Particle-hole representation

Apart from the occupation number operators for one-particle states we can introduce the operator 

According to (13.12), the eigenvalues of this operator will also be zero and unity, however, unity is now an eigenvalue of the one-determinant wave funotion in which this one-electron state a is vacant and zero is an eigenvalue of a function for which this state is filled. Thus, quantity /,na is the occupation number operator for the hole state a. 

For commutation relations of this operator with electron creation and annihilation operators, instead of (13.29) and (13.30), we obtain 

For some one-determinant state ai.ajy) we can completely change over from particle description to hole description if, instead of electron creation and annihilation operators, we introduce, respectively, annihilation and creation operators for holes 

Wave function /,0) is also a vacuum state for the particles, which are  

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Particle-hole representation 13.5 Particle-hole representation... 

In the theory of many-electron atoms, the particle-hole representation is normally used to describe atoms with filled shells. To the ground state of such systems there corresponds a single determinant, composed of one-electron wave functions defined in a certain approximation. This determinant is now defined as the vacuum state. In the case of atoms with unfilled shells, this representation can be used for the atomic core consisting only of filled shells. Then, the excitation of electrons from these shells will be described as the creation of particle-hole pairs. 

The utilization of the particle-hole representation to describe the wave functions of unfilled shells yields no substantial simplifications. As has been noted above, these wave functions can be expressed in terms of linear combinations of one-determinant wave functions. Selection of one of these determinants as the vacuum state yields a linear combination of one-determinant wave functions, now described using the particle-hole representation. 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

Underlined numerical indexes in (j. 2) are a set of indexes 1 = where a is the spinor index and v is the index of the particle-hole representation, numerating matrix elements in (20). Thus (12) is a matrix 8x8 with respect to discrete indexes aav. 

In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle-hole propagators, which have been at the heart of Jens electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9-15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions ... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

M. B. Ruskai, iV-representability problem particle-hole equivalence. J. Math. Phys. 11, 3218 (1970). 

The CSE allows us to recast A-representability as a reconstruction problem. If we knew how to build from the 2-RDM to the 4-RDM, the CSE in Eq. (12) furnishes us with enough equations to solve iteratively for the 2-RDM. Two approaches for reconstruction have been explored in previous work on the CSE (i) the explicit representation of the 3- and 4-RDMs as functionals of the 2-RDM [17, 18, 20, 21, 29], and (ii) the construction of a family of higher 4-RDMs from the 2-RDM by imposing ensemble representability conditions [20]. After justifying reconstmction from the 2-RDM by Rosina s theorem, we develop in Sections III.B and III.C the functional approach to the CSE from two different perspectives—the particle-hole duality and the theory of cumulants. 

Many-body problems in quantum mechanics are usually described by the number of particles N in the system and the probabilities of finding those particles at different locations in space. If the rank of the one-particle basis is a finite number r, an equally valid description of the system may be given by specifying the number of holes r N in the system and the probabilities of finding these holes at different locations in space. This possibility for an equivalent representation of the system by particles or holes is known as the particle-hole duality. By using the fermion anticommutation relation... 

In the Af-representability literature these positivity conditions are known as the D- and the g-conditions [5, 7, 63]. The two-particle RDM and the two-hole RDM are linearly related via the particle-hole duality,... 

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 basic elements of the diagrams are shown in Figure 1. Figure 1 (a) shows the diagrammatic representation of a one-electron operator matrix element. Figure 1 (b) shows the representation of a two-electron matrix which in the Brandow scheme includes permutation of the two electrons involved. Upward (downward) directed lines represent particles (holes) created above (below) the Fermi level when an electron is excited. 

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... 

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),... 

In this appendix we present the interaction part (at,a, y o, a,<) of the static particle-hole self energy discussed in Sec. VIC. We assume Coulomb interacting electrons with the usual position space representation (106) of the two-body interaction V. The expression for the interaction part of the static self particle-hole self energy can then be readily evaluated, either from the definitions of the extended states (1) or from Eq. (68) ... 

To obtain a spectral representation of the propagator that contains a unit metric, one must transform the set of particle-hole and hole-particle operators to the representation where they give a diagonal metric with unit elements. This transformation is carried out using the excitation operators defined below ... 

As all four blocks of the matrix propagator contain the same information regarding excitation energies and transition moments (as is clear from the spectral representation), one can choose to concentrate on the so-called particle-hole propagator, which satisfies... 

The complete set of density matrices (eq 1.2) may be subsequently calculated using the eigenvectors. " Only particle-hole and hole-particle components of Iv are computed in the restricted TDHF scheme (Appendix A). Therefore, this non-Hermitian eigenvalue problem of dimension 2M x 2M, M = A cc x Nvir = Nx K — N) in the MO basis set representation may be recast in the form ... 

This is an unusual scalar product. It can further be expressed through the particle—hole Xj and hole-particle (Y) components of the interband density matrix in MO representation as... 

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