Fixed-node diffusion Monte Carlo (DMC) is a stochastic algorithm for findingthe lowest energy many-fermion wave function with the same nodal surface as achosen trial function. It has proved itself among the most accurate methodsavailable for calculating many-electron ground states, and is one of the fewapproaches that can be applied to systems large enough to act as realisticmodels of solids. In attempts to use fixed-node DMC for excited-statecalculations, it has often been assumed that the DMC energy must be greaterthan or equal to the energy of the lowest exact eigenfunction with the samesymmetry as the trial function. We show that this assumption is not justifiedunless the trial function transforms according to a one-dimensional irreduciblerepresentation of the symmetry group of the Hamiltonian. If the trial functiontransforms according to a multi-dimensional irreducible representation,corresponding to a degenerate energy level, the DMC energy may lie below theenergy of the lowest eigenstate of that symmetry. Weaker variational bounds maythen be obtained by choosing trial functions transforming according toone-dimensional irreducible representations of subgroups of the full symmetrygroup.
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