13.2.3 Multireference Approaches

In summary, it can be said that multireference Cl methods provide a balanced description of static and dynamic correlation effects. This comes at the cost of a considerably increased computational demand compared to single-reference methods. Nevertheless, a multireference approach is inevitable for systems with a genuine multiconfigurational character such as symmetric transition metal compounds. 

Analytical gradients and Hessians are available for CASSCF, and it is expected that this technology will be extended to the MR-CI and MP2 methods soon. Further, by virtue of the multireference approach, a balanced description of ground and excited states is achieved. Unfortunately, unlike black boxes such as first-order response methods (e.g., time-dependent DFT), CAS-based methods require considerable skill and experience to use effectively. In the last section of this chapter, we will present some case studies that serve to illustrate the main conceptual issues related to computation of excited state potential surfaces. The reader who is contemplating performing computations is urged to study some of the cited papers to appreciate the practical issues. 

Alternatively, the convergence toward the precise solution of the Schrodinger equation can be accelerated by using a multireference approach. There is a corresponding multireference (MR) approach to each of the single reference... 

The non-relativistic CASPT2 method developed by Anderson et al. [11,12] is one of the most familiar multireference approaches. It is well established and has been applied to a large number of molecular systems with the non- or quasi-relativistic approaches. Because the CASPT2 method treats dynamic correlation effects pertur-batively, it is less expensive than the multireference Cl (MRCI) method. The... 

Another symmetry-breaking problem to which a number of single and multireference approaches have been applied is the nitrate radical, NO3 (47). Different experiments have been interpreted in terms of either a totally symmetric D3h equilibrium geometry, or a lower symmetry C2v structure with one N-O bond length different from the other two. However, as Kawaguchi and coworkers (48) comment, no one has advanced a model which explains all of the available spectroscopic data. 

A particular advantage of the Cl approach, as opposed to coupled-cluster or perturbation theoretic techniques, is that it can be readily formulated to handle the case in which is multiconfigurational in character. Corresponding multireference CC or perturbation theory approaches are much less well developed. As we shall see, there are many important chemical applications in which a multireference approach is mandatory, and we shall therefore concentrate mainly on the Cl method in this article. 

Comparison of correlation energies (all signs reversed) for the ground state of the S4DZP model (in mhartree). Multireference approaches use (2,2) and (4,4) model spaces. The AL-RMR-CCSD-z methods are simply designated as AL-RMR-i. 

As already pointed out in Ref. 13, the externally corrected CCSD is equivalent to (truncated) CCSDTQ with zero-iteration on and T4 amplitudes that are in turn obtained from some external sources. Depending on the source of these amplitudes, we usually deal with only a proper subset of all possible T3 and T4 amplitudes. This subset is fixed in the externally corrected CCSD calculations. The RMR CCSD is then a special case of the general externally corrected CCSD in which the MR CISD wave function is used as the external source. The RMR CCSD method represents in fact a multireference approach in the sense that it is uniquely defined by the choice of the reference space and the fact that the RMR CCSD wave function involves the same number of connected cluster amplitudes as the corresponding genuine MR CCSD, such as the state-universal CCSD employing the same reference space. 

Multiply excited WF includes important contributions from doubles, triples, and so on Multireference approaches necessary... 

Perturbation theory methods. A perturbation approach can be used to include efficiently high-order correlation effect into low-level Cl solutions, either from a singlereference or a multireference approach. This gives rise to... 

The importance of using a multiconfigurational reference has been demonstrated in the study of relative conformational energies for systems containing carbon-carbon double bonds. The multireference approach is needed to represent correctly the low lying n excitations. Similarly, multireference character is likely to play a significant role in the accurate description of transition states and systems containing transition metals. 

The intermediate Hamiltonian approaches presented here may be applied within any multiroot multireference infinite order method. Recently [43] we implemented the XIH scheme to another all-order relativistic multiroot multireference approach, the Hilbert space or state universal CC, which is the main alternative to and competitor of Fock-space CC. This approach will not be discussed here. We only mention that it allows mixing P-space sectors, which can interact strongly, e.g., 1-particle with 2-particle 1-hole [37]. 

The active-space SRCC methods and their EOMCC extensions are very promising and we will continue to develop them. They are relatively easy to use, although, in analogy to multireference approaches, they require choosing active orbitals, which in some cases may be a difficult thing to do. From this point of view, the active-space CC methods are not as easy-to-use as the noniterative perturbative methods, such as CCSD(T) or CCSD(TQf), or their response CC or EOMCC extensions. Undoubtedly, it would be desirable to have an approach that combines the simplicity of the noniterative CC schemes with the effectiveness with which the iterative active-space CC and EOMCC methods, such as CCSDt and EOMCCSDt, describe ground-and excited-state PESs. 

Quantum chemical predictions of tautomeric equilibria Ab initio multireference Cl studies of tautomers UPS spectra Combined approach to the tautomerism in azaaromatic heterocycles by N, C, and H NMR Fluoro-azoles Experimental data and MNDO studies Tautomerism A review... 

The present approach is one of the second-generation multireference perturbation treatments first opened by the CIPSI algorithm 20 years ago. Even if the spirit of these new treatments is different, mainly because the reference space is chosen on its completeness rather than on energetical criteria, it remains that the unavoidable problems of disk storage, bottleneck of variational approaches, can now be conveniently transferred to the problem of CPU time which is less restrictive. 

The methodology presented here expands the recent CASPT2 approach to more flexible zeroth-order variational spaces for a multireference perturbation, either in the Moller-Plesset scheme or in Epsein-Nesbet approach [70-72]. Furthermore, it allows for the use of a wide set of possible correlated orbitals. These two last points were discussed elsewhere [34]. 

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