VB wavefunctions

When working with atomic orbitals, it is usual to write the electron density in terms of a certain matrix called (not surprisingly) the electron density matrix. For the simple dihydrogen VB wavefunction, we have... 

The CASVB strategy [1-9] uses a very efficient algorithm for the transformation of CASSCF [10] structure spaces, for the interpretation of CASSCF wavefunctions, and for the fully-variational optimization of VB wavefunctions. Important features for the quality of the final description include the unbiased optimization of both the VB orbitals and the mode of spin coupling, and also flexibility in the choice of the form of wavefunction. 

Symmetry restrictions may also be placed on the active orbitals in order to determine the nature of the resulting modern valence bond solution. This is exemplified by the common use of o - n separation for planar molecules (c/ Section 5). In earlier applications to ozone and diborane [2,4] it was also seen that the distribution of active orbitals among the irreducible representations was the deciding factor for the types of VB solution possible. It should also be borne in mind here that the nature of the lowest-lying CASSCF solution may not always coincide with that of the optimal fully-variational modem VB wavefunction. 

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. 

An extraordinarily simple, but nevertheless quite useful procedure involves an analysis of the distribution of Pvb among the irreducible representations in the point group. In CASVB this is easily achieved by transforming the VB wavefunction to the CASSCF MO basis giving weights according to... 

For a detailed discussion of nonorthogonal weights in modem VB wavefunctions, see Ref. 16. 

We consider here a slightly different perspective to the one adopted in Section 2.1, more strongly associated with the fully-variational optimization of VB wavefunctions. 

As is the case for standard orthogonal-orbital MCSCF calculations, the optimization of VB wavefunctions can be a complicated task, and a program such as CASVB should therefore not be treated as a black box . This is tme, to a greater or lesser extent, for most procedures that involve orbital optimization (and, hence, non-linear optimization problems), but these difficulties are compounded in valence bond theory by the... 

Bianco et al. [23] proposed a direct VB wavefunction method combined with a PCM approach to study chemical reactions in solution. Their approach is based on a Cl expansion of the wavefunction in terms of VB resonance structures, treated as diabatic electronic states. Each diabatic component is assumed to be unchanged by the interaction with the solvent the solvent effects are exclusively reflected by the variation of the coefficients of the VB expansion. The advantage of this choice is related to its easy interpretability. The method has been applied to the study of the several SN1/2 reactions. 

To obtain geometries, 10-orbital 10-electron complete active space self-consistent field (CASSCF) [82-84] calculations were performed with the GAMESS-UK program [6], The occupied orbital order in an SCF for flat benzene is n,2c,2n. In the bent molecule, there is no clear distinction between a- and tt-orbitals and we want to include all the tt-orbitals in the CAS-space. Thus, 10 orbitals in the active space are required. Obviously, the 5 structure VB wavefunction would have been a preferable choice to use in the geometry optimisation. However, at that time, the VB gradients were not yet available. The energies of the VBSCF at the CASSCF geometries followed the CASSCF curve closely. 

As mentioned in Section 1, in a traditional VB treatment, a VB wavefunction is expressed as the linear combination of 2m Slater determinants, where m is the number of covalent bonds in the system. For some applications in which only a few bonds are involved in the reaction, it is too luxurious to adopt the PPD algorithm, as the number of Slater determinants is still not too large to deal with. It would be more efficient to use a traditional Slater determinant expansion algorithm than the PPD algorithm. Therefore, as a complement, a Slater determinant expansion algorithm is also implemented in the package. 

Usually, an individual VB structure assembled from the localized bonding components does not share the point group symmetry of the molecule anymore. However, the overall VB wavefunction, PVB, should retain the same symmetry properties as the MO wavefunction (in the sense of full Cl, they are in fact identical). Therefore, TVs can be classified by an irreducible representation associated with a given point group. In order to sort vFra by symmetry, a project operator can be introduced as follows ... 

In the MO formalism it is quite straightforward to deal with the excited states of a molecule. An adequate wavefunction of an excited state can be constructed according to the resultant configuration and its symmetry arising from electron promotion among MO series. Compared with numerous MO-based methods, VB approaches are far less employed to study excited states due to the difficulty in VB computations. Recently, by observing the correlation between MO theory and resonance theory, as well as the symmetry-adapted VB wavefunction described in the last section, we performed VB calculations on low-lying states of some molecules [71, 72],... 

The method has been used to study the LiH system [13,14,15] for which the main interest was in the first excited state, which governs the dynamical behaviour of the neutral LiH molecule in the presence of a naked proton. Various nuclear configurations have been sampled, both in the subreactive [14] and reactive regions of the configuration space [13]. It turned out that a simple two-reference VB wavefunction was sufficient for the subreactive study, while the stretching of the LiH bond in the reactive regions required the use of an additional reference function. For this system, the ground state SC wavefunction has the form ... 

The right-hand panel of Figure 1 compares the excited curve obtained from the final MR-SCVB (or, MRVB for short) calculation with that from the full Cl. In this study, starting with the double-reference SC functions, we optimized a set of 4 pairs of virtuals for each reference and, at the end, we built a VB wavefunction consisting of 84 spatial configurations for a total of only 125 VB structures. This... 

The virtual orbitals are then employed to construct the set of singly- and doubly-excited configurations which provide the final MO-VB wavefunction, where the SCF-MI determinant represents the zero order state. The final wavefunction has a general Molecular Orbital-Valence Bond form ... 

The accuracy of the MO-VB wavefunction is expected to be close to that of a full SD-CI wavefunction involving excitations to the full virtual spaces of each monomer (vertical excitations). Very recently, a new version of the MO-VB optimization scheme has been developed that is apt to guarantee that the wavefunction approaches as close as possible the full SD-CI limit, via saturation of the optimal virtual space. Explorative calculations on the very challenging helium dimer system are encouraging. 

The whole procedure can be repeated n times generating - for each pair of occupied orbitals - n optimised virtual orbital pairs, whose contribution to the interaction energy is strictly decreasing up to saturation of the space, i.e. up to the full use of the SCF-MI virtual orbital space. Consistent with the employed basis set, the final MO-VB wavefunction (16) can be so improved to the desired degree of accuracy. 

---