Elements of the Hamiltonian and Overlap

In the spin-free VB theory, the many-electron wavefunction for a system is expressed as a linear combination of spin-free VB functions 

Clearly k may be a BT, defined by Eq. (13). The Hamiltonian and overlap matrix elements are now written as 

It is evident that the matrix elements of the Hamiltonian and overlap are independent of the index r of BT in Eq. (13) and only the first diagonal element of the irreducible representation matrix, D P), is required, which has been well discussed [31,33,42,43], and is easily determined. It is worthwhile to emphasize that Eqs. (21) and (22) are the unique formulas of the matrix elements in the spin-free approach, even though one can take some other forms of VB functions. For example, it is possible to construct VB functions by Young operator [2], but the forms of the matrix elements are identical to Eqs. (21) and (22) [44], 

Both of Eqs. (21) and (22) involves N terms due to N permutations of the symmetric group SN, which is similar to a determinant expansion or a permanent, except for different coefficients. If one-electron functions are orthogonal, only a few terms are non-zero and make contributions to the matrix elements [42], and consequently the matrix elements are conveniently obtained. However, the use of 

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The use of this perturbation approximation to compute optimized virtual orbitals introduces a major saving. We need only compute diagonal and first row elements of the hamiltonian and overlap matrices, because the relevant energy expression takes the form ... 

Slater s bond eigenfunctions constitute one choice (out of an infinite number) of a particular sort of basis function to use in the evaluation of the Hamiltonian and overlap matrix elements. They have come to be called the Heitler-London-Slater-Pauling (HLSP) functions. Physically, they treat each chemical bond as a singlet-coupled pair of electrons. This is the natural extension of the original Heitler-London approach. In addition to Slater, Pauling[12] and Eyring and Kimbal[13] have contributed to the method. Our following description does not follow exactly the discussions of the early workers, but the final results are the same. 

The DV-Xa molecular orbital calculational method used here utilizes basis sets of numerically calculated atomic orbitals, as well as those of analytical atomic orbitals such as Slater orbitals. Matrix element of the Hamiltonian and the overlap integral are calculated numerically by summing integrand at sampling points rk, the Diophantine points, which are distributed according to the weighted function, and expressed as. 

The main numerical effort in this scattering method is the calculation of matrix elements. Those in (2.53), (2.54), (2.56) have to be recalculated for each energy. The matrix elements of the Hamiltonian and the overlap matrix for the basis ui(f)n, / > 2, in (2.55) have to be calculated only once, but the inversion of M has to be repeated for each energy. 

This is a finite sum in squared powers of overlap integrals between the orbitals of the two atoms. A similar development can be carried through for the 1- and 2-electron contributions to the matrix elements of the hamiltonian, from which one obtains immediately an expression for the matrix elements of the interaction in the form... 

Coupling between the CT state and ground electronic state is expressed by the matrix element of the Hamiltonian (H) of the system (A,D) and shown to be proportional to the overlap integral (S) ... 

One of the oldest and most familiar such approaches is the Extended Hueckel Approximation (Hoffman, 1963.) Let us take a moment to examine this approach, though later we shall choose an alternative scheme. Detailed rationalizations of the approach are given in Blyholder and Coulson (1968), and in Gilbert (1970, p. 244) a crude intuitive derivation will suffice for our purposes, as follows. We seek matrix elements of the Hamiltonian between atomic orbitals on adjacent atoms, (j3 H a). If a) were an eigenstate of the Hamiltonian, we could replace H a) by fia a), where is the eigenvalue. Then if the overlap (j3 a) is written... 

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