On the hierarchy equations of the wave-operator for open-shell systems

Abstract

Starting with the open-shell analogue of the Gell Mann-Low theorem of many-body perturbation theory, a non-perturbative linear operator equation is derived for the linked part of the wave-operator W for open-shell systems. It is shown that, for a proper treatment of the linked nature of the wave-operator, a separation into its connected and disconnected components has to be made, and this leads to a hierarchy of equations for the various connected components. It is proved that the set of equations can be cast into a form equivalent to the non-perturbative equations of the wave-operator recently derived by Mukherjee and others in a coupled-cluster or exp (T) type formalism if a consistent use is made of a 'core-valence separability' condition introduced earlier. A comparison of the coupled-cluster representation of W with the perturbative representation reveals that various alternative forms of W in the coupled-cluster representation are possible and these reflect alternative ways of realising the core-valence expansion of the wave-operator. In particular it is emphasised how the use of Mandelstam block-ordering simplifies the coupled-cluster theories to a considerable extent and a comparison is made with coupled-cluster methods for open-shells put forward very recently by Ey and Lindgren. Finally, it is shown how difference energies of interest may be derived in a compact manner using the Mandelstam block-ordering of the wave-operator.