CASVB

We discuss all of the key features of our current CASVB methodology for modem valence bond calculations on ground and excited states. The CASVB strategy may be used to generate compact representations of CASSCF wavefunctions or, alternatively, to perform the fully-variational optimization of various general types ofVB wavefunction. We report also a new application, namely to the fourteen % electrons of a planar dimethylenecyclobu-tadiene chain with three rings. 

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. 

In an early application to butadiene [16], and later to the ground and excited states of benzene [17], Berry analyzed MO-based wavefunctions using valence bond concepts, simply by considering the overlaps with nonorthogonal VB structures. Somewhat closer than this to a CASVB type of approach, are the procedures employed by Linnett and coworkers, in which small Cl wavefunctions were transformed (exactly) to nonorthogonal representations [18-20]. The main limitation in their case was on the size of systems that may be treated (the authors considered no more than four-electron systems), both because this non-linear transformation must exist, and because it must be possible to obtain it with reasonable effort. 

Hirao has also recently considered the transformation of CASSCF wavefrmctions to valence bond form [24, 25]. An orthogonal VB orbital basis was first considered, in which case the CASSCF Cl vector may be found by re-solving the Cl problem. Later he considered also the transformation to a classical VB representation. The transformation of the CASSCF space was achieved by calculating all overlap terms, (oCASscFj cASVB gjjjj golving the subsequent linear problem, using a Davidson-like iterative scheme. 

The main purpose of the present article is to provide, in one place, an overview of all of the key features of our current CASVB methodology. The CASVB code has already been applied to a wide range of problems [1-9,32,33] and its use in further fully-variational modem-VB calculations are reported elsewhere in this Volume. Although we have chosen to concentrate here on methodology, we do present also a new application (to a fourteen-electron system). 

An Overview of the CASVB Approach to Modem Valence Bond Calculations... 

We consider in this Section particular aspects relating to the optimization of a CASVB wavefunction. As for most procedures involving the optimization of orbitals, special attention should be given to the choice of optimization strategy. The optimization problem is in this case non-linear, so that an exact second-order scheme is preferable in order to ensure reliable convergence. A particularly useful account of various second-order optimization schemes has been presented by Helgaker [46]. 

We have always aimed in CASVB for the simplest possible elimination procedure, bearing in mind that the number of variational prameters can be quite considerable in practical applications. 

Here, Cys is the Cl vector in the basis of VB structures, projected such that it transforms according to the irreducible representation phi. Because even the standard CASVB approach involves an expansion of Fvb in terms of structures formed from orthogonal molecular orbitals (the transformation given in Eq. (41)), this implementation is completely straightforward. 

As is the case for standard orthogonal-orbital MCSCF calculations, the optimization of VB wavefimctions can be a complicated task, and a program such as CASVB should therefore not be treated as a black box . This is true, 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... 

In practice, the success of a CASVB calculation depends strongly on the way in which the optimization is carried out, and we emphasize in this Section some particular aspects that have not been considered in Sections 3.1 and 3.2 ... 

For the valence bond orbitals themselves, it is generally natural to specify a starting guess in the AO basis. Such a guess might, of course, not lie entirely inside the space spanned by the active space, and it must therefore be projected onto the space of the active MOs. This is achieved trivially in CASVB, by multiplication by the inverse of the matrix of MO coefficients. 

The various schemes for evaluating nonorthogonal weights all require various components of the structure overlap matrix, which in the CASVB strategy may be obtained from... 

In the implementation of Eq. (48) within CASVB, we use the fact that the effect of an orbital permutation is very straightforward to realize in the determinant basis. Just as for more general transformations, the permutation may be decomposed into separate a and P parts, and the transformation P x pP carried out either in two steps, or as a single pass through all the determinants. This procedure is quite inexpensive, even for a Cl vector in the complete CASSCF space. In our implementation of the full-Cl stmcture transformation (described in Section 2.1), we have employed a decomposition of O with full pivoting, in order to improve numerical accuracy. 

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... 

Table 2 Summary of (14, 14, S) CASSCF and 14-electron CASVB results for the system show in Figure 2.

Our CASVB code is incorporated as a standard feature in molpro [41], and we are actively seeking ways to make all of the methodology more widely available via other packages. 

T. Thorsteinsson, D.L. Cooper, J. Gerratt and M. Raimondi A New Approach to Valence Bond Calculations CASVB, in R. McWeeny, J. Mamani, Y.G. Smeyers and S. Wilson (Eds.), Quantum Systems in Chemistry and Physics Trends in Methods and Applications, Kluwer Academic Publishers, Dordrecht (1997). 

As we have indicated, the fully-variational SC calculations along the 6 in 6 CASSCF IRC curves were performed with CASVB, as implemented in MOLPRO [19]. An older SC code [27] was employed only at the transition structures and at the ends of the calculated IRC segments, mainly in order to assist orbital plotting. 

An overview of the CASVB approach to modern valence bond calculations 303... 

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