Hartree-Fock method general equations

But alas most of what has been described so far concerning density theory applies in theory rather than in practice. The fact that the Thomas-Fermi method is capable of yielding a universal solution for all atoms in the periodic table is a potentially attractive feature but is generally not realized in practice. The attempts to implement the ideas originally due to Thomas and Fermi have not quite materialized. This has meant a return to the need to solve a number of equations separately for each individual atom as one does in the Hartree-Fock method and other ab initio methods using atomic orbitals. 

Open-shell Pseudohamiltonians.—The majority of atoms do not have valence structures which can be represented by the fully closed-shell wavefunction of equation (14), and consequently ab initio pseudopotentials cannot be derived directly from the theory outlined above. Acceptable wavefunctions for such atoms require either more than one determinant or the use of the symmetry-equivalenced or generalized Hartree-Fock method, and usually include partially filled shells. The total all-electron wavefunction may be symbolically expressed in terms of four subspaces,... 

As shown in Section 9.1, the A -electron wavefunction is more correctly represented by a Slater determinant of spinorbitals (9.1) rather than a Hartree product of orbitals (9.20), thus accounting automatically for the exclusion principle and the indistinguishability of electrons. The Hartree-Fock method, developed in 1930, is a generalization of the SCF based on Slater determinant wavefimctions. The Hartree-Fock (HF) equations for the spinorbitals (pa have the form... 

An alternative computational scheme which has proven very successful is based on a hybrid between DFT and the Hartree-Fock method. The Kohn-Sham equations are a generalization of the HF equations (9.21)... 

A concrete example of the variational principle is provided by the Hartree-Fock approximation. This method asserts that the electrons can be treated independently, and that the -electron wavefimction of the atom or molecule can be written as a Slater determinant made up of orbitals. These orbitals are defined to be those which minimize the expectation value of the energy. Since the general mathematical form of these orbitals is not known (especially in molecules), then the resulting problem is highly nonlinear and formidably difficult to solve. However, as mentioned in subsection (A 1.1.3.2). a common approach is to assume that the orbitals can be written as linear combinations of one-electron basis functions. If the basis functions are fixed, then the optimization problem reduces to that of finding the best set of coefficients for each orbital. This tremendous simplification provided a revolutionary advance for the application of the Hartree-Fock method to molecules, and was originally proposed by Roothaan in 1951. A similar form of the trial function occurs when it is assumed that the exact (as opposed to Hartree-Fock) wavefimction can be written as a linear combination of Slater determinants (see equation (A 1.1.104 ) ). In the conceptually simpler latter case, the objective is to minimize an expression of the form... 

Optimal Spinorbitals Are Solutions of the Fock Equation (General Hartree-Fock Method) Unrestricted Hartree-Fock (UHF) Method... 

We have derived the general Hartree-Fock method (GHF, p. 407) providing completely free variations for the spinorbitals taken from Eq. (8.1). As a result, the Fock equation [Eq. (8.27)] was derived. 

What has been said previously about the Hartree-Fock method is only a sort of general theory. The time has now arrived to show how the method works in practice. We have to solve the Haitree-Fock-Roothaan equation (ef. Chapter 8, pp. 431 and 531). 

In general, the Hartree-Fock method indicates this SCF-based method. Despite the simplicity of the procedure, it soon became clear that solving this equation is not-trivial for usual molecular electronic systems. The Hartree-Fock equation essentially cannot be solved for molecules without computers. Actually, solving the Hartree-Fock equation for molecules had to await the appearance of general-purpose computers. 

The Fock equation for optimal spinorbitals (General Hartree-Fock method - GHF)... 

Excited states may be studied using the general post-Hartree-Fock methods listed above, or some specialized techniques, such as configmation interaction with single substitutions (CIS) (Foresman et al. 1992), time-dependent density functional theory (TDDFT) (Dreuw and Head-Cordon 2005 Elliott et al. 2009), equations-of-motion coupled cluster (EOM-CC) (Kowalski and Piecuch 2004 Wloch et al. 2005). 

The main importance of Cl is that the FCI calculations provide results that are used as benchmarks for testing other post-Hartree-Fock methods. Less important seems to be the use of Cl as a post-Hartree-Fock method in routine chemical applications, because results of about the same accuracy may be obtained more economically by other methods. The size inconsistency of CI-SD may also be a drawback in some applications. Still, the recent progress in the development of Cl programs indicates that Cl might regain its importance even in this field. The traditional domain of Cl has been in electronic spectroscopy and excited electronic states in general. This is still true for semiempirical calculations. For ab initio calculations, however, it may be preferable to use multi-reference second-order perturbation theory, SAC-CI, or the equation-of-motion CC approach. 

Dirac s one-particle equation was soon generalized to an equation for an electron in the self-consistent field of the other electrons in an atom - the relativistic analogue of the Hartree-Fock approximation for many-electron atoms. These equations - together with the proper bookkeeping to account for the notion of the Dirac sea - - are the basis of what is nowadays called Dirac-Hartree-Fock methods in relativistic electronic structure theory. ... 

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