Hartree, self consistent field calculations

By this method one obtains a simple analytic expression for the shielding potential. With the above values of pi and p2 this expression is hardly different from Hartree s self-consistent field calculated via incomparably difficult numerical techniques, and is even perhaps a bit more exact, as it lies between the self-consistent field with and without exchange in the case of the sodium atom. ... 

Hartree-Fock and Self-Consistent-Field Calculations... 

With the success of these calculations for isolated molecules, we began a systematic series of supermolecule calculations. As discussed previously, these are ab initio molecular orbital calculations over a cluster of nuclear centers representing two or more molecules. Self-consistent field calculations include all the electrostatic, penetration, exchange, and induction portions of the intermolecular interaction energy, but do not treat the dispersion effects which can be treated by the post Hartree-Fock techniques for electron correlation [91]. The major problems of basis set superposition errors (BSSE) [82] are primarily associated with the calculation of the energy. 

There is generally little ambiguity associated with calculations of molecular orbitals for closed-shell molecules. The Hartree-Fock method (as appoximated by self-consistent field calculations in necessarily finite atomic orbital basis sets11) provides a solution that obeys some of the symmetry properties that must... 

A different approach to treat correlation effects which are not well described within the LSDA consists in incorporating self-interaction corrections (SIC) [111-114] in electron structure methods for solids, Svane et al. [115-120]. In the Hartree-Fock (HF) theory the electron-electron interactions are usually divided into two contributions, the Coulomb term and the exchange term although they both are Coulomb interactions. The separation though, is convenient because simplifications of self-consistent-field calculations can be obtained by including in both terms the interaction of the electron itself. In the HF theory this has no influence on the solutions because these selfinteractions in the Coulomb and exchange terms exactly cancel each other. However, when the exchange term is treated... 

The first of a series of more extensive molecular-orbital calculations used Hartree-Fock self-consistent field calculations, thereby deriving a value of... 

One of the more successful attempts at constructing an interatomic potential is from Tsune5uiki, et al. [1] who proposed a new interatomic potential for silica. They employed Hartree-Fock self-consistent-field calculations to examine elusters of Si02. From these calculations, they fit pairwise interatomic potentials to the structural energies of the clusters. The form of the interatomie potentials they employed is given by... 

Hartree-Fock self-consistent field calculations indicate that the energy of an electron in the 4s orbital of vanadium lies above that of the 3d orbital in the ground state configuration, [Ar]3dMs. Explain why [Ar]3d 4s and [Ar]3d are less stable configurations than the ground state. 

Our study was extended to the binary system silica-calcia. The parameters of the Born-Mayer potential (Equation 3.4.2) were taken from Ref. [15]. The interatomic potential was calculated from the potential used for pure silica performed by TTAM [3] and derived from the ab initio Hartree-Fock self-consistent field calculations for model clusters of silica. The parameters used in Equation 3.4.2 are summarized in Table 3.4.2. 

A configuration interaction calculation uses molecular orbitals that have been optimized typically with a Hartree-Fock (FIF) calculation. Generalized valence bond (GVB) and multi-configuration self-consistent field (MCSCF) calculations can also be used as a starting point for a configuration interaction calculation. 

Introductory descriptions of Hartree-Fock calculations [often using Rootaan s self-consistent field (SCF) method] focus on singlet systems for which all electron spins are paired. By assuming that the calculation is restricted to two electrons per occupied orbital, the computation can be done more efficiently. This is often referred to as a spin-restricted Hartree-Fock calculation or RHF. 

Ab initio calculations can be performed at the Hartree-Fock level of approximation, equivalent to a self-consistent-field (SCF) calculation, or at a post Hartree-Fock level which includes the effects of correlation — defined to be everything that the Hartree-Fock level of approximation leaves out of a non-relativistic solution to the Schrodinger equation (within the clamped-nuclei Born-Oppenhe-imer approximation). 

In the RISM-SCF theory, the statistical solvent distribution around the solute is determined by the electronic structure of the solute, whereas the electronic strucmre of the solute is influenced by the surrounding solvent distribution. Therefore, the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. It is noted that SCF (self-consistent field) applies not only to the electronic structure calculation but to the whole system, e.g., a self-consistent treatment of electronic structure and solvent distribution. The MO part of the method can be readily extended to the more sophisticated levels beyond Hartree-Fock (HF), such as configuration interaction (Cl) and coupled cluster (CC). 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

For planar unsaturated and aromatic molecules, many MO calculations have been made by treating the a and n electrons separately. It is assumed that the o orbitals can be treated as localized bonds and the calculations involve only the tt electrons. The first such calculations were made by Hiickel such calculations are often called Hiickel molecular orbital (HMO) calculations Because electron-electron repulsions are either neglected or averaged out in the HMO method, another approach, the self-consistent field (SCF), or Hartree-Fock (HF), method, was devised. Although these methods give many useful results for planar unsaturated and aromatic molecules, they are often unsuccessful for other molecules it would obviously be better if all electrons, both a and it, could be included in the calculations. The development of modem computers has now made this possible. Many such calculations have been made" using a number of methods, among them an extension of the Hiickel method (EHMO) and the application of the SCF method to all valence electrons. ... 

The metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

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