t -expansion and connected-moment expansion: A generalization to excited states

Abstract

The $t$-expansion method for calculating ground-state energies of Hamiltonian systems is generalized to excited states. We introduce quantities ${M}_{n}(t)$ and ${D}_{n}(t)$ which for $t\ensuremath{\rightarrow}\ensuremath{\infty}$ converge to ${\ensuremath{\Sigma}}_{n}={E}_{0}+{E}_{1}+\dots{}+{E}_{n}$ and ${\ensuremath{\Delta}}_{n}={E}_{n+1}\ensuremath{-}{E}_{n}$, respectively. Like in the standard $t$-expansion, the $t$-series for ${M}_{n}(t)$ and ${D}_{n}(t)$ are constructed from moments of the Hamiltonian. The values of ${\ensuremath{\Sigma}}_{n}$ and ${\ensuremath{\Delta}}_{n}$ can then be estimated using (e.g.) a generalization of the connected-moment expansion of Cioslowski.