Properties of One-Dimensional Correlated Gaussian Wave Functions

The properties of a certain class of unsymmetrized one-dimensional correlated Gaussian wave functions—those which are ground-state eigenfunctions of some coupled harmonic-oscillator Hamiltonian—are investigated in detail. It is shown that a properly symmetrized wave function constructed from these may be used to calculate the expectation value $E_0$ of the Hamiltonian appropriate to a system of interacting one-dimensional atoms and that this energy is, to a high degree of accuracy, equal to the value obtained when the unsymmetrized wave function is used. A method is given by which correction terms to $E_0$ may be obtained. In addition, it is found that even though the number of particles be very large, the necessary multivariable integrals may be performed quite simply.