Polarization propagator approximation

One somewhat displeasing detail in the approximate polarization propagator methods discussed in the previous section is the fact that concern needs to be made as to which formulation of wave mechanics that is used. This point has been elegantly resolved by Christiansen et al. in their quasi-energy formulation of response theory [23], in which a general and unified theory is presented for the evaluation of response functions for variational as well as nonvariational electronic structure methods. 

Approximate polarization propagator methods are thus obtained by truncating the set of operators and by using an approximate reference state Mpller-Plesset... 

It is now clear that the determination of the poles of the matrix R(o>), and hence of the approximate polarization propagator, is equivalent to finding the free-oscillation solutions of the MC TDHF equation (12.7.8a) for R(a>) will become infinite at the m-values that make the determinant of the matrix on the right in (13.5.30) vanish— and these are simply the eigenvalues of... 

The simplest polarization propagator corresponds to choosing an HF reference and including only the h2 operator, known as the Random Phase Approximation (RPA). For the static case oj = 0) the resulting equations are identical to those obtained from a Time-Dependent Hartree-Fock (TDHF) analysis or Coupled Hartree-Fock approach, discussed in Section 10.5. 

Figure 4 Conduction band levels and excitation levels of infinite periodic hydrogen chains by using different approximations of the polarization propagator. The left part refers to the crystalline orbital energy differences, namely, the Hartree-Fock excitation energies the right part refers to the random phase approximation results obtained by using 41 k-points in half the first Brillouin zone.

We have employed the second-order polarization propagator approximation (SOPPA) in this study, a method which was mainly developed by Jens Oddershede and his co-workers [3,4,20,51-56]. Barone et al. [32] have recently shown that SOPPA reproduces the vicinal F-F couplings reasonably well in 1,2-difluoroethene. 

In order to calculate the GOS, one requires the excitation energies and the generalized transition moments. Oddershede and Sabin had already started in 1992 the investigation of the GOS and the stopping cross section in the first Bom approximation by means of the polarization propagator method [78]. 

However, the assumption on the idea behind the use of the polarization propagator is based on the use of a complete basis set. In practical terms that is not possible and one has to resort to truncated basis set. The question that arises then is how large is the angular momentum required in the basis set to satisfy the Bethe sum rule within the polarization propagator in the RPA approximation ... 

Finally - and equally important - Jens contribution to the formal treatment of GOS based on the polarization propagator method and Bethe sum rules has been shown to provide a correct quantum description of the excitation spectra and momentum transfer in the study of the stopping cross section within the Bethe-Bloch theory. Of particular interest is the correct description of the mean excitation energy within the polarization propagator for atomic and molecular compounds. This motivated the study of the GOS in the RPA approximation and in the presence of a static electromagnetic field to ensure the validity of the sum rules. 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

J. Schirmer. Beyond the random-phase approximation - a new approximation scheme for the polarization propagator, Phys. Rev. A, 26 2395-2416 (1982). 

Up to this point, our main concern was to reformulate the results of the LD ligand influence theory in the DMM form. Its main content was the symmetry-based analysis of the possible interplay between two types of perturbation substitution and deformation, controlled by the selection rules incorporated in the polarization propagator of the CLS. The mechanism of this interplay can be simply formulated as follows substitution produces perturbations of different symmetries which are supposed to induce transition densities of the same symmetries. In the frontier orbital approximation, only those densities among all possible ones can actually appear, which have the symmetry which enters into decomposition of the tensor product TH TL to the irreducible representations. These survived transition densities then induce the geometry deformations of the same symmetry. 

SOPPA Secondary order polarization propagator approximation... 

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