Random phase approximation calculations

Random Phase Approximation Calculations of Gamow-Teller /3-Strength Functions in the A = 80-100 Region with Woods-Saxon Wave Functions... 

At a higher computational level, the Kirkwood model gave [M]d+ 11.8° for the enantiomer. At a still higher level, Random Phase Approximation calculations were used to obtain an extensive catalog of transitions for (R,R)-( —)-31, which reproduced the... 

K. J. >ilkanen, P. J. Stephens, P. Lazzeretti, and R. Zanasi, /. Pt s. Chem., 93,6583 (1989). Random Phase Approximation Calculations of Vibrational Circular Dichroism trans-2,3 Dideuteriooxirane. 

Kim JK, Kimishima K, Hashimoto T (1993) Random-phase approximation calculation of the scattering function for multicomponent polymer systems. Macromolecules 26 125-136... 

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. 

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... 

In order to calculate in the framework of Random Phase Approximation the intensity 1(6) of scattering at angle 6 of the incident radiation with wavelength X. recourse should be made to the formula [31]... 

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... 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

Many-body calculations which go beyond the Hartree-Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects superposition-of-configurations (or configuration interaction (Cl)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree-Fock (MCHF) approach, etc. However, to date only Cl and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions. 

Here q is a wavevector (eqn 1.6), ip(q) is the Fourier transform of />(r), and S(q) is the structure factor (Fourier transform of the two-point correlation function). The cubic term, ft, is zero for a symmetric system and otherwise may be chosen to be positive. The quartic term, y, is then positive to ensure stability. For block copolymers, these coefficients may be expressed in terms of vertex functions calculated in the random phase approximation (RPA) by Leibler (1980). The structure factor is given by... 

Lowenhaupt and Hellmann (1991) have determined whether microphase separation or macrophase separation occurs in blends of a PS-PMMA diblock with PMMA homopolymer with a < 1 and a > 1 using TEM. They found that the transition between purely microphase separation and macrophase separation occurs for a lower diblock content for blends with a smaller a, as supported by calculations of the instability limit using the random phase approximation. Blends with a < 1.4 were always initially microphase separated, although in a blend with a - 1.4 this was followed by macrophase separation. However, the macrophase-separated structure took the form of aggregates of micelles (see Fig. 6.1), suggesting a nucleation and growth mechanism for the secondary... 

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.

To account for the interchannel coupling, or, which is the same, electron correlation in calculations of photoionization parameters, various many-body theories exist. In this paper, following Refs. [20,29,30,33], the focus is on results obtained in the framework of both the nonrelativistic random phase approximation with exchange (RPAE) [55] and its relativistic analogy the relativistic random phase approximation (RRPA) [56]. RPAE makes use of a nonrelativistic HF approximation as the zero-order approximation. RRPA is based upon the relativistic Dirac HF approximation as the zero-order basis, so that relativistic effects are included not as perturbations but explicitly. Both RPAE and RRPA implicitly sum up certain electron-electron perturbations, including the interelectron interaction between electrons from... 

A comparison between experimental and theoretical values for the J (2p) parameter in neon is shown in Fig. 2.14. (The corresponding comparison between experimental and theoretical values for the partial cross section experimental data are given by the solid curve surrounded by a hatched area which takes into account the error bars. Theoretical results from advanced photoionization theories (many-body perturbation theory, R-matrix theory, and random-phase approximation) are represented by the other lines, and they are in close agreement with the experimental data (for details see [Sch86]). The theoretical / (2p) data of Fig. 2.13 are also close to the experimental values, except in the threshold region. 

Next the results from the relativistic random-phase approximation (RRPA) and the many-body perturbation theory (MBPT), also shown in Table 5.1, will be discussed. Because both calculations include basically the same electron-electron interactions, rather good agreement exists, and it is sufficient to concentrate only on the RRPA model. 

A different analysis applies to the LR approach (in either Tamm-Dancoff, Random Phase Approximation, or Time-dependent DFT version) where the excitation energies are directly determined as singularities of the frequency-dependent linear response functions of the solvated molecule in the ground state, and thus avoiding explicit calculation of the excited state wave function. In this case, the iterative scheme of the SS approaches is no longer necessary, and the whole spectrum of excitation energies can be obtained in a single run as for isolated systems. 

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