Multi-reference function perturbation

Today, there remain a number of problems in molecular electronic structure theory. The most outstanding of these is undoubtedly the development of a robust theoretical apparatus for the accurate description of dissociative processes which usually demand the use of multi-reference functions. This requirement has recently kindled a renewal of interest in the Brillouin-Wigner perturbation theory and its application to such problems. This contribution describes the application of... 

We turn now to the Brillouin-Wigner perturbation theory for a system described in zero order by a multi-reference function. The multi-reference formalism closely parallels that given in the previous section for the case of a single-reference function. Let us begin by defining a reference space V. Let... 

We are now in a position to obtain perturbation expansions by expanding the inverse operator in the effective Hamiltonian, the wave operator and the reaction operator. We begin, as we did in our discussion of the partitioning technique, by considering the case of a single-reference function and then turn our attention to the multi-reference function case. 

The tautomerism of furoxan (l,2,5-oxadiazole-2-oxide) has been investigated by different computational methods comprising modern density functions as well as single-reference and multi-reference ab initio methods. The ring-opening process to 1,2-dinitrosoethylene is the most critical step of the reaction and cannot be treated reliably by low-level computations (Scheme 2). The existence of cis-cis-trans- 1,2-dinitrosoethylene as a stable intermediate is advocated by perturbational methods, but high-level coupled-cluster calculations identify this as an artifact <2001JA7326>. 

The reference (zeroth-order) function in the CASPT2 method is a predetermined CASSCF wave function. The coefficients in the CAS function are thus fixed and are not affected by the perturbation operator. This choice of the reference function often works well when the other solutions to the CAS Hamiltonian are well separated in energy, but there may be a problem when two or more electronic states of the same symmetry are close in energy. Such situations are common for excited states. One can then expect the dynamic correlation to also affect the reference function. This problem can be handled by extending the perturbation treatment to include electronic states that are close in energy. This extension, called the Multi-State CASPT2 method, has been implemented by Finley and coworkers.24 We will briefly summarize the main aspects of the Multi-State CASPT2 method. 

When the reference wave function contains substantial multi-reference character, a perturbation expansion based on a single determinant will display poor convergence. If the reference wave function suffers from symmetry breaking (Section 3.8.3), the... 

In order to go beyond the (6/6)CASSCF level, while maintaining the ability of a (6/6)CASSCF wave function to describe the two possible diradical extremes for the Cope TS, CASPT2 calculations proved ideal. This method, which had been developed by Roos and co-workers [19], applies multi-reference, second-order perturbation theory to CASSCF wave functions and in 1993 CASPT2 had become available in the MOLCAS suite of at initio calculations from the Roos group [20]. 

Subsequently, Kozlowski et al. [24] also revisited the Cope rearrangement with inclusion of dynamic correlation between the active and inactive electrons. However, they used Davidson s own version of multi-reference, second-order perturbation theory [25], which allows the coefficients of the configurations in the CASSCE wave function to be recalculated after inclusion of dynamic electron correlation. Kozlowski et al. found that the addition of dynamic correlation to the (6/6)CASSCE wave function for the Cope TS causes the weight of the RHE configuration to increase at the expense of the pair conhgurations that are necessary to describe the two diradical extremes in Eig. 30.1. Thus, without the inclusion of dynamic electron correlation in the wave function, (6/6) CASSCF overestimates the diradical character of the C2 wave function [24]. 

---