A finite‐difference solution of the Hartree–Fock equations for diatomic molecules

Abstract

AbstractA more general application of the self‐consistent field iteration is coupled with a finite‐difference Newton–Raphson algorithm to solve the set of coupled second‐order integro‐differential equations with split boundary conditions which constitutes the Hartree–Fock problem for diatomic molecules. The N orbitals are assumed to be of the form ψα = Lα(λ) Mα (μ)eimαϕ (2π)−1/2, (α = 1, …︁, N), where λ, μ, and ϕ are the usual confocal elliptical coordinates. Requiring the expectation value of the electronic Hamiltonian to be stationary with respect to independent variations of the functions Lα and Mα, subject to constraints of orthonormality, leads to a set of coupled one‐dimensional differential equations for the functions Lα and Mα. In the new method a corresponding set of finite‐difference equations including the split boundary conditions for each function, as well as the Lagrange multipliers and associated constraints on normalization and orthogonality, are incorporated into a large system of nonlinear algebraic equations which is solved by means of a coupled self‐consistent field‐generalized Newton–Raphson iteration. As examples, calculations of the (1)2 1Σ and (1) (2pσu) 3Σ states of H2 are presented. The calculated energy for the 1Σ state of H2 is 99.985% of the three‐dimensional Hartree–Fock limit. The discrepancy is due to the assumed factored form of the orbitals ψα, and a generalization of the finite‐difference method is suggested to improve the results.