Correlation term

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself... 

The first term on the right-hand side is a contribution from external fields, usually zero. The second term is the contribution from the kinetic energy and the nuclear attraction. The third term is the Coulomb repulsion between the electrons, and the final term is a composite exchange and correlation term. 

We often split the exchange-correlation term into a sum of one part for exchange effects and one part for correlation effects. 

This fitting does not reduce the formal scaling, since the exchange-correlation term already is of order M. ... 

Ffactor The vapor kinetic-energy parameter, often used as a correlating term for flooding velocity, foam density, etc. 

We note that the virial theorem is automatically fulfilled in the Hartree-Fock approximation. This result follows from the fact that the single Slater determinant (Eq. 11.38) built up from the Hartree-Fock functions pk x) satisfying Eq. 11.46 is the optimum wave function of this particular form, and, since this wave function cannot be further improved by scaling, the virial theorem must be fulfilled from the very beginning. If we consider a stationary state with the nuclei in their equilibrium positions, we have particularly Thf = — Fhf, and for the correlation terms follows consequently that... 

Concepts of Coherence (1.18-21) The correlation term in the expression for the intensity of superimposed helds, and in general the ergodic mean of the product of held amplitudes sampled at different points in space and time, is called mutual intensity. If the conditions are temporally stationary - which they would be if the observed sources are stable with time - it is the time difference r which matters. 

We are now ready to derive an expression for the intensity pattern observed with the Young s interferometer. The correlation term is replaced by the complex coherence factor transported to the interferometer from the source, and which contains the baseline B = xi — X2. Exactly this term quantifies the contrast of the interference fringes. Upon closer inspection it becomes apparent that the complex coherence factor contains the two-dimensional Fourier transform of the apparent source distribution I(1 ) taken at a spatial frequency s = B/A (with units line pairs per radian ). The notion that the fringe contrast in an interferometer is determined by the Fourier transform of the source intensity distribution is the essence of the theorem of van Cittert - Zemike. 

Development in recent years of fast-response instruments able to measure rapid fluctuations of the wind velocity (V ) and of fhe tracer concentration (c ), has made it possible to calculate the turbulent flux directly from the correlation expression in Equation (41), without having to resort to uncertain assumptions about eddy diffusivities. For example, Grelle and Lindroth (1996) used this eddy-correlation technique to calculate the vertical flux of CO2 above a foresf canopy in Sweden. Since the mean vertical velocity w) has to vanish above such a flat surface, the only contribution to the vertical flux of CO2 comes from the eddy-correlation term c w ). In order to capture the contributions from all important eddies, both the anemometer and the CO2 instrument must be able to resolve fluctuations on time scales down to about 0.1 s. 

Goh, S. K., Gallant, R. T., St-Amant, A., 1998, Towards Linear Scaling for the Fits of the Exchange-Correlation Terms in the LCGTO-DF Method via a Divide-and-Conquer Approach , Int. J. Quant. Chem., 69, 405. 

Finally, in the third generation of DFT schemes, a portion of the exact Hartree-Fock (HF) exchange energy is mixed to the DFT exchange-correlation term, using the adiabatic con-... 

Since turbulent fluctuations not only occur in the velocity (and pressure) field but also in species concentrations and temperature, the convection diffusion equations for heat and species transport under turbulent-flow conditions also comprise cross-correlation terms, obtained by properly averaging products of... 

For the correlation terms the gradient corrections given by Perdew [39] are used, with the Vosko parameterization of the local part. 

Once the coefficients for the expansion of the exchange-correlation term have been evaluated, all matrix elements can be calculated analytically. The Obara and Saika [47] recursive scheme has been used for the evaluation of the one and the two electron integrals. The total energy is therefore expressed in terms of the fitting coefficients for the electronic density and the exchange-correlation potential. 

An alternative procedure consists in using a numerical integration scheme to evaluate the exchange-correlation contribution. In this case, no auxiliary basis set is needed for the exchange-correlation terms, and numerically more reliable results can be obtained. 

In homogeneous, locally isotropic turbulence the velocity-gradient correlation terms in (2.127) simplify to (Pope 2000)... 

For example, the vortex-stretching term is a triple-correlation term that corresponds to the rate at which dissipation is created by spectral energy passing from the inertial range to the dissipative range of the energy spectrum (see (2.75), p. 43). Letting /cdi 0.1 denote... 

The triple-correlation term (u xix/)") and the molecular-transport term T -, defined by... 

At this point, the next step is to decompose the velocity into its mean and fluctuating components, and to substitute the result into the left-hand side of (6.42). In doing so, the triple-correlation term (UiUjUk) will appear. Note that if the joint velocity PDF were known (i.e., by solving (6.19)), then the triple-correlation term could be computed exactly. This is not the case for the RANS turbulence models discussed in Section 4.4 where a model is required to close the triple-correlation term. 

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