Schrodinger equation Ritz method

Variational methods [6] for the solution of either the Schrodinger equation or its perturbation expansion can be used to obtain approximate eigenvalues and eigenfunctions of this Hamiltonian. The Ritz variational principle,... 

In quantum calculations, the Rayleigh-Ritz variational method is widely used to approximate the solution of the Schrodinger equation [86], To obtain exact results, one should expand the exact wave function in a complete basis set... 

If the problem were solved exactly, then the solution of the Schrodinger equation could be sought e.g., by using the Ritz method (p. 238). Then we have to decide what kind of basis set to use. We could use two auxiliary complete basis sets one that depended on the electronic coordinates i fk(r)], and another that depended on the nuclear coordinates 0 (/ ). The complete basis set for the Hilbert space of our system could be constructed as a Cartesian product 0t(r) x 0 (R) i.e., all possible product-like functions iJfk(r) i(R). Thus, the wave function could be expanded in a series, as follows ... 

We will use the Ritz variational method (see Chapter 5, p. 238) to solve the Schrodinger equation. What should we propose as the expansion functions It is usually recommended that we proceed systematically and choose first a complete set of functions depending on R, then a complete set depending on R, and finally a complete set that depends on the f variables. Next, one may create the complete set depending on all five variables (these functions wUl be used in the Ritz variational procedure) by taking all possible products of the three functions depending on R, R,... 

If the problem were solved exactly, then the solution of the Schrodinger equation could be sought e.g., by using the Ritz method (p. 238). Then we have to decide what kind of basis set to use. We could use two auxiliary complete basis sets one that depended on the electronic coordinates and another that depended on the... 

Diabatic case. Imagine now a basis set (r R),i = 1, 2,3,... M of some particular electronic wave functions (we will call them diabatic) that also depend parametrically on R. There are two reasons for considering such a basis set. The first is that we are going to solve the Schrodinger equation H iti = by using the Ritz method (Chapter 5) and we need a basis... 

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