Free-energy correlations diagrams

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

The first thing we have to do is to establish the anchor points of the correlation diagram, that is to say the relative positions of the spectroscopic states under the extreme conditions of an infinitely weak interaction and of an infinitely strong interaction. Consider an ion with a d electronic configuration, the states of the free ion are, in order of increasing energy, F, P, 1G, and 1S. This order and the precise... 

A correlation diagram for a d ion (e.g. Vt+) in an octahedral environment is shown in Fig. 12-7.2. What this diagram does is to demonstrate how the energy levels of the free ion behave as a function of the strength (A0) of the ion s interaction with a set of octahedrally disposed ligands. 

For the free energy, disconnected vertices first contribute in two loop order (Fig. 5.20c). For correlation functions, however, disconnected vertices may contribute even to one loop order, and a calculation ignoring such contributions by keeping only naively connected diagrams would yield wrong results. 

The equations described earlier contain two unknown functions, h(r) and c(r). Therefore, they are not closed without another equation that relates the two functions. Several approximations have been proposed for the closure relations HNC, PY, MSA, etc. [12]. The HNC closure can be obtained from the diagramatic expansion of the pair correlation functions in terms of density by discarding a set of diagrams called bridge diagrams, which have multifold integrals. It should be noted that the terms kept in the HNC closure relation still include those up to the infinite orders of the density. Alternatively, the relation has been derived from the linear response of a free energy functional to the density fluctuation created by a molecule fixed in the space within the Percus trick. The HNC closure relation reads... 

Fig. 3. Schematic correlation diagram showing excitation energies for Co " in various environments. Left to right free ion ion in point charge crystal field energies from cluster calculation in bulk site energies from cluster calculations in surface site experimental bulk excitation energies. The inset shows schematic one-electron 3d energy levels in bulk and surface sites. Adapted from ref. 91.

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