Multireference wavefunctions

We have so far examined the performance of the canonical transformation theory when paired with a suitable multireference wavefunction, such as the CASSCF wavefunction. As we have argued, because the exponential operator describes dynamic correlation, this hybrid approach is the way in which the theory is intended to be used in general bonding situations. However, we can also examine the behavior of the single-reference version of the theory (i.e., using a Hartree-Fock reference). In this way, we can compare in detail with the related... 

A Modern First-Principles View on Ligand Field Theory Through the Eyes of Correlated Multireference Wavefunctions... 

Although the MCSCF wavefunction for SF(a S ) is dominated by the HF configuration for all interatomic separations from the well outward, it is instructive to consider the NOs and GVB orbitals for a simple multireference wavefunction, one that is restricted to double excitations into the first a virtual (9o). Again keeping nonbonding electrons in singly or doubly occupied orbitals as they are in S and F, the spatial wavefunction for the state is... 

As for the diatomics, each orbital pair indicates the inclusion of three configurations in the MCSCF wavefunction. With the two orbital couplings present in aU three states of SF2, there are thus nine configurations in the minimal multireference wavefunction. We now discuss the nature of the bonding in each of the three states of SF2 in turn. We then discuss these states in OF2. Unless otherwise noted, all results in this section are at the RCCSD(T)/AVQZ level. 

Regardless of whether a single- or multireference wavefunction is used for a CISD calculation, the effects of higher excitations (e.g., triples and quadruples) are obviously not included. Quadruple excitations are particularly important because it is their absence from CISD wavefunctions for closed-shell molecules that results in these wavefunctions not being size consistent. Methods for estimating the effect of quadruple excitations have been developed, both by Davidson and by Pople. The Davidson corrections for quadruples can easily be computed by hand, but they are also automatically calculated as part of the CISD modules in the MOLPRO and MOLCAS programs. 

CASPT2 is most useful for calculations on excited states and diradicals, where multireference wavefunctions are required. However, there are methods available for including electron correlation for radicals and radical ions for which single-determinantal wavefunctions represent good zero-order approximations, without resorting to multideterminantal (i.e., CASSCF) reference wavefunctions. Two of these methods are discussed in the following sections, and we recommend them over CASPT2 for most calculations on molecules with just one unpaired electron. 

As is indicated in Table 3, almost all molecular properties that can be evaluated at the HF level can also be determined at the MP2 level since analytical first and second energy derivatives are available. This applies to RHF, UHF, and ROHF wavefunctions, however not to GVB or other multireference wavefunctions. Analytical first derivatives have... 

Electron Correlation. Given the open-shell nature of the ground and excited states of lanthanide impurity ions in crystals, electron correlation is exU cmely important. Currently, electronic structure methods based on the use of multireference wavefunctions appear to be the fittest to respond to the requirements. Furthermore, their current evolution towards allowing more and more flexible definitions of the active space makes them even more adequate. Even though the methods used in the applications contained in this chapter are well known and their performance has been proven and documented in many highly correlated systems, we summarize here how to adapt them to the impurity lanthanide ion electronic structure demands. 

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