Optimized random phase approximation

Whether these requirements can be met depends on the model considered and on the closure relation involved for the calculation of the correlation functions. Examples for which Eq. (7.54) has actually been used pertain to the class of simple QA systems, that is, QA systems with no rotational degree of freedom where the interaction potentials contain a spherical hard-core contribution plus (at most) an attractive perturbation. For such sj stems, the free energy has been calculated on the basis of correlation functions in the mean sphericfxl approximation (or an optimized random-phase approximation) [114, 298). 

The LOGA is identical to the optimized random phase approximation (ORPA) of Andersen and Chandler. The optimized v they consider is defined by using (2.138) for rd. 

H. C. Andersen, D. Chandler, and J. D. Weeks, Roles of repulsive and attractive forces in liquids The optimized random phase approximation, J. Chem. Phys. 56, 3812 (1972). 

An alternative theory for describing systems that contain strong repulsive and attractive interactions is the recent range optimized random phase approximation (RO-RPA) of Donley, Heine, and Wu [147]. This theory is much closer in spirit to RISM and PRISM, yet, for the cases examined to date, seems to describe well the properties of polyelectrolyte solutions at high charge densities and interaction energies while also handling attractive interactions. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

Random Phase Approximation (RPA) method, SINDO modelT84 Synchronous reaction, 356 Updated Hessian, in optimization methods. 

A proof of Eq. (14) is given in several places [27, 66-68]. The advantage of this equation is that it can be used to systematically improve the approximate Hamiltonian and the free energy F by optimizing a set of variable parameters or functions contained in Hq. Specific forms chosen for lead to the Self-Consistent Phonon (SCP) method, the Mean-Field (MF) method and the Time-Dependent Hartree (TDH) or Random-Phase Approximation (RPA). 

To find an approximation to the optimal control function we collect all successful realizations (qfc(t),qfc(t), J esc(f)) that move it from the CA to 0fl. An approximate solution u(t) is then found as an ensemble average over the corresponding realizations of the random force l c c(t)) (the exact solution is u t) = lim/) o u t)). The results of this procedure are shown in the upper trace of Fig. 18. To remove the irrelevant high-frequency component left after averaging, we filtered through a zero-phase low-pass filter with frequency cutoff coc = 1.9. 

Apb is the scattering length density difference, Q is Porod s invariant, and Y the mean chord length. For the calculation of Yo(r) we approximated I(q) hy a cubic spline. The equations used for the calculation of " pore and " soUd are to be found in [8,30,39-41,47]. Analytical expressions for the descriptors of RES were published in [10,11,13,42,43]. In its most simple variant, the stochastic optimization procedure evolves the two-point probability S2 (r) of a binary representation of the sample towards S2(r) by randomly excWiging binary ceUs of different phases, starting from a random configuration which meets the preset volume fractions. After each exchange the objective function... 

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