Derivation of a formula for the resonance integral for a nonorthogonal basis set

Abstract

In a self-consistent field calculation, a formula for the off-diagonal matrix elements of the core Hamiltonian is derived for a nonorthogonal basis set by a polyatomic approach. A set ofparameters is then introduced for the repulsion integral formula of Mataga-Nishimoto to fit the experimental data. The matrix elements computed for the nonorthogonal basis set in the aielectron approximation are transformed to those for an orthogonal basis set by the I~wdin symmetrical orthogonalization.Semiempirical calculations using the Hartree-Fock equation (1) with a zero differential overlap (2-7) neglect the overlap integrals and assume orthogonality of a set of atomic orbitals. These approaches have been rationalized through the use ofthe Lowdin orthogonalized atomic orbital (8). Explanations of the earlier Pariser-Parr method (2-4) have been given by Gladney (9), Fischer-Hjalmars (10), Berthier et al. (11), and several other authors (12-14); a systematic treatment of the zero differential overlap assumption using the nonorthogonal atomic orbitals has been compared with the procedure using the nonorthogonal Lowdin atomic orbitals by Fischer-Hjalmars (15-18); and further application of the orthogonal transformation has been carried out by Adams and Miller (19,20). Linderberg (21,22) has shown that the equivalence of the dipole length and velocity form of oscillator strength places a condition on the parameters used in the Pariser-Parr model. For the case of two orbitals centered at positions A and B and having a vanishing gradient perpendicular to the axis connecting them, Linderberg has derived Eq. 1. h2 L aS4 mRv OR' [a1] where 4A and 1k are the atomic orbitals on atoms A and B, SAV = (4+,j4,), R AV = (R, -Rv), and f3A is a resonance integral between the orbitals ',, and 4p. This procedure is a diatomic approach and neglects the higher order corrections that would occur in polyatomic molecules. There is not, however, a proper limit when B approaches A in this formula. To compute a, for the diagonal matrix elements of the core Hamiltonian, one needs to know / values for the nonorthogonal basis set, but we now only have /8 values for the orthogonal basis set, as given by Eq. 1. The /8 values for the nonorthogonal basis set can be obtained by the procedure described below. Derivation of the off-diagonal matrix element of the core Hamiltonian The fundamental equations for the self-consistent field method are FC = ESC, IF -ESI = 0, Ci s Cj = 8ij, [2] where F is the Fock matrix, E is the eigenvalue matrix of F, C is the eigenmatrix of F, Bij is the Kronecker delta, and S is the overlap matrix. The zero differential overlap approximation leads to the familiar contradiction of neglecting the overlap integral on the one hand and taking a nonzero value of the resonance integral on the other hand.The L6wdin orthogonalized atomic orbitals and the Mulliken approximation (23) can be combined into a computational method in a self-consistent field calculation. The nonorthogonal orbitals, 4i,, are transformed into the orthogo...