Higher-order correlations

The thennodynamic properties of a fluid can be calculated from the two-, tln-ee- and higher-order correlation fiinctions. Fortunately, only the two-body correlation fiinctions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy... 

Eq. (5) is useful when analyzing different approximations in the theory of inhomogeneous fluids. In particular, if all the terms involving third- and higher-order correlations in the right-hand side of Eq. (5) are neglected, and if Pi(ro))P2( o)i )Pv( o) are chosen as the densities of species for a uniform system at temperature T and the chemical potentials p,, the singlet hypemetted chain equation (HNCl) [50] results... 

Notice that in this general case, correlation functions cannot be solved for directly instead, there is an entire hierarchy of lower-order correlations expressed as functions of higher-order correlations. For example if we take an average of equation 7.79 over all space-time histories, and assume that we have a steady-state so... 

P being multiplicative over P(Jk,Jk-i,t)- Higher order correlation functions (rim=i A. )) are found by analogy. 

In this generalized oscillator equation, the frequency is related to the restoring force acting on a particle and Q is a friction constant. The key quantity of the theory is the memory kernel mq(l — t ), which involves higher order correlation functions and hence needs to be approximated. The memory kernel is expanded as a power series in terms of S(q, t)... 

The full characterization of the stochastic properties of a surface requires consideration of higher order correlations of the height function. However, it can be difficult to construct surfaces in this manner without experimental input. As an approximation, it may be reasonable to neglect the higher order terms. 

The inclusion of both three and four-particle correlations in nuclear matter allows not only to describe the abundances oft, h, a but also their influence on the equation of state and phase transitions. In contrast to the mean-field treatment of the superfluid phase, also higher-order correlations will arise in the quantum condensate. 

Other variants of G3 theory have been proposed that use alternate geometries, zero-point energies, or higher-order correlation methods. G3... 

The Dunning cc-pVnZ basis sets can be used with our PNO extrapolations to form a potent new combination. We shall consider the SCF energy first, then the MP2 correlation energy, and finally higher-order correlation energy through CCSD(T). 

The determination of electron affinities (EAs) is one of the most serious problems in quantum chemistry. While the Hartree-Fock electron affinity can be easily evaluated, most anions turn out to be unbound at this level of theory. Thus, the correlation effects are extremely crucial in evaluating EAs. At this point, lithium hydride and lithium hydride anion make up a very good benchmark system because they are still small enough yet exhibit features of more complicated systems. Four and five electrons, respectively, give rise to higher-order correlation effects that are not possible in H2. 

In two-site systems, there is only one correlation function which characterizes the cooperativity of the system. In systems with more than two identical sites, for which additivity of the higher-order correlations is valid, it is also true that the pair correlation does characterize the cooperativity of the system. This is no longer valid when we have different sites or nonadditivity effects. In these cases there exists no single correlation that can be used to characterize the system, hence the need for a quantity that measures the average correlation between ligands in a general binding system. There have been several attempts to define such a quantity in the past. Unfortunately, these are valid only for additive systems, as will be shown below. 

By a straightforward generalization we can write the triplet and higher-order correlations. For instance, the triplet correlation 2 3( i = ot> P Y) obtained from the corresponding triplet probability... 

For higher-order correlations, the particular correlation depends on m and on whether the chain is open or closed. For instance, when m = 3... 

The dependence of the higher-order correlations on h, K, and T is more complicated. One example has already been examined in Section 5.5. We shall not examine this aspect here. 

We turn now to the finite open and closed chain and compare the pair correlations obtained in the different systems. First, we note that in the m —> °o limit all the sites become identical in the weak sense, i.e., there is only one intrinsic binding constant, but different pair (and higher-order) correlations as shown in Eq. (7.4.28). It should be noted, however, that owing to the translational invariance of the infinite system there is only one nn pair correlation, only one second nn pair correlation, etc. In other words, it does not matter where in the chain we choose the pair of nn neighbors, or the second nn neighbors, etc. This translational invariance is lost in the finite open system. 

These are actually the first members of a coupled hierarchy of equations for the successively higher order correlations. 

For a 0, Eqs. (8) and (9) for the mean values qf contain terms in higher-order correlations, a familiar and frequently frustrating type of event in statistical mechanics. However, as shown in the Appendix, when the system is not too far from equilibrium (i.e., small perturbations only) the terms (a (a, Oj) may be replaced by < j)d l, when d is the dimensionality of the system. 

These higher-order correlation functions play a large role in determining many physical properties of polyatomic systems. For example, the vibrational relaxation can, in some cases, be expressed in terms of the rotational kinetic energy autocorrelation function.27... 

Likewise all higher-order correlation functions, can also be computed. 

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