Correlation functions static correlations

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

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. 

Often such correlation functions are time dependent, and measure how the correlation between two quantities changes over time. They may be normalized by the corresponding static (i.e. t = to) limit. 

The shape of the static contour is readily calculated from Eq. (2.13) with the correlation function... 

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

Accounting for electron correlation in a second step, via the mixing of a limited number of Slater determinants in the total wave function. Electron correlation is very important for correct treatment of interelectronic interactions and for a quantitative description of covalence effects and of the structure of multielec-tronic states. Accounting completely for the total electronic correlation is computationally extremely difficult, and is only possible for very small molecules, within a limited basis set. Formally, electron correlation can be divided into static, when all Slater determinants corresponding to all possible electron populations of frontier orbitals are considered, and dynamic correlation, which takes into account the effects of dynamical screening of interelectron interaction. 

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

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. 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

Westlund developed also a theory for PRE in the ZFS-dominated limit for S = 1, which included a stringent Redfield-limit approach to the electron spin relaxation in this regime (118). Equations (35) and (38) were used as the starting point also in this case. Again, the correlation function in the integrand of Eq. (38) was expressed as a product of a rotational part and the spin part. However, since it is in this case appropriate to work in the principal frame of the static ZFS, the rotational part becomes proportional to exp(—t/3tb) (if Tfl is the correlation time for reorientation of rank two spherical harmonics, then 3t is the correlation time for rank one spherical... 

The electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

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. 

There is an important case which is intermediate between small bounded systems and macroscopic fully extended systems, namely the description of the surface region of a macroscopic metal. The correlation functions which describe density fluctuations in the surface region are extremely anisotropic and of long range, very unlike their counterparts in the bulk, and the thermodynamic limit must be taken with sufficient care. Consider the static structure factor for a large system of N particles contained within a volume Q,... 

Consequently, discontinuities in certain correlation functions are not uncommon in the thermodynamic limit. Other examples are known. For example, Kirzhnits made a similar point concerning the static dielectric function [6]. The mathematical reason why such discontinuities are not prohibited is that the commutation rule, [JV, H] = 0, becomes meaningless in the thermodynamic limit. The reader is referred to the literature for additional discussion [7, 8]. 

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. 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

In this section some details of the static and dynamic structure factors and on the first cumulant of the time correlation function are given. Hie quoted equations are needed before the cascade theory can be applied. This section may be skipped on a first reading if the reader is concerned only with the application of the branching theory. 

For the discussion of the properties of the static structure factors, it is often more convenient to write the scattering functions in terms of a space correlation function y(r)4wr2dr. 

Some experiments only measure the power spectrum of correlation functions over very restricted frequency ranges. Hence, the correlation functions themselves cannot be reconstructed from the experimental data. This is the case in static transport coefficient measurements where only the power spectra of specific correlation functions at zero frequency are measured.12... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

The phase space correlation function can be expressed in terms of Maxwellian distribution and static pair correlation function and can be written as... 

Mode coupling theory provides the following rationale for the known validity of the Stokes relation between the zero frequency friction and the viscosity. According to MCT, both these quantities are primarily determined by the static and dynamic structure factors of the solvent. Hence both vary similarly with density and temperature. This calls into question the justification of the use of the generalized hydrodynamics for molecular processes. The question gathers further relevance from the fact that the time (t) correlation function determining friction (the force-force) and that determining viscosity (the stress-stress) are microscopically different. 

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