Kohn-Sham/Hartree-Fock model

All calculations were carried out with the Gaussian code, using a recent hybrid Kohn-Sham/Hartree-Fock (KS/HF) model hereafter referred to as PBEO. This approach is obtained casting the PBE exchange and correlation functional in a hybrid scheme HF/DFT, where the HF exchange ratio (1 4) is fixed apriorf . ... 

Here, /i are the so-called Kohn-Sham orbitals and the summation is carried out over pairs of electrons. Within a finite basis set (analogous to the LCAO approximation for Hartree-Fock models), the energy components may be written as follows. 

In solid-state physics, theorists are dealing with many-electron systems where "many mean billions not just dozens as in molecular theories. This means that methods based on the electron density are much more widely used and much more intuitively appealing. Their constant efforts to develop such methods have been rewarded by a series of amazing theorems showing that it is possible to obtain the exact electron density without having recourse to the wavefunction. Naturally, these results have been taken up with some enthusiasm by workers in the field of molecular electronic structure. In this chapter the celebrated Kohn-Hohenberg-Sham approach is developed and its close relationship to the Hartree-Fock model is used to indicate how it can be implemented. The very different intuitions" of chemists and physicists about electronic structure generates some tensions in the interpretation of the results of these theories. 

Abstract. The paper by Kohn and Sham (KS) is important for at least two reasons. First, it is the basis for practical methods for density functional calculations. Second, it has endowed chemistry and physics with an independent particle model with very appealing features. As expressed in the title of the KS paper, correlation effects are included at the level of one-electron equations, the practical advantages of which have often been stressed. An implication that has been less widely recognized is that the KS molecular orbital model is physically well-founded and has certain advantages over the Hartree-Fock model. It provides an excellent basis for molecular orbital theoretical interpretation and prediction in chemistry. 

Bienati, M., Adamo, C., Barone, V., 1999, Performance of a New Hybrid Hartree-Fock/Kohn-Sham Model (B98) in Predicting Vibrational Frequencies, Polarisabilities and NMR Chemical Shifts , Chem. Phys. Lett., 311, 69. 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

An alternative approach to conventional methods is the density functional theory (DFT). This theory is based on the fact that the ground state energy of a system can be expressed as a functional of the electron density of that system. This theory can be applied to chemical systems through the Kohn-Sham approximation, which is based, as the Hartree-Fock approximation, on an independent electron model. However, the electron correlation is included as a functional of the density. The exact form of this functional is not known, so that several functionals have been developed. 

The Xa (X = exchange, a is a parameter in the Xa equation) method gives much better results [23, 24]. It can be regarded as a more accurate version of the Thomas-Fermi model, and is probably the first chemically useful DFT method. It was introduced in 1951 [25] by Slater, who regarded it [26] as a simplification of the Hartree-Fock (Section 5.2.3) approach. The Xa method, which was developed mainly for atoms and solids, has also been used for molecules, but has been replaced by the more accurate Kohn-Sham type (Section 7.2.3) DFT methods. 

Becke and Johnson200,201 proposed recently a new method to treat van der Waals complexes, to be used in association with either Hartree-Fock or Kohn-Sham orbitals. In this model, the total energy is expressed as ... 

The first derivative of the density matrix with respect to the magnetic induction (dPfiv/dBi) is obtained by solving the coupled-perturbed Hartree-Fock (or Kohn-Sham) equations to which the first derivative of the effective Fock (or Kohn-Sham) operator with respect to the magnetic induction contributes. Due to the use of GIAOs, specific corrections arising from the effective operator Hcnv describing the environment effects will appear. We refer to Ref. [28] for the PCM model and to Ref. [29] for the DPM within either a HF or DFT description of the solute molecule. 

An important aspect of the MO model is the choice of the molecular orbitals that are used for discussing the properties of molecules. Early work considered either qualitative sketches of MOs or molecular orbitals, which came from EHT calculations. Later work used Hartree-Fock (HF) orbitals for the MO models. The latter have the disadvantage that the correlation energy is not considered although correlation effects can be very important for the stmcture and reactivity of molecules. The advent of DFT calculations introduced also the use of Kohn-Sham (KS) orbitals for MO models. The advantage of KS orbitals is that correlation effects are... 

C. Density functional theory Density functional theory (DFT) is the third alternative quantum mechanics method for obtaining chemical structures and their associated energies.Unlike the other two approaches, however, DFT avoids working with the many-electron wavefunction. DFT focuses on the direct use of electron densities P(r), which are included in the fundamental mathematical formulations, the Kohn-Sham equations, which define the basis for this method. Unlike Hartree-Fock methods of ab initio theory, DFT explicitly takes electron correlation into account. This means that DFT should give results comparable to the standard ab initio correlation models, such as second order M(j)ller-Plesset (MP2) theory. 

---