Some further properties of density matrices

In Chapter 5 we discussed various properties of density matrices such as p(jc, xj). We must now turn to the density operators, of which these matrices—or more correctly kernels— provide representations. Let us first consider the general form (5.3.5), namely 

In other words, we change the variable from jti to x[, multiply by the kernel, and integrate to get rid of xi—the result being a new function of Xi. Such operators are linear operators, just like the various differential operators we have used, and play an important part in the general formulation of quantum mechanics. We call p the 1-electron density operator. 

It has been noted (footnote on p.l24) that p(xi xl) formally resembles a matrix element, in which Xi andxj play the part of (continuous) row and column indices in this sense it provides a particular representation of p. We now note that, on introducing any orthonormal set i/ r(JCi) the array of coefficients appearing in (6.4.1) simply provides a true matrix representation of the operator p, in which Xj, x[ are replaced by the discrete indices r, s. This follows easily from the definition of the matrix elements of an operator since, using the orthonormality property, 

the full array p is the matrix representing the density operator p, with kernel (6.4.1), referred to the basis i/ ,. An exactly similar interpretation may be placed upon the 2-electron density kernel x, X2, x[,x, the operator n being defined by an equation analogous to (6.4.2) and the matrix representing n simply being the array of coefficients n ,m when we write 

The Fock-Dirac density matrix (5.3.7) corresponding to a 1-determinant wavefunction has a very special property. We regard the N occupied spin-orbitals as the first N members of a complete orthonormal 1-electron basis and consider 

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