Multiconfigurational self-consistent field orbitals

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 ... 

It is possible to divide electron correlation as dynamic and nondynamic correlations. Dynamic correlation is associated with instant correlation between electrons occupying the same spatial orbitals and the nondynamic correlation is associated with the electrons avoiding each other by occupying different spatial orbitals. Thus, the ground state electronic wave function cannot be described with a single Slater determinant (Figure 3.3) and multiconfiguration self-consistent field (MCSCF) procedures are necessary to include dynamic electron correlation. 

Finally, we note that if we retain two-particle operators in the effective Hamiltonian, but restrict A to single-particle form, we recover exactly the orbital rotation formalism of the multiconfigurational self-consistent field. Indeed, this is the way in which we obtain the CASSCF wavefunctions used in this work. 

Nondynamical electron correlation effects are generally important for reaction path calculations, when chemical bonds are broken and new bonds are formed. The multiconfiguration self-consistent field (MCSCF) method provides the appropriate description of these effects [25], In the last decade, the complete active space self-consistent field (CASSCF) method [26] has become the most widely employed MCSCF method. In the CASSCF method, a full configuration interaction (Cl) calculation is performed within a limited orbital space, the so-called active space. Thus all near degeneracy (nondynamical electron correlation) effects and orbital relaxation effects within the active space are treated at the variational level. A full-valence active space CASSCF calculation is expected to yield a qualitatively reliable description of excited-state PE surfaces. For larger systems, however, a full-valence active space CASSCF calculation quickly becomes intractable. 

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

K. Ruud, T. Helgaker, R. Kobayashi, P. Jorgensen, K. Bak, H. Jensen, Multiconfigurational self-consistent field calculations of nuclear shieldings using london atomic orbitals, J. Chem. Phys. 100 (1994) 8178. 

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. 

Gilbert, T. L. (1972). Multiconfiguration self-consistent-field theory for localized orbitals. I. The orbital equation. Phys. Rev. A6, 580-600. 

The computationally viable description of electron correlation for stationary state molecular systems has been the subject of considerable research in the past two decades. A recent review1 gives a historical perspective on the developments in the field of quantum chemistry. The predominant methods for the description of electron correlation have been configuration interactions (Cl) and perturbation theory (PT) more recently, the variant of Cl involving reoptimization of the molecular orbitals [i.e., multiconfiguration self-consistent field (MCSCF)] has received much attention.1 As is reasonable to expect, neither Cl nor PT is wholly satisfactory a possible alternative is the use of cluster operators, in the electron excitations, to describe the correlation.2-3... 

Shadwick, 1976) and thus by extrapolation to the fully correlated problem via application to the multiconfiguration self-consistent field (MCSCF) problem. In spite of the fact that the mathematics of simultaneously averaging a nonlocal potential and proceeding to self-consistency is cumbersome at present, this approach seems a most promising avenue toward Slater s goal of using a local potential for orbital generation in Cl calculations. 

A more balanced description thus requires multiconfiguration self-consistent field (MCSCF)-based methods, where the orbitals are optimized for each particular state or optimized for a suitable average of the desired states (state-averaged MCSCF). In semiempirical methods, however, an MCSCF procedure is normally not required due to the limited flexibility of the minimal valence atomic orbital basis commonly used in these methods. Instead, a multireference Cl method including a limited number of suitably chosen configurations will be appropriate. 

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