Particle-hole formalism

The essence of the particle-hole formalism lies in the redefinition of the vacuum state that it permits. Consider now the closed-shell ground state Slater determinant l o) which can be expressed, using eq. (3.75), in the form 

In using the particle-hole formalism, we adopt o) as a new vacuum state. We call this reference state the Fermi vacuum state. With respect to this Fermi vacuum state l o), we can now define new creation and annihilation operators. To distinguish them from the operators X+ and X described above, the creation and annihilation operators in the particle-hole formalism shall be designated Y+ and Y, respectively. If we label the occupied spin-orbitals as A ) and the virtual spin-orbitals as A ), then the creation (Y+) and annihilation (F) operators in the particle-hole formalism are then defined in the following manner  

From this definition, it is evident that application of creation operator Y, to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in l o). The effect of Y, on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in o) The effect of Y on the Fermi vacuum is the creation of a particle in the virmal spin-orbitals and finally, the effect of Y on o) is the annihilation of a particle in virtual spin-orbitals. Thus, for example, a singly excited Slater determinant ) can be described as 

The reference state f o) corresponds to the state in which both the number of particles and the number of holes equal zero. The determinant ) corresponds to a single particle and a single hole. Similarly, a doubly excited Slater determinant can be described in terms of two particles and two holes  

For particle-hole creation and annihilation the Y operators satisfy the same anticommutation relations as those given above for the X operators. Specifically, we have 

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

The particle-hole formalism has been introduced as a simplihcation of many-body perturbation theory for closed-shell states, for which a single Slater determinant dominates and is hence privileged. One uses the labels i,j, k,... for spin orbitals occupied in <1> and a,b,c,... for spin orbitals unoccupied virtual) in . 

This result represents the most important advantage of the particle-hole formalism. Many-body perturbation theory (MBPT) consists mainly in the evaluation of expectation values (with respect to the physical vacuum) of products of excitation operators. This is easily done by means of Wick s theorem in the 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. 

W. Kutzelnigg, Quantum chemistry in Fock space. III. Particle-hole formalism. J. Chem. Phys. 80, 822 (1984). 

Although the material contained in this book concerns the theory of many-electron atoms and ions, its many ideas and methods (e.g., graphical methods, quasispin and isospin techniques, particle-hole formalism, etc.) are fairly universal and may be easily applied (or already are) to other domains of physics (nuclear theory, elementary particles, molecular, solid state physics, etc.). 

Analysis of the products of field operators in these equations leads to a representation of the wave function and of the level shift in terms of diagrams of the type first introduced by Feynman. These diagrams provide a simple pictorial description of electron correlation effects in terms of the particle-hole formalism. 

The matrix elements of the generators, Sab, can be evaluated using the graphical methods of spin algebra.174-176 The particle-hole formalism introduced by Flores and Moshinsky177 and recently discussed by Paldus and Boyle178 is most suitable for such developments. 

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. 

Ppor an explanation of -creation and -annihilation operators, see the earlier discussion of the particle-hole formalism in the section on The Fermi Vacuum and Particle-Hole Formalism. 

For ground states and low-lying excited states it is convenient to adopt a particle-hole formalism. We use the indices... 

The diagrams are interpreted in terms of the particle-hole formalism. The Fermi level is defined such that all single particle states lying below it are occupied and all above it are unoccupied. In the particle-hole picture, the reference state is taken to be a vacuum state, containing no holes below the Fermi level and no particles above it. Excitation leads to the creation of particle-hole pairs, with particles above the Fermi level and holes below it. 

In conclusion, a few words should be said about the equivalence between the ket-bra formalism frequently used in this article and the particle-hole formalism based on the ideas of second quantization T commonly used in the special propagator theories and the EOM method. Both formalisms are used to construct a basis for the operator space, and the essential difference is that the latter treats particles having specific symmetry properties—i.e., fermions or bosons—whereas the former is not yet adapted to any particular symmetry. In order to get a connection between the two schemes, it may be convenient in the ket-bra formalism to introduce a so-called Fock space for different numbers of particles... 

Goldstone used a second quantized particle-hole formalism based on an arbitrary choice of vacuum state. The interaction representation, which is intermediate between the Schrddinger and Heisenberg pictures, was employed and the energy was evaluated by the Gell-Mann-Low formalism78 with Hamiltonian... 

In this particle-hole formalism, the normal product is written... 

Figure 4 Interpretation of diagrams in terms of the particle-hole formalism...

In the second-quantized operators (31) and (32), the summation over the particle indices i,j,... runs over all the electron states of the (complete one-electron) spectrum. If these operators act to the right upon the reference state, i.e. the many-electron vacuum of the particle-hole formalism, some of these (strings of) creation and annihilation operators create excitations while other gives simply zero, i.e. no contribution. For the pure vacuum, in particular, the behavior of the second-quantized operators can be read off quite easily because the creation operators appear left of the annihilation operators in expressions (31) and (32), respectively. 

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]

In words, for virtual orbitals the holes operators bi", bj act exactly in the same manner as the particle operators aj do, while their role is reversed for occupied orbitals. Operator creates an electron in the virtual space, while it annihilates an electron in the Fermi see. This is equivalent to saying that it creates a hole in HF>. Similarly, operator bj creates an electron in 1HF>, while it annihilates one in the virtual subspace. This particle-hole formalism is in analogy with that of quantum field theory where, for instance, the holes correspond to positrons while the particles are electrons. 

Figure 3.2. Simple illustration of the particle-hole formalism. This figure shows the states given in the particle formalism in Figure 3.1 when depicted in the particle-hole picture, (a) shows the reference configuration or vacuum state with no particles above the Fermi level and no holes below it. (6) corresponds to a single excitation which creates a hole below the Fermi level and a particle above it. (c) is a doubly excited state with two holes and two particles and (d) is associated with a triply excited state with three holes and three particles. The particle-hole formalism focusses attention on the excitation process the particles and holes created during an excitation. The other electrons in the studied many-body system are merely spectators to the excitation process.

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