Comparison of fully internally and strongly contracted multireference configuration interaction procedures

Multireference (MR) methods occupy an important class of approaches in quantum chemistry. In many instances, for example, in studying complex magnetic properties of transition metal complexes, they are actually the only physically satisfactory choice. In traditional MR approaches, single and double excitations are performed with respect to all reference configurations (or configuration state functions, CSFs), which leads to an explosive increase of computational cost for larger reference spaces. This can be avoided by the internal contraction scheme proposed by Meyer and Siegbahn, which effectively reduces the number of wavefunction parameters to their single-reference counterpart. The “fully internally contracted” scheme (FIC) is well known from the popular CASPT2 approach. An even shorter expansion of the wavefunction is possible with the “strong contraction” (SC) scheme proposed by Angeli and Malrieu in their NEVPT2 approach. Promising multireference configuration interaction formulations (MRCI) employing internal contraction and strong contraction have been reported by several authors. In this work, we report on the implementation of the FIC-MRCI and SC-MRCI methodologies, using a computer assisted implementation strategy. The methods are benchmarked against the traditional uncontracted MRCI approach for ground and excited states of small molecules (N2, O2, CO, CO+, OH, CH, and CN). For ground states, the comparison includes the “partially internally contracted” MRCI based on the Celani-Werner ansatz (PC-MRCI). For the three contraction schemes, the average errors range from 2% to 6% of the uncontracted MRCI correlation energies. Excitation energies are reproduced with ∼0.2 eV accuracy. In most cases, the agreement is better than 0.2 eV, even in cases with very large differential correlation contributions as exemplified for the d-d and ligand-to-metal charge transfer transitions of a Cu[NH3]42+ model complex. The benchmark is supplemented with the investigation of typical potential energy surfaces (i.e., N2, HF, LiF, BeH2, ethane C–C bond stretching, and the ethylene double bond torsion). Our results indicate that the SC-scheme, which is successful in the context of second- and third-order perturbation theory, does not offer computational advantages and at the same time leads to much larger errors than the PC and FIC schemes. We discuss the advantages and disadvantages of the PC and FIC schemes, which are of comparable accuracy and, for the systems tested, also of comparable efficiency.