Active-space single-reference

Hamiltonian proposed by Muller and Plesset gives rise to a very successful and efficient method to treat electron correlation effects in systems that can be described by a single reference wave function. However, for a multireference wave function the Moller-Plesset division can no longer be made and an alternative choice of B(0> is needed. One such scheme is the Complete Active Space See-ond-Order Perturbation Theory (CASPT2) developed by Anderson and Roos [3, 4], We will briefly resume the most important definitions of the theory one is referred to the original articles for a more extensive description of the method. The reference wave function is a CASSCF wave function that accounts for the largest part of the non-dynamical electron correlation. The zeroth-order Hamiltonian is defined as follows and reduces to the Moller-Plesset operator in the limit of zero active orbitals ... 

In practice, one often has to restrict the active space to include only a few of the highest occupied and lowest unoccupied molecular orbitals. The correlation effects not included in the CASSCF calculation can then be recovered by a multireference Cl (MRCI) calculation, in which all single and double excitations from the CASSCF reference are taken into account [27], A computationally more efficient way of including dynamical electron correlation effects is perturbation theory with respect to the CASSCF reference. The most widely employed method of this type is the CASPT2 method developed by Roos and collaborators [28], The CASSCF and CASPT2 methods have been essential tools for the calculations described in this contribution. 

State average orbitals are not optimized for a specific electronic state. Normally, this is not a problem and a subsequent CASPT2 calculation will correct for most of it because the first order wave function contains CFs that are singly excited with respect to the CASSCF reference function. However, if the MOs in the different excited states are very different it may be needed to extend the active space such that it can describe the differences. A typical example is the double shell effect that appears for the late first row transition metals as described above. 

We have above discussed the CASSCF method and how we can choose the active space. We noted that this choice was closely connected to the method we use to compute the effects of dynamic correlation, in this case the CASPT2 method. The development of this approach was inspired by the success of the Mpller-Plesset second order perturbation theory (MP2), which has been used for a long time to treat electron correlation for ground states, where the reference function is a single determinant. It was assumed that such an approach would be even more effective with the more accurate CASSCF reference function. A first attempt was made soon... 

For molecules that require a multireference description, use of a single reference post-HF method can often fail since the dynamic correlation space is insufficient. Multireference post-HF methods are quite taxing in terms of computational resources and comprise a very active area of theoretical development." A method that has shown some recent promise is multireference coupled cluster (MRCC) theory, and the implementation proposed by Mukheijee and coworkers" (often labeled as MkCC or MkMRCC) has garnered much interest." ... 

This procedure suffers from a high degree of arbitrariness in the choice of just which configurations are deemed important. The calculation can be made somewhat more objective by including all excitations between a subset of occupied MOs and a subset of vacant orbitals. (These excitations are subject to certain restrictions as to multiplicity or order of excitation.) The orbitals chosen for the excitations are referred to as the active space , and the method is dubbed Complete Active Space Self Consistent Field (CASSCF) - ". Both MCSCF and CASSCF provide a certain fraction of the correlation energy, relative to a single configuration, Hartree-Fock, calculation. 

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