Propagators in Quantum Chemistry, 2nd Edition

Chapter 4

Double-Time Green’s Functions

The proper electron interaction hamiltonian is

with

This hamiltonian is correct because it gives the correct expectation values. The average value with respect to a two-electron state is kl\H_int\kl = (kk\ll) – (kl\lk) exactly as the result (coulomb minus exchange) of calculating the average of 1/|₁ – ₂| with respect to a two-electron Slater determinant with the spin orbitals u_k and u_l 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 a^† operators, ...