Polarization propagator static

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. 

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. 

We saw in Section III that the polarization propagator is the linear response function. The linear response of a system to an external time-independent perturbation can also be obtained from the coupled Hartree-Fock (CHF) approximation provided the unperturbed state is the Hartree-Fock state of the system. Thus, RPA and CHF are the same approximation for time-independent perturbing fields, that is for properties such as spin-spin coupling constants and static polarizabilities. That we indeed obtain exactly the same set of equations in the two methods is demonstrated by Jorgensen and Simons (1981, Chapter 5.B). Frequency-dependent response properties in the... 

S.P.A. Sauer, J. Oddershede, Correlated Polarization Propagator Calculations of Static Dipole Polarizabilities, Int. J. Quantum Chem. 50 (1994) 317. 

The static and dynamic polarizability of the polyyne (C2 H2) series is treated in the TDHF and correlated second order polarization propagator methods by Dalskov et al.2i2 The calculated polarizabilities are extrapolated to the infinitely... 

In the case of the application of the polarization propagator method127 the static and dynamic chain length. Their saturation with a 3-21G basis was not reached even after 14 H2 molecules, though their values were close to the values belonging to those of an infinite chain (especially with increasing frequency ). 

For a polyyne chain142 the static al and dynamic a(— co)L polarizabilities have been computed using non-linear sequence transformations for the extrapolation and besides RPA the SOPPA (correlated second order polarization propagator approximation) method. In this way the authors have obtained for a C2 iH2 (polyyne) chain quite stable extrapolated values for both quantities. 

The complex polarization propagator method has been applied to the calculations of dipole-dipole dispersion coefficients of pyridine, pyrazine, and r-tetrazine. These calculations refer to the electronic ground states as well as the first excited states of 7t—>7t character <2004MI321>. Calculations of static and dynamic polarizabilities of excited states by means of DFT have been performed <2004JCP9795>. 

This equation, which is the perturbation theory expansion of an expectation value in the presence of a static field P in static response functions or polarization propagators, is another way of writing Eq. (3.36). Therefore, we can identify the static response function as the first derivative of the first-order correction to a perturbed expectation value, i.e. 

In the previous sections it was shown that frequency-dependent linear response prop>-erties, such as frequency-dependent polarizabilities, can be obtained as the value of the polarization propagator for the appropriate operators. Furthermore, all static second-order properties discussed in Chapters 4 and 5 can be calculated as the value of a polarization propagator for zero frequency. 

Table 13.2 Comparison of different polarization propagator (Dalskov and Sauer, 1998) and analytical derivative methods (McDowell et al., 1995) for the calculation of static dipole polarizabihties a (in units of e aoE ) using the medium-size polarized basis sets (Sadlej, 1988, 1991a Andersson and Sadlej,...

Figure 11 To the medium under investigation is rigidly attached the laboratory Cartesian reference system x, y, z, whereas to the molecule under consideration is attached the Cartesian system of axes x, Xj, Xj. The static electric field E, which polarizes the medium non-linearly, acts along the z-axis. The light wave which serves for analysing the induced non-linearity propagates along y...

The first contribution to the polarization induces a modification of the wave propagation in the material, for both its amplitude and phase, but without any frequency change. This phenomenon is known as the optical Kerr effect, by analogy with the magneto-optic and electro-optic Kerr effects where the medium refractive index varies proportionally with the square of the applied magnetic or electric static field. The second contribution corresponds to the third harmonics generation (THG). 

The unpolarized incident light beam is described as an electromagnetic wave consisting of two circularly polarized components of equal amplitude, propagating in the z-direction. The medium is subject to a static magnetic field in the same direction. We can write the electric field of the light as... 

---