Expectation Values of Singular Operators in Variational Calculations of Atomic P-States

Abstract

Expectation values of singular operators evaluated in the framework of the Rayleigh-Ritz variational method in quantum mechanics may show slow convergence with increasing the number of basis functions, K. An example of such commonly used operator in the case of high-accuracy calculations of few-electron atoms and molecules is the Dirac delta function dependent on interparticle distances, δ(rij). One way to improve the convergence is to adopt the expectation value identities, in which the singular operator is replaced by an certain non-singular operator so that the expectation value is the same in the limit when the trial wave function approaches the exact solution to the Schrödinger equation. However, when the wave function is approximate, which takes place for any finite K, the convergence of the expectation value of this equivalent non-singular operator is usually improved, often by orders of magnitude. In this thesis, we provide the derivation of formulas for such expectation value identities and implement them into a computer code for the case of atomic P-states, whose wave function is expanded in terms of all-particle explicitly correlated Gaussian basis functions.