Schrodinger equation molecular integrals

Now that you know the mathematical form, you can solve the independent-electron Schrodinger equation for the molecular orbitals. First substitute the LCAO form above into equation (47) on page 193, multiply on the left by and integrate to represent... 

Since an analytical solution of the Schrodinger equation is no longer possible, the molecular orbitals must be constructed as linear combinations of basis functions using the variation principle. An exact solution would require an infinite set of basis functions, leading to an infinite set of m.o.s. Thus an additional approximation, at the operational level, is introduced which is dependent on the dimension of that set. Further approximations are encountered in the calculation of the m.o.s, depending on the estimation of the various integrals as matrix elements as required by the variation theorem. We shall develop this subject at the end of Chapter 7. 

H. Conroy, Molecular Schrodinger equation. VIII. A new method for the evaluation of multidimensional integration. J. Chem. Phys., 1967, 47, 5307-5318. 

In principle, owing to its generality, the Schrodinger equation contains the solution of all problems of molecular structure. Unfortunately, it is only integrable in a few special cases, the hydrogen atom for instance. 

Hartree-Fock and post-Hartree-Fock wavefunctions, which do not explicitly contain many-body correlation terms lead to molecular integrals that are substantially more convenient for numerical integration. For this reason, the vast majority of (non-Monte Carlo) work is done with such independent-particle-type functions. However, given the flexibility of Monte Carlo integration, it is very worthwhile in VMC to incorporate many-body correlation explicitly, as well as incorporating other properties a wavefunction ideally should possess. For example, we know that because the true wavefunction is a solution of the Schrodinger equation, the local energy... 

For a trial function, standard procedure is to substitute the approximate function into the Schrodinger equation, and then obtain the energy expression, by left multiplying by the trial function and integrating over the whole of the atomic or molecular space. 

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