perhaps be described as a combination of the Coulson-Fisher method and techniques used in the early calculations of the Weinbaum.[7] The latter are characterized by using differently scaled orbitals in different VB structures. The BOVB does not use direct orbital scaling, of course, but forms linear combinations of AOs to attain the same end. Any desired combination of orbitals restricted to one center or allowed to cover more than one is provided for. These workers suggest that this gives a simple wave function with a simultaneous effective relative accuracy. 

Each such Lewis structure is represented by a single VB spin-eigenfunction (HVH n), hereafter called a "VB structure". These VB structures are linear combinations of Slater determinants involving the same occupied AOs as the corresponding Lewis structures, as in eqs 10-12. 

Having defined a diabatic state as a unique VB structure, or more generally as a linear combination of a subset of the full VB structure set that describes the adiabatic state, in the next step one has to specify the orbitals needed to construct the VB structure(s) of this diabatic state. One first possibility is to keep for the diabatic state the same orbitals that optimize the adiabatic state. This has the advantage of simplicity. Practically, once the orbitals have been determined at the end of the BOVB orbital optimization process, the hamiltonian matrix is constructed in the space of the VB structures and the... 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

In general, the orbitals in this method are expanded as linear combinations of a basis set of Slater functions in the same spirit as in the LCAO-MO-SCF method. However, in the present case the orbitals are essentially localized, and a description such as equation (67) is clearly equivalent to a linear combination of a great many VB structures, both covalent and ionic. Thus, in the case of methane this should provide a very good description of the ground state, particularly of the potential surfaces for such processes as... 

The validity of a perfect-pairing approximation depends on the particular choice of paired orbitals and may be improved by introducing new orbitals, as linear combinations of the original set, so as to lower the energy of the structure. Mixing of the orbitals in this way, in the context of VB theory, is usually referred to as hydidiza-... 

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