Phase approximation

Keller G 1986 Random-phase-approximation study of the response function describing optical second-harmonic generation from a metal selvedge Rhys. Rev. B 33 990-1009... 

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... 

Pordand cement clinker is formed by the reactions of calcium oxide and acidic components to give C S, C2S, C A, and a ferrite phase approximating... 

In fact, (3.22) is the usual stationary phase approximation, performed however for an infinitedimensional path integral, which picks up the trajectories with classical action S. Further, at fixed time t we take the integral over Xi again in the stationary phase approximation, which gives... 

The amount of inhibitor required to treat the free water, as given by Equation 4-3, plus the amount of inhibitor lost to the vapor phase and the amount that is soluble in the hydrocarbon liquid will be the total amount required. Figure 4-9 is a chart for determining the amount of methanol that will be lost to the vapor phase. Approximately 0.5% will be soluble in the hydrocarbon liquid. 

All three metals form a wide variety of binary chalcogenides which frequently differ both in stoichiometry and in structure from the oxides. Many have complex structures which are not easily described, and detailed discussion is therefore inappropriate. The various sulfide phases are listed in Table 22.4 phases approximating to the stoichiometry MS have the NiAs-type structure (p. 556) whereas MS2 have layer lattices related to M0S2 (p. 1018), Cdl2, or CdCl2 (p. 1212). Sometimes complex layer-sequences occur in which the 6-coordinate metal atom is alternatively octahedral and trigonal prismatic. Most of the phases exhibit... 

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. 

Coupled Hartree-Fock values within the Random Phase approximation... 

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.

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

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... 

Stirred tanks are modeled assuming that both phases are well mixed. Tray columns are usually modeled as well mixed on each tray so that the overall column is modeled as a series of two-phase, stirred tanks. (Distillation trays with tray efficiencies greater than 100% have some progressive flow within a tray.) When reaction is confined to a single, well-mixed phase, the flow regime for the other phase makes little difference but when the reacting phase approximates piston flow, the flow regime in the other phase must be considered. The important cases are where both phases approximate piston flow, either countercurrent or cocurrent. 

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]. 

Petke JD, Maggiora GM, Christoffersen RE (1992) Ab-initio configuration interaction and random phase approximation caclulations of the excited singlet and triplet states of uracil and cytosine. J Phys Chem 96 6992... 

Intestinal phase approximately 10% of gastric secretory response to a meal... 

Using the random phase approximation (RPA), the coherent scattering intensity Icoh(Q, t) of a polymer blend/solvent or a diblock copolymer/solvent system can... 

In the random phase approximation (RPA), that is, when the dephasing is rapid in the timescale under consideration, we have... 

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]... 

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