Effects of Electron Correlation

This style of analysis was followed up by another group of coworkers who replaced the ZH3 molecule by water and also included the effects of electron correlation . The 6-31G set was used, as was the larger -l-(VP) (2d) containing two sets of d-functions, plus a diffuse sp-set . As in the earlier cases, the absolute minimum is of cyclic type, with a weak H-bond between the H2O and the second HF molecule, as illustrated in Fig. 5.27. Since HjO is a poorer proton acceptor than is NH3, one would expect the H-bond to HF to be weaker, and the cooperativity within the trimer to be less extensive. On the other hand, the better proton-donating ability of H2O should strengthen the last H-bond that completes the cyclic character of the trimer. 

The geometric features of the binary complexes are listed in Table 5.28, along with the 1 2 complex in the last two rows. The expected contraction of the O—F distance as the second HF molecule is added is immediately apparent. This contraction amounts to 0.10 A at the SCF level, less than that in the HjN—HF- HF complex where the shrinkage was 0.13 A. Correlation has little influence on the H-bond reduction the MP2 contraction is 0.11 A. From the perspective of adding the proton acceptor molecule to the FIF dimer, R(F F) is reduced by 0.12 A for both H2O and NH3, at the SCF level. The distinction between the latter two molecules is perhaps most clearly seen in the bond length of the inner HF molecule, ij. This bond stretches by 0.017 A when the outer HF molecule is added to HjN- HF, but by only 0.009 A if H N is replaced by HjO. Note, however, that when correlation is added, this stretching doubles. 

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DFT methods are attractive because they include the effects of electron correlation—the fact that electrons in a molecular system react to one another s motion and attempt to keep out of one another s way—in their model. Hartree-Fock calculations consider this effect only in an average sense—each electron sees and... 

As we have seen throughout this book, the Hartree-Fock method provides a reasonable model for a wide range of problems and molecular systems. However, Hartree-Fock theory also has limitations. They arise principally from the fact that Hartree-Fock theory does not include a full treatment of the effects of electron correlation the energy contributions arising from electrons interacting with one another. For systems and situations where such effects are important, Hartree-Fock results may not be satisfactory. The theory and methodology underlying electron correlation is discussed in Appendix A. 

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 ... 

In the last few years, methods based on Density Functional Theory have gained steadily in popularity. The best DFT methods achieve significantly greater accuracy than Harttee-Fock theory at only a modest increase in cost (far less than MP2 for medium-size and larger molecular systems). They do so by including some of the effects of electron correlation much less expensively than traditional correlated methods. 

The semi-empirical methods have better MAD s than th Hartree-Fock-based methods, indicating that their parametrization ha accounted for some of the effects of electron correlation. However, thei maximum errors are very large. Semi-empirical methods are especiall poor at predicting ionization potentials and proton affinities. 

The HF wave funetion eontains equal amounts of ionie and eovalent eontributions (Section 4.3), For covalently bonded systems, like H2O, the HF wave funetion is too ionie, and the effect of electron correlation is to increase the covalent contribution. Since the ionic dissociation limit is higher in energy than the covalent, the effect is that the equiUbrium bond length increases when correlation methods are used. For dative bonds, such as metal-ligand compounds, the situation is reversed. In this case the HF wave function dissociates correctly, and bond lengths are normally too long. Inclusion of... 

It should also be noted that the effect of electron correlation at the MP2 level (relative to HF) is largely independent of the basis set, but there is a significant coupling between... 

It should be noted that the above conclusions have been reached on strictly electrostatic grounds a spin property has not been invoked for the two electrons. From the variation of i/i along the box it can be shown that the singlet state is of higher energy than the triplet because the two electrons are more crowded together for (S-state) than for (T-state). Thus there is less interelectronic repulsion m the T-state. The quantity 2J j. is a measure of the effect of electron correlation which reduces the repulsive force between the two electron (Fermi correlation energy). 

The effects of electron correlation are investigted through the CIPSI (Configuration Interaction with Perturbatively Selected Configurations) calculations (4) of the molecular states. 

Boyd, R. J., and L.-C. Wang. 1989. The Effect of Electron Correlation on the Topological and Atomic Properties of the Electron Density Distributions of Molecules. J. Comp. Chem. 1, 367. 

Gatti, C., P. J. MacDougall, and R. F. W. Bader. 1988. Effect of Electron Correlation on the Topological Properties of Molecular Charge Distributions. J. Chem. Phys. 88, 3792. 

There have now been four experimental determinations of a silicon-carbon double bond length. The first of these was a gas phase electron diffraction study of 1,1-dimethylsilene (173). This study was the subject of much controversy since the experimentally determined bond length, 1.83 A, was much longer than the one predicted by ab initio calculations (1.69-1.71 A, see below) (159). Since the calculations were carried out at a relatively high level of theory and the effects of electron correlation on determining the Si=C bond length were considered, the validity of the data extracted from the electron diffraction study is in serious doubt. 

Despite the successful prediction of chemical shifts for a great structural variety of carbocations some difficulties have been encountered for vinyl cations.47 The effect of electron correlation, basis sets and geometry on calculated NMR spectra of vinyl cations has been studied in some detail also for the parent vinyl cation in its linear form.48 Comparative experimental and computational NMR studies, however, have... 

Geometries were fully optimized at the HF/6-31G level of theory, and single point energies were evaluated at the MP2/6-31G level to indude the effects of electron correlation. Transition states were characterized by harmonic frequency analysis. 

All stationary point geometries were fully optimized at the HF/6-31G level of theory and characterized by harmonic frequency analysis. Single point energies were evaluated at the MP2/6-31G level to account for the effects of electron correlation. Since experiments were carried out in a relatively low dielectric environment (chlorobenzene solvent), it is likely that the shape of the potential energy surface in the gas phase and solution would be comparable... 

Prior to stretching C-S bond, we optimized the geometry of the anionic Me-S-Me molecule and parent neutral molecule at the unrestricted second-order Mpller-Plesset (UMP2) perturbation level of theory (in order to take into account the effect of electron correlation) with aug-cc-pVDZ basis sets [9]. We also... 

Note that the energy is minimized with respect to all choices of the orbital basis and subject to the (1, conditions on p, = F, ,- this ensures that there exists an ensemble of Slater determinants with the desired electron density. Because an ensemble average of Slater determinants does not describe electron correlation, these variational energy expressions include a correlation functional, Ec p, which corrects the energy for the effects of electron correlation. Reasonable approximations for Ec[p] exist, though they tend to work only in conjunction with approximate exchange-energy functionals, Ex p. ... 

Dimensional scaling theory [109] provides a natural means to examine electron-electron correlation, quantum phase transitions [110], and entanglement. The primary effect of electron correlation in the D 00 limit is to open up the dihedral angles from their Hartree-Fock values [109] of exactly 90°. Angles in the correlated solution are determined by the balance between centrifugal effects, which always favor 90°, and interelectron repulsions, which always favor 180°. Since the electrons are localized at the D 00 limit, one might need to add the first harmonic correction in the 1/D expansion to obtain... 

Several issues remain to be addressed. The effect of the mutual penetration of the electron distributions should be analyzed, while the use of theoretical densities on isolated molecules does not take into account the induced polarization of the molecular charge distribution in a crystal. In the calculations by Coombes et al. (1996), the effect of electron correlation on the isolated molecule density is approximately accounted for by a scaling of the electrostatic contributions by a factor of 0.9. Some of these effects are in opposite directions and may roughly cancel. As pointed out by Price and coworkers, lattice energy calculations based on the average static structure ignore the dynamical aspects of the molecular crystal. However, the necessity to include electrostatic interactions in lattice energy calculations of molecular crystals is evident and has been established unequivocally. 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

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