Multi configurational self consistent field

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. 

When deriving the Hartree-Fock equations it was only required that the variation of the energy with respect to an orbital variation should be zero. This is equivalent to the first derivatives of the energy with respect to the MO expansion coefficients being equal to zero. The Hartree-Fock equations can be solved by an iterative SCF method, and 

RHF to UHF, or to a TCSCF, is almost pure static correlation. Increasing the number of configurations in an MCSCF will recover more and more of the dynamical correlation, until at the full Cl limit, the correlation treatment is exact. As mentioned above, MCSCF methods are mainly used for generating a qualitatively correct wave function, i.e. recovering the static part of the correlation. 

The full Cl expansion within the active space severely restricts the number of orbitals and electrons that can be treated by CASSCF methods. Table 4.3 shows how many singlet CSFs are generated for an [n, n]-CASSCF wave function (eq. (4.13)), without any reductions arising from symmetry. 

The structure on the left is biradical, while the two others are ionic, corresponding to both electrons being at the same carbon. The simplest CASSCF wave function which qualitatively can describe this system has two electrons in two orbitals, giving the three configurations shown above. The dynamical correlation between die two active electrons will tend to keep them as far apart as possible, i.e. favouring the biradical structure. Now 

It is particularly desirable to use MCSCF or MRCI if the HF wave function yield a poor qualitative description of the system. This can be determined by examining the weight of the HF reference determinant in a single-reference Cl calculation. If the HF determinant weight is less than about 0.9, then it is a poor description of the system, indicating the need for either a multiple-reference calculation or triple and quadruple excitations in a single-reference calculation. 

Unfortunately, these methods require more technical sophistication on the part of the user. This is because there is no completely automated way to choose which configurations are in the calculation (called the active space). The user must determine which molecular orbitals to use. In choosing which orbitals to include, the user should ensure that the bonding and corresponding antibonding orbitals are correlated. The orbitals that will yield the most correlation 

An MCSCF calculation in which all combinations of the active space orbitals are included is called a complete active space self-consistent held (CASSCF) calculation. This type of calculation is popular because it gives the maximum correlation in the valence region. The smallest MCSCF calculations are two-conhguration SCF (TCSCF) calculations. The generalized valence bond (GVB) method is a small MCSCF including a pair of orbitals for each molecular bond. 

As mentioned above, full Cl is impossible, except for very small systems. The only general applicable method is CISD. Consider now a series of CISD calculations in order to construct the interaction potential between two Ho molecules as a function of the distance between them. Relative to the HF wave function, there will be determinants 

MCSCF wave function optimizations afe the energy to second order in the variational parameters (orbital and configurational coefficients), analogously to the second-order SCF procedure described in Section 3.8.1, using Newton-Raphson based motbods descrihed in Chapter 14 tO force -convergence to a minimum.  

MCSCF methods are rarely used for calculating large fractions of the correlation energy. The orbital relaxation usually, does not recover much electron correlation, it is 

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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. 

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. 

The Multi-Configuration Self-Consistent Field (MCSCF) method includes configurations created by excitations of electrons within an active space. Both the coefficients ca of the expansion in terms of CSFs and the expansion coefficients of the... 

By calculating A.U (R) and Al/ (i ) separately, we can straightforwardly calculate the total adiabatic correction V (R) for any isotopes of A and B. The adiabatic corrections are calculated by numerical differentiation of the multi-configurational self-consistent field (MCSCF) wave functions calculated with Dalton [23]. The nurnerical differentiation was performed with the Westa program developed 1986 by Agren, Flores-Riveros and Jensen [22],... 

Spin-restricted and multi-configuration self-consistent-field methods differ in the assumed func-... 

A modification of this scheme has been used for optimizing excited states of multi-configurational self-consistent field wave functions, see H. J. Aa. Jensen and H. Agren, Chem. Phys. 104, 229 (1986). 

MCSCF multi-configuration self-consistent field... 

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. 

Veillard, A., and E. Clementi Complete multi-configuration self-consistent field theory. Theoret. Chim. Acta (Berlin) 7, 133 (1967). 

Linear Combination of Atomic Orbitals Many Body Perturbation Theory Multi-configuration Self Consistent Field Molecules in Molecules... 

Henne Hettema, Hans Jorgen Aa. Jensen, Poul Jorgensen, and Jeppe Olsen (1992). Quadratic response functions for a multi-configurational self-consistent-field wave-function. J. Chem. Phys. 97, 1174-1190. 

MCSCF Multi-Configuration Self-Consistent Field. A means of variationally minimizing the energy of several electron configurations of a given system simultaneously, so as to provide a better description of its electronic structure. 

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