  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
two-electron, 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.
