On the Coulson–Fischer wave function for the X¹A′ H⁺₃ molecular ion: parametrisation using distributed Gaussian basis sets

Abstract

Coulson–Fischer theory is applied to the ground electronic state of the simplest polyatomic molecule, the H⁺₃ molecular ion. The Coulson–Fischer orbitals are parametrised in terms of a distributed Gaussian basis set of s-type functions with variationally optimised exponents and positions. The efficacy of these basis set is demonstrated by performing matrix Hartree–Fock calculations which can be compared with previous studies using alternative methods of basis set construction. The ground-state potential energy surface is explored for a range of equilateral, isosceles and scalene triangular geometries for which a number of authors have reported the results of accurate calculations. It is demonstrated that the Coulson–Fischer wave function supports a global approximation to the ground-state potential energy surface of the H⁺₃ system. The Coulson–Fischer orbitals afford a simple picture of the molecular wave function which provide a simple description of the bonding for all geometries. The distributed Gaussian basis set affords a more efficient method of approximation than some other techniques, such as, for example, the more widely used expansions in terms of atom-centred basis functions.

Keywords:

• Coulson–Fischer theory
• distributed Gaussian basis sets
• H₃⁺ ion
• potential energy surface

Notes

The extended basis set Coulson–Fischer wave function [78] may be viewed as a prototype for a number of approaches to the molecular many-electron problem including the multiconfiguration self-consistent field (mcscf) optimised double configuration (odc) method [80], the spin-coupled theory [81–86] the pair function model of Hurley, Lennard–Jones and Pople [87] and others [88–90], and the generalised valence bond (gvb) method [91–93].

For a historical account of the use of Gaussian basis functions within the context of configuration interaction (ci) theory, see [98].

See, for example [47,49,99–101] .

In the original Coulson–Fischer description of H₂, the orbitals have C_∞v symmetry and are related by a reflection σ_h whereas the molecular orbital has D_∞h, the full molecular point symmetry group.

In Jensen’s work, partial derivatives of the electronic energy with respect to the exponents were obtained numerically by finite central differences.

The functions contributing to the Hartree–Fock ground-state energy have σ symmetry and number (21 + 11 + 8 + 5 + 3 + 1) × 3 = 147 or 29.3 % of the total primitive basis set of atom-centred Gaussian-type functions.

Supported by a total of (21 + 11 × 3 + 8 × 5 + 4 × 7) × 3 = 366 primitive Gaussian-type functions of which (21 + 11 + 8 + 4) × 3 = 132 functions contribute to the Hartree–Fock energy.

Supported by a total of (21 + 11 × 3 + 8 × 5 + 5 × 7 + 3 × 9) × 3 = 468 primitive Gaussian-type functions of which (21 + 11 + 8 + 5 + 3) × 3 = 144 functions contribute to the Hartree–Fock energy.