Hartree Fock generalized

Besides the traditional scheme, AIMD using semiempirical, Hartree-Fock, generalized valence bond (GVB), complete active space (CASSCF), and configuration interaction (CI) electronic structure methods have been realized. Several different variations concerning the basis set... 

Although orbital wave functions, such as Hartree-Fock, generalized valence bond, or valence-orbital complete active space self-consistent field wave functions, provide a semi-quantitative description of the electronic structure of molecules, accurate predictions of molecular properties cannot be made without explicit inclusion of the effects of dynamical electron correlation. The accuracy of correlated molecular wave functions is determined by two inter-related expansions the many-electron expansion in terms of antisymmetrized products of molecular orbitals that defines the form of the wave function, and the basis set used to expand the one-electron molecular orbitals. The error associated with the first expansion is the electronic structure method error the error associated with the second expansion is the basis set error. Only by eliminating the basis set error, i.e., by approaching the complete basis set (CBS) limit, can the intrinsic accuracy of the electronic structure method be determined. 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

HyperChein perforins ab initio. SCK calculations generally. It also can calculate the coi relation energy (to he added to the total -SCK energy) hy a post Hartree-Fock procedure call. M P2 that does a Moller-Plesset secon d-order perturbation calculation. I he Ml 2 procedure is on ly available for sin gle poin t calculation s an d on ly produces a single tiuin ber, th e Ml 2 correlation energy, to be added to the total SCF en ergy at th at sin gle poin t con figuration of th e ti iiclei. 

Drowicz F W and W A Goddard IB 1977. The Self-Consistent Field Equations for Generalized Valence Bond and Open-Shell Hartree-Fock Wave Functions. In Schaeffer H F III (Editor). Modem Theoretical Chemistry III, New York, Plenum, pp. 79-127. 

DFT is generally faster than Hartree-Fock for systems with more than 10-... 

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. 

Not all iterative semi-empirical or ab initio calculations converge for all cases. For SCF calculations of electronic structure, systems with a small energy gap between the highest occupied orbital and the lowest unoccupied orbital may not converge or may converge slowly. (They are generally poorly described by the Hartree-Fock method.)... 

Each cell in the chart defines a model chemistry. The columns correspond to differcni theoretical methods and the rows to different basis sets. The level of correlation increases as you move to the right across any row, with the Hartree-Fock method jI the extreme left (including no correlation), and the Full Configuration Interaction method at the right (which fuUy accounts for electron correlation). In general, computational cost and accuracy increase as you move to the right as well. The relative costs of different model chemistries for various job types is discussed in... 

Hartree-Fock theory does quite a poor job of predicting the structures and frequencies for these compounds. It produces highly distorted structures in all three cases, and its computed frequencies bear little resemblance to the experimental observations. MP2 theory generally does better for the structures, although it fails to located a distorted structure for Na F3. The frequencies computed at the MP2 level also vary widely from experiment. 

Note that the frequency calculation produces many more frequencies than those listed here. We ve matched calculated frequenices to experimental frequencies using symmetry types and analyzing the normal mode displacements. The agreement with experiment is generally good, and follows what might be expected of Hartree-Fock theory in the ground state. ... 

In general, DFT calculations proceed in the same way as Hartree-Fock calculations, with the addition of the evaluation of the extra term, This term cannot be evaluated analytically for DFT methods, so it is computed via numerical integration. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

General expressions for the force constants and dipole derivatives of molecules are derived, and the problems arising from their practical application are reviewed. Great emphasis is placed on the use of the Hartree-Fock function as an approximate wavefunction, and a number... 

In general, the basis set should be in balance with the computational method A highly sophisticated method [e.g., CCSD(T)] in combination with a small basis or a low-level method [e.g., Hartree-Fock (HE)] in combination with a very large basis may be useful only in very specific cases. Consequently, increasing the basis set should be done while increasing the quality of the post-HF approach for a better representation of the electron correlation. 

Regarding current ab initio calculations it is probably fair to say that they are not really ab initio in every respect since they incorporate many empirical parameters. For example, a standard HF/6-31G calculation would generally be called "ab initio", but all the exponents and contraction coefficients in the basis set are selected by fitting to experimental data. Some say that this feature is one of the main reasons for the success of the Pople basis sets. Because they have been fit to real data these basis sets, not surprisingly, are good at reproducing real data. This is said to occur because the basis set incorporates systematical errors that to a large extent cancel the systematical errors in the Hartree-Fock approach. These features are of course not limited to the Pople sets. Any basis set with fixed exponent and/or contraction coefficients have at some point been adjusted to fit some data. Clearly it becomes rather difficult to demarcate sharply between so-called ab initio and semi-empirical methods.4... 

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. 

The idea of constructing a good wave function of a many-particle system by means of an exact treatment of the two-particle correlation is also underlying the methods recently developed by Brueck-ner and his collaborators for studying nuclei and free-electron systems. The effective two-particle reaction operator and the self-consistency conditions introduced in this connection may be considered as generalizations of the Hartree-Fock scheme. 

A common feature of the Hartree-Fock scheme and the two generalizations discussed in Section III.F is that all physical results depend only on the two space density matrices p+ and p, which implies that the physical and mathematical simplicity of the model is essentially preserved. The differences lie in the treatment of the total spin in the conventional scheme, the basic determinant is a pure spin function as a consequence of condition 11.61, in the unrestricted scheme, the same determinant is a rather undetermined mixture of different spin states, and, in the extended scheme, one considers only the component of the determinant which has the pure spin desired. 

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... 

Eden, R. J., Phys. Rev. 99, 1418, "Nuclear saturation a generalized Hartree-Fock method."... 

Fischer-type carbene complexes, generally characterized by the formula (CO)5M=C(X)R (M=Cr, Mo, W X=7r-donor substitutent, R=alkyl, aryl or unsaturated alkenyl and alkynyl), have been known now for about 40 years. They have been widely used in synthetic reactions [37,51-58] and show a very good reactivity especially in cycloaddition reactions [59-64]. As described above, Fischer-type carbene complexes are characterized by a formal metal-carbon double bond to a low-valent transition metal which is usually stabilized by 7r-acceptor substituents such as CO, PPh3 or Cp. The electronic structure of the metal-carbene bond is of great interest because it determines the reactivity of the complex [65-68]. Several theoretical studies have addressed this problem by means of semiempirical [69-73], Hartree-Fock (HF) [74-79] and post-HF [80-83] calculations and lately also by density functional theory (DFT) calculations [67, 84-94]. Often these studies also compared Fischer-type and... 

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. 

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