Self-consistent field, Hartree-Fock multiconfigurational

C. Froese Fischer, Self-Consistent-Field (SCF) euid Multiconfiguration (MC) Hartree-Fock (HF) Methods in Atomic Calculations Numerical Integration Approaches, Comput. Phys. Rep. 3 (1986) 273-325. 

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 order to make up for those imperfections one needs to turn to post-Hartree-Fock methods. Two variational techniques are worth discussing due to their popularity the configuration-interaction (SCF Cl) method and the multiconfiguration self-consistent-field (MC SCF) method. 

Complete multiconfiguration-self consistent-field (CMC-SCF) technique designates the method where a given occupied molecular orbital of the set is excited to all unoccupied molecular orbitals. If an occupied orbital is excited to one or more, but not all, of the unoccupied orbitals, the technique is described as incomplete MC-SCF (IMC-SCF). The reader is referred to refs. 13 and 14 for details of the derivation. The CMC-SCF formalism differs from most many body techniques presented to date insofar as the Hartree-Fock energy is not assumed to be the zero order energy. 

The adiabatic potential energy curves for these electronic states calculated in the Born-Oppenhelmer approximation, are given in Figure 1. Since we have discussed the choice of basis functions and the choice of configurations for these multiconfiguration self-consistent field (MCSCF) computations (12) previously (] - ), we shall not explore these questions in any detail here. Suffice it to say that the basis set for Li describes the lowest 2s and 2p states of the Li atom at essentially the Hartree-Fock level of accuracy, and includes a set of crudely optimized d functions to accommodate molecular polarization effects. The basis set we employed for calculations involving Na is somewhat less well optimized than is the Li basis in particular, so molecular orbitals are not as well described for Na2 (relatively speaking) as they are for LI2. 

Hurley proposed a simple, sufficient condition for the applicability of the Hellmann-Feynman theorem. " If within a variational framework, the family of trial functions is invariant to changes in parameter a, then the Hellmann-Feynman theorem is satisfied by the optimum trial function. In variational approaches involving Lagrange multipliers, for example, in the Hartree-Fock and multiconfigurational self-consistent field methods, Hurley s condition is fulfilled. ... 

A. Hartree-Fock, Configuration-interaction and Multiconfiguration Self-consistent Field Calculations... 

Configuration Interaction Density Functional Theory Equation of Motion CCSD Multiconfiguration-SCF Moller Plesset 2nd Order Multireference Cl Multistate CASPT2 Restricted Active Space SCF Restricted Hartree Fock Symmetry Adapted Cluster-CI Self Consistent Field Singlestate CASPT2... 

In this substection we will shortly discuss the computational methods used for calculation of the spin-spin coupling constants. Two main approaches available are ab initio theory from Hartree-Fock (or self-consistent field SCF) technique to its correlated extensions, and density function theory (DFT), where the electron density, instead of the wave function, is the fundamental quantity. The discussion here is limited to the methods actually used for calculation of the intermolecular spin-spin coupling constants, i. e. multiconfigurational self consistent field (MCSCF) theory, coupled cluster (CC) theory and density functional theory (DFT). For example, the second order polarization propagator method (SOPPA) approach is not... 

Fig. 4 Potential energy curves for the hydrogen molecule obtained from (a) the Hartree-Fock function [96], (b) the ODC multiconfiguration self-consistent field flmction, (c) the generalized Coulson-Fischer function [77] and (d) the extended James-CooUdge flmction of Krfos and...

We have seen that the canonical Hartree-Fock orbitals are not the best choice of orbitals for use in Cl calculations. Let us consider a multideter-minantal wave function containing a relatively small number of configurations. What orbitals should we use in constructing these configurations so as to obtain the best possible result From the variation principle, it is clear that we should vary the orbitals so as to minimize the energy. This is the central idea of the multiconfiguration self-consistent-field (MCSCF) method. Thus the MCSCF wave function is a truncated Cl expansion... 

The configuration interaction (Cl) procedure is one of the commonly used methods for determination of electronically excited states [30]. Starting from a finite set l/j of orthonormal one-electron basis functions (which can be either Hartree-Fock (HF) or canonical multiconfigurational self-consistent field (MCSCF) orbitals) [30], a subset of all possible antisymmetrized products have to be constructed ... 

Results of quasirelativistic pseudopotential calculations at the unrestricted Hartree-Fock and multiconfiguration self-consistent field level for the equilibrium geometries of the lanthanide trihalides RX3 (X=F, Cl, Br, 1) have been presented by Cundari et al. (1995), Kobayashi et al. (1995) report Hartree-Fock and density-functional calculations of La2 Cgo. 

The calculations were performed with several different levels of correlation treatment Hartree-Fock (HF), configuration interaction with single and double excitations (SDCI), Multiconfiguration self consistent field (CAS), and multireference configuration interaction (MRCI). Relativistic efferts were accounted for using either the Douglas-Kroll method or a relativistic effective core potential approach (RECP). 

The Multi-Configuration Self-Consistent Field method combines the ideas of orbital optimization through a SCF technique as in the Hartree-Fock method, and a multiconfiguration expansion of the electronic wavefunction as in the configuration interaction method. In other words, the electronic wavefunction is still expressed as a linear combination of Slater determinants but now both the coeffi-... 

Calculations of core hole states and core electron binding energies have mostly concerned state-by-state optimization using open-shell Hartree-Fock, multiconfiguration self-consistent field, and configuration interaction techniques, while propagator techniques have been of restricted use owing to the relaxation problem ... 

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