Consistent inclusion of triple substitutions within a coupled cluster based static quantum embedding theory

TL;DR

The paper addresses the challenge of achieving CCSD(T)-level accuracy in systems with strong correlation by extending the MPCC static embedding framework to include a CCSDT solver for the fragment and perturbative treatments of environment triples. It introduces two main variants, $MPCCSDT(pt)$ (single-shot environment triples) and $MPCCSDT(it)$ (iterative environment triples with back-coupling to the fragment), and compares them with low-level MP embed schemes. Across N$_2$, F$_2$, transition metal hydrides, and W4-11-like strongly correlated systems, the results show that environment triples are essential for high accuracy, and second-order treatment of environment amplitudes significantly improves performance in challenging cases; in many energy-difference problems, $MP2CCSDT(pt)$ provides a robust and efficient default. The work demonstrates a viable post-$ CCSD(T)$ embedding strategy that systematically improves upon standard embedding by incorporating higher-order correlations where they matter, with potential for scaling improvements and extensions to even higher-order solvers in the future.

Abstract

We incorporate a solver for the fragment problem with accuracy beyond coupled cluster singles and doubles (CCSD) into the previously proposed static embedding framework, MPCC. To this end, we employ a CCSDT solver for the fragment subsystem. For the environment subsystem, we construct a perturbative estimate of the triples amplitudes, explicitly accounting for feedback from all fragment amplitudes. The resulting approach is denoted MPCCSDT(pt). We further introduce a more complete formulation in which feedback from the environment amplitudes to the fragment amplitudes is also included. This scheme involves an iterative treatment of the environment triples amplitudes and is denoted MPCCSDT(it). In addition, we assess the accuracy of the previously proposed low-level method by introducing a modified low-level approach that incorporates a lowest-order treatment of selected long-range effects, including spin fluctuations and charge polarization. All resulting approaches may be viewed as post-CCSD(T) methods. We therefore consider test cases for which CCSD(T) exhibits substantial deviations from CCSDT. Our results demonstrate that inclusion of triples amplitudes at the fragment level alone is insufficient; a perturbative treatment of the environment triples amplitudes is required. For many energy-difference applications, feedback from the environment triples amplitudes to the fragment amplitudes, is not essential, but it does play a role in the very challenging molecules. A very interesting finding from our study is that in some challenging cases, we need an improved (second-order) perturbative method for the SD amplitudes, going beyond the first-order one used in our earlier work.

Figures (3)

• Figure 1: Electronic energy error with respect to UCCSDT along a PEC for N$_2$ in the cc-pCVTZ basis set. AVAS active space is constructed with cc-pVDZ basis set. We set the energy at dissociation as the zero energy for each method.
• Figure 2: Electronic energy error with respect to UCCSDT along a PEC for F$_2$ in the cc-pCVTZ basis set. AVAS active space is constructed with cc-pVDZ basis set. We set the energy at dissociation as the zero energy for each method.
• Figure S1: Interaction energy error with respect to UCCSDT along a PEC for N$_2$ in the cc-pCVTZ basis set. AVAS active space is constructed with the cc-pVDZ basis set. Density-fitted integrals are used for all the methods described here.