Schrodinger equation solution using Gaussian basis sets

Exercise 6.6. Solution of the Schrodinger equation for the hydrogen molecule ion problem using Gaussian basis sets and the direct evaluation of integrals. 

The generator coordinate method (GCM), as initially formulated in nuclear physics, is briefly described. Emphasis is then given to mathematical aspects and applications to atomic systems. The hydrogen atom Schrodinger equation with a Gaussian trial function is used as a model for former and new analytical, formal and numerical derivations. The discretization technique for the solution of the Hill-Wheeler equation is presented and the generator coordinate Hartree-Fock method and its applications for atoms, molecules, natural orbitals and universal basis sets are reviewed. A connection between the GCM and density functional theory is commented and some initial applications are presented. 

Such a possible feature can be found, as an example, within a typical set of solutions of the Schrodinger equation. The harmonic oscillator provides an obvious particular case of such an EH space. It is well known that harmonic oscillator solutions constitute the set of Hermite polynomials [73], weighted by a gaussian function [65]. These polynomials can be considered related to the GTO basis functions most widely used in contemporary Quantum Chemistry. First derivatives of Hermite polynomials are always well defined, producing another polynomial of the same kind. 

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