Integrals for lower bounds to the exact energy

Abstract

Methods for calculating lower bounds to the exact energy using the variance of the upper bound energy are discussed and explored. All the matrix elements of the Hamiltonian squared are collected and considered, and those for which no known solutions could be found in the literature are derived for an explicitly correlated Gaussian (ECG) basis set. Analytical Solutions are determined for twoelectron, mono-nuclear systems, in addition to a one-dimensional integral expression which has use in polyatomic calculations. The newly derived integral expressions have been implemented in the integral library of the QUANTEN computer program. I. INTRODUCTIONSince the successful application of Schrödinger equation to the hydrogen atom in the mid 1920's, few-body systems have been of great interest to physicists and chemists. The ability to describe few-particle systems such as small atoms and molecules however becomes problematic due to a lack of analytical solutions. As such, quantum chemistry requires the use of approximations in order to simplify quantum problems, allowing for approximate solutions to be obtained for the physical properties of the systems in question.In atomic and molecular physics, one property of great interest is the energy of different eigenstates. The energy of a particular state can be calculated by approximating the wave function through linear and non-linear parameterisation. This approximate wave function is called the trial wave function for the system, and it can be shown that the energy expectation value associated with this trial wave function serves as an upper bound to the exact energy [1,2]. Calculating 'tight' upper bounds to the exact energy is well known and occurs frequently in modern quantum chemistry. However, the upper-bound value itself does not give information on how close it is to the exact energy. If a lower bound could also be calculated then this, along with the upper-bound value, would give an interval within which the exact energy can be found. Of course, the lower-bound value is only meaningful if the lower bound is of the same quality as the upper bound.As is true with upper bounds, improving the basis set with respect to the variational principle improves the lowerbound value, though the convergence of lower bounds is slower. Nevertheless, convergence of both upper and lower bounds leads to ever-tighter intervals, within which the exact energy lies. Two such methods for calculating lower bounds are the Weinstein criterion [3] and Temple's lower bound [4], presented by D. Weinstein and G. Temple respectively. Weinstein's bound tends to give lower bounds that are of too low a quality compared to the upper bounds, whereas Temple's bound is regarded to give better quality lower bounds [5,6]. Both the Weinstein criterion and Temple's bound make use of the variance of the upper bound energy, σ E 2 , which requires the matrix elements of the Hamiltonian operator squared, Ĥ2 .The trial wave function can be written as a linear combination of basis functions in the many-particle sta...