Structure and spin‐purity conditions for reduced density matrices of arbitrary order

For arbitrary k, the separation of spin variables is performed in the reduced density matrix of the kth order (RDM‐k) on the basis of the Fock coordinate function method. The independent spatial components of RDM‐k are analyzed. For RDM‐k of the total spin eigenstate, their number is proved never to exceed its spin multiplicity 2s + 1. Integral and other nontrivial interrelations between spatial components are established which turn out to be the necessary and sufficient conditions of spin purity of a wavefunction corresponding to a given RDM‐k. It is shown that the r‐rank k‐particle spin distribution matrix F kr, defined as a spatial coefficient at the spin‐tensorial operator of rank r in the RDM‐k expansion, can be obtained by reduction of the (k + r)‐particle charge density matrix F k+r0. In particular, all spatial components of RDM‐2 are explicitly expressed in terms of the four‐electron charge density matrix only. This allows us to purpose some approximative formulas for the McWeeny‐Mizuno spin–orbit and spin–spin coupling functions in the case of the weak spin contamination.