Correlation, of electrons

The correlation of electron motion in molecular systems is responsible for many important effects, but its theoretical treatment has proved to be very difficult. Thus many quantum valence calculations use wave functions which are adjusted to optimize kinetic energy effects and the potential energy of interaction of nuclei and electrons but which do not adequately allow for electron correlation and hence yield excessive electron repulsion energy. This problem may be subdivided into cases of overlapping and nonoverlapping electron distributions. Both are very important but we shall concern ourselves here with only the nonoverlapping case. 

Brickstock, A., and Pople, J. A., Phil. Mag. 43, 1090, The spatial correlation of electrons in atoms and molecules. II. Two-electron systems in excited states. ... 

Lennard-Jones, J. E., J. Chem. Phys. 20, 1024, Spatial correlation of electrons in molecules. Study of spatial probability function using the single determinant. 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

Kamlet, M. J., M. E. Jones, J.-L. M. Abboud, and R. W. Taft. 1979. Linear Solvation Energy Relationships. Part 2. Correlations of Electronic Spectral Data for Aniline Indicators with Solvent tt and 3 Values. J. Chem. Soc., Perkins Trans. 2, 342. 

Green, J. P., and Kang, S. (1970) The correlation of electronic structures of indole derivatives with their biological activities. In Molecular Orbital Studies in Chemical Pharmacology, edited by L. B. Kier, pp. 105-120. Springer-Verlag, New York. 

A classical Hansch approach and an artificial neural networks approach were applied to a training set of 32 substituted phenylpiperazines characterized by their affinity for the 5-HTiA-R and the generic arAR [91]. The study was aimed at evaluating the structural requirements for the 5-HTiA/ai selectivity. Each chemical structure was described by six physicochemical parameters and three indicator variables. As electronic descriptors, the field and resonance constants of Swain and Lupton were used. Furthermore, the vdW volumes were employed as steric parameters. The hydrophobic effects exerted by the ortho- and meta-substituents were measured by using the Hansch 7t-ortho and n-meta constants [91]. The resulting models provided a significant correlation of electronic, steric and hydro-phobic parameters with the biological affinities. Moreover, it was inferred that the... 

There is also a hierarchy of electron correlation procedures. The Hartree-Fock (HF) approximation neglects correlation of electrons with antiparallel spins. Increasing levels of accuracy of electron correlation treatment are achieved by Mpller-Plesset perturbation theory truncated at the second (MP2), third (MP3), or fourth (MP4) order. Further inclusion of electron correlation is achieved by methods such as quadratic configuration interaction with single, double, and (perturbatively calculated) triple excitations [QCISD(T)], and by the analogous coupled cluster theory [CCSD(T)] [8],... 

For correlations in which cr° values are considered relevant, the ordinary am and ap should be used (Section III.C). Similarly for correlations of electron-demanding processes in which the use of cr+ values is appropriate for — R substituents, the ordinary am and ap values of NO2 should be used (Section VI.C). When the use of a values is appropriate for +R substituents, the value 1.25 may often be used for NO2 (Section III.D, but see also Section V.A). 

Simultaneous and correlated excitations of electrons in both molecules lead to correlation of electron motions and to a general net stabilization of the complex. This effect is usually attributed to the so-called attractive dispersion forces and the corresponding energy is therefore called dispersion energy zJ dis-... 

Comparison of forms of atomic fragments limited by the zero flux surfaces in ESP and electron density (Fig. 7) displays the role of different factors in the formation of the crystal structure. So in crystals with NaCl-type structure the exchange and correlation of electrons decrease the size of the cation and enlarge the size of the anion which leads to the structureforming interactions anion-anion in the (001) plane of the electron density maps. In ESP-maps the big cations and small anions are seen. 

Correlated Models. Models which take implicit or explicit account of the Correlation of electron motions. Moller-Plesset Models, Configuration Interaction Models and Density Functional Models... 

Antisymmetrization results in T vanishing, not only if two electrons with the same spin occupy the same orbital, but also if electrons with the same spin have the same set of spatial coordinates. Thus, antisymmetrization of tp results not only in the Pauli exclusion principle but also in the correlation of electrons of the same spin. 

As with the summaries of the other sections, we mention a number of calculation parameters or variables that have been demonstrated to be of critical importance for accurate prediction of aspects of the interactions. Symmetry constraints on the clusters have been shown to introduce arti-factual behavior. Corrections to account for the correlation of electrons have become essential in a calculation, and they must be incorporated self-consistently rather than as postoptimization corrections. Basis sets need to have the flexibility afforded by double- or triple-zeta functionality and polarization functions to reproduce known parameters most accurately. The choice of the model cluster and its size affect the acid strength, and the cluster must be large enough not to spatially constrain reactants or transition states. The choice of cluster is invariably governed by the available resources, but a small cluster can still perform well. Indeed, some of the... 

Figure 85 Correlation of electronic spectral energies and g values for [Cu(bipy)2X]Y complexes in a structural...

Pearson et al. (1952) employed this approach to derive a series of substituent parameters for electron-deficient reactions of substituted benzenes. These constants, designated as sigmae, were based on a study of the Beckmann rearrangement of -substituted acetophenone oximes. These authors considered the rates of the rearrangement reaction of the oximes to deviate from the Hammett eq. (1). It is pertinent that, with the sole exception of the yi-OMe group, the deviations were not major. The entropy of activation, AS, for the -anisyl derivative was, however, 20 e.u. different from the essentially constant values for the other substituents. To remedy the deviations, Pearson and his associates suggested the sigmae constants. It was indicated that these constants were more suitable for the correlations of electron-deficient reactions than the conventional cr-values. 

F. B. Brown and D. G. Truhlar, Chem. Phys. Lett., 117, 307 (1985). A New Semi-Empirical Method of Correcting Large-Scale Configuration Interaction Calculations for Incomplete Dynamic Correlation of Electrons. 

We will use the basis vectors (1) where > i2 and apply equation (4) when needed. For two tlu electrons, our basis (1) consists of 15 different state vectors I/) (for two holes, the fivefold hu degeneracy leads to 45 states). In the following we will study the intramolecular correlations of electrons (holes) within a multipole expansion of the two-body Coulomb potential V(r, f) = 1/lr —1 (charge e = unity). In terms of real spherical harmonics YJ, where r stands for m = 0,... 

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