Particle-hole density matrix

One can further define a particle-hole density matrix, which also has the same two-particle cumulant... 

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. 

For the formulation of the generalized Wick theorem corresponding to the generalized normal ordering, we need the matrix element rf, Eq. (65), of the one-hole density matrix and the cumulants kjc, Eqs. (39)-(47), of the fc-particle density matrices. 

For the weakest occupied orbitals, it makes sense to interchange particles and holes. The resulting analog to the MCSCF Fock matrix is then 2f- and the hole density matrix is, as before, d = 2 1 - D. 

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. 

M. V. Mihailovic and M. Rosina, The particle-hole states in some light nuclei calculated with the two-body density matrix of the ground state. Nucl. Phys. A237, 229-234 (1975). 

M. Rosina and M. V. Mihailovic, The determination of the particle—hole excited states by using the variational approach to the ground state two-body density matrix, in International Conference on Properties of Nuclear States, Montreal 1969, Les Presses de I Universite de Montreal, 1969. 

In view of this interpretation of eqn ( 33.4) we might expect that this normalisation condition is essentially due to the exchange part of the exchange-correlation hole and that it might be useful to divide this density into an exchange hole and a correlation hole . Guided by the product form of the (real) two-particle density for the single-determinant model which involves the density function p and the density matrix pi ... 

Consequently, not all elements of the density matrix are independent. The number of degrees of freedom of dp subject to the condition eq A6 is precisely the number of its particle-hole matrix elements, and 71(1) can therefore be expressed in terms of using eq A6 and eq A5... 

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... 

Within the given hole-particle approach, D p is a counterpart of the corresponding EUE density matrix (6.5). Technical details for computing FCI and closed-shell CCSD (singles and doubles coupled cluster) approaches are given in [16, 47]. We write here only the simplest relation... 

Having at disposal density matrix (6.69) it is easy to perform the hole-particle analysis of the CIS method. In this case, Eqs. (6.40) and (6.43) are valid because the CIS states have no anomalies in the density matrix spectmm. Simple manipulations on Eq. (6.37) lead to... 

As a matter of fact, the hole and particle occupancies are identical for any bipartite networks treated within Jt-approximation, up to FCl/PPP. This is a simple corollary of the generalized pairing theorem of McLachlan [94] stating that the jr-electron charge density matrix of the alternant hydrocarbons is of the form... 

It is instructive to compare the approximate weak-coupling theory to essential exact, numerical (density matrix renormalization group) calculations on the same model (namely the Pariser-Parr-Pople model). The numerical calculations are performed on polymer chains with the polyacetylene geometry. Since these chains posses inversion symmetry the many-body eigenstates are either even (Ag) or odd By). As discussed previously, the singlet exciton wave function has either even or odd parity when the particle-hole eigenvalue is odd or even. Conversely, the triplet exciton wavefunction has either even or odd parity when the particle-hole eigenvalue is even or odd. As a consequence, we can express a B state as... 

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