The single-reference theoretical methods considered above, although they are the most commonly used, are not the only approaches available. The so-called multi-reference methods are less straightforward to apply and more computationally expensive, but they are more robust for nondynamical correlation problems. We were surprised to discover that these methods had not previously been calibrated against exact full CI potential energy curves for a dense set of points. In the first study of its kind, we again considered BH, HF, and CH₄ as test cases to assess the reliability of multi-reference methods such as complete-active-space self-consistent-field (CASSCF), second-order multi-reference perturbation theory based on a CASSCF reference function (CASPT2), and two types of multi-reference configuration interaction (MRCI). The two MRCI wavefunctions considered were second-order CI (SOCI), which is ``the ultimate`` MRCI wavefunction, including all single and double excitations out of every determinant present in the CASSCF procedure, and an approximation called CISD[TQ] which includes only partial triple and quadruple excitations. This work is described in our article, M. L. Abrams and C. D. Sherrill, J. Phys. Chem. A 107, 5611-5616 (2003). We have also compared some multi-reference methods against full CI for the more challenging C₂ molecule in another article, C. D. Sherrill and P. Piecuch, J. Chem. Phys. 122, 124104 (2005).

As mentioned above, multi-reference methods are not as straightforward to apply as single-reference methods. One difficulty is that the user has to select an ``active space'' of orbitals in which the reference determinants are formed. There is no clear-cut way to choose the active space. Some researchers pick the valence space (i.e., a number of orbitals of each symmetry type consistent with the MO's formed from a minimal basis of atomic orbitals). Others choose a one-to-one active space that has one antibonding orbital for each bonding orbital. Usually, researchers use neither of these spaces, which tend to be large, and they simply pick a few of the ``most important'' orbitals using ``chemical intuition.'' We wish to investigate in a more systematic way how this choice affects results in comparison to the exact values.

Results of the the nonparallelity errors for various multi-reference methods using the valence (val) or one-to-one (1:1) active spaces are given in the table below. Note that for BH, the valence space is larger than the 1:1 space, but for HF, the reverse is true. In the initial study of the reactions breaking bonds to hydrogen, we observed that the larger active space always works better, and the errors for the ``wrong'' choice of active space can be significant. In general, the multi-reference methods are much better than the single-reference methods, which is not a surprise except that the difference is larger than one might have expected for such simple reactions.

Generally speaking, the methods intended to capture dynamical electron correlation (CASPT2, CISD[TQ], SOCI) perform better than the basic CASSCF method, which comes as no surprise. SOCI performs very well indeed, except for the case of HF with a valence active space. Subsequent study suggested that this was an anomalous result due to an improper mixing in the CASSCF between valence and core orbitals, and it might be avoided by disallowing mixing between them. The CISD[TQ] approximation works well for the simpler test cases, but it not good for the C₂ molecule, where it performs worse (in terms of nonparallelity errors) than CASSCF.

Small errors vs Full CI for the lowest three singlet states of the difficult C₂ molecule, using the second-order CI method (labeled "MRCI")

In summary, our studies to date indicate that the bond breaking problem is more challenging than may be generally recognized. Single-reference methods have difficulty even for the simplest possible cases, and multi-reference methods can have a significant dependence on the choice of the active space.