The MCSCF Method

The MCSCF Method. As noted previously, the primary reason for poor convergence in Cl calculations is the use of SCF virtual orbitals in constructing excited configurations these orbitals are not determined variationally and so are rather poor approximations to the true virtual orbitals. The MCSCF method treats a linear combination of Slater determinants Pi, 

Because the variation principle is involved, certain matrix elements disappear as in ordinary SCF theory, and such relations have recently been referred to as generalized Brillouin theorems.38 A major review of the MCSCF method has been given by Wahl 39 the practical limit on the number of configurations seems to be around 50 at present, but energy results compare extremely favourably with those of traditional Cl calculations invoking many more configurations, and other calculated properties are encouraging. 

McLean and M. Yoshimine, personal communication to S. Green, reported in ref. 6. 

UHF Methods. A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this leads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,40 where two sets of singly occupied orbitals are constructed instead of the doubly occupied set. The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minimization,41 arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. 

A major difficulty is that there is no guarantee that the projected wavefunction will give better expectation values than the UHF wavefunction. Ideally the energy should be minimized after spin projection, which is an MCSCF-typc of problem. Also, the amount of correlation introduced is unknown. In spite of these difficulties, Smeyers and Delgado-Barrio have found that both the total energy and dipole moment variation with bond length in LiH can approach the best Cl results using a very simple UHF function with spin projection. 

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For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

The Cl procedure just described uses a fixed set of orbitals in the functions An alternative approach is to vary the forms of the MOs in each determinantal function O, in (1.300), in addition to varying the coefficients c,. One uses an iterative process (which resembles the Hartree-Fock procedure) to find the optimum orbitals in the Cl determinants. This form of Cl is called the multiconfiguration SCF (MCSCF) method. Because the orbitals are optimized, the MCSCF method requires far fewer configurations than ordinary Cl to get an accurate wave function. A particular form of the MCSCF approach developed for calculations on diatomic molecules is the optimized valence configuration (OVC) method. 

Applications of continuum solvation approaches to MCSCF wavefunctions have required a more developed formulation with respect to the HF or DFT level. Even for an isolated molecule, the optimization of MCSFCF wavefunctions represents a difficult computational problem, owing to the marked nonlinearity of the MCSCF energy with respect to the orbital and configurational variational parameters. Only with the introduction of second-order optimization methods and of the variational parameters expressed in an exponential form, has the calculation of MCSCF wavefunction became routine. Thus, the requirements of the development of a second-order optimization method has been mandatory for any successful extension of the MCSCF approach to continuum solvation methods. In 1988 Mikkelsen el ol. [10] pioneered the second-order MCSCF within a multipole continuum model approach in a spherical cavity. Aguilar et al. [11] proposed the first implementation of the MCSCF method for the DPCM solvation model in 1991, and their PCM-MCSCF method has been the basis of many extensions to more robust second-order MCSCF optimization algorithms [12],... 

CHa + CH2— C2H4.—-The coplanar approach of two methylenes to form ethylene was investigated by Basch268 using the MCSCF method. The states of bent methylene that correlate with the ground state of ethylene are the triplet states. It is found that for two closed-shell singlet-state methylenes, the reaction path is purely repulsive. 

Docken and Hinze108 have presented a very detailed study of the potential-energy curves for five valence excited states of LiH by the MCSCF method. In this type of calculation, the wavefunction, expressed in the form (5), is variationally optimized... 

One of the most powerful tools presently available for accurate electronic structure calculations is the multiconfiguration reference CI(SD) method. In MR-CI(SD) wavefunctions, all configurations that are singly or doubly excited relative to any of the reference eonfigurations are taken into account, and their coefficients are determined variationally. The reference wavefunctions are usually optimized by the MCSCF method. They should properly describe the dissociation of bonds and near-degeneracy effects. If the reference wavefunction includes the most important double excitations from the... 

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