Correlation function Terms Links

Linear response theory10 provides a link between the phenomenological description of the kinetics in term of reaction rate constants and the microscopic dynamics of the system [33]. All information needed to calculate the reaction rate constants is contained in the time correlation function... 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

Equation (55), expressing the conditional probability P B,t A, 0) in terms of a time correlation function, now permits to link the phenomenological and microscopic descriptions. By equating (51) and (55) and by noting that Nb)/N = (hfi) we find that... 

Silverstone and Sinanoglu [77] wrote the cluster expansion of fhe nonrela-f ivisfic N-elecf ron eigenfunction in terms of a zero-order reference wavefunc-fion fhaf is multicmfigurational, in accordance wifh the earlier suggestion of Wafson [31] and the study of H-ND in Be [75, 76]. In their formalism, the one-, two-, three-, etc. correlation functions (i.e., the virtual electron-excitations in the language of Cl) are linked fo spin orbitals from an extended zero-order set of occupied and unoccupied spin orbitals. This set was named the Hartree-Fock sea (H-F sea). Optimally, the H-F sea spin orbitals are supposed to be computed self-consistently. 

The scattering of radiation by particles is described in terms of an experimental structure factor S(K, o>), where K and wave-vector and angular frequency, respectively. The structure factor S(K, (o) is linked to the van Hove correlation function by a Fourier transformation ... 

Measurements [40, 41] obtained on radiation cross-linked polystyrene gels subsequently swollen in cyclohexane at different temperatures exemplify the type of results obtained. A typical correlation function is shown (Figure 12.13) in which the ordinate axis was calibrated directly in terms of osmotic modulus using data obtained by Scholte [42] from ultracentrifugation analysis of polystyrene solutions. Scaling relationships can be used to interpret the dependence of and on... 

In the parametrization of equ. (4.68) the terms associated with the Legendre polynomials Pk(cos ab) represent that part of the angular correlation which is independent of the light beam, while the terms associated with the bipolar harmonics are due to the multipole expansion of the interactions of the electrons with the electric field vector. The link between geometrical angular functions and dynamical parameters is made by the summation indices ku k2 and k. These quantities are related to the orbital angular momenta of the two individual emitted electrons, and they are subject to the following conditions ... 

This is the point at which to demonstrate the intimate connection between density functional and density matrix theory. The link will be forged by expressing the exchange-correlation potential VXc(r) of DFT in terms of the 1 and 2 DMs of the atom, molecule, or finite cluster under consideration. The result for Vxc r) in terms of low-order DMs is due to Holas and March [52],... 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

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