The Electron Propagator

The higher moments in the moment expansion of the propagator or the propagator matrix can become quite comphcated and approximations are necessary. The simplest approximation that yields useful results proceeds by approximating higher moments as powers of the first moment F = (X HX). Denote S = (X X) and obtain the approximation 

Show that the average value of Eq. (4.1) with respect to a two-electron state is kl Hint kl) — kk ll) — kl lk), and that it is the same as calculating the average of l/ r — r2 with respect to a two-electron Slater determinant with the spin orbitals Uk and ui in first-quantization . Also show that the average value of the electron interaction hamiltonian in Eq. (4.1) is zero with respect to a one-electron state, as it should be. 

The electron propagator is obtained when the basis field operators are restricted to the electron field operators A, = a and becomes 

The elements of the metric matrix S and the dynamical matrix F that occur in the geometric approximation axe readily calculated from the basic anticommutation relations (Eq. (3.49)) and the hamiltonian as given by Eqs. (4.3) and (4.4). The results are 

Diagonalization of F with the constraint that the first-order reduced density matrix 7 (the one-matrix) satisfies 7sr = (ar s) = (nr)Ssr with occupation numbers (n ) = 0 or 1 i.e., Tr y = N and 7 = 7) is done iteratively and converges to a single determinantal SCF approximation for the iV-electron ground state corresponding to the appropriate set of occupation numbers. 

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Good early overviews of the electron propagator (that is used to obtain IP and EA data) and of the polarization propagator are given in ... 

Ortiz J V 1997 The electron propagator picture of molecular electronic structure Computational Chemistry Reviews of Current Trends vo 2, ed J Leszczynski (Singapore World Scientific) pp 1-61... 

The physical meaning of the electron propagator rests chiefiy in its poles (energies where singularities lie) and residues (coefficients of the terms responsible for the singularities) [1]. In its spectral form, the r,s element of the electron... 

It is possible to use full or limited configuration interaction wavefunctions to construct poles and residues of the electron propagator. However, in practical propagator calculations, generation of this intermediate information is avoided in favor of direct evaluation of electron binding energies and DOs. 

Thus the matrix elements of the electron propagator are related to field operator products arising from the superoperator resolvent, El — H), that are evaluated with respect to N). In this sense, electron binding energies and DOs are properties of the reference state. 

According to equation 15, eigenvalues of the superoperator Hamiltonian matrix, H, are poles (electron binding energies) of the electron propagator. Several renormalized methods can be defined in terms of approximate H matrices. The... 

Spin Density Properties from the Electron Propagator Hyperfine and Nuclear Spin-Spin Couplings... 

When the operators A and B in Eq. (2.7) are sin q)le creation and annihilation operators the resulting propagator is called electron (nopagator or one-particte Green s function, and = -t-1. Collecting all these creation and amiihi-lation operators in a row vector a, the electron propagator can be expressed as. 

Thus, the electron propagator matrix elements can be written as,... 

The complete Hamiltonian of the molecular system can be wrihen as H +H or H =H +H for the commutator being linear, where is the Hamiltonian corresponding to the spin contribution(s) such as, Fermi contact term, dipolar term, spin-orbit coupling, etc. (5). As a result, H ° would correspond to the spin free part of the Hamiltonian, which is usually employed in the electron propagator implementation. Accordingly, the k -th pole associated with the complete Hamiltonian H is , so that El is the A -th pole of the electron propagator for the spin free Hamiltonian H . 

The matrix Hfj would be the transpose of Hf, if it were Hermitian. The Hermiticity of the superoperator Hamiltonian has been a concern since the beginnings of the electron propagator theory (46,129). For a Hermitian spin ftee Hamiltonian (// ) the following relation can be written describing the Hermiticity problem,... 

The Hi jCf tl-Hjj)" matrix is usually computed in the iterative pole search of the electron propagator (31,130). Thus, the implementation of the above expression for the splitting, Eq. (7.1.1), becomes simple, since it requires only the additional calculation of the Hf, and Hfj matrices. 

It is still necessary to perform an order analysis of the correlation potential in the calculation of. The usual implementation of the electron propagator is performed up to the third or partial fourth orders (31,32,129,130), which needs... 

In this book, the experts who have developed and tested many of the currently used electronic structure procedures present an authoritative overview of the theoretical tools for the computation of thermochemical properties of atoms and molecules. The first two chapters describe the highly accurate, computationally expensive approaches that combine high-level calculations with sophisticated extrapolation schemes. In chapters 3 and 4, the widely used G3 and CBS families of composite methods are discussed. The applications of the electron propagator theory to the estimation of energy changes that accompany electron detachment and attachment processes follow in chapter 5. The next two sections of the book focus on practical applications of the aforedescribed... 

A not-trivial ratchet effect can be observed when the injected charge density is voltage-independent, EL/R = Ep eV/2. Symmetry considerations require an asymmetric U (x) for a non-vanishing ratchet current in this case. Also an electron interaction must be present. Indeed, for free particles the reflection coefficient R(E) is independent of the electron propagation direction [14] and hence I(V) = —/(—V). 

J. Schirmer, A.B. Trofimov, G. Stelter, A non-Dyson third-order approximation scheme for the electron propagator, J. Chem. Phys. 109 (1998) 4734. 

Using superoperators in combination with (1.1), we get the electron propagator matrix... 

The proposals found here can be seen as the result of a two-way strategy for the treatment of large molecules. First, we improve on the accuracy of the very efficient second order approximation. In addition, we introduce approximations that lower considerably the required computer resources for the use of higher-order approximations to the electron propagator within the quasiparticle approach. 

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