Multireference Coupled-Cluster Methods: Recent Developments

Ring coupled cluster doubles at the multireference level

A ring approximation within an internally contracted multireference (MR) Coupled Cluster (CC) framework is worked out and tested. Derivation of equations utilizes MR based, generalized normal ordering and the corresponding generalized Wick-theorem (MR-GWT). Contractions among cluster operators are avoided by adopting a normal ordered exponential ansatz. The original version of the MR ring CC doubles (MR-rCCD) equations [Á. Szabados and Á. Margócsy, Mol. Phys. 115, 2731 (2017)] is rectified in two aspects. On the one hand, over-completeness of double excitations is treated by relying on the concept of frames. On the other hand, restriction on the maximal cumulant rank is lifted from two to four. This is found essential for obtaining reliable correlation corrections to the energy. The MR function underlying the approach is provided by the Generalized Valence Bond (GVB) model. The pair structure of the reference ensures a fragment structure of GVB cumulants. This represents a benefit when evaluating cumulant contractions appearing as a consequence of MR-GWT. In particular, cumulant involving terms remain less expensive than their traditional, pair-contracted counterpart, facilitating an O(N⁶) eventual scaling of the proposed MR-rCCD method. Pilot applications are presented for covalent bond breaking, deprotonation energies, and torsional potentials.

Iterative universal state selective correction for the Brillouin-Wigner multireference coupled-cluster theory

As a further development of the previously introduced a posteriori Universal State-Selective (USS) corrections [K. Kowalski, J. Chem. Phys. 134, 194107 (2011); J. Brabec et al., ibid. 136, 124102 (2012)], we suggest an iterative form of the USS correction by means of correcting effective Hamiltonian matrix elements. We also formulate USS corrections via the left Bloch equations. The convergence of the USS corrections with excitation level towards the full configuration interaction (FCI) limit is also investigated. Various forms of the USS and simplified diagonal USS corrections at the singles and doubles and perturbative triple levels are numerically assessed on several model systems and on the ozone and tetramethyleneethane molecules. It is shown that the iterative USS correction can successfully replace the previously developed a posteriori Brillouin-Wigner coupled cluster size-extensivity correction, while it is not sensitive to intruder states and performs well also in other cases when the a posteriori one fails, like, e.g., for the asymmetric vibration mode of ozone.

Potential energy surface studies via a single root multireference coupled cluster theory

We have employed complete active space based single root multireference coupled cluster method (the resulting method is referred to by the acronym sr-MRCC) to compute the potential energy surfaces (PESs) of some well studied "protypical model" systems for which a highly accurate and reliable database is available for comparison. As that of state-specific theory, the sr-MRCC approach focuses and correlates one state while using a multiconfigurational reference and thus it naturally avoids intruder states. The present method is structurally different from the well known state specific multireference coupled cluster (SS-MRCC) method introduced by Mahapatra et al. [Mol. Phys. 94, 157 (1998)]. As that of the SS-MRCC theory, the present method is also based on the Jeziorski-Monkhorst ansatz where a different exponential cluster operator exp(T(mu)) acts on its corresponding model function phi(mu). The final cluster finding equations contain coupling between the cluster operators for all the mu, which are mainly responsible to prove the extensivity of both the cluster amplitudes and the energy. The present sr-MRCC theory is size-extensive and size-consistent when localized orbitals are used. The systems considered here exhibit varying degrees of degeneracy at different regions of PES. The treatment of these systems via traditional effective Hamiltonian based methods suffers from divergence problems in the iterative solution of the CC equations (the issue termed as "intruder state"). The sr-MRCC results lie closer to the ones obtained by the SS-MRCC method for these systems. To judge the efficacy of the present method, we have compared our results with other previously published theoretical estimations, which clearly indicate that the present method is reliable in studying the dissociation PES of states plagued by electronic degeneracy as well as notorious intruder effects. The highly satisfactory performance of the sr-MRCC method, vis-a-vis the other sophisticated methods, in describing the lowest and the first excited singlet states of BeH(2) at points of high degeneracy is noticeable.

Diagnosis of the performance of the state-specific multireference coupled-cluster method with different truncation schemes

We have tested the linked version of a iterative (partial) triples correction for the Jeziorski-Monkhorst ansatz based state-specific multireference coupled cluster (SS-MRCC) approach with singles and doubles (SD) excitations [abbreviated as SS-MRCCSDT-1a and SS-MRCCSDT-1a+d]. The assessments of SS-MRCCSDT-1a and SS-MRCCSDT-1a+d schemes have been performed on the ground potential energy surface (PES) of P4, Li(2),Be(2) systems which demand the MR description, and on study of the excitation energy between the ground and first excited state for P4 system. Illustrations in the isomerization of cyclobutadiene also show the power of the schemes. One of the designed features of the SS-MRCCSDT-n methods introduced here is that they do not require storage of the triples amplitudes. In the entire range of geometries, we found a definite improvement provided by SS-MRCC with SDT-1a and SDT-1a+d schemes over the standard SD one. In the nondegenerate regions of PES, the closeness of the performance of the single-reference CC to the SS-MRCC methods increases after inclusion of even partial triple excitations. Generally, the performance of the SS-MRCCSDT-1a+d approach is closer to the corresponding full configuration interaction (FCI) one than to the SS-MRCCSDT-1a specially in the degenerate geometries (as is evident from nonparallelism error). The deviation from FCI for the first excited state of the P4 model using various SS-MRCC theories with different truncation schemes obtained by converging on the second root of the effective Hamiltonian has also been reported. We also compare our results with the current generation state-of-the-art single and multireference CC calculations to envisage the usefulness of the present approach. Initial implementation indicates that the SS-MRCCSDT-n formalism can provide not only reliable excitation energies and barrier height even when used in a relatively small model space, but also offers a considerable promise in generating the entire energy surface with low nonparallelity error.