Generalized normal ordering particle-hole formalism

We introduce the generalized normal ordering in various steps, starting with the traditional particle-hole formalism. 

The results of the last section, which are essentially a reformulation of the traditional particle-hole formalism for excitation operators, were first presented in 1984 [8]. At that time it was not realized that only two very small steps are necessary to generalize this formalism to arbitrary reference states. Only after Mukherjee approached the formulation of a generalized normal ordering on a rather different route [2], did it come to our attention how easy this generalization actually is, when one starts from the results of the last section. 

In general, a given sequence of creation and annihilation operators is said to be normal ordered, if all the creation operators appear left of all annihilation operators. Such an ordering of the operator strings simplifies the manipulation of operator products as well as the evaluation of their matrix elements, as the action of these operators can be read off immediately. In the particle-hole formalism, its hereby obvious that we can annihilate only those particles or holes which exist initially in fact, an existing hole is nothing else than that there is no electron in this hole state. In this formalism, therefore, an operator in second quantization is normal ordered with regard to the reference state [Pg.190]

A full relativistic theory for coupling tensors within the polarization propagator approach at the RPA level was presented as a generalization of the nonrelativistic theory. Relativistic calculations using the PP formalism have three requirements, namely (i) all operators representing perturbations must be given in relativistic form (ii) the zeroth-order Hamiltonian must be the Dirac-Coulomb-Breit Hamiltonian, /foBC, or some approximation to it and (iii) the electronic states must be relativistic spin-orbitals within the particle-hole or normal ordered representation. Aucar and Oddershede used the particle-hole Dirac-Coulomb-Breit Hamiltonian in the no-pair approach as a starting point, Eq. (18),... 

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