Hartree-Fock self-consistent-field calculations

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. 

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. 

Semiempirical (CNDO, MNDO, ZINDO, AMI, PM3, PM3(tm) and others) methods based on the Hartree-Fock self-consistent field (HF-SCF) model, which treats valence electrons only and contains approximations to simplify (and shorten the time of) calculations. Semiempirical methods are parameterized to fit experimental results, and the PM3(tm) method treats transition metals. Treats systems of up to 200 atoms. 

In earlier theoretical studies Shen and coworkers used Hartree-Fock self-consistent-field (HF) calculations with different basis sets to study water complexes of anionic ONO—0 Two stable ONO—O isomers, cis and trans, formed hydrogen bonds with... 

The Section on More Quantitive Aspects of Electronic Structure Calculations introduces many of the computational chemistry methods that are used to quantitatively evaluate molecular orbital and configuration mixing amplitudes. The Hartree-Fock self-consistent field (SCF), configuration interaction (Cl), multiconfigurational SCF (MCSCF), many-body and Mpller-Plesset perturbation theories,... 

In earlier theoretical studies Shen and coworkers used Hartree-Fock self-consistent-field (HF) calculations with different basis sets to study water complexes of anionic ONO—O-30. Two stable ONO—O isomers, cis and trans, formed hydrogen bonds with H2O molecules at different positions. Second-order Mpller-Plesset perturbation theory (MP2) with a 6-311+G(d,p) basis set has also been applied to the study of ONO—O-, (H2O), (n = 1 or 2) complexes31. Koppenol and Klasinc studied the cis and trans conformers as well as the transition state for torsional motion of ONO—O- at the HF/6-31(d) level32. In their calculations, the trans conformer is slightly more stable than the cis form, and the rotational barrier was thought to be quite high. However, correlated methods (MP2) were also used to study this molecule, and they predict that the cis conformer is more stable than the trans conformer33,34. 

Now we want to discuss IR optical spectra of the C60H36 synthesized at high-pressure. Results of this study were published in Bazhenov et al. (2008). There are a lot of publications devoted to theoretical and experimental study of C60H36. We should pay attention on the existing discrepancies in the results of theoretical calculations of the dipole-active spectra C60H36, compare, for example, papers Bini et al. (1998) and Bulusheva et al. (2001). There were used different theoretical models. Semiempirical method of the MNDO type (Dewar and Thiel 1977) was used in (Bini et al. 1998). Ab initio Hartree-Fock self-consistent field approximation was used in (Bulusheva et al. 2001). 

The accurate calculation of these molecular properties requires the use of ab initio methods, which have increased enormously in accuracy and efficiency in the last three decades. Ab initio methods have developed in two directions first, the level of approximation has become increasingly sophisticated and, hence, accurate. The earliest ab initio calculations used the Hartree-Fock/self-consistent field (HF/SCF) methodology, which is the simplest to implement. Subsequently, such methods as Mpller-Plesset perturbation theory, multi-configuration self-consistent field theory (MCSCF) and coupled-cluster (CC) theory have been developed and implemented. Relatively recently, density functional theory (DFT) has become the method of choice since it yields an accuracy much greater than that of HF/SCF while requiring relatively little additional computational effort. 

The choice of orbital exponent (a) to use for a particular atomic Slater orbital has been the subject of several investigations. Originally, Slater (9) proposed a set of empirical rules for choosing exponents however, these are not used frequently in modern calculations. Hartree-Fock self-consistent-field (SCF)... 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

Ab-initio molecular orbital calculations which use the Hartree-Fock self-consistent field theory with one-electron molecular orbitals. This method is based on the variation theorem to seek the nuclear geometry of the molecule or hydrogen-bonded complex with lowest energy [248-253]. It uses no experimental data. 

In Sections 7 and 8 we present results for the more successful approaches. The calculations presented are of the Hartree-Fock self consistent field type utilizing the INDO/1 [14,15] model Hamiltonian. Although the force constants from... 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born-Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrddinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree-Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ah initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis - indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

It is obvious by symmetry that the coefficients are related ca = cb, a = /b and Ca = =teB, but what about the ratios of ca to a to epP. I ll just mention for now that there is a systematic procedure called the Hartree-Fock self consistent field method for solving this problem. In the special case of the hydrogen molecular ion, which only has a single electron, we can calculate the variational integral and find the LCAO expansion coefficients by requiring that the variational integral is a minimum. Dickinson (1933) first did this calculation using Is and 2porbital exponents to be is = 1.246 and 2pa = 2.965 (See Table 3.2.)... 

In this procedure Hartree-Fock self-consistent field (S.C.F.) or other calculations are carried out on the molecular ground state and on the relevant ionic states, and the calculated energy differences are compared with experimental ionization energies. 

Bond lengths R (A), binding energies D. (eV) and vibrational constants a>e (cm ) of the homonuclear halogen dimers from dl-electron (AE) Douglas-Kroll-HeB (DKH) and valence-only energy-consistent pseudopotential (EC-PP) Hartree-Fock self-consistent field (SCF) calculations. The effects of static and dynamic core-polarization at the valence-only level are modelled by a core-polarization potential (CPP). 

The present quantum-chemical calculations of fullerenes deal with the optimized geometries [177-181] obtained at semiempirical, ab initio Hartree-Fock Self-Consistent Field (HF SCF) or density functional theory (DFT) level while ab initio... 

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