MCSCF model

The computational model capable of yielding accurate spin-spin coupling constants is the multiconfigurational self-consistent field (MCSCF) model, and before the advent of density functional theory, spin-spin coupling constants in small systems were often... 

The discussion of the various aspects of the MCSCF method requires the discussion of some background material. In this section, some of the elementary concepts of linear algebra are introduced. These concepts, which include the bracketing theorem for matrix eigenvalue equations, are used to define the MCSCF method and to discuss the MCSCF model for ground states and excited states. The details of V-electron expansion space represent-... 

The multiconfigurational SCF (MCSCF) model [43,44] is a generalization of the Hartree-Fock model to several configurations ... 

The effective Hartree-Fock matrix equation for a many-shell system has been derived in Chapter 14 and used in several applications open shells and some MCSCF models. So far, it has been seen simply as the formally correct equation to generate SCF orbitals for these many-sheU structures without any interpretation. In particular, the fact that the effective Hartree-Fock matrix (the McWeenyan ) contains many arbitrary parameters has not been addressed, nor has the practical problem of the actual grounds for the choice of values for these parameters been systematised. In looking at this problem we must bear two points in mind ... 

The extension of the above result to any variational approximation to the solution of the Schrodinger equation involving optimisation of orbitals is surprisingly straightforward. In the MCSCF model of molecular electronic structure the model wavefunction is written as a linear combination of antisymmetric terms (usually determinants) constructed from a set of MOs which are themselves vari-ationally optimised as linear combinations of some basis functions ... 

In practice, the simultaneous optimization of orbitals and Cl coefficients is a difficult nonlinear problem, which severely restricts the length of MCSCF expansions relative to those of Cl wave functions. By itself, the MCSCF model is therefore not suited to the treatment of dynamical correlation (for which large basis sets and long configuration expansions are needed) but it may be used in conjunction with a subsequent correlation treatment by a mote extensive multirefeience Cl wave function, providing the reference configurations and orbitals needed for such treatments. 

The MCSCF method is not a black-box method and cannot easily be applied by nonspecialists. The optimized MCSCF state should always be inspected carefully to ensure that it properly represents the desired electronic state. Nevertheless, the MCSCF model is a highly flexible one and it still constitutes the only approach that, in a balanced manner, can describe bond breakings and molecular dissociation processes, where an unbiased treatment of several electronic configurations is required. The small numbo- of active orbitals that can be treated in MCSCF theory usually makes it impossible to treat dynamical correlation and only a qualitatively correct description of the electronic system can therefore be expected. 

In Hartree-Fock theory, complications arise only for open-shell systems, where the active-active rotations are in some cases redundant, in other cases nonredundant For instance, for open-shell states constructed by distributing two electrons between two orbitals, we found in Section 10.1.2 that the active-active rotations are redundant for the triplet state but nonredundant for the singlet state. We can easily imagine that the situation becomes even more complicated in the MCSCF case, where the wave function is generated by optimizing simultaneously the orbital-rotation parameters and a (potentially) laige number of Cl coefficients. Fortunately, for the more common MCSCF models such as those based on the CAS and RAS concepts, the question of redundancies is simple and unexpected redundancies will only rarely arise. 

The CCR idea has been around for a long time, as reviewed in Refs. 389 and 391, and many applications to temporary anion resonances have been reported. Nevertheless, this technique has remained somewhat specialized. Within the context of electronic structure theory, what is required for a CCR calculation is to combine the complex-scaled Hamiltonian in Eq. [63] with the usual wave function ansdtze, and this involves extending quantum chemistry codes to handle complex-valued wave functions and energies and non-Hermitian matrices. CCR implementations of the Hartree-Fock, configuration interaction, and multiconfigurational SCF (MCSCF) models have been reported but are not available in standard... 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

In the first example of applications of the theory in this chapter, we made a point with respect to the polarizability of molecules and showed how the problem could have been handled by the RISM-SCF/MCSCF theory. However, the current level of our method has a serious limitation in this respect. The method can handle the polarizability of molecules in neat liquids or that of a single molecule in solution in a reasonable manner. But in order to be able to treat the polarizability of both solute and solvent molecules in solution, considerable generalization of the RISM side of the theory is required. When solvent molecules are situated within the influence of solute molecules, the solvent molecules are polarized differently depending on the distance from the solute molecules, and the solvent can no longer be neat. Therefore, the polarizable model developed for neat liquids is not valid. In such a case, solvent-solvent PCF should be treated under the solute... 

MCSCF theory is a specialist branch of quantum modelling. Over the years Jt has become apparent that there are computational advantages in treating all oossible excitations arising by promoting electron(s) from a (sub)set of the occu-orbitals to a (sub)set of the virtual orbitals. We then speak of complete active ace MCSCF, or CASSCF. 

A classical description of M can for example be a standard force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or any of the ab initio methods, i.e. HF, MCSCF, CISD, MP2 etc. Although the electrostatic potential can be derived directly from the electronic wave function, it is usually fitted to a set of atomic charges or multipoles, as discussed in Section 9.2, which then are used in the actual solvent model. 

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

The number No of occupied valence SCF orbitals in a molecule is typically less than the total number Nmb of orbitals in the minimal valence basis sets of all atoms. The full valence MCSCF wavefunction is the optimal expansion in terms of all configurations that can be generated from N b molecular orbitals. Closely related is the full MCSCF wavefunction of all configurations that can be generated from Ne orbitals, where Nc is the number of valence electrons, i.e. each occupied valence orbital has a correlating orbital, as first postulated by Boys (48) and also presumed in perfect pairing models (49,50), We shall call these two types of frill spaces FORS 1 and FORS 2. In both, the inner shell remains closed. 

Molecule/Model No of Dets SD-NO MCSCF Based on Optimized Energy lowering Due to MCSCF (mh)... 

The truncation procedure for fiill-valence-space and N-electrons-in-N-orbitals SDTQ MCSCF waveflmctions is based on choosing split-localized molecular orbitals as configuration generators since they lead to the greatest number of deadwood configurations that can be deleted. A quite accurate estimation method of identifying the latter has been developed so that the truncation can be performed a priori. The method has been shown to be effective in applications to the molecules HNO, OCO and NCCN where, for instance, the energies of the full SDTQ[N/N] calculations are recovered to better than 1 mh by truncated expansions that require only 11.8%, 10.9% and 6.3%, respectively, of the number of determinants in the full calculations. Similar trends are observed for the FORS 1 model. 

Bryce and Wasylishen196 have presented ab initio calculations of 2hJ(N, N) couplings using a MCSCF wave function taking the methyleneimine dimer [8] as a model system. 

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