Hartree-Fock method self-consistent

The Hartree-Fock or self-consistent field (SCF) method is a procedure for optimizing the orbital functions in the Slater determinant (9.1), so as to minimize the energy (9.4). SCF computations have been carried out for all the atoms of the periodic table, with predictions of total energies and ionization energies generally accurate in the 1-2% range. Fig. 9.2 shows the electronic radial distribution function in the argon atom, obtained from a Hartree-Fock computation. The shell structure of the electron cloud is readily apparent. 

B. The Hartree-Fock and Self-Consistent Field (SCF) Methods. 143... 

Hartree -Fock or Self-Consistent Field (SCF) Method Spin Optimized Self-Consistent Field Method Configuration Interaction Iterative Natural Orbital Method Multi-Configuration SCF Many Body Perturbation Theory Valence-Bond Method Pair-Function or Geminal Method... 

In this substection we will shortly discuss the computational methods used for calculation of the spin-spin coupling constants. Two main approaches available are ab initio theory from Hartree-Fock (or self-consistent field SCF) technique to its correlated extensions, and density function theory (DFT), where the electron density, instead of the wave function, is the fundamental quantity. The discussion here is limited to the methods actually used for calculation of the intermolecular spin-spin coupling constants, i. e. multiconfigurational self consistent field (MCSCF) theory, coupled cluster (CC) theory and density functional theory (DFT). For example, the second order polarization propagator method (SOPPA) approach is not... 

Table 1 contains some further information useful to characterize the different contributions to the molecule/surface interaction orientation dependence and the typical strength of the different contributions, and whether or not they can be understood on a purely classical basis. If one wants to calculate molecule/surface interactions by means of quantum-mechanical or quantum-chemical methods, the most important question is whether standard density functional (DPT) or Hartree-Fock theory (self consistent field, SCF) is sufficient for a correct and reliable description. Table 1 shows that all contributions except the Van der Waals interaction can be obtained both by DPT and SCF methods. However, the results might be connected with rather large errors. One famous example is that the dipole moment of the CO molecule has the wrong sign in the SCF approximation, with the consequence that SCF might yield a wrong orientation of CO on an oxide surface (see also below). In such cases, the use of post Hartree-Fock methods or improved functionals is compulsory. 

QSAR - Quantitative Structure-Activity (or Property) Relation RHF - Restricted Hartree Fock SCF - Self-Consistent Field Method STO - Slater-type Orbital... 

The problem to be solved is expressed by Parr [10] in the following way. For an electronic system containing an even number of electrons, how can we find the best single determinantal wave function of the form (6.36) We calculate the energy using (6.39) for successive sets of orthonormal trial functions until the minimum energy is obtained. This process is known as the Hartree-Fock or self-consistent-field (SCF) method [11,12]. 

The book s simple and illustrative presentation of eoneepts and analyses inelude both basic physical-chemical quantum principles and observabihty at each level of matter s oiganization as well as advanced (more abstract, thus most necessary) formalisms of density matrix, path integrals, Hartree-Fock (as self-consistent quantum methods), original Heisenberg uncertainty by a sub-quantum extension (with more quantum fluctuation insight for the free evolution modeling), eventually leading to a novel undulatory/corpuscular characterization of the Si-based nano/ mesosystems. 

Largest 2D neutral cluster We have investigated the neutral structures of Big isomers applying the Hartree-Fock (HF) self-consistent-field (SCF) methods using the small basis set ST0-3G. The preliminary all-electron calculations show that the most stable neutral boron cluster... 

The standard way of treating conjugated systems using the o-n separation approximation in the 1930s and into the 1960s was the so-called Hiickel methody In this method, the electron-electron interactions are not explicitly considered. Rather, the positions of the nuclei are fixed, and the electrons move in the field of the nuclei. Much of the error that results from neglecting the interactions of the electrons with one another can be circumvented with proper adjustment of empirical parameters. These parameters are also adjusted to allow (approximately) for the interaction from the o system, which is thus taken into account without specific calculations. This method is crude but does often give qualitative results that enable rationalization of many chemical phenomena of interest. It was a powerful and useful tool in its time. A better approximation is the Hartree-Fock or self-consistent field (SCF) method in which the electron-electron interactions are explicitly considered (Self-Consistent Field Method in Chapter 3). The quantum mechanical calculations on the n system in this case are carried out in a... 

In an interdisciplinary volume such as the present, it does not seem appropriate to give any sort of detailed coverage to the theoretical methods currently in use (, 7 ). It must be noted, however, that the Hartree-Fock or Self-Consistent-Field (SCF) method remains at the core of electronic structure theory. Although SCF theory is sometimes adequate in describing potential energy surfaces, this has turned out more often not to be the case. That is, electron correlation, which incorporates the instantaneous repulsions of pairs of electrons, can have a qualitative effect on the topology of fluorine hydrogen potential surfaces. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

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... 

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],... 

In the quantitative development of the structure in the self-consistent field approximation (S.C.F.) using the Hartree-Fock method the energy Ei is made up of three terms, one for the mean kinetic energy of the electron in one for its mean potential energy in the field of the nuclei, and a... 

Keywords strongly correlated electrons nondynamic correlation density matrix renormalization group post Hartree-Fock methods many-body basis matrix product states complete active space self-consistent field electron correlation... 

In most atomic programs (5) is actually solved self-consistently either in a local potential or by the relativistic Hartree-Fock method. There is, however, an important time-saving device that is often used in energy band calculations for actinides where the same radial Eq. (5) must be solved If (5.a) is substituted into (5.b) a single second order differential equation for the major component is obtained... 

The starting point of the creation of the theory of the many-electron atom was the idea of Niels Bohr [1] to consider each electron of an atom as orbiting in a stationary state in the field, created by the charge of the nucleus and the rest of the electrons of an atom. This idea is several years older than quantum mechanics itself. It allows one to construct an approximate wave function of the whole atom with the help of one-electron wave functions. They may be found by accounting for the approximate states of the passive electrons, in other words, the states of all electrons must be consistent. This is the essence of the self-consistent field approximation (Hartree-Fock method), widely used in the theory of many-body systems, particularly of many-electron atoms and ions. There are many methods of accounting more or less accurately for this consistency, usually named by correlation effects, and of obtaining more accurate theoretical data on atomic structure. 

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