Hartree-Fock creation operators

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

The form of the wave operators need not be defined, but, in principle, they can describe any type of wavefunction, for example, Hartree-Fock or coupled-cluster wavefunctions. However, at their core, they always consist of strings of creation operators. We define the supermolecular wavefunction as... 

Often configuration interaction calculations are performed after an initial Hartree-Fock calculation. In such cases the one-electron creation and annihilation operators would refer to eigenfunctions of the Fock operator, and the Fock operator would have a simple form similar to that shown in equation (53). 

In (1) a ic.b Kjaic and b f are boson creation and annihilation operators for the a and b Hartree-Fock particle states with momentum hn and kinetic energy K-h k /2m. pa and pb denote the densities of the component holon gases while pa and pb denote their respective chemical potentials. In equilibrium, pa=Pb-Pt where u is determined by the condition that the statistical average of A Eic (a icaic b Kbic) be equal to p=pa+pb=N/A, the total number of holons per unit area (N , A ). V is taken to satisfy V<pairing interaction. Finally, V is restricted to operate between holons with k[Pg.45]

Surveying the history of the theory of optical lanthanide spectroscopy, we can discern several main features the usefulness of Lie groups, following their introduction by Racah (1949) the relevance of the method of second quantization, as demonstrated by the use of annihilation and creation operators for electrons and the inability of the Hartree-Fock method and its various elaborations to provide accurate values (say to within 1%) of such crucial quantities as the Slater integrals F (4f,4f) and the Sternheimer correction factors R , for a free ion. The success of the formal mathematics is in striking contrast to the failure of the machinery of computation. This turn of events has happened over a period of time when... 

Avery, J., Creation and Annihilation Operators, McGraw-Hill, New York, 1976. Chapter 2 formulates a number of quantum mechanical approximations, including the Hartree-Fock approximation, in the language of second quantization. 

In these equations, f is the Hartree-Fock operator and X (Xg) are creation (annihilation) operators defined with respect to Fermi vacuum o) and N[ ] is the normal ordered product of creation and annihilation operators. 

Commutation occurs since the excitation operators always excite from the set of occupied Hartree-Fock spin orbitals to the virtual ones - see (13.1.2) for the double-excitation operators. The creation and annihilation operators of the excitation operators therefore anticommute. 

The operator may be the identity operator (in which case it does not affect the Hartree-Fock ket) or it may be a string of creation and annihilation operators for the occupied and virtual spin orbitals of system A, respectively (in which case it annihilates the Hartiee-Fock ket). The same considerations apply to Tg. Since operators referring to different noninteracting subsystems commute, we may write the matrix element in (14.2.56) as... 

---