Self-energies

2 Three-dimensional dipolar system F.2.1 Self-energy 

We now derive an expression for the self-contribution to the dipolar energy in Ewald formulation given in Eq. (6.32) by recalling that the corresponding 

Approximating erf (avij) /rij by its Taylor expansion for small distances ry given in Eq. (F.48), we obtain 

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The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

Fiorentini V and Baldereschi A 1995 Dielectric scaling of the self-energy scissor operator in semiconductors and insulators Phys. Rev. B 51 17 196... 

Godby R W, Schluter M and Sham L J 1988 Self-energy operators and exchange-correlation potentials in semiconductors Phys. Rev. B 37 10159-75... 

A set of properties of states (causality, resh ictions on the spectra of self-energies, existence or absence of certain isolated energy bands, etc.). 

If the simulated system uses periodic boundary conditions, the logical long-range interaction includes a lattice sum over all particles with all their images. Apart from some obvious and resolvable corrections for self-energy and for image interaction between excluded pairs, the question has been raised if one really wishes to enhance the effect of the artificial boundary conditions by including lattice sums. The effect of the periodic conditions should at least be evaluated by simulation with different box sizes or by continuum corrections, if applicable (see below). 

Representing a heteropolymer, the model tries to capture tliree contributions to the overall energy E of each conformation the self-energy e of each amino acid, a bond energy tenn Ji i+i between two neighboring residues, and a nonbonded Ki j interaction... 

Fig. 1. Born ion self energy and the problem of inorganic ion passage through a lipid barrier.

A. Hydration energy profile, using the Bom formalism (Eqn. 1), shows the drop of ion self energy as a function of the radius of a hydration sphere. Note that even with a hydration shell of 10 A radius not all of the hydration energy is obtained. 

In Eq. (11-120) Sc denotes the contribution to 8 from connected diagrams only, i.e., from the diagrams shown in Fig. 11-5. The energy E0 in this phase factor, exp (+iE0T), can be interpreted as the vacuum self-energy due to the interaction By redefining... 

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 ... 

Here, an effective one-electron operator matrix has Fock and energy-dependent, self-energy terms. Prom this matrix expression, one may abstract one-electron equations in terms of the generalized Fock and energy-dependent, self-energy operators ... 

The usual initial guess, Cp -I- Epp(cp), usually leads to convergence in three iterations. Relationships between diagonal self-energy approximations, the transition operator method, the ASCF approximation and perturbative treatments of electron binding energies have been analyzed in detail [17, 18]. 

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... 

From considerations on translational symmetry in the limit of a stereoregular polymer, which are more conveniently analyzed in terms of conservation constraints on momenta at interaction vertices and within self-energy diagrams (31), each Ih line can be easily shown (see e.g. Figure 4 for a second-order process)... 

The physical meaning of our final equation is best seen on eqn 39. The term containing w is essentially the self-energy correction introduced by Mulliken in his analysis of electronegativities to account for the average repulsion of electrons occupying the same orbital. In order to get an idea of the orders of magnitude, let us apply eqn 39 to a model computation of FeCO, made to compare the ClPSl results of Berthier et al. [11] with those of a simple orbital scheme. Consider one of the two x systems of FeCO, treated under the assumption of full localization (and therefore strict cr — x separation)... 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

Mohr, P.J. (1992) Self-energy correction to one-electron energy levels in a strong Coulomb field. Physical Review A, 46, 4421-4424. 

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