Single-reference methods problem

One final informative example is offered by the O3 molecule. As discussed O3 possesses resonance structures and biradical character, which usually recommends at least two reference funaions in its zeroth-order description. We can expeCT this to be a demanding case for single reference methods,although multireference methods have their own problems. These results are presented in Table 15. 

As with the single-reference methods, size extensivity does not guarantee size consistency (a proper dissociation limit in the case of potential curves). The reference space must be carefully chosen to contain important configurations at any point of the potential curve as it was recognized long time ago [72]. But this is not a specific problem of MR BWCCSD. 

All of the methods discussed above are based on a multi-reference (MR) approach to obtaining wavefunctions and properties. Such MR approaches are often necessary, because many chemical problems involve species with considerable configurational mixing due the existence of near degeneracies (diradical character). However, the amount of diradical character in a chemical system can span a very broad range, from essentially zero (e.g. HOMO occupancy 2, LUMO occupancy 0) to fully diradical (HOMO occupancy 1, LUMO occupancy 1). As one approaches fully diradical character, all single reference methods break down, but they do not break down at the same rate as one approaches this limit. In particular, there is considerable evidence that coupled cluster (CC) methods, particularly those like CCSD(T) that incorporate a triples correction, can overcome the deficiency of a single reference wavefunction for problems with non-trivial diradical character. This has been demonstrated, for example, by... 

Just as single reference Cl can be extended to MRCI, it is also possible to use perturbation methods with a multi-detenninant reference wave function. Formulating MR-MBPT methods, however, is not straightforward. The main problem here is similar to that of ROMP methods, the choice of the unperturbed Hamilton operator. Several different choices are possible, which will give different answers when the tlieory is carried out only to low order. Nevertheless, there are now several different implementations of MP2 type expansions based on a CASSCF reference, denoted CASMP2 or CASPT2. Experience of their performance is still somewhat limited. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

Note that it is not required that all the cf coefficients be very different from unity, and our experience is that even in the cases of having a second coefficient equal to unity (e.g., for a state dominated by two single excited spin partners), the method does not diverge, though the accuracy of the method is reduced. The problem of the applicability of single reference dressing methods to states which are not actually dominated by a single determinant has been discussed elsewhere [41]. 

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