Propagator, polarization

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

If the P/Q operators are number conserving operators, the propagator is called a Polarization Propagator (PP). It may be viewed as the response of property P to perturbation Q. For the case where P = Q = r (the position operator), the propagator describes the response of the dipole moment (4fo r 4fo) to a linear field F = Ft. 

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 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. 

The contributions of the second order terms in for the splitting in ESR is usually neglected since they are very small, and in feet they correspond to the NMR lines detected in some ESR experiments (5). However, the analysis of the second order expressions is important since it allows for the calculation of the indirect nuclear spin-spin couplings in NMR spectroscoi. These spin-spin couplings are usually calcdated via a closed shell polarization propagator (138-140), so that, the approach described here would allow for the same calculations to be performed within the electron Hopagator theory for open shell systems. 

Polarization propagator for computing the polarizability per unit cell of polymers... 

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

POLARIZATION PROPAGATOR FOR COMPUTING THE POLARIZABILITY PER UNIT CELL OF POLYMERS... 

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... 

P(E) is called the irreducible polarization propagator, and W( ) the dielectrically screened interaction. 

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. 

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]... 

Exeitation energies are readily obtained as poles of a polarization propagator [37-40], whereas the transition moments are known as first-order non-adiabatie... 

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