Electron propagator equations

According to equation 15, eigenvalues of the superoperator Hamiltonian matrix, H, are poles (electron binding energies) of the electron propagator. Several renormalized methods can be defined in terms of approximate H matrices. The... 

We will describe, in some detail, one such modification, an effective Dirac equation (EDE) which was derived in a number of papers [7, 8, 9, 10]. This new equation is more convenient in many applications than the original BS equation, and we will derive some general formulae connected with this equation. The physical idea behind this approach is that in the case of a loosely bound system of two particles of different masses, the heavy particle spends almost all its life not far from its own mass shell. In such case some kind of Dirac equation for the light particle in an external Coulomb field should be an excellent starting point for the perturbation theory expansion. Then it is convenient to choose the free two-particle propagator in the form of the product of the heavy particle mass shell projector A and the free electron propagator... 

It would be very useful if there were an equation of motion for the electron density or for the reduced density matrices corresponding to a pure state of a many-electron system from which these quantities could directly be determined. Unfortunately this is not the case. The quantity closest to the one-matrix that has an equation of motion from which it can be determined by well-defined approximations is the one-electron Green s function or electron propagator. We explore its connection to the electron density in the next section. 

In (10), an additional factor 2 accounts for the two symmetrical diagrams Fig. 1, (c) and (e). In these equations, Sp(x,y E)y0 denotes the time-independent Green function of a bound electron related to the four-dimensional electron propagator by... 

Analogous to the previous treatments /69/ the Fourier transformed equation of motion of the dilated electron propagator may now be written as... 

A formal analc y is evident between Eq. (2.5), describing electron propagation, and Eq. (2.9), which is the basic equation of lattice dynamics. 

Electron propagator calculations 25-28 of electron-binding energies (that is, electron-attachment and -detachment energies) may be based on one-electron equations which read... 

In the diagonal, second-order approximation to the self-energy of the electron propagator, solutions of the Dyson equation (with self-consistent pole energies, cvp) satisfy... 

Eq. (58) represents the starting point for all approximate propagator methods. Even though in the derivation we only discussed the linear response functions or polarization propagators, a similar equation holds for the electron propagator. The equation for this propagator has the same form but there are differences in the choice of h and in the definition of the binary product (Eq. (52)), which for non-number-conserving, fermion-like operators should be... 

Eq. (83) is the Dyson equation (Abrikosov etal., 1963) which is used in the Green s function method (Cederbaum and Domcke, 1977) and the two approximate schemes, the Green s function and the electron propagator methods, thus use the same starting equations. 

Obviously, those are the same considerations as we went through in order to obtain Eqs (90) and (91) and the electron propagator method and the ADC are thus equivalent methods. Using n = 2 in Eq. (93) we determine and n — 3 gives The U matrix in Eq. (93) corresponds to the transition matrix (cf. Eq. (75)). Both and only contain C and D terms (see Eqs (87), (88), (90) and (91), i.e. hj = hj alone. From Eq. (63) we see that we may classify the operators in hj as 2p-lh (two-particle, one-hole) and 2h-lp operators, and the n = 3 ADC approach, corresponding to the third-order electron propagator method, is therefore referred to as the extended 2p-lh Tamm-Dancoff approximation (TDA) (Walter and Schirmer, 1981). A fourth-order approximation to the ADC equations has also been described (Schirmer et ai, 1983) but not yet tested in actual applications. 

Dyson orbitals and electron binding energies (i.e., negative VDEs and VAEs) may be obtained from the Dyson equation, which, in its inverse form, relates the electron propagator matrix to its zeroth-order counterpart via... 

Most of these diagrams contain two intermediate electron propagators and, therefore, double summations over the whole spectrum of the Dirac equation in the external nuclear field. This makes their computation numerically intensive. Both the selfenergy and vacuum-polarization screening corrections are ultraviolet divergent and require renormalization to yield a finite result. 

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