Self-energy matrix

Corrections to the zeroth-order, inverse propagator in equation 26 are gathered together in a term known as the self-energy matrix, E(E). The Dyson equation may be written as... 

In the self-energy matrix, there are energy-independent terms and energy-dependent terms ... 

This choice produces asymmetric superoperator matrices. A simplified final form for the self-energy matrix that does not require optimization of cluster amplitudes is sought for large molecules the approximation... 

With this choice, several third-order terms that appeared with the usual metric are eliminated. The new self-energy matrix in third order is asymmetric and is expressed by... 

Neglect of third-order, 2p-h terms produces this self-energy matrix ... 

Comparison of the self-energy matrix elements of equation 45 with older, related methods [7,15] reveals the advantages of the P3 approximation. Among the intermediates required in third order is... 

This procedure requires analytical expressions for EPP(E) and its derivative with respect to E it usually converges in three iterations. Neglect of off-diagonal elements of the self-energy matrix also implies that the corresponding Dyson orbital is given by ... 

More satisfactory results are obtained from full third-order calculations [32, 33]. Diagonal elements of the full third-order, self-energy matrix are given by... 

Second-order and third-order results often bracket the true correction to pF - Three schemes that scale the third-order terms in various ways are known as the Outer Valence Green s Function (OVGF) [8], In OVGF calculations, one of these three recipes is chosen as the recommended one according to rules based on numerical criteria. These criteria involve quantities that are derived from ratios of various constituent terms of the self-energy matrix elements. Average absolute errors for closed-shell molecules are somewhat larger than for P3 [31]. 

Neglecting off-diagonal elements of the self-energy matrix in the canonical Hartree-Fock basis in (1.15) constitutes the quasiparticle approximation. With this approximation, the calculation of EADEs is simplified, for each KT result may be improved with many-body corrections that reside in a diagonal element of the self-energy matrix. 

AH matrices are of dimension 6Z and the harmonic propagator matrix G(0)(q m) is diagonal. The problem of calculating the phonon propagators thus reduces to the calculation of the self-energy matrices S(q ioif) that contain all anharmonic information. It is not difficult to demonstrate that the self-energy matrix is a Hermitian function of oif from which it follows that its analytic continuation in the complex frequency plane, in the neighborhood of the real axis, has the form... 

Renormalizations in the constant part of the self-energy matrix, X (oo), are often included in ADC(3) calculations. An algorithm for the evaluation of... 

We can apply for the diagonal elements of the self-energy matrix, X(ft>,) in the Moeller-Plesset (MP) many body perturbation theory (MBPT) in the second order (MP2) approximation... 

Here X is the self-energy matrix, and the diagonal matrices and eHF, respectively, contain the quasi particle (correlation corrected) and the HF one-... 

The matrix M(z) consists of two parts, in correspondence with the form (1). The solution of Eq. (12) or the diagonalization of M(z) implies the accurate construction and handling of the complex self-energy matrix A(z), which is the same as the complex quantity in the resonance formula in Feshbach s theory [2a, p. 367,2b, p. 304]. For a many-electron system, such a goal is very difficult to achieve rigorously. Therefore, one has to search for a practical computational method which produces the resonance wavefunction and the corresponding complex energy. 

The self-energy matrix in a spin orbital basis can be expressed as... 

This means that the self-energy matrix becomes... 

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