Hartree-Fock theory, solution

Reproducing the exact solution for the relevant n-electron problem a method ought to yield the same results as the exact solution to the Schrodinger equation to the greatest extent possible. What this means specifically depends on the theory underlying the method. Thus, Hartree-Fock theory should be (and is) able to reproduce the exact solution to the one electron problem, meaning it should be able to treat cases like HeH ... 

The first cell in the last tow of the table represents the Hartree-Fock limit the best approximation that can be achieved without taking electron correlation into account. Its location on the chart is rather far from the exact solution. Although in some cases, quite good results can be achieved with Hartree-Fock theory alone, in many others, its performance ranges from orfly fair to quite poor. We ll look at some these cases in Chapters 5 and 6. 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

When the SCRF method is employed in conjunction with Hartree-Fock theory for the solute, then the Fock operator is given by... 

The usual first ah initio approximation to the wave function leads to the Hartree-Fock theory, where V molecular spin orbitals (. with one for each electron. Then, asking the question what is the single determinant solution with the lowest possible energy, we obtain the Hartree-Fock equations and density, ... 

Orbital interaction theory forms a comprehensive model for examining the structures and kinetic and thermodynamic stabilities of molecules. It is not intended to be, nor can it be, a quantitative model. However, it can function effectively in aiding understanding of the fundamental processes in chemistry, and it can be applied in most instances without the use of a computer. The variation known as perturbative molecular orbital (PMO) theory was originally developed from the point of view of weak interactions [4, 5]. However, the interaction of orbitals is more transparently developed, and the relationship to quantitative MO theories is more easily seen by straightforward solution of the Hiickel (independent electron) equations. From this point of view, the theoretical foundations lie in Hartree-Fock theory, described verbally and pictorially in Chapter 2 [57] and more rigorously in Appendix A. 

Numerical solutions of the Schrodinger equation can be obtained within several degrees of approximation, for almost any system, using its exact Hamiltonian. Density functional theory has proven to be one of the most effective techniques, because it provides significantly greater accuracy than Hartree-Fock theory with just a modest increase in computational cost.io> 3-45 The accuracy of DFT method is comparable, and even greater than other much more expensive theoretical methods that also include electron correlation such as second and higher order perturbation theory. 

The two fundamental building blocks of Hartree-Fock theory are the molecular orbital and its occupation number. In closed-shell systems each occupied molecular orbital carries two electrons, with opposite spin. The occupied orbitals themselves are only defined as an occupied one-electron subspace of the full space spanned by the eigenfunctions of the Fock operator. Transformations between them leave the total HF wave function invariant. Normally the orbitals are obtained in a delocalized form as the solutions to the HF equations. This formulation is the most relevant one in studies of spectroscopic properties of the molecule, that is, excitation and ionization. The invariance property, however, makes a transformation to locahzed orbitals possible. Such localized orbitals can be valuable for an analysis of the chemical bonds in the system. 

There are a number of important differences between single-determinant root functions and multiconfigurational root functions. One of the most important is that a one-electron operator cannot in general have a multiconfigurational Pg as an eigenfunction. Consider first the ordinary Mpller-Plesset perturbation series The Hq operator is chosen to be Hq = F, the Fock operator of Hartree-Fock theory. The orbitals are usually chosen to diagonalize F. The solution of the perturbation... 

We can now go back to the QM aspects of the PCM model by considering the methods for approximated solution of the effective non-linear Schrodinger equation for the solute. In principle, any variationedly approximated solution of the effective Schrodinger equation can be obtained by imposing that first-order variation of G with respect to an arbitrary vaxiar tion of the solute wavefunction is zero. This corresponds to a search of the minimum of the free energy functional within the domain of the variar tional functional space considered. In the case of the Hartree-Fock theory,... 

At present, a newcomer to solid-state chemistry might therefore believe that this science must have been a key proponent in challenging quantum mechanics (and quantum chemistry, too) for the solution of solid-state chemical problems. Strangely, this is not at all the case. Let us remind ourselves that the puzzle of chemical bonding was ingeniously clarified in 1927, not for a crystalline solid but for the hydrogen molecule. The rapidly emerging scientific discipline, quantum chemistry, also focused on the molecular parts of chemistry both because of technical and "political" reasons first, the most important quantum-chemical workhorse (Hartree-Fock theory) has been particularly resistant to adaptation to the solid state (see Section 2.11.3) and, second, we surely must be aware of the fact that the solid-state chemical commimity is limited in size such that the number of "customers" for quantum chemists is relatively small. As a sad consequence, the solid-state chemists have been left alone for some... 

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