Variational self-consistent-field

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... 

Application of the variational self-consistent field method to the Haitiee-Fock equations with a linear combination of atomic orbitals leads to the Roothaan-Hall equation set published contemporaneously and independently by Roothaan and Hall in 1951. For a minimal basis set, there are as many matr ix elements as there are atoms, but there may be many more elements if the basis set is not minimal. 

The Pauli Hamiltonian is ideally suited for carrying out relativistic corrections as a first-order perturbation to a non-relativistic Hamiltonian. In recent years, several authors have considered inclusion of the Pauli terms in variational self-consistent field (SCF) calculations. Wadt, Hay and... 

Although the Pauli Hamiltonian was derived for use in perturbational computations as corrections to nonrelativistic Hamiltonian, these terms have been used in variational self-consistent field (SCF) calculations by Hay and Wadt, These authors have incorporated the relativistic terms into the ECPs, except that the spin-orbit effects are not included in their ECPs, and, thus, in the molecular calculations based on this scheme. 

A Self-Consistent Field Variational Calculation of IP for the Helium Atom... 

The multiconfigurational self-consistent field ( MCSCF) method in whiehthe expeetation value < T H T>/< T T>is treated variationally and simultaneously made stationary with respeet to variations in the Ci and Cy,i eoeffieients subjeet to the eonstraints that the spin-orbitals and the full N-eleetron waveflmetion remain normalized ... 

At the energy minimum, each electron moves in an average field due to the Other electrons and the nuclei. Small variations in the form of the orbitals at this point do not change the energy or the electric field, and so we speak of a self-consistent field (SCF). Many authors use the acronyms HF and SCF interchangeably, and I will do so from time to time. These HF orbitals are found as solutions of the HF eigenvalue problem... 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

Xie W, Song L, Trahlar DG, Gao J (2008) Incorporation of QM/MM buffer zone in the variational double self-consistent field method. J Phys Chem B 112(45) 14124-14131... 

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-... 

A selection of the predictions of the equilibrium structure of DPPC bilayers as found by numerical self-consistent-field calculations is given in the following figures. In a series of articles, the SCF predictions for such membranes were published, starting in the late 1980s. As discussed above, we will update these early predictions for the theory outlined above with updated parameter sets. The calculations are very inexpensive with respect to the CPU time, and thus variations of the parameter-set will also provide deeper insight into the various subtle balances that eventually determine the bilayer structure - the mechanical properties as well as the thermodynamic properties. 

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]. 

Density-Functional Theory. Transition metals pose a problem for classical quantum chemical methods like self-consistent field (SCF), perturbation theory, configuration interaction (Cl), and variations on these methods, because of the very large electron correlation. SCF underestimates binding substantially, and post-SCF methods are so expensive for transition metals that one can do a calculation only on models with few atoms. DFT on the other hand is relatively cheap it is about as expensive as SCF. Moreover, with the development of the generalized-gradient approximations it is also reasonably accurate. A large majority of quantum chemical... 

MCSCF calculations in practice require much more technical expertise than do singleconfiguration HF analogs. One particularly difficult problem is that spurious minima in coefficient space can often be found, instead of the variational minimum. Thus, convergence criteria are met for die self-consistent field, but the wave function is not really optimized. It usually requires a careful inspection of the orbital shapes and, where available, some data on relative energetics between related species or along a reaction coordinate to ascertain if this has happened. 

The simplest version of the self-consistent field approach is the Hartree method, in which the variational principle is applied to a non-symmetrized product of wave functions, and the orthogonality conditions for functions with different n are neglected. This leads to neglecting the exchange part of the potential, which causes errors in the results. 

The somewhat arbitrary use of the auxiliary inductive parameter in the Hiickel MO calculations has been questioned and the effect of the nonuniform distribution of a-electron densities, particularly in the CN bond,90 upon the 77-electron distribution has been discussed.65 Variable electronegativity self-consistent field (VESCF) molecular orbital calculations, which are an elaboration of the conventional SCF method and allow for the variation in electronegativity of an... 

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