Electron-correlation methods

Naively it may be expected that the correlation between pairs of electrons belonging to the same spatial MO would be the major part of the electron correlation. However, as the size of the molecule increases, the number of electron pairs belonging to different spatial MOs grows faster than those belonging to the same MO. Consider for example the valence orbitals for CH4. There are four intraorbital electron pairs of opposite spin, but there are 12 interorbital pairs of opposite spin, and 12 interorbital pairs of the same spin. A typical value for the intraorbital pair correlation of a single bond is 20kcal/ mol, while that of an interorbital pair (where the two MO are spatially close, as in CH4) is 1 kcal/mol. The interpair correlation is therefore often comparable to the intrapair contribution. 

As we ve noted several times, Hartree-Fock theory provides an inadequate treatment of the correlation between the motions of the electrons within a molecular system, especially that arising between electrons of opposite spin. 

When Hartree-Fock theory fulfills the requirement that 4 be invarient with respect to the exchange of any two electrons by antisymmetrizing the wavefunction, it automatically includes the major correlation effects arising from pairs of electrons with the same spin. This correlation is termed exchange correlation. The motion of electrons of opposite spin remains uncorrelated under Hartree-Fock theory, however. 

Any method which goes beyond SCF in attempting to treat this phenomenon properly is known as an electron correlation method (despite the fact that Hartree-Fock theory does include some correlation effects) or a post-SCT method. We will look briefly at two different approaches to the electron correlation problem in this section. 

Bond dissociation energies (BDEs) are at the heart of organic thermochemistry and many mechanistic considerations, so it is imperative to be able to compute and predict these quantities accurately. The problem is that, upon lengthening a bond in the early stages of cleaving 

Configuration interaction treatments can also be easily extended to multideterminant problems but these require even more care and an exorbitant computational effort [15]. Multi-reference Cl approaches are ideally suited to determining, for example, electronic spectra, as they can deal with excited states in a straightforward fashion [16]. 

The corrections in going from HF to MP2 are large (Table 7.3) and the computational effort increases dramatically for small improvements in energies and geometries. With a limited basis set such as DZP, it is clear that CCSD(T), which should be superior to MP4, cannot five up to its promise because the basis set is too small. A major conclusion is that there must be a good balance between attempted amount of electron correlation recovery and basis set size. It is advisable to use at least triple-f quality type basis sets for highly correlated methods such as coupled cluster theory. 

In a nutshell (electron correlated methods) Complete electronic description, straightforward interpretation Highly accurate methods that can rival experiment for small organic molecules Very time consuming Cl methods are not size-extensive CC methods are non-variational Slow convergence Strongly basis set dependent Systematic improvement straightforward. 

Introduction to Computational Chemistry, Second Edition. Frank Jensen. 2007 John Wiley Sons, Ltd 

MOs are spatially close, as in CHt) is 8kJ/mol. The interpair correlation is therefore often comparable to the intrapair contribution. 

Since the correlation between opposite spins has both intra- and interorbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or, equivalently, the antisymmetry of the wave function) has the consequence that there is no intraorbital correlation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, there is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, and the corresponding phenomenon for electrons of same spin is the Fermi hole. This hole picture is discussed in more detail in connection with density functional theory in Chapter 6. 

The HF method determines the energetically best one-determinant trial wave function (within the given basis set). It is therefore clear that, in order to improve on HF results, the starting point must be a trial wave function that contains more than one Slater determinant (SD) O. This also means that the mental picture of electrons residing in orbitals has to be abandoned and the more fundamental property, the electron density, should be considered. As the HF solution usually gives -99% of the correct answer, electron correlation methods normally use the HF wave function as a starting point for improvements. 

A generic multi-determinant trial wave function can be written as in eq. (4.1), where flo is usually close to one. 

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A very important difference between H2 and molecular orbital calculations is electron correlation. Election correlation is the term used to describe interactions between elections in the same molecule. In the hydrogen molecule ion, there is only one election, so there can be no election correlation. The designators given to the calculations in Table 10-1 indicate first an electron correlation method and second a basis set, for example, MP2/6-31 G(d,p) designates a Moeller-Plesset electron coiTclation extension beyond the Hartiee-Fock limit canied out with a 6-31G(d,p) basis set. 

These SVWN5 results are somewhat fortuitous. Be careful not to overgeneralize from their agreement to experiment. We will see a different result in Exercise 6.7. Several other excerises will also include comparisons of DFT methods to Hartree-Fock theory, MP2 and other electron correlation methods. 

I even the higher order electron correlation methods. 

Chapter 6, Selecting an Appropriate Theoretical Method, discusses the model chemistry concept introduced in Chapter 1 in detail. It covers the strengths, computational cost and limitations of a variety of popular methods, beginning with semi-empirical models and continuing through Hartree-Fock, Density Functional Theory, and electron correlation methods. 

Such MO integrals are required for all electron correlation methods. The two-electron AO mtegrals are the most numerous and the above equation appears to involve a computational effect proportional to M AO integrals each multiplied by four sets of M MO coefficients). However, by performing the transformation one index at a time, the computational effort can be reduced to. ... 

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