Excited-state exchange in solidHe3

A general derivation of the energy of solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is given which uses nonorthogonal perturbation theory. Certain terms are identified with the exchange energy and two of these, not previously derived, are discussed in detail. Both terms arise from the "operator" nature of exchange, i.e., they involve transitions to excited intermediate states in phonon space. The first term examined is second order in the perturbation but first order in exchange and is the generalization of the "double-occupation" exchange mode discussed by Guyer, Mullin, and McMahan. It is argued that such terms, which arise from the use of approximations to "home-base" functions may contribute substantially to the pair exchange integral. The second term examined is second order in exchange and gives rise to four-spin terms analogous to those considered recently by Guyer. The complete expression for such terms is presented here and estimates of the size and range of such terms, similar to those of Guyer, are given. Future theories of exchange will apparently need to include both these new terms and possibly may need a summation to all orders in the perturbation theory presented.