Electron Propagator Concepts

A special case of this approach is represented by the Hartree-Fock equations, where the effective operator Heff contains the usual kinetic (T), nuclear attraction (U), Coulomb (J), and exchange (K) components such that 

Since the J and K operators depend on the occupied orbitals, the pseudoeigenvalue problem must be solved iteratively until consistency is achieved between orbitals that determine J and K and those that emerge as eigenfunctions of Heff, which in this approximation is known as the Fock operator F. 

Electron propagator formalism allows for generalizations that include the effects of correlation. Here the pseudoeigenvalue problem has the following structure 

Now the Fock operator is supplemented by the self-energy operator E(E). This operator depends on an energy parameter E and is nonlocal. All 

Eigenfunctions that accompany these eigenvalues have a clear physical meaning that corresponds to electron attachment or detachment. These functions are known as Dyson orbitals, Feynman-Dyson amplitudes, or generalized overlap amplitudes. For ionization energies, they are given by 

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Electron propagator theory generates a one-electron picture of electronic structure that includes electron correlation. One-electron energies may be obtained reliably for closed-shell molecules with the P3 method and more complex correlation effects can be treated with renormalized reference states and orbitals. To each electron binding energy, there corresponds a Dyson orbital that is a correlated generalization of a canonical molecular orbital. Electron propagator theory enables interpretation of precise ab initio calculations in terms of one-electron concepts. 

The concept of order in the perturbation expansion of the electron propagator ultimately means order in terms of the electron-electron interaction, or equivalently, two-electron integrals. The inclusion of electron correlation through first order in the reference state is achieved with the double excitation terms K2, whereas the Ki terms are also needed for second-order corrections. 

This chapter deals with the discussion and interpretation of approximate molecular electronic structure methods in terms of propagator concepts. Only situations with fixed nuclear frameworks are considered, and the discussion is limited to the description of states that are close in energy to the normal state of the system. We adopt the view that the main features of the electronic structure of such states can be developed in terms of atomic orbital representations of operators and that only valence shell orbitals need be considered. 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

Quantum mechanics had exploded between 1923 and 1927. A. H. Compton, in 1923, had discovered the change in frequency of X-rays scattered from the electrons (the Compton effect).25 Compton and, independently, Debye had underlined the importance of this discovery in support of the Einstein conception of light-quanta or photon propagation in space.26... 

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