5 - The Variation Method

Abstract

The methods that help in the evaluation of approximations to the energy of some states of the various physical systems are: the variation method due to Rayleigh and the perturbation method introduced by Schroedinger and known as the Rayleigh–Schroedinger (RS) perturbation theory. The RS perturbation theory makes it possible to deal with the problem of electric properties of molecules and intermolecular forces. This chapter outlines the principles of the Wentzel–Kramers–Brillouin method that establishes in a direct way the connection between the classical and quantum mechanics. The variation method is extremely useful in finding bounds to the energy of ground and excited states of atomic and molecular systems. The variation method allows finding variational approximations to energy and wavefunction of ground and excited states of the system. As the variation method is based on the average value of the energy, the variational method privileges the space regions near to the nucleus, where the potential energy is larger (r small). The usage of variationally optimized wavefunctions can give poor results for operators different from the variables such as the dipole moment operator μ =er that takes large values far from the nucleus.