Correlation, static

There are two types of electron correlations static and dynamic. Static correlation refers to a near degeneracy of a given state a dynamic correlation refers to the in stantaneous avoidance of electrons with each other. 

RHF to UHF, or to a TCSCF, is almost pure static correlation. Increasing the number of configurations in an MCSCF will recover more and more of the dynamical correlation, until at the full Cl limit, the correlation treatment is exact. As mentioned above, MCSCF methods are mainly used for generating a qualitatively correct wave function, i.e. recovering the static part of the correlation. 

Those Warren-Cowley parameters have been determined in situ above the order-disorder transition temperature by diffuse neutron scattering. From these experimentally determined static correlations, the first nine effective pair interactions have been deduced using inverse Monte Carlo simulations. 

Let us note now, that the results obtained can be easily expressed via the static correlation function... 

Conventional presentaticsis of DFT start with pure states but sooner w later encounter mixed states and d sities (ensemble densities is the usual formulation in the DFT literature) as well. These arise, for example in formation or breaking of chemical bonds and in treatments of so-called static correlation (situations in which several different one-electron configurations are nearly degenerate). Much of the DFT literature treats these problems by extension and generalization from pure state, closed shell system results. A more inclusively systematic treatment is preferable. Therefore, the first task is to obtain the Time-Dependent Variational Principle (TDVP) in a form which includes mixed states. 

CASSCF wave function includes only the static correlation only a small number of electrons spanning frontier orbitals are correlated between them, while... 

The Dependencies of Radius of Gyration Rg, Static Correlation Length Hydrodynamic Screening Length Viscosity r, Self-Translational Diffusion Coefficient D, Cooperative Diffusion Coefficient Dc, Coupled Diffusion Coefficient Df, and Electrophoretic Mobility p on c and N for Various Regimes of Polyelectrolyte and Salt Concentrations... 

As shown in Ref. 48, l is intimately related to the static correlation length In infinitely dilute solutions is proportional to Rg. In semidilute solutions, is proportional to and respectively, in low and high salt limits. In... 

It is to be noted that is proportional to the static correlation length and the numerical prefactor is not unity, is proportional to and... 

The scattering function g k) is a function of static correlation length as given by Eqs. (225)-(227). For semidilute solutions at high salt concentrations, Dc follows from Eqs. (226) and (282) in the —> 0 limit. 

In infinitely dilute solutions we have > cxd and the static correlation = Rg/V3 as already discussed. In this limit, Dc of Eq. (175) is therefore recovered. On the other hand, in the Rouse regime, Eq. (283) yields... 

To summarize, the results presented for five representative examples of nonadiabatic dynamics demonstrate the ability of the MFT method to account for a qualitative description of the dynamics in case of processes involving two electronic states. The origin of the problems to describe the correct long-time relaxation dynamics as well as multi-state processes will be discussed in more detail in Section VI. Despite these problems, it is surprising how this simplest MQC method can describe complex nonadiabatic dynamics. Other related approximate methods such as the quantum-mechanical TDSCF approximation have been found to completely fail to account for the long-time behavior of the electronic dynamics (see Fig. 10). This is because the standard Hartree ansatz in the TDSCF approach neglects all correlations between the dynamical DoF, whereas the ensemble average performed in the MFT treatment accounts for the static correlation of the problem. 

In the previous section it has been shown that the temporal evolution of the pair correlations at the interchain peak is governed by the structural relaxation. If we move now towards more local scales - i.e. higher Q-values - we see that the static correlations observed in Spair(Q) correspond to pair correlations along a given chain. It is then natural to think that their time dependence might relate to dynamic processes other than the structural relaxation. 

Typically, scaling approaches are employed to explain the behavior in the semidilute regime. By examining static correlations near the temperature, Daoud and Jannick( ) have expressed the density-density correlation function in terms of a correlation length that is inversely proportional to concentration. Since the diffusion coefficient is inversely proportional to the correlation length it is directly proportional to the concentration. 

The above equation shows that C i) is governed by static correlations contained in ((5A ) ) and dynamical correlations contained in G f). I will discuss here the roles of both types of correlations, starting with the dynamical ones. 

C. The Role of Static Correlations in Solvation Dynamics and Approximations to C(t)... 

A surprising aspect of SD is how rapidly C i) in highly polar solvents decays relative to other relaxation processes such as reorientation of solvent dipoles. This very rapid time scale cannot be ascribed to dynamical solvent-solvent correlations, which, as illustrated in Fig. 6, are modest even for the longest ranged A . Thus the key to imderstanding the reasons for the rapid decay of C i) is in examining how solvent-solvent correlations contribute to it and to what extent their contributions can be accounted for in terms of static correlations measured by ((5A ) ), Eq. (32). The initial cmvature of C(t), which characterizes its short-time Gaussian-like behavior is often characterized in terms of the solvation frequency co o/v... 

A convenient measme of the the relative importance static correlations is the initial value of Qs(i) ... 

Equation (13) shows that the complete temperature and field dependence of the strains can be calculated from static correlation functions (J Jj )7-,h (y, y — 1.2,3 label the cartesian components of the angular momentum J) where O7- h denote thermal expectation values (Callen and Callen 1965). As already mentioned above, a mean field theory may be used to evaluate (13) and calculate the magnetostriction. 

Correlation effects in molecules are normally partitioned into near-degeneracy effects (static correlation) and dynamic correlation. Qualitatively they differ in the way they separate the electrons. Static correlation leads to a large separation in space of the two electrons in a pair, for example on two different atoms in a dissociation process. Dynamic correlation on the other hand deals... 

In the above expression, yf (f) is the short-time part that arises from the static correlations and r spp(t) is the long-time part that arises from the density mode contribution. 

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