Self-consistent-field treatments

SCF, see Self-consistent field treatment (SCF) Schroedinger equation, 2,4,74 Secular equations, 6,10, 52 solution by matrix diagonalization, 11 computer program for, 31-33 Self-consistent field treatment (SCF), of molecular orbitals, 28 Serine, structure of, 110 Serine proteases, 170-188. See also Subtilisin Trypsin enzyme family comparison of mechanisms for, 182-184, 183... 

Recently a new analytic approach has been initiated by Edwards36 using a self-consistent field treatment similar to that of Hartree for atomic wave functions. (Edwards also gives a more rigorous justification in terms of functional integration.) The resulting formula for mean square end-to-end distance is... 

SCF, see Self-consistent field treatment (SCF) Schroedinger equation, 2,4, 74 Secular equations, 6,10, 52... 

However, beyond this overt similarity, there are differences. For example, Covalon by the nature of covalency would have to operate under a much more stringent correlation than that existing in the Frohlich s model between one paired n-electron and all other such pairs along the chain. This is a natural consequence of distortion in the alternating single double bonds. This treatment also differs from that of self-consistent field treatment [19] of a linear chain and that of Little [20] in our inclusion of bond vibration. Covalon also differs from polaron treatments [21] in the consideration of the movement of spin-paired correlated electrons in a covalent bond, instead of movement of spin-uncorrelated electrons in the zeroth order. 

The result that the decay length , of the order parameter oscillations remains finite at yc is also suggested by Shull [20] using a numerical self-consistent field treatment. On the other hand, for not too strong Hj and %<%c his treatment is in good numerical agreement with the result of Fredrickson [12], and hence one can conclude that the extension to thin films [61] described above is sensible. 

Many ionization potentials have now been calculated for simple and complex molecules using more sophisticated self-consistent field treatments and, when the effect of electron correlation is considered, extremely good results may be obtained (e.g. Hush and Pople, 1954). Because ionization is rapid, the Franck-Condon principle applies in the calculation of ionization potentials, and the structure of the ion immediately after formation is essentially that of the molecule. On vibration, the geometry of the ion may change. 

In a more sophisticated self-consistent field treatment, Boer et al. (1968) have discussed aspects of the McLafferty rearrangement. For... 

Ellison, F. O. Shull, H. (1955). Molecular Calculations. I. LCAO MO self-consistent field treatment of the ground state of HjO. f. Chem. Phys. 23, 2348-57. [5, 7]... 

In summary, the computer simulation of micelles with shells formed by an annealed PE is a combination of MC with the self-consistent field treatment of electrostatic forces. However, it goes beyond the mean-field approximation. It is also evident from simulation snapshots (see below) that the a priori assumption of the spherical symmetry of the electrostatic field (which is an inherent feature of studied micelles) does not impose a strong constraint on instantaneous chain conformations. 

Amovilli et al. [24] essentially replaced the use of TF statistical theory in March [1] with HF theory for the fullerenes C50, Qq, C70, and Cg4 with almost spherical C cages. At this point, it is relevant to make a brief digression to relate to the lower dimensionality example of planar ring clusters. The work of Amovilli and March [36] is, essentially, the two-dimensional analogue of the TF self-consistent field treatment of March [1],... 

Obtaining the wavefunction of a given electron, however, requires assuming the wavefunctions of all others. Thus, a self-consistent field treatment must rely on a method of successive approximation for each electronic wavefunction, and consequently, choosing the initial wavefunction form is once again critical. An intuitive... 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

One approach to the problem of the ri2 temi is a variational self-consistent field approximation. Our treatment here follows that by Rioux (1987), in which he starts... 

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). 

In the bibliography, we have tried to concentrate the interest on contributions going beyond the Hartree-Fock approximation, and papers on the self-consistent field method itself have therefore not been included, unless they have also been of value from a more general point of view. However, in our treatment of the correlation effects, the Hartree-Fock scheme represents the natural basic level for study of the further improvements, and it is therefore valuable to make references to this approximation easily available. For atoms, there has been an excellent survey given by Hartree, and, for solid-state, we would like to refer to some recent reviews. For molecules, there does not seem to exist something similar so, in a special list, we have tried to report at least the most important papers on molecular applications of the Hartree-Fock scheme, t... 

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

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. 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

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],... 

The complete treatment of solvation effects, including the solute selfpolarization contribution was developed in the frame of the DFT-KS formalism. Within this self consistent field like formulation, the fundamental expressions (96) and (97) provide an appropriate scheme for the variational treatment of solvent effects in the context of the KS theory. The effective KS potential naturally appears as a sum of three contributions the effective KS potential of the isolated solute, the electrostatic correction which is identified with the RF potential and an exchange-correlation correction. Simple formulae for these quantities have been presented within the LDA approximation. There is however, another alternative to express the solva-... 

Scattering, Reactive, in Molecular Beams (Herschbach). Self-Consistent Field Molecular-Orbital Treatment of 10 319... 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

Just before returning to Europe in 1929, Slater generalized into an N-electron system the wave function used by Pauling in the treatment of helium in the 1928 Chemical Reviews essay. The title of Slater s paper, "The Self-Consistent Field and the Structure of Atoms," shows his debt to Hartree, although Slater s method turned out to be a great deal more practical than Hartree s, as well as consistent with the methods of Heitler, London, and Pauling.70... 

Although the MEG model is essentially ionic in nature, it may also be used to evaluate interactions in partially covalent compounds by appropriate choices of the wave functions representing interacting species. This has been exemplified by Tossell (1985) in a comparative study in which MEG treatment was coupled with an ab initio Self Consistent Field-Molecular Orbital procedure. In this way, Tossell (1985) evaluated the interaction of C03 with Mg in magnesite (MgC03). Representing the ion by a 4-31G wave function, holding its... 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

In the hands of Lipscomb and others 76,143, 145a, 153,189, 190), such localized bond schemes, particularly when obtained via self-consistent field (SCF) molecular orbital (MO) treatments, have proved particularly valuable for rationalizing the shapes and interatomic distances and estimating the charge distribution in many higher boranes. 

The Hartree-Fock or self-consistent-field approximation is a simplification useful in the treatment of systems containing more than one electron. It is motivated partly by the fact that the results of Hartree-Fock calculations are the most precise that still allow the notion of an orbital, or a state of a single electron. The results of a Hartree-Fock calculation are interpretable in terms of individual probability distributions for each electron, distinguished by characteristic sizes, shapes and symmetry properties. This pictorial analysis of atomic and molecular wave functions makes possible the understanding and prediction of structures, spectra and reactivities. 

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