Hartree-Fock model, ground

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

It may be concluded thus that the Half-Projected Hartree-Fock model proposed more than two decades ago for introducing some correlation effects in the ground state wave-function [1,2], could be employed advantageously for the direct determination of the lowest triplet and singlet excited states, in which Ms = 0. This procedure could be especially suitable for the singlet excited states of medium size molecules for which no other efficient procedure exists. 

Except for systems where the difference in energy between the conformers is very small, even the STO-3G Hartree-Fock model properly assigns ground-state conformation. However, conformational energy differences from STO-3G calculations show large errors in some cases. Results from 3-2IG calculations are generally even worse, and the simplest Hartree-Fock model to provide a reliable (and for the most part quantitative) account of conformational energy differences is the 6-3IG model. Except for formic acid and methyl... 

Density functional theory (DFT) is based on the Hohenberg-Kohn theorem, which states that there is a functional which gives the exact ground-state energy for the exact electron density. DFT models have become very popular because they are not more costly than Hartree-Fock models. The reason is that in the HF, Cl and MP models, a wavefunction for an N-electron system depends on 3N coordinates, whereas in the DFT approach, the electron density depends on only three coordinates, irrespective of the number of electrons. The problem is that the exact functional would be the Schrodinger equation itself Several approximate functionals have been developed by many authors (Becke, Parr, Perdew, and others) and different forms of the functional can yield slightly different results. Some of the most common DFT models are ... 

The idea behind the Hartree-Fock model is that each electron should be assigned for a single spin-orbital. These spin orbitals are then multiplied together and antisymmetrized to form a single Slater determinant. Most stable molecules have singlet ground states and may be described by a Hartree-Fock wavefunction in which each spacial orbital occurs twice once with a spin and once with /J. Such systems are called closed shell systems. The Hartree-Fock wavefunction for a closed shell system may be written in the form ... 

Another manifestation of the reciprocity of densities in r- and p-space is provided by Fig. 19.2. It shows the radial electron number density D r) = Aiir pir) and radial momentum density /(p) = Aitp nip) for the ground state of the beryllium atom calculated within the Hartree-Fock model in which the Be ground state has a ls 2s configuration. Both densities show a peak arising from the Is core electrons and another from the 2s valence electrons. However, the origin of the peaks is reversed. The sharp,... 

The CEO computation of electronic structure starts with molecular geometry, optimized using standard quantum chemical methods, or obtained from experimental X-ray diffraction or NMR data. For excited-state calculations, we usually use the INDO/S semiempirical Hamiltonian model (Section IIA) generated by the ZINDO code " however, other model Hamiltonians may be employed as well. The next step is to calculate the Hartree— Fock (HE) ground state density matrix. This density matrix and the Hamiltonian are the Input Into the CEO calculation. Solving the TDHE equation of motion (Appendix A) Involves the diagonalization of the Liouville operator (Section IIB) which is efficiently performed using Kiylov-space techniques e.g., IDSMA (Appendix C), Lanczos (Appendix D), or... 

Calculation of accurate excited states of molecules is an order of magnitude more difficult than calculations of the ground state. The Hartree-Fock model can be used... 

TABLE 6. A comparison of calculated correlation energies for the ground state of the water molecule at the optimized geometry determined with the matrix Hartree-Fock model and the cc-pVDZ basis set. See text for further details."... 

Having considered the Hartree-Fock description of the helium atom, let us now turn to the exact wave function for this system. Figure 7.2 shows, for the exact ground-state wave function, a plot similar to that for the Hartree-Fock wave function in Figure 7.1. As seen from the distortion of the concentric contour lines close to the fixed electron, the amplitude of the free electron now depends on its position relative to the nucleus as well as to the fixed electron. Moreover, a careful comparison of Figures 7.1 and 7.2 reveals that the probability of finding the two electrons close to each other is overestimated at the Hartree-Fock level. Still, the description afforded by the Hartree-Fock model is reasonably accurate, differing from that of the exact wave function only in the details. 

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. 

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