Generalized random phase methods

At the fifth level of the Jacob s ladder classification, the full information of the KS orbitals is employed, i.e. not only the occupied but also the virtual orbitals are included. The formalism here becomes similar to those used in the random phase approximation (Section 10.9), but very little work has appeared on such methods. Inclusion of the virtual orbitals is expected to significantly improve on, for example, dispersion (such as van der Waals) interactions, which is a significant problem for almost all current functionals. 

Whether one should consider the OEP method as a density or wave functional theory is an open question, as it clearly tries to combine the best of both worlds. It has the advantage of being able to systematically improve the results towards the exact limit, but inherits also the wave function disadvantages of a slow convergence with respect to basis set size. 

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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). 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

In describing polarization propagator methods it is instructive to start out with the simplest consistent method of the kind, namely the random-phase approximation (RPA). Within the framework we use here, RPA is described as the approximation to the general equation of motion (Eq. (58)) in which we set h = hj and assume 0> = HF>, that is, use the simplest truncation in both Eqs (64) and (89). It is convenient to split hj up into p-h and h-p excitation operators... 

Although the possibility of the order-disorder transition was recognized in most of the block copolymer theories, it is Leibler who has expressedly addressed this problem. He derived the free energy of a block copolymer system in a series expanded in powers of the order parameter j denoting the deviation of the local density from the mean. The coefficients of this expansion up to the fourth ordef term were evaluated by a method which is a generalization of the random phase approximation method described above (Equation (16) was, in fact, derived as the second order term in the... 

The Pd closed-shell ground state (4d ° S) was treated by some further methods, comprising also relativistic ones a = 2.9 (coupled HF approximation 0 = 1.00073) [9], 3.1 (relativistic random-phase approximation (RPA) = generalization of coupled HF) [1], 4.928 (relativistic Hartree-Fock-Slater (HFS) = 2.086) [10]. 

We have used the basis set of the Linear-Muffin-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed the Vosko-Wilk-Nusair parametrization for the exchange-correlation energy density and potential. In conjunction with this we have treated the alloying effects for random and partially ordered phases with a multisublattice generalization of the coherent potential approximation (CPA). 

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

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