The Polarization Propagator

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

Although eq. (10.103) for the propagator appears to involve the same effort as the perturbation approach (sum over all excited states, eq. (10.18)), the actual calculation of the propagator is somewhat different. Returning to the time representation of the polarization propagator, it may be written in terms of a commutator. 

The main quantity providing the dynamic longitudinal polarizability of closed-shell infinite periodic systems is the polarization propagator which at the RPA level takes the form [23-25] ... 

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.

Since the excited levels are determined by searching the poles of the polarization propagator, the poles corresponding to very small dipole transition strengths are difficult to identify. This is particularly the case for the highest energy poles and, consequently, it is not possible to determine by such procedure the top of the conduction band. 

When Jens Oddershede was elected a Fellow of the American Physical Society in 1993, the citation read For contribution to the theory, computation, and understanding of molecular response properties, especially through the elucidation implementation of the Polarization Propagator formalism. Although written more than a decade ago, it is still true today. The common thread that has run through Jens work for the past score of years is development of theoretical methods for studying the response properties of molecules. His primary interest has been in the development and applications of polarization propagator methods for direct calculation of electronic spectra, radiative lifetime and linear and non-linear response properties such as dynamical dipole polarizabilities and... 

Excited triplet states n) with energy E are included in the sum for the FC and SD terms, while excited singlet states contribute to the OP term. Recalling the spectral representation of the polarization propagator for zero frequency w [60]... 

From the Orbital Implementation of the Kinetic Theory to the Polarization Propagator Method in the Study of Energy Deposition Problems... 

Orbital Implementation of the Kinetic Theory and the Polarization Propagator Method to equation (5) was cast in terms of CAB contributions, i.e.,... 

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. 

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

The above comprises the derivation of the expression for the PES of the complex system which is not only free from the necessity to recalculate the wave function of the classical subsystem in each point, but formally not requiring any wave function of the M-system at all, since the result is expressed in terms of the generalized observables - one-electron Green s functions and the polarization propagator of the free M-system. Reality is of course more harsh as the necessary quantities must be known for a system we know too little about, except the initial assumption that its orbitals do exist. Section 3.5 will be devoted to reducing this uncertainty. 

It is remarkable that the supermatrix A 1 is nothing [42] but the polarization propagator II for the CLS subsystem calculated for the symmetric molecule. With this we get ... 

As mentioned previously, the specifics of the central atoms in CCs are determined by the structure of the supermatrix II, which is in its turn predefined by the structure of the carrier space of the CLS group and by the number of electrons in it. Indeed, the supermatrix II of the polarization propagator is particularly simple in the basis of the eigenstates of the Fock operator Fq. Its matrix elements then are ... 

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. 

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