Wave functions Multireference

There are many more error correction methods, which are reviewed in detail by Duch and Diercksen. They also discuss the correction of other wave functions, such as multireference methods. In their tests with various numbers of Be atoms, the correction most closely reproducing the full Cl energy is... 

Figure 2. (rel) = AEf2 and C(nonrel) = AE12.A, iiA, at each iteration of the solution of Eq. (15b) for OH + H2 using multireference configuration interaction wave functions. 

An example of a multireference technique is the multiconfigurational SCF (MCSCF) approach, where the wave function is obtained by simultaneously optimizing both the molecular orbitals and the configuration coefficients, thereby blending the different resonance structures together. [28] Historically, the MCSCF approach has been used extensively to provide qualitatively accurate representations of surfaces however, this method still suffers two primary drawbacks (1) the ambiguous choice of configurations and (2) the lack of dynamical correlation. 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

Another class of methods uses more than one Slater determinant as the reference wave function. The methods used to describe electron correlation within these calculations are similar in some ways to the methods listed above. These methods include multiconfigurational self-consistent field (MCSCF), multireference single and double configuration interaction (MRDCI), and /V-clcctron valence state perturbation theory (NEVPT) methods.5... 

Within the multireference BWPT [45], the exact wave functions for o = 1,..., d can be expanded in the Brillouin-Wigner (BW) perturbation series as follows... 

Several different versions of second-order perturbation theory for multireference wave functions have been implemented but the one currently in widest use is probably the CASPT2 method. This method was developed by Roos and co-workers in Lund, Sweden, and it is available in their MOLCAS package of computer programs. 

CCSD is similarly sensitive to multireference character, aldiough it is less obvious that this should be so based on the formalism presented above. However, inclusion of triples in the CCSD wave function is usually very effective in correcting for a single-reference treatment of a weakly to moderately multireference problem. Of course, die most common way to include die triples is by perturbation theory, i.e., CCSD(T), and as noted above, this level too can be unstable if singles amplitudes are large. In such an instance, BD(T) calculations, which eliminate die singles amplitudes, can be efficacious. 

One way of achieving size-consistency for a dissociation process is to use an MCSCF wave function as the reference. Unfortunately, as noted above, there are as yet no general multireference perturbation theory or multireference coupled-cluster treatments that can be applied to such an MCSCF reference function. For rather few electrons, as we shall see, the MRCI approach performs acceptably. 

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