Nonorthogonality of Basis States

The repulsive term in the pseudopotential was introduced in Chapter 15 as an approximate correction for the extra kinetic energy due to the presence of core states. This was in direct analogy with the way it was included in the overlap interaction. We also saw, in Appendix B, that this extra kinetic energy is directly related to the nonorthogonality of basis states. Indeed the rigorous formulation of pscudopotentials has been based upon the required orthogonality of the valence-band (or conduction-band) states to the core wave functions. Let us use that approach here. For extensive discussion of the formulation, as well as applications, see Harrison (1966a). 

For the benzene molecule with six 7r-orbitals the 5 = 0 subspace is spanned by five basis states. A particular nonorthogonal representation of these basis states is illustrated in Fig. G.l. There are two equivalent Kekule structures and three equivalent Dewar structures (Coulson 1961). 

It is of some note that many of the models may be (and often were) obtained by-passing the derivational approach here. Basically each model may be viewed as represented by the first terms in a graph-theoretic cluster expansion [80]. Once the space on which the model to be represented is specified, the interactions in the orthogonal-basis cases are just the simplest additive few-site operators possible. For the nonorthogonal bases the overlaps are just the simplest multiplicative operators possible, while the associated Hamiltonian operators are the simplest associated derivative operators. These ideas lead [80] to proper size-consistency and size extensivity. Similar sorts of ideas apply in developing wavefunction Ansatze or ground-state energy expansions for the various models. 

The preceding discussion has been based on the assumption of orthonormal electronic states (see Eq. 14). In some cases it is advantageous to employ a non-orthogonal diabatic basis, especially when Tg is evaluated directly in terms of variationally determined charge-localized SCF [34] (or, in some cases, correlated [116a]) wavefunctions, which are generally nonorthogonal ... 

In this subsection, we will briefly discuss how one may construct a basis

carrier space which is adapted not only to the treatment of the ground state of the Hamiltonian H but also to the study of the lowest excited states. In molecular and solid-state theory, it is often natural and convenient to start out from a set of n linearly independent wave functions = < > which are built up from atomic functions (spin orbitals, geminals, etc.) involved and which are hence usually of a nonorthogonal nature due to the overlap of the atomic elements. From this set O, one may then construct an orthonormal set tp = d>A by means of successive, symmetric, or canonical orthonormalization.27 For instance, using the symmetric procedure, one obtains... 

For CAS and RAS expansions, the matrix elements between nonorthogonal states are conveniently obtained using the procedure of Section 11.9. For each pair of states, we determine a biorthonormal basis - see Section 1.9.3. The overlap and Hamiltonian matrix elements of this basis are obtained by transforming the integrals and Cl vectors as described in Sections 1.9.3 and 11.9, respectively. A subsequent diagonalization of the resulting small Hamiltonian matrix (suitably weighted by... 

The discussion in this section follows closely that for the exponential orbitals in Section 6.5. First, a complete set of orthonormal functions with Gaussian radial forms and a fixed exponent a is introduced. From these orbitals, we next arrive at a more flexible and useful set of nonorthogonal basis functions by simplifying the polynomial part and by introducing variable exponents. As for the exponential functions in Section 6.5, the performance of the Gaussian-based functions is illustrated by carrying out simple expansions of the ground-state orbitals of the carbon atom. We... 

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