We have begun our investigation into the reliability of standard approximations for bond breaking reactions by starting with the simplest possible case, breaking bonds to hydrogen atoms. One might suppose that this would not pose too great a difficulty and that the usual approximations would perform adequately. In one paper [A. Dutta and C. D. Sherrill, J. Chem. Phys. 118, 1610-1619 (2003)], we compared many single-reference theoretical methods to the exact full CI results for potential energy curves removing a hydrogen in BH, HF, and CH4 using aug-cc-pVQZ, 6-31G**, and 6-31G* basis sets, respectively. In principle, these should be among the easiest kinds of bond-breaking reactions for theory to treat because of the smaller number of electrons involved in bonds to hydrogen. We have also compared single-reference methods to full CI for the much more challenging case of the C2 molecule, which is known to be challenging for electronic structure theory even at its equilibrium geometry [M. L. Abrams and C. D. Sherrill, J. Chem. Phys. 121, 9122 (2004)].
In these comparisons, absolute errors in potential energies are not relevant, only relative errors are --- if an approximate potential energy curve is shifted up from the exact curve by a uniform amount across the entire curve, then all molecular properties (equilibrium bond length, vibrational frequencies, dissociation energy, etc) will be correctly predicted. However, a small error in one part of the curve compared to a large error in another part of the curve will lead to a potential with the wrong shape and therefore wrong molecular properties. A measure of these relative errors is the nonparallelity error (NPE), which is the difference between the maximum and minimum errors along the curve. The NPE's for several single-reference approximations are given below.
| Method | BH | HF | CH4 | C2 |
| RHF | large | large | large | large |
| MP2 | large | large | large | large |
| CCSD | 8.1 | 13.0 | 10.3 | 24.3 |
| CCSD(T) | 23.3 | 26.8 | large | large |
| B3LYP | large | large | large | n/a |
| UHF | 28.1 | 47.8 | 27.4 | 48.7 |
| UMP2 | 17.9 | 25.9 | 17.1 | 40.7 |
| UCCSD | 4.7 | 6.0 | 5.1 | 27.0 |
| UCCSD(T) | 3.1 | 3.7 | 3.2 | 21.6 |
| UB3LYP | n/a | 5.9 | 11.4 | n/a |
Our study also employed our optimized-orbital coupled cluster program [Sherrill et al., J. Chem. Phys. 109, 4171 (2003)]. We found that errors are generally reduced with this approach, but the largest errors along the curve remain about the same. In summary, even ``gold-standard'' single-reference methods like (U)CCSD(T) can have nonparallelity errors of several kcal/mol for bond breaking reactions even in very simple cases, and more complicated cases like C2 are not adequately handled by any single-reference method.
