In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of N antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order k has two sets of indices of each k variables, and it is further antisymmetric in each set of these indices. Every normalizable antisymmetric wave function may be expanded in a series of determinants of order N over all ordered configurations formed from a basic complete set of one-particle functions ₊, which gives a representation of the wave function and its density matrices also in the discrete k-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly N spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence. In case the basic set is of finite order M, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.
Per‐Olov Löwdin (1955) studied this question.