The State-Universal Multi-Reference Coupled-Cluster Theory: An Overview of Some Recent Advances

Abstract

Some recent advances in the area of multi-reference coupled-cluster theory of the state-universal type are overviewed. An emphasis is placed on the following new developments: (i) the idea of combining the state-universal multi-reference coupled-cluster singles and doubles method (SUMRCCSD) with the multi-reference many-body perturbation theory (MRMBPT), in which cluster amplitudes of the SUM-RCCSD formalism that carry only core and virtual orbital indices are replaced by their first-order MRMBPT estimates; and (ii) the idea of combining the recently proposed method of moments of coupled-cluster equations with the SUMRCC formalism. It is demonstrated that the new SUMRCCSD(1) method, obtained by approximating the SUMRCCSD cluster amplitudes carrying only core and virtual orbital indices by their first-order MRMBPT values, provides the results that are comparable to those obtained with the complete SUMRCCSD approach.

Multireference coupled-cluster Ansatz

“…The MRexpT method of Hanrath [112,[139][140][141][142] and the C-conditions method of Li and Paldus [92][93][94][95][96][97] seem to offer a promising solution of the first problem. On should also point out that the incomplete model spaces and the C-conditions can also be used in the state-selective framework -in the BW-CC [119] and [60,151], who approximated the core-virtual bi-excited amplitudes by perturbation theory and obtained very good results with a greatly reduced number of variables in the nonlinear CC equations. In fact, this core-virtual bi-excited amplitudes are, through the first order of perturbation theory, independent of µ and one may contemplate to determine them nonperturbatively from MRCC equation assuming that they are the same for all µ's through all orders [152].…”

“…The difficulties to converge the solutions of MRCC equations may be overcome by using better convergence acceleration or equation regularization techniques [134], or by including higher excitations in T µ , which should reduce the number of unphysical solutions [60,86] hampering convergence to the physical ones. A promising way of solving these convergence difficulty, successfully applied in the context of the Fock space CC theory [42], is to use the intermediate-Hamiltonian approach to reformulate the SU-CC equations [153].…”

“…This is certainly not a situation conducive to black-box type applications of the SU-CC theory. There is some hope, however, that the inclusion of higher than doubly-excited clusters will help to reduce the number of multiple solutions [60,87,88] and will alleviate problems with converging the SU-CC equations in physically relevant applications.…”

New classes of non-iterative energy corrections to multi-reference coupled-cluster energies

“…, M). Aside from various mathematical difficulties that this assumption creates, the requirement of having a separate cluster operator T ð pÞ for each reference configuration jÈ p i leads to an excessively large number of cluster amplitudes when the dimension of the model space (M) is large, particularly when we are only interested in a few lowlying states whose number is much less than M. We have recently addressed an issue of an excessively large number of cluster amplitudes in the SUMRCCSD theory by proposing the SUMRCCSD(1) approach, which is based on the idea of combining the SUMRCCSD method with the MRMBPT approach [133,134]. In the SUMRCCSD(1) method, the most numerous doubly excited cluster amplitudes, which carry only core and virtual orbital indices, are approximated by their firstorder MRMBPT estimates.…”

“…The SUMRCC methods are also capable of providing the spectroscopically accurate description of electronic energy separations in small molecular systems, as has been illustrated by the calculations of the singlettriplet (A 1 A 1 À X 3 B 1 ) [125,126] and singlet-singlet (2 1 A 1 À 1 1 A 1 ; 1 1 A 1 A 1 A 1 ) [124,132] energy gaps in methylene. Unfortunately, apart from the earlier advances in formulating, implementing, and testing the spin-adapted and spin-orbital SUMRCCSD methods [6,8,[111][112][113][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128][129][130] and apart from the limited recent activities in our group (see, e.g., [131][132][133][134]), the group of Pal [135], who formulated the response SUMRCC theory, and the Paldus Waterloo group [136][137][138][139], who introduced the generalized version of the SUMRCC method enabling the systematic use of incomplete model spaces in SUMRCC calculations, the development of the genuine SUMRCC method has practically stopped. We believe that this situation should change, since the SUMRCC theory has the potential of offering an elegant and, at the same time, well balanced description of many classes of ground and excited electronic states, particularly when the quasi-degeneracies and the degree of non-dynamic correlation are too severe to be handled by other CC methods.…”