Multi-reference methods correlation

Unfortunately, in many situation of interest, both static and dynamic electronic correlation need to be taken into account. This is particularly true for the study of processes involving excited electronic states as in UV spectroscopy or in photochemistry. In this case, methods capable of taking into account the dynamic electronic correlation on top of a multi-determinantal wavefunction of the MCSCF type are needed. These methods are usually called multi-reference methods. The two standard methods that are able to account for both the static and dynamic electronic correlation are the multi-reference configuration interaction (MRCI) and several variants of second-order multi-reference perturbation theory (MRPT). 

Electron correlation was treated by the CIPSI multi-reference perturbation algorithm ([24,25] and refs, therein). The Quasi Degenerate Perturbation Theory (QDPT) version of the method was employed, with symmetrisation of the effective hamiltonian [26], and the Maller-Plesset baricentric (MPB) partition of the C.I. hamiltonian. 

However, if this is not the case, the perturbations are large and perturbation theory is no longer appropriate. In other words, perturbation methods based on single-determinant wavefunctions cannot be used to recover non-dynamic correlation effects in cases where more than one configuration is needed to obtain a reasonable approximation to the true many-electron wavefunction. This represents a serious impediment to the calculation of well-correlated wavefunctions for excited states which is only possible by means of cumbersome and computationally expensive multi-reference Cl methods. 

Having decided about the basis set, we now turn to the problem of choosing an appropriate form of the wave function. Since we are going to calculate an energy difference between two electronic states, it is clear that we have to include as much correlation effects as possible. We shall do that in two steps In the first step we try to account for the near degeneracy effects by using the CASSCF method with an appropriate choice of the active space. On top of that we shall have to perform multi-reference Cl calculations to account for the dynamical correlation energy. 

In order to get more detailed information about, e.g., bond strengths and equilibrium geometries in transition metal systems it is necessary to include electron correlation. This can be done either by traditional ab initio quantum chemistry models, e.g., Cl-methods and coupled cluster methods, or by density functional theory (DFT) based methods. Correlated ab initio methods are often computationally very demanding, especially in cases where multi-reference based treatments are needed. Also, the computational cost of these methods increases dramatically with the size of the system. This implies that they can only be applied to rather small systems. 

The CASSCF method itself is not very useful for anything else than systems with few electrons unless an effective method to treat dynamical correlation effects could be developed. The Multi-Reference Cl (MRCI) method was available but was limited due to the steep increase of the size of the Cl expansion as a function of the number of correlated electrons, the basis set, and the number of active orbitals in the reference function. The direct MRCI formulation by P. Siegbahn helped but the limits still prevented applications to larger systems with many valence electrons [20], The method is still used with some success due to recent technological developments [21], Another drawback with the MRCI approach is the lack of size-extensivity, even if methods are available that can approximately correct the energies. Multi-reference coupled-cluster methods are studied but have not yet reached a state where real applications are possible. 

Most studies limit the Cl space to single and double substitutions from a single reference (CISD). Occasionally, when one reference does not provide a sufficient zeroth-order wavefunction, the multi-reference CISD method will be employed. The applications of such methods are too numerous to discuss here their general performance has already been described in section 2. We consider studies which go beyond CISD for one or a few references such highly correlated wavefunctions are useful when very accurate results are desired or when several electron configurations are needed for a qualitatively correct reference wavefunction (such as when multiple bonds are broken). We will limit our attention to methods which select the Cl space in an a priori fashion based on the distribution of electrons among various orbital subspaces. 

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