Calculation of One-Electron Charge, Momentum, and Spin Densities: The Statistical Exchange Approximation

Abstract

In these notes we will present a certain view of the status of the theory of one-electron densities, using concrete examples to illustrate the applicability of theoretical models to experimentally interesting systems. Our task is made easier by the excellent lectures on the fundamental theory given by Professor March and other participants of this school. Let us begin by remembering the connection between the basic N-electron quantum state ${{\psi }_{l}}\left( 1,2,..N \right)$ and the density matrices whose properties we eventually measure: The p^th order density matrix can be defined as /1,2/

$$\begin{array}{l} \rho _\ell ^p(1,2, \ldots p|1'2' \ldots p) = \left( {\begin{array}{*{20}{c}} N\\ p \end{array}} \right)\smallint \psi _\ell ^*(1,2, \ldots N)\psi \ell (1'2' \ldots p',p + 1, \ldots N)\\ d(p + 1)d(p + 2) \ldots d(N) \end{array}$$

(1)

Evidently, integration over the coordinates (p+l)...N suppresses a great deal of information contained in the state ψ so that one eventually arrives at the simple one-particle and two-particle distributions or densities which are accessible to experiment. We can best display the hierarchy of density matrices and their final profection onto various “islands of experiment” in Fig. 1, borrowing from the Dick Weiss’ delightful model.