Hartree-Fock theory post-theories

In the present chapter, we review the pericyclic reactions studied with DFT methods to date. Local, nonlocal, and hybrid DFT methods have been used to study the parent systems of the most important pericyclic reactions. These results are compared with results of Hartree-Fock theory, post-Hartree-Fock calculations, and available experimental data. Our aim is to provide an overview... 

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. 

Density functional theory (DFT),32 also a semi-empirical method, is capable of handling medium-sized systems of biological interest, and it is not limited to the second row of the periodic table. DFT has been used in the study of some small protein and peptide surfaces. Nevertheless, it is still limited by computer speed and memory. DFT offers a quantum mechanically based approach from a fundamentally different perspective, using electron density with an accuracy equivalent to post Hartree-Fock theory. The ideas have been around for many years,33 34 but only in the last ten years have numerous studies been published. DFT, compared to ab initio... 

Computational costs of IET are much lower than that of standard molecular simulations. This is crucial when considering the combination of IET and the time consuming, highly sophisticated post-Hartree-Fock ab initio theory. The QM/MM method requires much computational power because the quantum chemical calculation must be repeated many times. 

The set of atomic orbitals Xk is called a basis set, and the quality of the basis set will usually dictate the accuracy of the calculations. For example, the interaction energy between an active site and an adsorbate molecule might be seriously overestimated because of excessive basis set superposition error (BSSE) if the number of atomic orbitals taken in Eq. [4] is too small. Note that Hartree-Fock theory does not describe correlated electron motion. Models that go beyond the FiF approximation and take electron correlation into account are termed post-Flartree-Fock models. Extensive reviews of post-HF models based on configurational interaction (Cl) theory, Moller-Plesset (MP) perturbation theory, and coupled-cluster theory can be found in other chapters of this series. ... 

A good first approach to a quantum mechanical system is often to consider one-electron functions only, associating one such function, a spin-orbital , with one electron. Most popular are the one-electron functions which minimize the energy in the sense of Hartree-Fock theory. Alternatively one can start from a post-HF wave function and consider the strongly occupied natural spin orbitals (i.e. the eigenfunctions of the one-particle density matrix with occupation numbers close to 1) as the best one-electron functions. Another possibility is to use the Kohn-Sham orbitals, although their physical meaning is not so clear. 

We shall in this section give a historic overview of how the electronic structure theory for transition metal complexes in their ground state has evolved from the 1950s to the present time. The account will include a discussion of wave function methods based on Hartree Fock and post-Hartree Fock approaches as well as Kohn-Sham density functional theory (KS-DFT). 

Table 1 contains some further information useful to characterize the different contributions to the molecule/surface interaction orientation dependence and the typical strength of the different contributions, and whether or not they can be understood on a purely classical basis. If one wants to calculate molecule/surface interactions by means of quantum-mechanical or quantum-chemical methods, the most important question is whether standard density functional (DPT) or Hartree-Fock theory (self consistent field, SCF) is sufficient for a correct and reliable description. Table 1 shows that all contributions except the Van der Waals interaction can be obtained both by DPT and SCF methods. However, the results might be connected with rather large errors. One famous example is that the dipole moment of the CO molecule has the wrong sign in the SCF approximation, with the consequence that SCF might yield a wrong orientation of CO on an oxide surface (see also below). In such cases, the use of post Hartree-Fock methods or improved functionals is compulsory. 

The definition of the gas-phase acidity through reaction (7.3) implies that this quantity is a thermodynamic state function. Thus, one could use quantum chemical approaches to obtain gas-phase acidities from the theoretically computed enthalpies of the species involved. However, two points must be noted before one proceeds A chemical bond is being broken and an anion is being formed. Thus, one may anticipate the need for a proper treatment of electronic correlation effects and also of basis sets flexible enough to allow the description of these effects and also of the diffuse character of the anionic species, what immediately rules out the semi-empirical approaches. Hence, our discussion will only consider ab initio (Hartree-Fock and post-Hartree-Fock) and DFT (density functional theory) calculations. 

The Amsterdam Density Functional package (ADF) is software for first-principles electronic structure calculations (quantum chemistry). ADF is often used in the research areas of catalysis, inorganic and heavy-element chemistry, biochemistry, and various types of spectroscopy. ADF is based on density functional theory (DFT) (see Chapter 2.39), which has dominated quantum chemistry applications since the early 1990s. DFT gives superior accuracy to Hartree-Fock theory and semi-empirical approaches, especially for transition-metal compounds. In contrast to conventional correlated post-Hartree-Fock methods, it enables accurate treatment of systems with several hundreds of atoms (or several thousands with QM/MM)." ... 

Perturbation methods are size consistent in other words, they lead to total energies that scale linearly with the size of the system. However, they have drawbacks. First, their energies are not upper bounds to the exact energy of the system (because the energy expression is not of the expectation value form). Second, the wavefunction is expressed in terms of corrections to a presumed dominant reference function of a single determinant. Therefore, when Hartree-Fock theory presents a major problem, MP2 may not be an appropriate method of rectification. For further discussion of post-Hartree-Fock methods, see, for example, a recent, excellent review by Bartlett and Stanton. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

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