Addressing Strong Correlation by Approximate Coupled-Pair Methods with Active-Space and Full Treatments of Three-Body Clusters

TL;DR

This paper tackles the failure of standard single-reference CC methods in strongly correlated regimes by extending approximate coupled-pair (ACP) approaches to include leading three-body correlations through active-space triples (ACCSDt/ACCSDT) and $n_{ m o}$/$n_{ m u}$-dependent scaling of $(T_2)^2$ diagrams. The authors introduce ACCSD(1, 3 × n_o/(n_o+n_u) + 4 × n_u/(n_o+n_u)), ACCSDt, and ACCSDT variants, with active spaces designed to capture dominant $T_3$ contributions, and benchmark them on symmetric dissociations of H$_6$ and H$_{10}$ rings and the H$_{50}$ chain. The results show that active-space triples combined with selective diagram scaling remove CCSD/CCSDt/CCSDT instabilities in strongly correlated regions and reproduce (near-)exact energies relative to FCI or DMRG, at a fraction of the cost. This work points to a practical pathway toward scalable, accurate treatment of strong correlation in larger ab initio systems using ACP-based schemes augmented with targeted $T_3$ physics.

Abstract

When the number of strongly correlated electrons becomes larger, the single-reference coupled-cluster (CC) CCSD, CCSDT, etc. hierarchy displays an erratic behavior, while traditional multi-reference approaches may no longer be applicable due to enormous dimensionalities of the underlying model spaces. These difficulties can be alleviated by the approximate coupled-pair (ACP) theories, in which selected $(T_2)^2$ diagrams in the CCSD amplitude equations are removed, but there is no generally accepted and robust way of incorporating connected triply excited ($T_3$) clusters within the ACP framework. It is also not clear if the specific combinations of $(T_2)^2$ diagrams that work well for strongly correlated minimum-basis-set model systems are optimum when larger basis sets are employed. This study explores these topics by considering a few novel ACP schemes with the active-space and full treatments of $T_3$ correlations and schemes that scale selected $(T_2)^2$ diagrams by factors depending on the numbers of occupied and unoccupied orbitals. The performance of the proposed ACP approaches is illustrated by examining the symmetric dissociations of the $\text{H}_6$ and $\text{H}_{10}$ rings using basis sets of the triple- and double-$ζ$ quality and the $\text{H}_{50}$ linear chain treated with a minimum basis, for which the conventional CCSD and CCSDT methods fail.

Figures (4)

• Figure 1: Goldstone--Brandow orbital diagrams for the $(T_2)^2$ contributions $\Lambda_k^{(2)}(AB,IJ;S_r)$, $k = 1\text{--}5$, to the CCD or CCSD equations projected on the singlet pp--hh coupled orthogonally spin-adapted doubly excited $\ket{\Phi_{IJ}^{AB}}_{S_r}$ states. The intermediate spin quantum number $S_r$ in the definition of the $\ket{\Phi_{IJ}^{AB}}_{S_r}$ states and the associated doubly excited cluster amplitudes $t_{AB}^{IJ}(S_r)$ in the definition of the orthogonally spin-adapted $T_{2}$ operators, represented by the Brandow-type, oval-shaped vertices, is 0 or 1. The occupied orbital indices $M$ and $N$, the unoccupied orbital indices $E$ and $F$, and the intermediate spin quantum numbers $\widetilde{S}_r$ and $\widetilde{S}_r^{\prime}$ are summed over. The ${\mathscr S}_{AB} = 1 + (AB)$ and ${\mathscr S}_{IJ} = 1 + (IJ)$ operators at the diagrams are index symmetrizers that translate into the symmetrizers or antisymmetrizers, ${\mathscr S}_{AB}(S_{r}) = 1 + (-1)^{S_{r}} (AB)$ and ${\mathscr S}_{IJ}(S_{r}) = 1 + (-1)^{S_{r}} (IJ)$, respectively, in the resulting algebraic expressions.
• Figure 2: Ground-state PECs [panels (a)--(c)] and errors relative to FCI [panels (d)--(f)] for the symmetric dissociation of the $\text{H}_{6}$ ring resulting from the CCSD and various ACCSD [panels (a) and (d)], CCSDt and various ACCSDt [panels (b) and (e)], and CCSDT and various ACCSDT [panels (c) and (f)] calculations using the cc-pVTZ basis set. The CCSDt and ACCSDt approaches employed a minimum active space consisting of three occupied and three lowest-energy unoccupied MOs that correlate with the 1$s$ shells of the individual hydrogen atoms. The FCI PEC included in panels (a)--(c) is shown to facilitate comparisons.
• Figure 3: Ground-state PECs [panels (a)--(c)] and errors relative to FCI [panels (d)--(f)] for the symmetric dissociation of the $\text{H}_{10}$ ring resulting from the CCSD and various ACCSD [panels (a) and (d)], CCSDt and various ACCSDt [panels (b) and (e)], and CCSDT and various ACCSDT [panels (c) and (f)] calculations using the DZ basis set. The CCSDt and ACCSDt approaches employed a minimum active space consisting of five occupied and five lowest-energy unoccupied MOs that correlate with the 1$s$ shells of the individual hydrogen atoms. The FCI PEC included in panels (a)--(c) is shown to facilitate comparisons. Note that the CCSD, CCSDt, and CCSDT calculations failed to converge in the $R_\text{H--H} \ge 1.8$ Å region.
• Figure 4: Ground-state PECs for the symmetric dissociation of the $\text{H}_{50}$ linear chain obtained in the (a) CCSD and $\text{ACCSD}(1{,}\tfrac{3+4}{2})$ and (b) CCSDT and $\text{ACCSDT}(1{,}\tfrac{3+4}{2})$ calculations using the STO-6G basis set. In this case $n_{\rm o} = n_{\rm u} = N_{\rm o} = N_{\rm u}$, so that $\text{ACCSD}(1{,}\tfrac{3+4}{2})$ = DCSD is equivalent to $\text{ACCSD}(1{,} 3 \times \tfrac{n_\text{o}}{n_\text{o} + n_\text{u}} + 4 \times \tfrac{n_\text{u}}{n_\text{o} + n_\text{u}})$ and $\text{ACCSDT}(1{,}\tfrac{3+4}{2})$ is equivalent to $\text{ACCSDt}(1{,}\tfrac{3+4}{2})$, $\text{ACCSDt}(1{,} 3 \times \tfrac{n_\text{o}}{n_\text{o} + n_\text{u}} + 4 \times \tfrac{n_\text{u}}{n_\text{o} + n_\text{u}})$, and $\text{ACCSDT}(1{,} 3 \times \tfrac{n_\text{o}}{n_\text{o} + n_\text{u}} + 4 \times \tfrac{n_\text{u}}{n_\text{o} + n_\text{u}})$. The LDMRG(500) PEC included for comparison purposes is based on the data reported in Ref. Hachmann2006. The insets show the errors relative to LDMRG(500). Note that the CCSD calculations failed to converge in the $R_\text{H--H} > 2.0$ bohr region. We could not converge the CCSDT equations beyond $R_\text{H--H} = 1.8$ bohr.