Evaluating Multiconfigurational Trials for Accurate Phaseless Auxiliary-Field Quantum Monte Carlo on 3d Transition Metal Complexes

TL;DR

This work tackles the challenge of accurately predicting ionization potentials for transition metal complexes by leveraging phaseless AFQMC with multiconfigurational trial wavefunctions. It benchmarks three scalable MC-trial protocols (CASSCF small active space, HCIx2, and HCI-CASSCF) on the 3dTMV set and metallocenes, employing TZ and CBS extrapolation to approach experimental accuracy. The results identify HCI-CASSCF as a highly practical default for large systems, while DLPNO-CCSD(T1)/UB3LYP CBS corrections provide reliable, cost-effective extrapolations; ph-AFQMC with CISD trials also delivers excellent accuracy at higher cost. The findings enable scalable, high-accuracy quantum-chemical modeling of TM complexes and offer concrete CBS strategies to balance precision and computational effort.

Abstract

In this study, we evaluate multi-configurational trial wave function protocols for phaseless auxiliary field quantum Monte Carlo (ph-AFQMC) on transition metal containing systems. First, we benchmark vertical ionization potentials for 22 3d transition metal complexes against published high-accuracy ph-AFQMC values in a double zeta basis set. We then compute the vertical ionization potential for a set of six metallocenes using our best-performing protocol, alongside ph-AFQMC using a configuration interaction singles and doubles (CISD) trial state. We also analyze the performance of canonical coupled-cluster theory with singles, doubles and perturbative triples (CCSD(T)), as well as its local approximation using domain-based local pair natural orbitals (DLPNO-CCSD(T1)) using different reference orbitals. To reach the complete-basis-set (CBS) limit, we examine several extrapolation schemes and report CBS-limit ph-AFQMC and CCSD(T) values alongside experimental results. We find that ph-AFQMC with the best-performing trial in a triple zeta basis, followed by CBS correction from DLPNO-CCSD(T1) with unrestricted B3LYP reference orbitals, yields small deviations from experiment at modest cost. Using a CISD trial state in ph-AFQMC gives the closest agreement with experiment (errors < 2 kcal/mol), albeit with lower scalability.

Figures (4)

• Figure 1: Mean absolute deviation (MAD, top), root-mean-square deviation (RMSD, middle), and maximum absolute deviation (Max $|\mathrm{Dev.}|$, bottom) for ph-AFQMC with different trials for vertical IPs of the SR, SR/MR, and the entire 3dTMV set. ph-AFQMC/HCISCF from Ref. Neugebauer2023 serve as the reference. UHF and UPBE0 are single determinant trials; the remaining three trials are multiconfigurational. "QMC" denotes ph-AFQMC.
• Figure 2: Local error of DLPNO-CCSD(T1) with UHF and UB3LYP reference orbitals for vertical IPs of metallocenes in cc-pVTZ-DKH basis. The shaded gray bar indicates canonical CCSD(T) $\pm3.0~\mathrm{kcal/mol}$.
• Figure 3: Comparison of CBS corrections, calculated using different low level methods shown in the legends, for metallocene vertical IPs. The top two panels refer to errors (compared to experiment) in CBS-corrected AFQMC/HCI-CASSCF energies, and the bottom ones to errors in CBS-corrected AFQMC/CISD energies. Panels on the left show errors for each metallocene, and those on the right show aggregated errors. The grey bar in the left panels indicates the $\pm 2.31~\mathrm{kcal/mol}$ range of experimental error. "QMC" is short for ph-AFQMC, and "DLPNO" is short for DLPNO-CCSD(T1)/UB3LYP.
• Figure 4: Vertical IPs of metallocenes at the CBS limit from DLPNO-CCSD(T1) and canonical CCSD(T) using UHF reference orbitals (top 2 panels) and UB3LYP reference orbitals (bottom 2 panels). The canonical CCSD(T) values are extrapolated with the indicated low-level methods (“CC/low method” in the legend), whereas DLPNO-CCSD(T1) values use direct TZ/QZ extrapolation. The grey band in the left panels indicates the experimental uncertainty ($\pm 2.31~\mathrm{kcal/mol}$). “QMC” denotes ph-AFQMC, “CC” denotes canonical CCSD(T), and “DLPNO” denotes DLPNO-CCSD(T1)/UB3LYP.