Random phase approximation , for

In this contribution, we have shown that the Bethe sum rule, like the Thomas-Reiche-Kuhn sum rule, is satisfied exactly in the random phase approximation for a complete basis. Thus, in calculations that are related to the generalized oscillator strengths of a system, the Bethe sum rule may be used as an indicator of completeness of the basis set, much as the Thomas-Reiche-Kuhn... 

Geldart DJW (1964) Corrections to the Random Phase Approximation for an Interacting Electron Gas. Thesis, McMaster University, Hamilton... 

Fig. 6.15 Spinodal lines calculated using the random phase approximation for macrophase separation (solid lines) and microphase separation (dashed lines) for blends of a PS-PI diblock (Af = 100kg mol-1./PS = 0.46) with homopolymers with M /kg mol 1 = (a) 62, (b) 200, (c) 580 (Koizumi et al. 1992). For the blends with a 1, macrophase separation occurs first on lowering the temperature (increasing jV) for most compositions.

Appendix A Multicomponent Random Phase Approximation for Homopolymer... 

Yan ZD, Perdew JP, Kurth S (2000) Density functional for short-range correlation Accuracy of the random-phase approximation for isoelectronic energy changes, Phys Rev B, 61 16430-16439... 

It is reasonable to attempt an interpretation of depolarization for the state 2Pm within the energetically-isolated-state approximation. As the contribution to or 3/2) comes from large impact parameters, one may assume a rectilinear trajectory ( = 0) and use the random-phase approximation for R < Rm,Rm being the matching distance. The value of Rm itself is determined by the condition that the Coriolis coupling between molecular terms A% and is equal to the adiabatic splitting U B ) — t/04f). From equation (34) on the condition that AK Ae, we get U = U(B ) — U(A%) = AK(R), and using the polarization interaction equation (36) we finally obtain... 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

Balsara et al. [219] studied by SANS the thermodynamic behavior of a blend of AB-diblock with A and B homopolymers in the homogeneous disordered melt. They found the random phase approximation for multicomponent systems [220,221] to work for their systems of polyolefines, where only van-der-Waals interactions are present. 

Sakurai S, Mori K, Okawara A, Kimishima K, Hashimoto T (1992) Evaluation of segmental interaction by small-angle X-ray scattering based on the random-phase approximation for asymmetric, polydisperse tiiblock copolymers. Macromolecules 25 2679-2691... 

A second method is to use a perturbation theory expansion. This is formulated as a sum-over-states algorithm (SOS). This can be done for correlated wave functions and has only a modest CPU time requirement. The random-phase approximation is a time-dependent extension of this method. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

RPA (random-phase approximation) ah initio method used for computing nonlinear optical properties... 

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. 

In the random phase approximation, the transition amplitude from state 0) to l) for any one electron operator O may be written as... 

Lohse et al. have summarized the results of recent work in this area [21]. The focus of the work is obtaining the interaction parameter x of the Hory-Huggins-Stavermann equation for the free energy of mixing per unit volume for a polymer blend. For two polymers to be miscible, the interaction parameter has to be very small, of the order of 0.01. The interaction density coefficient X = ( y/y)R7 , a more relevant term, is directly measured by SANS using random phase approximation study. It may be related to the square of the Hildebrand solubility parameter (d) difference which is an established criterion for polymer-polymer miscibility ... 

Saue and Jensen used linear response theory within the random phase approximation (RPA) at the Dirac level to obtain static and dynamic dipole polarizabilities for Cu2, Ag2 and Au2 [167]. The isotropic static dipole polarizability shows a similar anomaly compared with atomic gold, that is, Saue and Jensen obtained (nonrelativ-istic values in parentheses) 14.2 for Cu2 (15.1 A ), 17.3 A for Ag2 (20.5 A ), and 12.1 A for Au2 (20.2 A ). They also pointed out that relativistic and nonrelativistic dispersion curves do not resemble one another for Auz [167]. We briefly mention that Au2 is metastable at 5 eV with respect to 2 Au with a barrier to dissociation of 0.3 eV [168, 169]. 

Diblock copolymers represent an important and interesting class of polymeric materials, and are being studied at present by quite a large number of research groups. Most of the scientific interest has been devoted to static properties and to the identification of the relevant parameters controlhng thermodynamic properties and thus morphologies [257-260]. All these studies have allowed for improvements to the random phase approximation (RPA) theory first developed by Leibler [261]. In particular, the role of the concentration fluctuations, which occur and accompany the order-disorder transition, is studied [262,263]. 

The exact high- and low-density limits can be found from arguments given in Ref. [57]. For = 0 in the high-density limit (where the random phase approximation becomes exact), the parallel-spin and anti-parallel-spin correlation energies are equal [57], so... 

The vertical IPs of CO deserve special attention because carbon monoxide is a reference compound for the application of photoelectron spectroscopy (PES) to the study of adsorption of gases on metallic surfaces. Hence, the IP of free CO is well-known and has been very accurately measured [62]. A number of very efficient theoretical methods specially devoted to the calculation of ionization energies can be found in the literature. Most of these are related to the so-called random phase approximation (RPA) [63]. The most common formulations result in the equation-of-motion coupled-cluster (EOM-CC) equations [59] and the one-particle Green s function equations [64,65] or similar formalisms [65,66]. These are powerful ways of dealing with IP calculations because the ionization energies are directly obtained as roots of the equations, and the repolarization or relaxation of the MOs upon ionization is implicitly taken into account [59]. In the present work we remain close to the Cl procedures so that a separate calculation is required for each state of the cation and of the ground state of the neutral to obtain the IP values. 

Explicit calculations of B Ano) were first carried out by Ma and Brueckner [12] and by Sham [13] for the correlation and exchange contributions, respectively, in the high density limit r < 1). The evaluation of the required Feynman graphs in the metallic and intermediate density range and the extension to include iterations of the scattering processes was given in a self-consistent random phase approximation [17, 18]. The results can be expressed as... 

In the 1950s, many basic nuclear properties and phenomena were qualitatively understood in terms of single-particle and/or collective degrees of freedom. A hot topic was the study of collective excitations of nuclei such as giant dipole resonance or shape vibrations, and the state-of-the-art method was the nuclear shell model plus random phase approximation (RPA). With improved experimental precision and theoretical ambitions in the 1960s, the nuclear many-body problem was born. The importance of the ground-state correlations for the transition amplitudes to excited states was recognized. 

---