Hartree-Fock theory electron correlation methods

Even if no integral parametrizations are introduced and the HF equations are all correctly solved, the method eventually turns out to be theoretically incomplete. Despite the correct treatment of electronic exchange (X) within Hartree-Fock theory, electronic correlation (C) is totally missing. This is easily shown for the case of the H2 molecule in which we use the bonding solution of the H2 molecular ion ( + = cr from Equation (2.15)) to build up an antisymmetrized molecular wave function. This means that we put both electrons (ri and rz) of the H2 molecule into the same ip+ orbital, and Pauli s principle is obeyed by means of the ct/ spinors. Neglecting orbital overlap and any pre-factors, for simplicity, the so-called Hund-Mulliken [124] (another name... 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

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. 

The DFT and MP2 calculations produce very similar structures, although the BLYP bond length is again longer than those of the other functionals. Hartree-Fock theory predicts a bond length which is significantly shorter than the methods including electron correlation. 

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. 

Despite these comparisons to Hartree-Fock theory, the O-Singles method does include some electron correlation. 

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. 

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. 

DFT has come to the fore in molecular calculations as providing a relatively cheap and effective method for including important correlation effects in the initial and final states. ADFT methods have been used, but by far the most popular approach is that based on Slater s half electron transition state theory [73] and its developments. Unlike Hartree-Fock theory, DFT has no Koopmans theorem that relates the orbital energies to an ionisation potential, instead it has been shown that the orbital energy (e,) is related to the gradient of the total energy E(N) of an N-electron system, with respect to the occupation number of the 2th orbital ( , ) [74],... 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

For anything but the most trivial systems, it is not possible to solve the electronic Schrodinger equation exactly, and approximate techniques must instead be used. There exist a variety of approximate methods, including Hartree-Fock (HF) theory, single- and multireference correlated ab initio methods, semiempirical methods, and density functional theory. We discuss each of these in turn. In Hartree-Fock theory, the many-electron wavefunction vF(r1, r2,..., r ) is approximated as an antisymmetrized product of one-electron wavefunctions, ifijfi) x Pauli principle. This antisymmetrized product is known as a Slater determinant. 

Numerical solutions of the Schrodinger equation can be obtained within several degrees of approximation, for almost any system, using its exact Hamiltonian. Density functional theory has proven to be one of the most effective techniques, because it provides significantly greater accuracy than Hartree-Fock theory with just a modest increase in computational cost.io> 3-45 The accuracy of DFT method is comparable, and even greater than other much more expensive theoretical methods that also include electron correlation such as second and higher order perturbation theory. 

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