Multi-configuration self-consistent field MCSCF method

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. 

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... 

We will now briefly review some of the methods used to calculate the nonadiabatic coupling elements. As can be easily siumised, the computational scheme of nonadiabatic coupling elements depends heavily on the methods of electronic state calculation such as configuration interaction, multi configurational self consistent field (MCSCF) method, and so on. 

PHF methods can, in turn, be classified as the variational and nonvariational ones. In the former gronp of methods the coefficients in linear combination of Slater determinants and in some cases LCAO coefficients in HF MOs are optimized in the PHF calculations, in the latter such an optimization is absent. To the former group of PHF methods one refers different versions of the configuration interaction (Cl) method, the multi-configuration self-consistent field (MCSCF) method, the variational coupled cluster (CC) approach and the rarely used valence bond (VB) and generaUzed VB methods. The nonvariational PHF methods inclnde the majority of CC reaUza-tions and many-body perturbation theory (MBPT), called in its molecular realization the MoUer-Plessett (MP) method. In MP calculations not only RHF but UHF MOs are also used [107]. 

Mulliken population analysis, 218 Multi-Configuration Self Consistent Field (MCSCF) methods, 117 Multi-Reference Configuration Interaction (MRCI) methods, 122 Multiple minima, 339... 

For variational methods, such as Hartree-Fock (HF), multi-configurational self-consistent field (MCSCF), and Kohn-Sham density functional theory (KS-DFT), the initial values of the parameters are equal to zero and 0) thus corresponds to the reference state in the absence of the perturbation. The A operators are the non-redundant state-transfer or orbital-transfer operators, and carries no time-dependence (the sole time-dependence lies in the complex A parameters). Furthermore, the operator A (t)A is anti-Hermitian, and tlie exponential operator is thus explicitly unitary so that the norm of the reference state is preserved. Perturbation theory is invoked in order to solve for the time-dependence of the parameters, and we expand the parameters in orders of the perturbation... 

FIGURE 17.10 Excited states of BH calculated using a multi-configurational-self-consistent-field (MCSCF) complete-active-space method using the MOLPRO program, the state-of-the-art ... 

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

Another type of approach uses experimental information for molecule-dependent empirical corrections. Melius et al. have used MP4/6-31G(d) calculations along with empirical bond additivity corrections based on bond types and bond distances (BAC-MP4). The correction factors are determined by fitting to well known enthalpies of formation. They have applied the method to a series of silicon hydrides and other systems. Along the same lines. Sax and Kalcher used multi-configuration self-consistent field (MCSCF) calculations with experimental input (enthalpies of formation of Si2, SiH4, SiaHs) to obtain enthalpies of formation of Si Hm hydrides. While these methods have been successful there are problems for cases such as clusters where the bonding may be unconventional or systems where the bonding is delocalized so that a local correction may not be valid. Also, such methods cannot be used for pairs of atoms that have not been calibrated. 

A number of types of calculations begin with a HF calculation and then correct for correlation. Some of these methods are Moller-Plesset perturbation theory (MPn, where n is the order of correction), the generalized valence bond (GVB) method, multi-configurational self-consistent field (MCSCF), configuration interaction (Cl), and coupled cluster theory (CC). As a group, these methods are referred to as correlated calculations. 

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. 

A conceptually straightforward improvement on the CI approximation is to reoptimize the molecular orbitals for a truncated CI expansion. This approach is called multi-configuration self-consistent field method (MCSCF) and its most prominent variant is the complete active space SCF method (CASSCF) [64]. In the first generation of MCSCF methods [65, 66], the CI coefficients C/ in Eq. 

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