Summary of Electron Correlation Methods

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative 

In terms of accuracy with a medium-sized basis set the following order is often 

All of these are single-determinant-based methods. Multi-reference methods cannot easily be classified as the quality of the results depends heavily on the size of the 

MP3 has not been included in the above comparison. As already mentioned, MP3 results are often inferior to those at MP2. In fact, MP2 often gives surprisingly good results, especially if large basis sets are used. Furthermore, it should be kept in mind that the MP perturbation series in many cases may actually be divergent, although corrections carried out to low order (i.e. 2-4) rarely display excessive oscillations. 

The use of Cl methods has been declining in recent years at the expense of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl- or MCSCF-based methods have been the preferred methods here. More recently linear response methods (Section 10.9) have been developed for coupled cluster wave functions, and which allow calculation of excited state properties. 

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

We have so far been careful to used the wording formal scaling . As already discussed, HF is formally an method but in practice the scaling may be reduced all the way down to Similarly, MP2 is formally an method. However, an MP2 calculation consists of three main parts the HF calculation, the AO to MO integral 

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Linear Scaling Techniques Semi-Empirical Methods 3.9.1 Neglect of Diatomic Differential Overlap Approximation (NDDO) 3.9.2 Intermediate Neglect of Differential Overlap Approximation (INDO) 80 81 82 83 4.13 Locahzed Orbital Methods 4.14 Summary of Electron Correlation Methods 4.15 Excited States References 5 Basis Sets 144 144 147 148 150... 

In summary, the calculations on this special impurity system, with only one open-shell electron and simple manifolds, in which the assigmnents of the 5/<-> 6d absorption/emission bands were specially clear, even though not complete, and where absorption/emission bands built on a single origin and recorded with an extremely hi resolution exist, allow to be initially optimistic in the evaluation of the applicability of present day ab initio methods of the Quantum Chemistry in structural and spectroscopic studies of actinide ion impurities in ionic crystals. The quality of the approximate Hamiltonians and of the approximate procedures for decoupling the treatment of electron correlation and spin-orbit seems to be acceptable. The performance in other impurities with several open-shell electrons and large manifolds will be presented in the next two sections. 

The composition of this review is as follows Section 2 describes the numerical examples of the rules for degenerate excitations. The data in the next section are obtained by highly correlated methods, since the effects of electron correlations are essential for accurate descriptions of the excited states. Section 3 demonstrates the interpretation of the rules by using the simplified model that corresponds to the frozen-orbital approximation (FZOA) [4]. In the excitation energy formulas to which the FZOA leads, the splitting schemes are related to the specific two-electron integrals, whose values are qualitatively analyzed by the relevant orbital characters. Finally, the summary is addressed in Section 4. 

In this work, the proton exchange reactions of four small alkanes, that is, ethane, propane, /-butane, and normal-butane (w-butane), in ZSM-5 were studied by means of quantum chemical methods. Effects of the size of the cluster model, the quahty of basis set, and the importance of the level of electron correlation were systematically investigated. The proton exchange barriers and the regioselectivity of these alkanes sorbed in ZSM-5 were addressed. The summary of types of proton exchange reactions of smdied alkanes is given in Table 1. 

In a (rather crude) summary, it might be said that DFT does not rigorously solve the problem of correlation energy, but displaces the problem to a well defined location, the Sex(p), where a parametric ambush can be set for its solution. The method, whose computational demand is similar or sometimes more modest than those of MO methods, has been very successful in applications to isolated molecules, where the introduction of electron correlation corrections has been seen to improve, for example, calculated optimized molecular geometries, but has not yet proved completely satisfactory for the calculation of intermolecular interaction energies in systems where coulombic contributions are not overwhelmingly dominant. Molecular crystals are a typical example. 

In summary the results observed in these studies [160] of poly(Sty-co-DVB) swelling in aromatic liquids serve to show that the method of measuring a is so sensitive that it can detect an effect caused by even the smallest modification in the molecular geometry of attached substituents, and that these differences correlate qualitatively with expectation based on the known principles of physico-organic chemistry of aromatic compounds. Since the observed a is the net effect of electronic attraction and steric hindrance between the sorbed molecule and the adsorption site, i.e. the monomer unit of the polymer, it would be impossible to separate quantitatively the electronic and steric contributions of a particular substituent. The ability to make such a differentiation, however, appears to be more promising with liquids that comprise homologous series of the type Z(CH2)nH (where Z is a phenyl, chloro, bromo or iodo substituent), since the added electronic contribution to Z by each additional methylene group is well known to be extremely small when n becomes >3 [165],... 

Figure 4 Summary of the method for obtaining the structure of a bilayer from an oriented multilamellar sample. The diffraction pattern (shaded circles) is that expected from a multibilayer sample oriented perfectly parallel to the glass slide that is rotated in the x-ray beam. The intensities of the spots are proportional to square of the structure factors F h), which causes the phases (+1 or-1 in this case see text) to be lost. If the phases can be determined, then the electron density profile p z) can be determined by Fourier transformation, as shown. The correlations of the peaks and valleys of p(z) with the features of the bilayer are illustrated. This figure is based upon Fig. 1 of an article by White et al. (1986).

In summary, full wavefunction methods are the most accurate ab initio methods, if electronic correlation is accounted for properly and the basis set for expansion of the electronic wavefunction is sufficiently complete — this might be difficult to judge by a non-expert. Furthermore, if a cluster model is used, it needs to be of sufficient size or be properly embedded for conclusions to be representative of an extended M/C interface. 

In summary, the recommended method for the inclusion of spin-orbit coupling and other relativistic effects for molecules containing heavy elements, considering computational complexity and accuracy factors, is one based on ab initio REPs. Future developments should include more extensive spin-orbit and Cl procedures in the A -S framework, the development and implementation of Cl computational codes in to-co coupling, the incorporation of reliable methods for the treatment of core-valence polarization and correlation effects, and selected benchmark all-electron calculations. 

In the preceding chapters we have seen how to define and describe the structure of a molecule or molecular fragment (Chapters 1 and 2), how to retrieve the relevant information from electronic databases (Chapter 3) and how to analyze this information with a variety of statistical methods (Chapter 4). Fragment definition, data retrieval and statistical analysis in structure correlation have their analogs in ordinary chemical experiments First a problem is identified, then the data are acquired from an appropriate experiment, filtered to distinguish the relevant from the irrelevant, and processed into a clear summary form. However, this is not the final step. The data still need to be understood. 

CO populations on the NMR and voltammetric timescale is very slow. In summary, these observations demonstrate that there are two major populations of CO on Pt/Ru surfaces prepared hy the spontaneous deposition method COs located on Ru islands undergoing fast, thermally activated diffusion, and COs on Pt sites further away from Ru, undergoing slower diffusion. " With increasing Ru coverage, the f -LDOS for the clean metal surface and the 2n orbital of chemisorbed CO decrease, indicating there are strong electronic perturbations caused by the Ru addition. The increase in the catalytic activity of the Ru modified Pt nanoparticles appears to have a direct correlation with these electronic modifications. 

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