Electron Hartree-Fock method

In a sense, CIS is a certain counterpart of the (1-electron) Hartree-Fock method for excited state. Indeed, the Hartree-Fock (HF) variational parameters can be packed into an 1-electron matrix, C, of the conventional occupied MO expansion coefficients. In turn, the CIS configurational coefficients comprise the matrix which is just the first variational derivative of the C-matrix. More exactly, given the ground state tV-electron Slater determinant ) = (1... A)). This ) is the antisymmetrized product of the spin-orbitals, the latter being the standard spinless spatial MOs, (p°), equipped with spin variables. As usually, MOs °) are... 

While the equations of the Hartree-Fock approach can he rigorously derived, we present them post hoc and give a physical description of the approximations leading to them. The Hartree-Fock method introduces an effective one-electron Hamiltonian. as in equation (47) on page 194 ... 

Within the periodic Hartree-Fock approach it is possible to incorporate many of the variants that we have discussed, such as LFHF or RHF. Density functional theory can also be used. I his makes it possible to compare the results obtained from these variants. Whilst density functional theory is more widely used for solid-state applications, there are certain types of problem that are currently more amenable to the Hartree-Fock method. Of particular ii. Icvance here are systems containing unpaired electrons, two recent examples being the clci tronic and magnetic properties of nickel oxide and alkaline earth oxides doped with alkali metal ions (Li in CaO) [Dovesi et al. 2000]. 

A variation on the HF procedure is the way that orbitals are constructed to reflect paired or unpaired electrons. If the molecule has a singlet spin, then the same orbital spatial function can be used for both the a and P spin electrons in each pair. This is called the restricted Hartree-Fock method (RHF). 

Another way of constructing wave functions for open-shell molecules is the restricted open shell Hartree-Fock method (ROHF). In this method, the paired electrons share the same spatial orbital thus, there is no spin contamination. The ROHF technique is more difficult to implement than UHF and may require slightly more CPU time to execute. ROHF is primarily used for cases where spin contamination is large using UHF. 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

Not all iterative semi-empirical or ab initio calculations converge for all cases. For SCF calculations of electronic structure, systems with a small energy gap between the highest occupied orbital and the lowest unoccupied orbital may not converge or may converge slowly. (They are generally poorly described by the Hartree-Fock method.)... 

Each cell in the chart defines a model chemistry. The columns correspond to differcni theoretical methods and the rows to different basis sets. The level of correlation increases as you move to the right across any row, with the Hartree-Fock method jI the extreme left (including no correlation), and the Full Configuration Interaction method at the right (which fuUy accounts for electron correlation). In general, computational cost and accuracy increase as you move to the right as well. The relative costs of different model chemistries for various job types is discussed in... 

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. 

So far, we have considered only the restricted Hartree-Fock method. For open shell systems, an unrestricted method, capable of treating unpaired electrons, is needed. For this case, the alpha and beta electrons are in different orbitals, resulting in two sets of molecular orbital expansion coefficients ... 

The Hartree-Fock description of the hydrogen molecule requires two spinorbitals, which are used to build the single-determinant two-electron wave function. In the Restricted Hartree-Fock method (RHF) these two spinorbitals are created from the same spatial... 

Table 5.3 Contributions of -orbitals to the total electron density at the iron nucleus (in a.u. ) as a function of oxidation state and configuration. Calculations were done with the spin-averaged Hartree-Fock method and a large uncontracted Gaussian basis set. (17 1 Ip 5d If)... 

The Hartree-Fock method is modified by mixing some important valence electron configurations with the ground-state one 20>. This is called the OVC optimized valence configurations) method. 

In order to find a good approximate wave function, one uses the Hartree-Fock procedure. Indeed, the main reason the Schrodinger equation is not solvable analytically is the presence of interelectronic repulsion of the form e2/r. — r.. In the absence of this term, the equation for an atom with n electrons could be separated into n hydrogen-like equations. The Hartree-Fock method, also called the Self-Consistent-Field method, regards all electrons except one (called, for instance, electron 1), as forming a cloud of electric charge... 

When the Hartree-Fock method is applied to molecules, molecular orbitals are used instead of atomic orbitals. To construct the molecular orbitals, one widely used approximation is LCAO (linear combinations of atomic orbitals). According to molecular orbital theory, the total wave function of the system is written as a combination of molecular orbitals, spin functions describing electrons in terms of spin j(a) or — j p). 

In the former, electrons are assigned to orbitals in pairs, the total spin is zero, so the multiplicity is 1. In this case, the restricted Hartree-Fock method (RHF) can be applied. For radicals with doublet or triplet states, the unrestricted Hartree-Fock (UHF) has to be applied. In this method, a and, 3 electrons (spin up and spin down) are assigned to different spatial orbitals, so there are two distinct sets I and FJf... 

The Hartree-Fock method adequately describes the ground state of most molecules. However, the exact wave function itself should take into account the fact that electrons repel each other and need breathing space. The electrons should be allowed to make use of energy levels which are normally empty in the ground state to maintain this breathing space. In other words, to add terms describing excited states in the ground state wave function. 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

In most work reported so far, the solute is treated by the Hartree-Fock method (i.e., Ho is expressed as a Fock operator), in which each electron moves in the self-consistent field (SCF) of the others. The term SCRF, which should refer to the treatment of the reaction field, is used by some workers to refer to a combination of the SCRF nonlinear Schrodinger equation (34) and SCF method to solve it, but in the future, as correlated treatments of the solute becomes more common, it will be necessary to more clearly distinguish the SCRF and SCF approximations. The SCRF method, with or without the additional SCF approximation, was first proposed by Rinaldi and Rivail [87, 88], Yomosa [89, 90], and Tapia and Goscinski [91], A highly recommended review of the foundations of the field was given by Tapia [71],... 

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