Approaching the Complete Basis Set Limit for Spin-State Energetics of Mononuclear First-Row Transition Metal Complexes

Convergence to the complete basis set (CBS) limit is analyzed for the problem of spin-state energetics in mononuclear first-row transition metal (TM) complexes by taking under scrutiny a benchmark set of 18 energy differences between spin states for 13 chemically diverse TM complexes. The performance of conventional CCSD(T) and explicitly correlated CCSD(T)-F12a/b calculations in approaching the CCSD(T)/CBS limits is systematically studied. An economic computational protocol is developed based on the CCSD-F12a approximation and (here proposed) modified scaling of the perturbative triples term (T#). This computational protocol recovers the relative spin-state energetics of the benchmark set in excellent agreement with the reference CCSD(T)/CBS limits (mean absolute deviation of 0.4, mean signed deviation of 0.2, and maximum deviation of 0.8 kcal/mol) and enables performing canonical CCSD(T) calculations for mononuclear TM complexes sized up to ca. 50 atoms, which is illustrated by application to heme-related metalloporphyrins. Furthermore, a good transferability of the basis set incompleteness error (BSIE) is demonstrated for spin-state energetics computed using CCSD(T) and other wave function methods (MP2, CASPT2, CASPT2/CC, NEVPT2, and MRCI + Q), which justifies efficient focal-point approximations and simplifies the construction of multimethod benchmark studies.

Benchmarks for transition metal spin-state energetics: why and how to employ experimental reference data?

Accurate prediction of energy differences between alternative spin states of transition metal complexes is essential in computational (bio)inorganic chemistry-for example, in characterization of spin crossover materials and in the theoretical modeling of open-shell reaction mechanisms-but it remains one of the most compelling problems for quantum chemistry methods. A part of this challenge is to obtain reliable reference data for benchmark studies, as even the highest-level applicable methods are known to give divergent results. This Perspective discusses two possible approaches to method benchmarking for spin-state energetics: using either theoretically computed or experiment-derived reference data. With the focus on the latter approach, an extensive general review is provided for the available experimental data of spin-state energetics and their interpretations in the context of benchmark studies, targeting the possibility of back-correcting the vibrational effects and the influence of solvents or crystalline environments. With a growing amount of experience, these effects can be now not only qualitatively understood, but also quantitatively modeled, providing the way to derive nearly chemically accurate estimates of the electronic spin-state gaps to be used as benchmarks and advancing our understanding of the phenomena related to spin states in condensed phases.

How does multi-reference computation change the catalysis chemistry? DFT and CASPT2 studies of the Cu-catalysed coupling reactions between aryl iodides and β-diketones

The molecular mechanism of a Cu-catalysed coupling reaction was theoretically studied using density functional theory (DFT) and the complete active space self-consistent field method followed by the second-order perturbation theory (CASSCF/CASPT2) to investigate the effects of the strong electron correlation of the Cu centre on the reaction profile. Both DFT and CASSCF/CASPT2 calculations showed that the catalytic cycle proceeds via an oxidative addition (OA) reaction, followed by a reductive elimination (RE) reaction, where OA is the rate-determining step. Although the DFT-calculated activation energies of the OA and RE steps are highly dependent on the choice of functionals, the CASSCF/CASPT2 results are less affected by the choice of DFT-optimised geometries. Therefore, with a careful assessment based on the CASSCF/CASPT2 single-point energy evaluation, an optimal choice of the DFT geometry is of good qualitative use for energetics at the CASPT2 level of theory. Based on the changes in the electron populations of the 3d orbitals during the OA and RE steps, the characteristic features of the DFT-calculated electronic structure were qualitatively consistent with those calculated using the CASSCF method. Further electronic structure analysis by the natural orbital occupancy of the CASSCF wavefunction showed that the ground state is almost single-reference in this system and the strong electron correlation effect of the Cu centre can be dealt with using the MP2 or CCSD method, too. However, the slightly smaller occupation numbers of the 3dπ orbital in the course of reactions suggested that the electron correlation effect of the Cu(III) centre appears through the interaction between the 3dπ orbital and the C-I antibonding σ* orbital in the OA step, and between the 3dπ orbital and the Cu-C antibonding σ* orbital in the RE step.

The Nature of the Chemical Bonds of High-Valent Transition-Metal Oxo (M=O) and Peroxo (MOO) Compounds: A Historical Perspective of the Metal Oxyl-Radical Character by the Classical to Quantum Computations

This review article describes a historical perspective of elucidation of the nature of the chemical bonds of the high-valent transition metal oxo (M=O) and peroxo (M-O-O) compounds in chemistry and biology. The basic concepts and theoretical backgrounds of the broken-symmetry (BS) method are revisited to explain orbital symmetry conservation and orbital symmetry breaking for the theoretical characterization of four different mechanisms of chemical reactions. Beyond BS methods using the natural orbitals (UNO) of the BS solutions, such as UNO CI (CC), are also revisited for the elucidation of the scope and applicability of the BS methods. Several chemical indices have been derived as the conceptual bridges between the BS and beyond BS methods. The BS molecular orbital models have been employed to explain the metal oxyl-radical character of the M=O and M-O-O bonds, which respond to their radical reactivity. The isolobal and isospin analogy between carbonyl oxide R2C-O-O and metal peroxide LFe-O-O has been applied to understand and explain the chameleonic chemical reactivity of these compounds. The isolobal and isospin analogy among Fe=O, O=O, and O have also provided the triplet atomic oxygen (3O) model for non-heme Fe(IV)=O species with strong radical reactivity. The chameleonic reactivity of the compounds I (Cpd I) and II (Cpd II) is also explained by this analogy. The early proposals obtained by these theoretical models have been examined based on recent computational results by hybrid DFT (UHDFT), DLPNO CCSD(T0), CASPT2, and UNO CI (CC) methods and quantum computing (QC).

Metastable Charged Dimers in Organometallic Species: A Look into Hydrogen Bonding between Metallocene Derivatives

Multiply charged complexes bound by noncovalent interactions have been previously described in the literature, although they were mostly focused on organic and main group inorganic systems. In this work, we show that similar complexes can also be found for organometallic systems containing transition metals and deepen in the reasons behind the existence of these species. We have studied the structures, binding energies, and dissociation profiles in the gas phase of a series of charged hydrogen-bonded dimers of metallocene (Ru, Co, Rh, and Mn) derivatives isoelectronic with the ferrocene dimer. Our results indicate that the carboxylic acid-containing dimers are more strongly bonded and present larger barriers to dissociation than the amide ones and that the cationic complexes tend to be more stable than the anionic ones. Additionally, we describe for the first time the symmetric proton transfer that can occur while in the metastable phase. Finally, we use a density-based energy decomposition analysis to shine light on the nature of the interaction between the dimers.

Advances in Parallel Heat Bath Configuration Interaction

Heat-bath configuration interaction (HCI) is a deterministic method that approaches the full CI limit at greatly reduced computational cost. In this work, computational improvements to the HCI algorithm are introduced targeting speed, parallel efficiency, and memory requirements. The new implementation introduces a hash function to distribute determinants and takes advantage of MPI and OpenMP for parallelism allowing for a (22e,168o) active space to be studied, which explicitly includes 2.39 × 10⁷ variational determinants and 8.95 × 10¹⁰ perturbative determinants. Benchmarks show up to 86% parallel efficiency of the perturbative step on 32 nodes (4096 cores) and a total efficiency of 74%. The new HCI implementation is benchmarked for accuracy against prior results and applied to study the triplet-quintet gap in the challenging [FeO(NH₃)₅]²⁺ complex.

An iron(ii)-based metalloradical system for intramolecular amination of C(sp2)-H and C(sp3)-H bonds: synthetic applications and mechanistic studies

A catalytic system for intramolecular C(sp2)-H and C(sp3)-H amination of substituted tetrazolopyridines has been successfully developed. The amination reactions are developed using an iron-porphyrin based catalytic system. It has been demonstrated that the same iron-porphyrin based catalytic system efficiently activates both the C(sp2)-H and C(sp3)-H bonds of the tetrazole as well as azide-featuring substrates with a high level of regioselectivity. The method exhibited an excellent functional group tolerance. The method affords three different classes of high-value N-heterocyclic scaffolds. A number of important late-stage C-H aminations have been performed to access important classes of molecules. Detailed studies (experimental and computational) showed that both the C(sp2)-H and C(sp3)-H amination reactions involve a metalloradical activation mechanism, which is different from the previously reported electro-cyclization mechanism. Collectively, this study reports the discovery of a new class of metalloradical activation modes using a base metal catalyst that should find wide application in the context of medicinal chemistry, drug discovery and industrial applications.

Reconciling Local Coupled Cluster with Multireference Approaches for Transition Metal Spin-State Energetics

Spin-state energetics of transition metal complexes remain one of the most challenging targets for electronic structure methods. Among single-reference wave function approaches, local correlation approximations to coupled cluster theory, most notably the domain-based local pair natural orbital (DLPNO) approach, hold the promise of bringing the accuracy of coupled cluster theory with single, double, and perturbative triple excitations, CCSD(T), to molecular systems of realistic size with acceptable computational cost. However, recent studies on spin-state energetics of iron-containing systems raised doubts about the ability of the DLPNO approach to adequately and systematically approximate energetics obtained by the reference-quality complete active space second-order perturbation theory with coupled-cluster semicore correlation, CASPT2/CC. Here, we revisit this problem using a diverse set of iron complexes and examine several aspects of the application of the DLPNO approach. We show that DLPNO-CCSD(T) can accurately reproduce both CASPT2/CC and canonical CCSD(T) results if two basic principles are followed. These include the consistent use of the improved iterative (T₁) versus the semicanonical perturbative triple corrections and, most importantly, a simple two-point extrapolation to the PNO space limit. The latter practically eliminates errors arising from the default truncation of electron-pair correlation spaces and should be viewed as standard practice in applications of the method to transition metal spin-state energetics. Our results show that reference-quality results can be readily achieved with DLPNO-CCSD(T) if these principles are followed. This is important also in view of the applicability of the method to larger single-reference systems and multinuclear clusters, whose treatment of dynamic correlation would be challenging for multireference-based approaches.

On the fly swapping algorithm for ordering of degrees of freedom in density matrix renormalization group

Density matrix renormalization group (DMRG) and its time-dependent variants have found widespread applications in quantum chemistry, includingab initioelectronic structure of complex bio-molecules, spectroscopy for molecular aggregates, and charge transport in bulk organic semiconductors. The underlying wavefunction ansatz for DMRG, matrix product state (MPS), requires mapping degrees of freedom (DOF) into a one-dimensional topology. DOF ordering becomes a crucial factor for DMRG accuracy. In this work, we propose swapping neighboring DOFs during the DMRG sweeps for DOF ordering, which we term 'on the fly swapping' (OFS) algorithm. We show that OFS is universal for both static and time-dependent DMRG with minimum computational overhead. Examples are given for one dimensional antiferromagnetic Heisenberg model,ab initioelectronic structure of N₂molecule, and the S₁/S₂internal conversion dynamics of pyrazine molecule. It is found that OFS can indeed improve accuracy by finding better DOF ordering in all cases.

Calculation of Metallocene Ionization Potentials via Auxiliary Field Quantum Monte Carlo: Toward Benchmark Quantum Chemistry for Transition Metals

The accurate ab initio prediction of ionization energies is essential to understanding the electrochemistry of transition metal complexes in both materials science and biological applications. However, such predictions have been complicated by the scarcity of gas phase experimental data, the relatively large size of the relevant molecules, and the presence of strong electron correlation effects. In this work, we apply all-electron phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) utilizing multideterminant trial wave functions to six metallocene complexes to compare the computed adiabatic and vertical ionization energies with experimental results. We find that ph-AFQMC yields mean absolute errors (MAEs) of 1.69 ± 1.02 kcal/mol for the adiabatic energies and 2.85 ± 1.13 kcal/mol for the vertical energies. We also carry out density functional theory (DFT) calculations using a variety of functionals, which yields MAEs of 3.62-6.98 kcal/mol and 3.31-9.88 kcal/mol, as well as one variant of localized coupled cluster calculations (DLPNO-CCSD(T₀) with moderate PNO cutoffs), which has MAEs of 4.96 and 6.08 kcal/mol, respectively. We also test the reliability of DLPNO-CCSD(T₀) and DFT on acetylacetonate (acac) complexes for adiabatic energies measured in the same manner experimentally, and we find higher MAEs, ranging from 4.56 to 10.99 kcal/mol (with a different ordering) for DFT and 6.97 kcal/mol for DLPNO-CCSD(T₀). Finally, by utilizing experimental solvation energies, we show that accurate reduction potentials in solution for the metallocene series can be obtained from the AFQMC gas phase results.

Coupled Cluster Benchmark of New DFT and Local Correlation Methods: Mechanisms of Hydroarylation and Oxidative Coupling Catalyzed by Ru(II, III) Chloride Carbonyls

We have evaluated a set of accurate canonical CCSD(T) energies for stationary points on the potential energy surface for Ru(II, III) chloride carbonyl catalysis of two competing reactions between benzene and methyl acrylate (MA), namely, hydroarylation and oxidative coupling. We have then applied this set to evaluate the performance of localized orbital coupled-cluster methods and several new and common density functionals. We find that (a) DLPNO-CCSD(T) with TightPNO cutoffs is an acceptable substitute for full canonical CCSD(T) calculations on this system; (b) for the closed-shell systems where it could be applied, LNO-CCSD(T) with tight convergence criteria is very close to the canonical results; (c) the recent ωB97X-V and ωB97M-V functionals exhibit superior performance to commonly used DFT functionals in both closed- and open-shell calculations; (d) the revDSD-PBEP86 revision of the DSD-PBEP86 double hybrid represents an improvement over the original, even though transition metals were not involved in its parametrization; and (e) DSD-SCAN and DOD-SCAN show comparable efficiency. Most tested (meta)-GGA and hybrid density functionals perform better for open-shell than for closed-shell complexes; this is not the case for the double hybrids considered.

Redox Isomerism in the S3 State of the Oxygen-Evolving Complex Resolved by Coupled Cluster Theory

The electronic and geometric structures of the water-oxidizing complex of photosystem II in the steps of the catalytic cycle that precede dioxygen evolution remain hotly debated. Recent structural and spectroscopic investigations support contradictory redox formulations for the active-site Mn4 CaOx cofactor in the final metastable S3 state. These range from the widely accepted MnIV 4 oxo-hydroxo model, which presumes that O-O bond formation occurs in the ultimate transient intermediate (S4 ) of the catalytic cycle, to a MnIII 2 MnIV 2 peroxo model representative of the contrasting "early-onset" O-O bond formation hypothesis. Density functional theory energetics of suggested S3 redox isomers are inconclusive because of extreme functional dependence. Here, we use the power of the domain-based local pair natural orbital approach to coupled cluster theory, DLPNO-CCSD(T), to present the first correlated wave function theory calculations of relative stabilities for distinct redox-isomeric forms of the S3 state. Our results enabled us to evaluate conflicting models for the S3 state of the oxygen-evolving complex (OEC) and to quantify the accuracy of lower-level theoretical approaches. Our assessment of the relevance of distinct redox-isomeric forms for the mechanism of biological water oxidation strongly disfavors the scenario of early-onset O-O formation advanced by literal interpretations of certain crystallographic models. This work serves as a case study in the application of modern coupled cluster implementations to redox isomerism problems in oligonuclear transition metal systems.

On the Use of Normalized Metrics for Density Sensitivity Analysis in DFT

In the past years, there has been a discussion about how the errors in density functional theory might be related to errors in the self-consistent densities obtained from different density functional approximations. This, in turn, brings up the discussion about the different ways in which we can measure such errors and develop metrics that assess the sensitivity of calculated energies to changes in the density. It is important to realize that there cannot be a unique metric in order to look at this density sensitivity, simultaneously needing size-extensive and size-intensive metrics. In this study, we report two metrics that are widely applicable to any density functional approximation. We also show how they can be used to classify different chemical systems of interest with respect to their sensitivity to small variations in the density.