Correlation corrections

The orbitals from which electrons are removed and those into which electrons are excited can be restricted to focus attention on correlations among certain orbitals. For example, if excitations out of core electrons are excluded, one computes a total energy that contains no correlation corrections for these core orbitals. Often it is possible to so limit the nature of the orbital excitations to focus on the energetic quantities of interest (e.g., the CC bond breaking in ethane requires correlation of the acc orbital but the 1 s Carbon core orbitals and the CH bond orbitals may be treated in a non-correlated manner). 

A variety of theoretical methods have been developed which include some effects of electron correlation. Traditionally, such methods are referred to as post-SCF methods because they add correlation corrections to the basic Hartree-Fock model. As of this writing, there are many correlation methods available in Gaussian, including the following ... 

The ab initio methods used by most investigators include Hartree-Fock (FFF) and Density Functional Theory (DFT) [6, 7]. An ab initio method typically uses one of many basis sets for the solution of a particular problem. These basis sets are discussed in considerable detail in references [1] and [8]. DFT is based on the proof that the ground state electronic energy is determined completely by the electron density [9]. Thus, there is a direct relationship between electron density and the energy of a system. DFT calculations are extremely popular, as they provide reliable molecular structures and are considerably faster than FFF methods where correlation corrections (MP2) are included. Although intermolecular interactions in ion-pairs are dominated by dispersion interactions, DFT (B3LYP) theory lacks this term [10-14]. FFowever, DFT theory is quite successful in representing molecular structure, which is usually a primary concern. 

Hurley, A. C., Proc. Phys. Soc. A69, 49, (i) On the method of atoms in molecules. II. An intra-atomic correlation correction." Modification of Moffitt s method. 

The ab initio calculations of various three-electron hemibonded systems [122, 123] indicated that the inclusion of electron correlation corrections is extremely important for the calculation of three-electron bond energies. The Hartree-Fock (HF) error is found to be nonsystematic and always large, sometimes of the same order of magnitude as the bond energy. According to valence bond (VB) and MO theories, the three-electron bond is attributed to a resonance between the two Lewis structures... 

When is an eigenvalue of r(.B),. E is a pole. The corresponding operator, r(JS), is nonlocal and energy-dependent. In its exact limit, it incorporates all relaxation and differential correlation corrections to canonical orbital energies. A normalized DO is determined by an eigenvector of T Epou) according to... 

The inclusion of the correlation corrections to the spin-spin coupling calculation via electron propagator is quite straightforward since at third order it can be written as... 

Our results fit also with a previous investigation (9) on polyenes based on a version of the 2h-lp Cl scheme restricted to the virtual one-electron states generated by a minimal basis. In our case, however, the fragmentation of lines into satellites is much more pronounced. The reason lies in the size-consistency of the ADC[3] approach (as contrasted with the size-inconsistency of any truncated form of Cl (27d), in the full handling of the virtual space, and (10) in the inclusion of correlation corrections to the reference ground state, leading to (37) a net reduction of the quasi-particle band gap of conjugated polymers. 

In this paper we present the relative energies of the isomers of the phenylenediamines, dihydroxybenzenes and difluorobenzenes from ab initio calculations using large basis sets and including correlation corrections at the MP-2 level. These calculations were done at the geometry optimized structures. We also include zero-point energy corrections based on our calculated force fields. 

The various total energies are given in Table I and the relative energies are given in Table II. The best calculations are those done with the DZ+D basis set at the optimized geometry and including a correlation correction. These values were used to calculate Ah differences which are given in Table III. 

The results of the simple DHH theory outlined here are shown compared with DH results and corresponding Monte Carlo results in Figs. 10-12. Clearly, the major error of the DH theory has been accounted for. The OCP model is greatly idealized but the same hole correction method can be applied to more realistic electrolyte models. In a series of articles the DHH theory has been applied to a one-component plasma composed of charged hard spheres [23], to local correlation correction of the screening of macroions by counterions [24], and to the generation of correlated free energy density functionals for electrolyte solutions [25,26]. The extensive results obtained bear out the hopeful view of the DHH approximation provided by the OCP results shown here. It is noteworthy that in... 

Becke, A. D., 1992b, Density Functional Thermochemistry. II. The Effect of the Perdew-Wang Generalized-Gradient Correlation Correction , J. Chem. Phys., 97, 9173. 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

Contributions Methods Electrostatics Exch.-rep. Correlation corrections Induction Exchange- induction Ind. and exch-Ind. Correlation corrections Dispersion ... 

Electronic (UV-Vis) spectroscopy has not been utilized in a routine manner. This is perhaps due to the fact that the nature of heterocycles is rather difficult to correlate correctly, with the chromophoric absorption. The UV-Vis absorption spectra of 23b (Table 2) have been studied in detail with R = H, C6H13, CgH17, C20H21, C4H9, and 4-( I, 11. With the exception of 23 (R = 4-C6Hi3C6H4) and which contains conjugated A -phenyl ring, the... 

To verify the mechanism presented, the quantum-chemical calculations of proton affinity, Aa, were carried out for modifiers, since the corresponding experimental data are quite rare. The calculations were performed for isolated molecules, since the properties of species in the interlayer space are probably closer to the gas phase rather than the liquid. The values of Ah were calculated as a difference in the total energy between the initial and protonated forms of the modifier. Energies were calculated using the TZV(2df, 2p) basis and MP2 electron correlation correction. Preliminarily, geometries were fully optimized in the framework of the MP2/6-31G(d, p) calculation. The GAMESS suite of ah initio programs was employed [10]. Comparison between the calculated at 0 K proton affinities for water (7.46 eV) and dioxane (8.50 eV) and the experimental data 7.50 eV and 8.42 eV at 298 K, respectively (see [11]), demonstrates a sufficient accuracy of the calculation. 

The GIAO-MP2/TZP calculated 13C NMR chemical shifts of the cyclopropylidene substituted dienyl cation 27 show for almost all carbon positions larger deviations from the experimental shifts than the other cations 22-26. The GIAO-MP2/TZP method overestimates the influence of cr-delocalization of the positive charge into the cyclopropane subunit on the chemical shifts. Electron correlation corrections for cyclopropylidenemethyl cations such as 27 and 28 are too large to be adequately described by the GIAO-MP2 perturbation theory method and higher hierarchies of approximations such as coupled cluster models are required to rectify the problem. 

Equation (96) shows that the effective KS potential may be simply obtained by adding to the standard KS potential of the isolated solute, an electrostatic correction which turns out to be the RE potential Or, and the exchange- correlation correction 8vxc. It is worth mentioning here, that Eq (96) is formally equivalent to the effective Fock operator correction bfteffi defined in the context of the self consistent reaction field (SCRF) theory [2,3,14] within the HF theory, the exchange contribution is exactly self-contained in Or, whereas correlation effects are completely neglected. As a result, within the HF theory 8v = Or, as expected. 

The complete treatment of solvation effects, including the solute selfpolarization contribution was developed in the frame of the DFT-KS formalism. Within this self consistent field like formulation, the fundamental expressions (96) and (97) provide an appropriate scheme for the variational treatment of solvent effects in the context of the KS theory. The effective KS potential naturally appears as a sum of three contributions the effective KS potential of the isolated solute, the electrostatic correction which is identified with the RF potential and an exchange-correlation correction. Simple formulae for these quantities have been presented within the LDA approximation. There is however, another alternative to express the solva-... 

In general, correlation corrections are larger for a holes than for ir holes. It is not unusual for these differential correlation effects to change the predicted order of final states. Heterocyclic organic molecules with nitrogen-centered, nonbonding electrons are not alone in this respect. Organometallics, transition metal complexes, and clusters of metal oxides and metal halides also require this kind of theoretical interpretation. 

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