Self-energy third-order

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... 

Perturbative expressions for the self-energy operator can achieve this goal for large, closed-shell molecules. In this review, we will concentrate on an approximation developed for this purpose, the partial third-order, or P3, approximation. P3 calculations have been carried out for a variety of molecules. A tabulation of these calculations is given in Table 5.1. 

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]. 

The present contribution concerns an outline of the response tlieory for the multiconfigurational self-consistent field electronic structure method coupled to molecular mechanics force fields and it gives an overview of the theoretical developments presented in the work by Poulsen et al. [7, 8, 9], The multiconfigurational self-consistent field molecular mechanics (MCSCF/MM) response method has been developed to include third order molecular properties [7, 8, 9], This contribution contains a section that describes the establisment of the energy functional for the situation where a multiconfigurational self-consistent field electronic structure method is coupled to a classical molecular mechanics field. The second section provides the necessary background for forming the fundamental equations within response theory. The third and fourth sections present the linear and quadratic, respectively, response equations for the MCSCF/MM response method. The fifth 283... 

The zeroth order degeneracy of K and T-states is lifted in the spectral representation of the fully interacting Green s function. The exact eigenstates decouple to yield A -particle states in the second component and tensor products of W — 1 and N -L 1-particle states in the third component. We will come back to discuss the meaning of the degenerate states A >) in connection with the lowest order of the static self energy S oo) in Sec. V. 

In Fig. 6.2, we list the nonvanishing third-order self-energy diagrams. These may, of course, also be identified with corresponding terms of the... 

FIG. 6.2. All third-order Hugenholtz self-energy diagrams. 

The third-order contributions to the self-energy can also readily be calculated. These terms lead to fivefold summations and are more cumbersome to compute. ... 

Restricting f to only some simple fermion-like products of electron field operators will generate only certain types of diagrams that can then be summed to all orders. For instance, self-energy diagrams in third order of the ring and ladder types, which can easily be generalized in any order. It is notable that between consecutive interaction lines, there only occur one hole line and two particle lines or vice versa [see Fig. 9.1]. 

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