The density matrix in many-electron quantum mechanics II. Separation of space and spin variables; spin coupling problems

Abstract

The spin-independent properties of an electronic system are completely determined by spinless density functions, P 1 (1; 1') and P 2 ( 1,2; 1',2 '), derived from the 1-and 2-electron density matrices, p 1 (1; 1') and p 2 (1,2 ; 1',2'). These functions, and their physical inter­-pretation, have been dealt with in part I: the present paper contains the generalizations necessary when the Hamiltonian contains spin terms (as in discussions of electron and nuclear spin resonance). The density matrices, and more generally the transition density matrices p 1 (kλ|1; 1') and p 2 (kλ|l,2; 1',2') associated with states and Φ k and Φ λ are resolved into spinless components associated with different spin situations: p 1 has four such components (one giving the density of a-spin electrons, a second the density of β-spin electrons, and two with less direct physical significance), while p 2 has sixteen components. It is shown that all components, for states k=( S, M ),λ= ( S',M' ) of given S , S' , are determined by three 2-electron functions.For k=λ these reduce to p 2 , the spinless pair function, from which the electron density p 1 may be derived by integration; Q 2 a ‘conditional' spin density, from which the (1-electron) spin density Q 1 may be derived (as in the case of p 1 and P 2 ); Q c. a., a ‘coupling anisotropy function', which describes the anisotropy of spin-spin coupling associated with quantization of the spin z -component. It is also useful to introduce (though it is not an independent function). Q c , a ‘coupling density function', which measures the coupling between the spins of two electrons at different points in space. All spin-dependent effects, including the usual spin-spin and spin-orbit couplings and also (indirect) nuclear and electron-nuclear spin-spin couplings, may be related to the many- electron wave function through the above density functions. All the densities have a physical interpretation, independent of the precise nature of the wave functions (which may, for example, be the exact eigenfunctions of a spinless Hamiltonian or merely approximations of orbital form). The derivation of the density functions from approximate wave functions is considered in some detail. For functions describing weakly coupled electron groups (e. g. inner-shell and valence electrons) the densities may be expressed in terms of those of the separate groups. This reduction may be followed by an orbital description of each group; and subsequently, by a linear-combination-of-atomic-orbitals (l. c. a. o.) approximation. In the simpler approxi­mations, all the coupling effects discussed are determined by electron and spin ‘populations'. Preliminary calculations suggest that such approximations may in many cases be adequate.