Spinfree formulation of reduced density matrices, density cumulants and generalised normal ordering

Abstract

The k-particle reduced density matrices γ _k for a singlet state or a spin-orientation averaged ensemble are invariant under the spin rotation group SU(2), i.e. they are independent of spin, even if they are expressed in terms of spin orbitals. The γ _k contain the same information as their spinfree counterparts Γ _k expressed in terms of spinfree orbitals. However, the relation between the two quantities is, by no means, elementary. The summation over spin is particularly demanding and surprisingly complicated for the k-particle density cumulants λ _k and for the formulation of a generalised normal ordering with respect to an arbitrary reference state, including the generalised Wick theorem. Plausible extrapolations from small k turned out to be unjustified and some published expressions for k = 4 need to be revised. A compact general result, valid for all k has not yet been found.

Keywords:

• reduced density matrices
• density cumulants
• elimination of spin
• generalised Wick theorem
• irreducible k-particle Brillouin conditions