Diagrammatic formalism

The presented form of the master equation (2.3.67) permits us to employ the diagrammatic technique of the perturbation theory [44, 108-110, 113]. The free Hamiltonian could be written as 

A procedure similar to the condensate separation in the imperfect Bose gas was employed by Lifshitz and Pitaevski [78], The diagrammatic technique allows us to calculate the reaction rate and steady-state joint correlation functions. A separation of a condensate from terms with k = 0 cannot be done without particle production p = 0), in which case nA,nB 0 as f oo. In this respect the formalism presented by Lushnikov [111] for the non-stationary processes is of certain interest. 

From Pair Kinetics Toward the Many-Reactant Problem 

---

Many varieties of diagrams have used throughout the chemical physics literature for many years (e.g., see Refs. 1, 2, 80, 117, and 119). The diagrammatic formalism we have chosen here has been frequently used in work by the Bartlett group among others and is particularly straightforward for conventional coupled cluster and many-body perturbation theories. 

The organization of this chapter is as follows. In Sect. 5.1 we present the basic formalism and work out the Feynman rules for the grand canonical ensemble. Diagrammatic representations valid in the thermodynamic limit are derived for both thermodjmamic quantities and correlation functions. The proof of the Linked Cluster Theorem is given in Appendix A 5.1. Section 5.2... 

The whole problem of calculating fc/ (at least up to the third order) is now reduced to the calculation of individual terms (55)—(58). The second quantization formalism has the advantage that these terms can be calculated easily by making use of the diagrammatic technique which will be demonstrated in Section IV.B. 

We have already shown that excitation energies can be diagrammatically decomposed to yield simpler quantities such as ionization potentials and electron affinities plus some remaining diagrams. MB-RSPT permits the use of this treatment for even more complex processes. In this section, we present the applicability of the theory to double ionizations observed in Auger spectra as well as excitations accompanying photoionization (shake-up processes) observed in ESCA and photoelectron spectroscopy. A detailed description of this approach is given in Refs.135,136. Here we shall present only the formal description. 

The superoperator formalism that has been used in previous publications is outlined here [2, 9, 29], The alternative diagrammatic and algebraic-diagrammatic representations can be found in other works [6],... 

We make use of the particle-hole formalism in diagrammatic analyses by drawing upward- and downward-directed lines that identify those orbitals which differ from those in the reference determinant, Oq j as shown in Figure 1. 

In Volume 5 of this series, R. J. Bartlett and J. E Stanton authored a popular tutorial on applications of post-Hartree-Fock methods. Here in Chapter 2, Dr. T. Daniel Crawford and Professor Henry F. Schaefer III explore coupled cluster theory in great depth. Despite the depth, the treatment is brilliantly clear. Beginning with fundamental concepts of cluster expansion of the wavefunction, the authors provide the formal theory and the derivation of the coupled cluster equations. This is followed by thorough explanations of diagrammatic representations, the connection to many-bodied perturbation theory, and computer implementation of the method. Directions for future developments are laid out. 

---