Diffusion Monte Carlo method with symmetry-constrained variational principle for degenerate excited states

Abstract

Fixed-node diffusion Monte Carlo (FN-DMC) is an accurate and useful method for estimating the wave function and the energy of the quantum ground state of a many-fermion system. However, it has been shown that difficulties with the method may occur when it is applied to a degenerate excited state because the nodal surface of the degenerate trial function is generally insufficient to impose the complete symmetry properties of the trial function on the FN-DMC wave function. As a result, the tiling theorem and the symmetry-constrained variational principle may be violated by FN-DMC when the excited state is degenerate. There are two practical consequences for the study of degenerate excited states: The FN-DMC energy may lie below the energy of the lowest stationary state that transforms according to the same degenerate irreducible representation as the trial function; and the convergence of the FN-DMC energy with improvements in the trial function may not be quadratic. In this paper a diffusion Monte Carlo method for degenerate excited states is presented. It provides a direct generalization of the FN-DMC method, and when applied to the study of degenerate excited states, it has the support of the tiling theorem and the symmetry-constrained variational principle. The method is applied to the lowest degenerate state of a simple test problem in which FN-DMC has been shown to violate both the tiling theorem and the symmetry-constrained variational principle. The numerical results support the assertion that this method for degenerate excited states satisfies both the tiling theorem and the symmetry-constrained variational principle.