Multi-Reference CI Selection Schemes

In the discussion of excitation based selection schemes provided above, it was assumed that a single reference configuration was dominant and that the importance of all the remaining configurations may be judged relative to that reference alone. Whereas a single reference picture is acceptable for most molecules near their equilibrium geometries, it has been found to be inadequate when attempting to map complete potential energy surfaces (PESs) and when examining chemical changes. In the case of the former, it is not possible for a single reference or even CISD wave functions based on that reference to adequately describe the electronic states associated with the multitude of nuclear configurations a molecule may take over its PES. For example, consider the ring opening of cyclopropane to propene on the C₃H₆ PES. At the start, the dominant configuration or a small number of excited determinants must describe a molecule with three equivalent C-C bonds, while at the finish one must describe a molecule with only two C-C bonds, one of which is a double bond. As a matter of fact, along one of the most energetically favorable pathways, the cyclopropane ring must open to form a diradical species which, in and of itself, cannot be described by just one reference due to the near degeneracy of configurations with the radical electrons in and out of the plane of the molecule. In the case of chemical changes, the problem grows worse because a chemical reaction has, by definition, reactants and products which differ in composition and thus in their electronic configurations; therefore, it is unreasonable and, in fact, impossible to construct a method which emphasizes only a single reference and expect it to describe both products and reactants equally or well. For example, one obviously has two configurations which change in importance as a bond breaks: one configuration with and one configuration without the bond being broken.

The obvious solution to these problems is to construct a CI method in which a number of important configurations are chosen as references and additional configurations are included based upon their relationship to these references. For obvious reasons, such a procedure is known as a multi-reference CI (MRCI). The selection of a suitable set of references for the section of the PES under investigation is not trivial and must often be based only on chemical intuition. One reason that reference selection becomes so complicated stems from the fact that configurations which are important at one point on the PES may be unimportant elsewhere, but must be included throughout for the method to remain consistent. Thus, there is no way to judge the importance of a configuration based only on one or two points on the PES; instead, a reasonable understanding of the entire region one is attempting to describe is required. For example, when determining a reaction path, it is not enough to know the dominant configurations of the reactants and products, one must also know all the intermediates the molecule passes through along the path and the important configurations of each. Lastly, it is important to note that the need for multiple references decreases as higher and higher excitations from each reference are permitted into the CI method; recall, however, that the computational cost of the CI method also increases. On the other hand, the more references one adds, the more expensive the calculation becomes. Therefore, the goal is to choose as small a number of references as necessary and only include a small number of excited configurations relative to each reference.

Two of the most commonly employed MRCI methods are the first and second order CI, denoted FOCI and SOCI[7], respectively. In these methods the references are generated as all possible distributions of electrons within an ``active'' space of orbitals and then all single (FOCI) or single and double (SOCI) excitations relative to the references are included in the CI. As stated above the choice of active space is not trivial, although if one is only interested in a single point on the PES it is common to select the active space based on the natural orbital populations of the orbitals in a CASSCF or CISD wave function at that point. As a matter of fact, it is common and desirable to use both the CASSCF active space and orbitals in the FOCI and SOCI methods. The SOCI method is a specific example of the more general class of MR-CISD methods in which a set of references is selected and all single and double excitations out of those references are included in the CI. A second example of such an MR-CISD method will be presented in Chapter 5.