Valence bond approach exploiting Clifford algebra realization of Rumer–Weyl basis

Abstract

A detailed algorithm is described that enables an implementation of a general valence bond (VB) method using the Clifford algebra unitary group approach (CAUGA). In particular, a convenient scheme for the generation and labeling of classical Rumer-Weyl basis (up to a phase) is formulated, and simple rules are given for the evaluation of matrix elements of unitary group generators, and thus of any spin-independent operator, in this basis. The case of both orthogonal and nonorthogonal atomic orbital bases is considered, so that the proposed algorithm can also be exploited in molecular orbital configuration interaction calculations, if desired, enabling a greater flexibility for N-electron basis-set truncation than is possible with the standard Gel'fandTsetlin basis. Finally, an exploitation of this formalism for the VB method, based on semiempirical Pariser-Parr-Pople (ppp)-type Hamiltonian and nonorthogonal overlap-enhanced atomic orbital basis, and its computer implementation, enabling us to carry out arbitrarily truncated or full VB calculations, is described in detail.

Bond length alternation in cyclic polyenes. VII. Valence bond theory approach

“…'1 (26) and consider Aii and AiSj. to be equivalent if the intersite separation i -j and i' -j' are identical.…”

“…A very nice and convincing demonstration of the importance of this concept, as well as its historical origins, may be found in a paper by Professor McWeeny [29], to whom we dedicate this study. Following this idea and exploiting CAUGA formalism [44,45], we have formulated the so-called PPP-VB method [25,26] and tested it on a number…”

An efficient algorithm for complete active space valence bond self‐consistent field calculation

“…In VB theory, a many‐electron wave function is expressed in terms of a set of VB functions, called VB structures, which are essential for the description of particular chemical problem, where ϕ may be either a spin‐coupled form, called Heitler‐London‐Slater‐Pauling (HLSP) function, or a tableau function,[15] if the spin‐free form of quantum chemistry is used. A HLSP function is expressed as a linear combination of Slater determinants, which are constructed from strictly localized HAOs, while a spin‐free VB wave function is defined by using the projection operator of the symmetric group.…”

“…A HLSP function is expressed as a linear combination of Slater determinants, which are constructed from strictly localized HAOs, while a spin‐free VB wave function is defined by using the projection operator of the symmetric group. [5, 15] In any case, the VB energy can be written as, In the VBSCF method, the structural coefficients C and the VB orbitals ϕ m are optimized simultaneously by minimizing the total energy, which is determined by solving following secular equation: Here, H VB and M VB are the Hamiltonian and overlap matrices between VB structures, defined as, The VBSCF structure weights can be evaluated by using Coulson‐Chirgwin formula,[25] as For a CAS, where all independent structures are involved in VBSCF calculation, the VBSCF(CAS) wave function is invariant under transformation of the active orbitals. Thus, nonorthogonal active VB orbitals and the inactive orbitals can be orthogonalized without loss of generality, and the procedure keeps the VBSCF wave function invariant.…”

“…Though VB computations are time demanding, VB theory has always attracted chemists owing to its conceptual lucidity and its ability of capturing the most important characters of chemical bonding. [1] In the last 3 decades, various VB methods[2–20] and computer programs[21–24] have been developed, and thus more and more molecules and reactions were studied by means of VB methods, not only for qualitative analyses but also for quantitative verifications and predictions of properties of chemical species and reactions.…”