Hartree-Fock theory correlation problem

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. 

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. 

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 boud dissociatiou euergy of hydrogeu peroxide has been accurately predicted by high-level ab initio theory . It was disclosed very early that Hartree-Fock theory, in the absence of electron-correlation correction, simply cannot be applied to problems involving 0-0 bond dissociation. For example, the predicted 0-0 bond energy in peroxyformic acid is only 1.0 kcalmoD by Hartree-Fock theory, whereas at the... 

As a matter of fact, the choice of the best IPM for open-shell systems raises a number of problems there are many possible extended forms of the Hartree-Fock theory in addition to the standard RHF method. All these schemes include both spin polarization and correlation to some extent, but in an ill-defined way. The best way to include spin polarization is probably to take RHF as a starting point and to admit single excitations afterwards. 

The problem of size-consistency is particularly frightful for restricted Cl calculations on solids where the correlation energy falsely arrives at zero, which is the worst possible result [126]. The difficulties of Hartree-Fock theory for the solid state or, more correctly, for solid metals, however, show up much earlier. When Hartree-Fock theory is applied to a gas of noninteracting electrons (free electrons) which comes close to the electronic situation in simple metals, such as the alkali elements, the band energy takes the following analytical form [131] ... 

Modem solutions of the time independent Schroedinger eqimtion (equation 1) follow two very different theories wavefimction-based (ab initio Hartree-Fock and correlated methods) and electron density-based (density functional theory, DFT) (15). We will outline these approaches and then describe inq)lementations suitable for nanoscopic problems. 

Basis Sets Correlation Consistent Sets Configuration Interaction Coupled-cluster Theory Density Functional Applications Density Functional Theory Applications to Transition Metal Problems G2 Theory Integrals of Electron Repulsion Integrals Overlap Linear Scaling Methods for Electronic Structure Calculations Localized MO SCF Methods Mpller-Plesset Perturbation Theory Monte Carlo Quantum Methods for Electronic Structure Numerical Hartree-Fock Methods for Molecules Pseudospectral Methods in Ab Initio Quantum Chemistry Self-consistent Reaction Field Methods Symmetry in Hartree-Fock Theory. 

Prior to the widespread usage of methods based on Density Functional Theory, the MP2 method was one of the least expensive ways to improve on Hartree-Fock and it was thus often the first correlation method to be applied to new problems. It can successfully model a wide variety of systems, and MP2 geometries are usually quite accurate. Thus, MP2 remains a very useful tool in a computational chemist s toolbox. We ll see several examples of its utility in the exercises. 

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. 

Lowdin, P.-O., Phys. Rev. 97, 1474, 1490, 1509, Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals and convergence problems in the method of configuration interaction. II. Study of the ordinary Hartree-Fock approximation. III. Extension of the Har-tree-Fock scheme to include degenerate systems and correlation effects. ... 

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