Some properties of the electron propagator

In the last section two special propagators (the electron propagator and the polarization propagator) were introduced and related to the 1- and 2-electron density matrices. The polarization propagator is by now a familiar object, and can indeed often be calculated rather efficiently by the variational methods of the previous chapter. The same cannot be said, however, of the electron propagator (or 1-electron Green s function), and its calculation will require the development of various new techniques. First of all, however, we shall want to know more about its potential value. 

Let US therefore examine G (o x x ), the electron propagator in frequency form, introduced in (13.3.8) et seq. and consider some of the reasons for its great importance. The matrix elements in the numerators involve wavefunctions for the N-electron system, and for its negative and positive ions (with N l electrons), and are of two types 

In other words, d,(N) must strike out the spin-orbital xjfr wherever it appears (in any expansion over a complete set of determinants), destroying any term not containing xpr, and then multiply by (N ) in 

The effect of the field operator xp(x) on any N-electron Schrodinger wavefunction is to replace the Nth variable x by the parameter x, multiplying the result by 

Similar rules may be derived for creation operators (Problem 13.7), but are less simple and, since a creation operator dJ(N) working on a ket function is equivalent to its adjoint (destruction operator), transferred to the bra, it is sufficient to know (13.4.5). For number-conserving operators, which contain equal numbers of d and d factors, the result is always N-independent Slater s rules are thus valid whatever the number of particles, and no N-dependence appears in the second-quantization Hamiltonian (3.6.9). 

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