Hartree-Fock theory, description

The Dirac equation can be readily adapted to the description of one electron in the held of the other electrons (Hartree-Fock theory). This is called a Dirac-Fock or Dirac-Hartree-Fock (DHF) calculation. 

Moller-Plesset perturbation theory (MPPT) aims to recover the correlation error incurred in Hartree-Fock theory for the ground state whose zero-order description is Moller-Plesset zero-order Hamiltonian is the sum of Fock operators, and the zero-order wave functions are deteiminantal wave functions constructed from HF MOs. Thus the zero-order energies are simply the appropriate sums of MO energies. The perturbation is defined as the difference between the sum of Fock operators and the exact Hamiltonian ... 

P. C. Hiberty, S. Humbel, D. Danovich, and S. Shaik,/. Am. Chem. Soc., 117, 9003 (1995). What Is Physically Wrong with the Description of Odd-Electron Bonding by Hartree-Fock Theory A Simple Nonempirical Remedy. 

When discussing the electronic structure of molecules and solids, one-electron descriptions, such as the molecular orbitals of Equation 8.1, are quite intuitive. It is common to talk about individual electrons occupying particular states. For example, reactions often occur by the mixing of the highest occupied molecular orbital (HOMO) of one species and the lowest unoccupied molecular orbital (LUMO) of another. In such a reaction the electrons in the HOMO state move into the new mixed orbital, lowering their energy. The HOMO and LUMO states are each pairs of one-electron molecular orbitals, since in the simplest case an orbital giving the spahal distribution for a spin up electron has an identical partner for spin down. Mulh-electron wavefunctions that describe the whole electronic structure in this picture are constructed from the one-electron states. So, for example, in a four-electron system in which all the electronic states are doubly occupied (spin up and spin down), based on Hartree-Fock theory we can write ... 

In 2002, Nakai [24] presented a non-Bom-Oppenheimer theory of molecular structure in which molecular orbitals (MO) are used to describe the motion of individual electrons and nuclear orbitals (NO) are introduced each of which describes the motion of single nuclei. Nakai presents an ab initio Hartree-Fock theory, which is designated NO+MO/HF theory , which builds on the earlier work of Tachikawa et al. [25]. In subsequent work published in 2003, Nakai and Sodeyama [26] apply MBPT to the problem of simultaneously describing both the nuclear and electronic components of a molecular system. Their approach will be considered in some detail in this paper as a first step in the development of a literate quantum chemistry program for the simultaneous description of electronic and nuclear motion. 

It is clear that, in the Ba+ problem, with an nf wavefunction poised on the knife edge between the two potential wells, there is an extremely sensitive dependence on the accuracy of the effective mean field. Hartree-Fock theory provides a first-order description of the phenomena involved, but... 

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. 

Section 3.5 contains a detailed illustration of the closed-shell ab initio SCF procedure using two simple systems the minimal basis set descriptions of the homonuclear (H2) and heteronuclear (HeH" ) two-electron molecules. We first describe the STO-3G minimal basis set used in calculations on these two molecules. We then describe the application of closed-shell Hartree-Fock theory to H2. This is a very simple model system, which allows one to examine the results of calculations in explicit analytical form. Finally, we apply the Roothaan SCF procedure to HeH. Unlike H2, the final SCF wave function for minimal basis HeH is not symmetry determined and the HeH example provides the simplest possible illustration of the iterative SCF procedure. The description of the ab initio HeH calculation given in the text is based on a simple FORTRAN program and the output of a HeH calculation found in Appendix B. By following the details of this simple but, nevertheless, real calculation, the formalism of closed-shell ab initio SCF calculations is made concrete. 

In Subsection 2.2.5 we presented our minimal basis H2 model, which has only one occupied molecular orbital and one virtual molecular orbital. With the description of the Is minimal STO-3G basis set given in the last subsection we are now in a position to illustrate ah initio Hartree-Fock calculations on H2. The model is simple but extension to larger basis sets is relatively straightforward and most of the aspects of Hartree-Fock theory that we wish to illustrate here are independent of the actual size of the basis set. Unfortunately, however, the model is too simple to be able to illustrate the iterative nature of the SCF procedure. In the next subsection we describe a minimal basis calculation on HeH", in order to illustrate this aspect of Hartree-Fock theory. 

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