Scalar Relativistic All-Electron and Pseudopotential Ab Initio Study of a Minimal Nitrogenase [Fe(SH)4H]− Model Employing Coupled-Cluster and Auxiliary-Field Quantum Monte Carlo Many-Body Methods

Nitrogenase is the only enzyme that can cleave the triple bond in N2, making nitrogen available to organisms. The detailed mechanism of this enzyme is currently not known, and computational studies are complicated by the fact that different density functional theory (DFT) methods give very different energetic results for calculations involving nitrogenase models. Recently, we designed a [Fe(SH)4H]− model with the fifth proton binding either to Fe or S to mimic different possible protonation states of the nitrogenase active site. We showed that the energy difference between these two isomers (ΔE) is hard to estimate with quantum-mechanical methods. Based on nonrelativistic single-reference coupled-cluster (CC) calculations, we estimated that the ΔE is 101 kJ/mol. In this study, we demonstrate that scalar relativistic effects play an important role and significantly affect ΔE. Our best revised single-reference CC estimates for ΔE are 85–91 kJ/mol, including energy corrections to account for contributions beyond triples, core–valence correlation, and basis-set incompleteness error. Among coupled-cluster approaches with approximate triples, the canonical CCSD(T) exhibits the largest error for this problem. Complementary to CC, we also used phaseless auxiliary-field quantum Monte Carlo calculations (ph-AFQMC). We show that with a Hartree–Fock (HF) trial wave function, ph-AFQMC reproduces the CC results within 5 ± 1 kJ/mol. With multi-Slater-determinant (MSD) trials, the results are 82–84 ± 2 kJ/mol, indicating that multireference effects may be rather modest. Among the DFT methods tested, τ-HCTH, r2SCAN with 10–13% HF exchange with and without dispersion, and O3LYP/O3LYP-D4, and B3LYP*/B3LYP*-D4 generally perform the best. The r2SCAN12 (with 12% HF exchange) functional mimics both the best reference MSD ph-AFQMC and CC ΔE results within 2 kJ/mol.

Table S1.Specification of the basis sets used in the present study.The (-DK) basis sets were used together with either the DKH2 or the X2C scalar relativistic Hamiltonian.2) and LR-CCSD(TQ)-1 approaches using the various Hamiltonians and double-ζ basis sets.The canonical CC and extrapolated results are given in the first two and last two columns (prefixed via -ex), respectively.
Table S6.All-electron CC energy difference ΔE computed using the scalar relativistic DKH2 Hamiltonian and polarized core-valence cc-pCVXZ-DK basis set with either 50 or 10 frozen core electrons.For each CC method reported, the energy correlation effect (CV) between subvalence core -semi-core plus valence electrons is estimated as net difference in CC ΔE computed keeping frozen 50 and 10 electrons.The resulting CV energy corrections are 1.3 and 1.2 kJ/mol for the UBCCD(T), and LR-CCSD(T) methods, respectively.
Table S8.The a posteriori HOC corrected ΔE (kJ/mol) computed from the extrapolated CC energies using the various Hamiltonians and mixed-ζ VXZ basis sets.
Table S9.Total energies and energy differences computed at the HF and DFT HFLYP levels with various basis sets.The results denoted "KS HFLYP orbitals" correspond to the non-self-consistent ROHF energies computed using molecular orbitals results from solving the ROKS DFT equations self-consistently.The results denoted "HF orbitals" and "DFT HFLYP" correspond to fully selfconsistent calculations.Note that the results reported for both the cc-pVDZ-DK and cc-pVXZ-DK basis sets were computed using the X2C scalar relativistic Hamiltonian.

Table S3 .
HF and CC energies (a.u.) employing ccECP pseudopotentials and corresponding basis sets.Note that the restricted open-shell CR-CC(2.3) and CC(t;3) results were computed without the h-type basis functions for the ccECP-pVXZ basis set, i.e., the original ccECP-pVXZ basis set was truncated to the spdfg subset.However, the LR-CCSD(T) results were computed with the nontruncated original ccECP-pVXZ basis set.Therefore, two sets of ROCCSD/ccECP-pVXZ energies are reported, viz.without and with h-type functions, marked with * and ** , respectively.

Table S4 .
HF and CC energies (a.u.) employing the second-order Douglas-Kroll Hamiltonian and the corresponding scalar relativistic Dunning's basis sets.Note that the restricted open-shell CR-CC(2.3) and CC(t;3) results were computed without the h-type basis functions for the cc-pVXZ-DK basis set, i.e., the original cc-pVXZ-DK basis set was truncated to the spdfg subset.However, the LR-CCSD(T) results were computed with the non-truncated original cc-pVXZ-DK basis set.Therefore, two sets of ROCCSD/cc-pVXZ-DK energies are reported, viz.without and with h-type functions, marked with * and ** , respectively.

Table S10 .
Reference heat bath configuration interaction energies (a.u.) computed employing different basis sets and ε != 1 ⨯ 10 "# .The resulting number of determinants (ndets) is also reported for each structure and basis set.

Table S11 .
Unrestricted CC energies (a.u) and energy differences (kJ/mol) computed employing the cc-pVXZ-DK basis set with the DKH2 and X2C scalar relativistic Hamiltonians.

Table S12 .
Energy difference ΔE (kJ/mol) computed using ph-AFQMC with different Hamiltonians and trial wavefunctions, and the double-ζ quality basis sets.ndets is the number of Slater determinants.

Table S14 .
Energy difference ΔE (kJ/mol) computed using ph-AFQMC with different Hamiltonians and trial wavefunctions, and the mixed-ζ quality basis sets.ndets is the number of Slater determinants.

Table S16 .
The ph-AFQMC total electronic energies (a.u.) computed using the ccECP pseudopotentials and corresponding ccECP-pVXZ basis set.The spin-projected 2 ph-AFQMC/UHF total electronic energies (a.u.) computed using an original approach by Zhang et al.2

Table S18 .
Total ph-AFMQC electronic energies and ΔE energy differences (kJ/mol) computed with the X2C Hamiltonian, the MSD trial wavefunctions, and the mixed-ζ quality cc-pVXZ-DK basis set.The number of propagated blocks was set to 6500.

Table S21 .
The extrapolated energies to the complete basis set limit (CBS) for ex-UBCCD(T) and ex-LR-CCSD(T) employing Dunning's DK sets.Note that the ex-BCCD(T) and ex-LR-CCSD(T) results were computed using the X2C and DKH2 scalar relativistic Hamiltonian, respectively.In the case of Riemann zeta function extrapolation scheme, a unified-single-parameter-extrapolation scheme by Varandas et al. was used to extrapolate Hartree-Fock energies individually.3

Table S22 .
The extrapolated energies to the complete basis set limit (CBS) for ex-UBCCD(T) and ex-LR-CCSD(T) employing ccECP basis sets.In the case of Riemann zeta function extrapolation scheme, a unified-single-parameter-extrapolation scheme by Varandas et al. was used to extrapolate Hartree-Fock energies individually.3