Double-Time Greens Functions

This hamiltonian is correct because it gives the correct expectation values. The average value with respect to a two-electron state is (kl Hint kl) = kk ll) — kl lk) exactly as the result (coulomb minus exchange) of calculating the average of l/ ri — f2 with respect to a two-electron Slater determinant with the spin orbitals Uk and u in first-quantization . The full electronic hamiltonian can be expressed as 

It is common to use an alternative notation for the electron-electron interaction integrals (rr ss ) = (rs r s ), i.e., with the complex conjugate spin orbitals, belonging to the operators, on the left. One can introduce the compact notation (rs tu) = (rs fn) — rs vt) for an antisymmetric interaction integral and write 

The state 0) is the A -electron ground state (including electron interaction) and satisfies /f 0) = Eq (N) 0). The electron propagator can then be expressed as 

Following Zubarev, the double bracket notation introduced above is used to define a general double-time Green s function or propagator 

Double-time Green s functions or propagators are often studied in terms of their Fourier transforms 

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D.N. Zubarev, Double-time Green Functions in statistical physics, Sov. Phys. Usp. 3 (1960) 320. 

The double-time Green s function associated to operators A(t) and B(t ) is defined as (21,127)... 

For a pure state density operator, the Fourier transform of this double-time Green s function yields the spectral representation of the propagator (21)... 

The response of the system concerned to an external electromagnetic field is conveniently described in terms of double-time Green s function (GF) which can be introduced in a variety of representations.144,218 221 In what follows we will involve the representation in Matsubara s frequency space218 which is accepted in the theory of anharmonic crystals197 and provides a number of exact solutions in the case of a single adsorbed molecule.I50,1 2 In this approach, the spectral line shape for high-frequency vibrations can be determined as follows 184... 

D. N. Zubarev has written an interesting account of the theory and application of general double-time Green s functions, published in Soviet Physics Uspekhi 3, 320 (1960) (English translation). 

This section deals with the calculation of double-time Green s functions for the study of electron propagation through a system with small interactions between atoms. 

It is often also called the polarization propagator in the time domain (Zubarev, 1974 Linderberg and Ohrn, 1973 J0rgensen and Simons, 1981), while mathematically it is a double-time Green s function. 

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