Self-consistent field methods, correlation

MCSCF (multiconfigurational self-consistent field) a correlated ah initio method... 

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... 

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... 

This expression excludes self-interaction. There have been a number of attempts to include into the Hartree-Fock equations the main terms of relativistic and correlation effects, however without great success, because the appropriate equations become much more complex. For a large variety of atoms and ions both these effects are fairly small. Therefore, they can be easily accounted for as corrections in the framework of first-order perturbation theory. Having in mind the constantly growing possibilities of computers, the Hartree-Fock self-consistent field method in various... 

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). 

Direct ab initio methods, in which data are recomputed when required, rather than being stored and retrieved, provide an alternative that seems more useful for parallel development. The simplest level of ad initio treatment (self-consistent field methods) can be readily parallelized when direct approaches are being exploited. Experience demonstrates, however, that data replication methods will not lead to truly scalable implementations, and several distributed-data schemes (described later) have been tried. These general approaches have also been used to develop scalable parallel implementations of density functional theory (DFT) methods and the simplest conventional treatment of electron correlation (second-order perturbation theory, MP2) by several groups. 3-118... 

The third entry refers to the self-consistent field method, developed by Hartree. Even for the best possible choice of one-electron functions V (r), there remains a considerable error. This is due to failure to include the variable rn in the wave-function. The effect is known as electron correlation. The fourth entry, containing a simple correction for correlation, gives a considerable improvement. Hylleraas in 1929 extended this approach with a variational function of the form... 

Hyperfine couplings, in particular the isotropic part which measures the spin density at the nuclei, puts special demands on spin-restricted wave-functions. For example, complete active space (CAS) approaches are designed for a correlated treatment of the valence orbitals, while the core orbitals are doubly occupied. This leaves little flexibility in the wave function for calculating properties of this kind that depend on the spin polarization near the nucleus. This is equally true for self-consistent field methods, like restricted open-shell Hartree-Fock (ROHF) or Kohn-Sham (ROKS) methods. On the other hand, unrestricted methods introduce spin contamination in the reference (ground) state resulting in overestimation of the spin-polarization. 

Looking at the history of correlation from the fifties to the seventies, one may be led to ask whether correlation has been a scientific fashion or a real problem. Twenty years ago, almost everybody seemed to accept the idea that the simple molecular orbital method (MO) must be completed by configuration interaction (Cl), in order to obtain reliable prediction for the physical properties of atoms and molecules. Ten years ago, electron correlation was considered as the central problem of Quantum Chemistry (7). Nowadays, about 90% of the quantum-mechanical calculations on molecules are performed by the self-consistent-field method (SCF) using more or less extended sets of basis functions, without any consideration of the possible effects of correlation. 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

There is, in principle, nothing which limits the self-consistent field method to any particular form of the exchange-correlation potential, and the procedure outlined above has been used in connection with several approximations for exchange and correlation. Most notable in this respect is SLATER S Xa method [1.4] which has been applied to all atoms in the periodic table, to some molecules, and in the majority of the existing electronic-structure calculations for crystalline solids. 

However, although, starting from this point, many sophisticated methods for wave function expansion, for example, the coupled cluster approach, multi-configuration self-consistent-field method or multi-reference Cl methods, have been developed, the correlation problem faced many computational limitation, some of them almost insurmountable, due to the immense number of integrals to be evaluated. 

In order to make up for these defects of the self-consistent field method some empirical or semi-empirical corrections must be introduced, the calibration of which are obtained by a close comparison of the theoretical results with experimental data for some fundamental compounds. The resulting procedure in its current form is the so-called Pariser-Parr-Pople approximation of the SCF method. The results obtained can still be improved by configuration mixing correcting for the residual correlation error. 

W. Kutzelnigg, Theor. Chim. Acta, 80, 349 (1991). Error Analysis and Improvements of Coupled-Cluster Theory. W. Kutzelnigg, in Modern Theoretical Chemistry, Vol. Ill, Methods of Electronic Structure Theory H. F. Schaefer 111, Ed., Plenum, New York, 1977, pp. 129-188. Pair Correlation Theories. A. C. Wahl and G. Das, in Modem Theoretical Chemistry, Vol. Ill, Methods of Electronic Structure Theory, H. F. Schaefer 111, Ed., Plenum, New York, 1977, pp. 51-78. The Multiconfiguration Self-Consistent Field Method. 1. Shavitt, in Modem Theoretical Chemistry Vbl. Ill, Methods of Electronic Structure Theory, H. F. Schaefer III, Ed., Plenum, New York, 1977, pp. 189-276. The Method of Configuration Interaction. 

Self-consistent-field and correlated calculations have now been made for a very large number of systems. The best way to judge the capabilities of these methods is to survey some of the results. 

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