Minimum principles and level splitting in quantum Monte Carlo excitation energies: Application to diamond

A diffusion and variational quantum Monte Carlo (DMC and VMC) study of excitation energies in diamond is reported. Good agreement is found between the DMC results and results from the $\mathrm{GW}$ approximation and, where available, with experiment. A group theoretical analysis is provided of the level splittings arising from the use of arbitrary degenerate single-particle orbitals in the construction of ground- and excited-state determinants. The results are discussed in light of the recently demonstrated variational theorems for excited states in the DMC study.