The factorial increase in the number of CSFs effectively limits the active space for CASSCF wave functions to fewer than 10-12 electrons/orbitals. Selecting the "important" orbitals to correlate therefore becomes very important. The goal of MCSCF methods is usually not to recover a large fraction of the total correlation energy, but rather to recover all the changes that occur in the correlation energy for the given process. Selecting the active space for an MCSCF calculation requires some insight into the problem. There are a few rules of thumb that may be of help in selecting a proper set of orbitals for the active space:
(1) For each occupied orbital, there will typically be one corresponding virtual orbital. This leads naturally to [n,m]-CASSCF wave functions where n and m are identical or nearly so.
(2) Including all the valence orbitals, i.e. the space spanned by a minimum basis set, leads to a wave function that can correctly describe all dissociation pathways. Unfortunately, a full valence CASSCF wave function rapidly becomes unmanageably large for realistic-sized systems.
(3) The orbital energies from an RHF calculation may be used for selecting the important orbitals. The highest occupied and lowest unoccupied are usually the most important orbitals to include in the active space. This can be partly justified by the formula for the second-order perturbation energy correction (Section 4.8.1): the smaller the orbital energy difference, the larger contribution to the correlation energy. Using RHF orbital energies for selecting the active space may be problematic in two situations.The first is when extended basis sets are used, where there will be many virtual orbitals with low energies, and the exact order is more or less accidental. Furthermore, RHF virtual orbitals basically describe electron attachment (via Koopmans' theorem, Section 3.4), and are therefore not particularly well suited for describing electron correlation. An inspection of the form of the orbitals may reveal which to choose: they should be the ones that resemble the occupied orbitals in terms of basis function contribution. The second problem is more fundamental. If the real wave function has significant multi-configurational character, then the RHF may be qualitatively wrong, and selecting the active orbitals based on a qualitatively wrong wave function may lead to erroneous results. The problem is that we wish to include the important orbitals for describing the multi-determinant nature, but these are not known until the final wave function is known. (4) An attempt to overcome this self-referencing problem is to use the concept of natural orbitals. The natural orbitals are those that diagonalize the density matrix, and the eigenvalues are the occupation numbers. Orbitals with occupation numbers significantly different from 0 or 2 (for a closed shell system) are usually those that are the most important to include in the active space. An RHF wave function will have occupation numbers of exactly 0 or 2, and some electron correlation must be included to obtain orbitals with non-integer occupation numbers. This may for example be done by running a preliminary MP2 or CISD calculation prior to the MCSCF. Alternatively, a UHF (when different from RHF) type wave function may also be used. The total UHF density, which is the sum of the a and b density matrices, will also provide fractional occupation numbers since UHF includes some electron correlation. The procedure may still fail. If the underlying RHF wave function is poor, the MP2 correction may also give poor results, and selecting the active MCSCF orbitals based on MP2 occupation number may again lead to erroneous results. In practice, however, selecting active orbitals based on for example MP2 occupation numbers appears to be quite efficient, and better than using RHF orbital energies.
In a CASSCF type wave function the CI coefficients do not have the same significance as for a single-reference CI based on HF orbitals. In a full CI (as in the active space of the CASSCF), the orbitals may be rotated among themselves without affecting the total wave function. A rotation of the orbitals, however, influences the magnitude of the coefficients in front of each CSF. While the HF coefficient in a single-reference CISD gives some indication of the "multi-reference" nature of the wave function, this is not the case for a CASSCF wave function, where the corresponding CI coefficient is arbitrary.
It should be noted that CASSCF methods inherently tend to give an unbalanced description, since all the electron correlation recovered is in the active space, with none in the inactive space, or between the active and inactive electrons.8 This is not a problem if all the valence electrons are included in the active space, but this is only possible for small systems. If only part of the valence electrons are included in the active space, the CASSCF method tends to overestimate the importance of "biradical" structures. Consider for example acetylene where the hydrogens have been bent 60° away from linearity (this may be considered a model for ortho-benzyne). The in-plane "n-orbital" now acquires significant biradical character. The true structure may be described as a linear combination of the following three configurations.
The structure on the left is biradical, while the two others are ionic, corresponding to both electrons being at the same carbon. The simplest CASSCF wave function that can qualitatively describe this system has two electrons in two orbitals, giving the three
Figure 4.12 Important configurations for a bend acetylene model configurations shown above. The dynamical correlation between the two active electrons will tend to keep them as far apart as possible, i.e. favouring the biradical structure. Consider now a full valence CASSCF wave function with ten electrons in ten orbitals. This will analogously tend to separate the two electrons in each bond with one being at each end. The correlation of the electrons in the C—H bonds, for example, will place more electron density on the carbon atoms. This in turn favours the ionic structures in Figure 4.12 and disfavours the biradical, i.e. the dynamical correlation of the other electrons may take advantage of the empty orbital in the ionic structures but not in the biradical structure. These general considerations may be quantified by considering the natural orbital occupancies for increasingly large CASSCF wave functions, as shown in Table 4.4 with the 6-31G(d,p) basis.
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