Communication: DFT treatment of strong correlation in 3d transition-metal diatomics

Intersystem Crossing Control of the Nb+ + CO2 → NbO+ + CO Reaction

The transfer of an oxygen atom from carbon dioxide (CO2) to a transition metal cation in the gas phase offers atomic level insights into single-atom catalysis for CO2 activation. Given that these reactions often involve open-shell transition metals, they may proceed through intersystem crossing between different spin manifolds. However, a definitive understanding of such spin-forbidden reaction requires dynamical calculations on multiple global potential energy surfaces (PESs) coupled by spin-orbit couplings. In this work, we report global PESs and spin-orbit couplings for three low-lying spin (quintet, triplet, and singlet) states for the reaction between the niobium cation (Nb+) and CO2, which are used to investigate the nonadiabatic reaction dynamics and kinetics. Comparison with experimental data of kinetics and collision dynamics shows satisfactory agreement. This reaction is found to be very similar to that between Ta+ + CO2. Specifically, our theoretical findings suggest that the rate-limiting step in this reaction is intersystem crossing, rather than potential barriers.

Symmetry breaking and self-interaction correction in the chromium atom and dimer

Density functional approximations to the exchange-correlation energy can often identify strongly correlated systems and estimate their energetics through energy-minimizing symmetry-breaking. In particular, the binding energy curve of the strongly correlated chromium dimer is described qualitatively by the local spin density approximation (LSDA) and almost quantitatively by the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA), where the symmetry breaking is antiferromagnetic for both. Here, we show that a full Perdew-Zunger self-interaction-correction (SIC) to LSDA seems to go too far by creating an unphysical symmetry-broken state, with effectively zero magnetic moment but non-zero spin density on each atom, which lies ∼4 eV below the antiferromagnetic solution. A similar symmetry-breaking, observed in the atom, better corresponds to the 3d↑↑4s↑3d↓↓4s↓ configuration than to the standard 3d↑↑↑↑↑4s↑. For this new solution, the total energy of the dimer at its observed bond length is higher than that of the separated atoms. These results can be regarded as qualitative evidence that the SIC needs to be scaled down in many-electron regions.

Activation of Methane by Zr+: A Deep-Dive into the Potential Surface via Pressure- and Temperature-Dependent Kinetics with Statistical Modeling

The kinetics of Zr⁺ + CH₄ are measured using a selected-ion flow tube apparatus over the temperature range 300-600 K and the pressure range 0.25-0.60 Torr. Measured rate constants are small, never exceeding 5% of the Langevin capture value. Both collisionally stabilized ZrCH₄⁺ and bimolecular ZrCH₂⁺ products are observed. A stochastic statistical modeling of the calculated reaction coordinate is used to fit the experimental results. The modeling indicates that an intersystem crossing from the entrance well, necessary for the bimolecular product to be formed, occurs faster than competing isomerization and dissociation processes. That sets an upper limit on the lifetime of the entrance complex to crossing of 10^-11 s. The endothermicity of the bimolecular reaction is derived to be 0.09 ± 0.05 eV, in agreement with a literature value. The observed ZrCH₄⁺ association product is determined to be primarily HZrCH₃⁺ not Zr⁺(CH₄), indicating that bond activation has occurred at thermal energies. The energy of HZrCH₃⁺ relative to separated reactants is determined to be -0.80 ± 0.25 eV. Inspection of the statistical modeling results under best-fit conditions reveals reaction dependences on impact parameter, translation energy, internal energy, and angular momentum. Reaction outcomes are heavily affected by angular momentum conservation. Additionally, product energy distributions are predicted.

Molecular-orbital-based machine learning for open-shell and multi-reference systems with kernel addition Gaussian process regression

We introduce a novel machine learning strategy, kernel addition Gaussian process regression (KA-GPR), in molecular-orbital-based machine learning (MOB-ML) to learn the total correlation energies of general electronic structure theories for closed- and open-shell systems by introducing a machine learning strategy. The learning efficiency of MOB-ML(KA-GPR) is the same as the original MOB-ML method for the smallest criegee molecule, which is a closed-shell molecule with multi-reference characters. In addition, the prediction accuracies of different small free radicals could reach the chemical accuracy of 1 kcal/mol by training on one example structure. Accurate potential energy surfaces for the H₁₀ chain (closed-shell) and water OH bond dissociation (open-shell) could also be generated by MOB-ML(KA-GPR). To explore the breadth of chemical systems that KA-GPR can describe, we further apply MOB-ML to accurately predict the large benchmark datasets for closed- (QM9, QM7b-T, and GDB-13-T) and open-shell (QMSpin) molecules.

del Campo J. Charge delocalization error in Piris Natural Orbital Functionals

Piris Natural Orbital Functionals (PNOF) have been recognized as a low-scaling alternative to study strong correlated systems. In this work, we address the performance of the fifth functional (PNOF5) and the seventh functional (PNOF7) to deal with another common problem, the charge delocalization error. The effects of this problem can be observed in charged systems of repeated well-separated fragments, where the energy should be the sum of the charged and neutral fragments, regardless of how the charge is distributed. In practice, an energetic overstabilization of fractional charged fragments leads to a preference for having the charge delocalized throughout the system. To establish the performance of PNOF functionals regarding charge delocalization error, charged chains of helium atoms and the W4-17-MR set molecules were used as base fragments and their energy, charge distribution and correlation regime were studied. It was found that PNOF5 prefers localized charge distributions, while PNOF7 improves the treatment of interpair static correlation and tends to the correct energetic limit for several cases, although a preference for delocalized charge distributions may arise in highly strong correlation regimes. Overall, it is concluded that PNOF functionals can simultaneously deal with static correlation and charge delocalization errors, resulting in a promising choice to study charge-related problems.

Effect of Intersystem Crossings on the Kinetics of Thermal Ion-Molecule Reactions: Ti+ + O2, CO2, and N2O

A selected-ion flow tube apparatus has been used to measure rate constants and product branching fractions of ²Ti⁺ reacting with O₂, CO₂, and N₂O over the range of 200-600 K. Ti⁺ + O₂ proceeds at near the Langevin capture rate constant of 6-7 × 10^-10 cm³ s^-1 at all temperatures to yield ⁴TiO⁺ + O. Reactions initiated on doublet or quartet surfaces are formally spin-allowed; however, the 50% of reactions initiated on sextet surfaces must undergo an intersystem crossing (ISC). Statistical theory is used to calculate the energy and angular momentum dependences of the specific rate constants for the competing isomerization and dissociation channels. This acts as an internal clock on the lifetime to ISC, setting an upper limit on the order of τ_ISC < 1e^-11 s. ²Ti⁺ + CO₂ produces ⁴TiO⁺ + CO less efficiently, with a rate constant fit as 5.5 ± 1.3 × 10^-11 (T/300)^{-1.1 ± 0.2} cm³ s^-1. The reaction is formally spin-prohibited, and statistical modeling shows that ISC, not a submerged transition state, is rate-limiting, occurring with a lifetime on the order of 10^-7 s. Ti⁺ + N₂O proceeds at near the capture rate constant. In this case, both Ti⁺ON₂ and Ti⁺N₂O entrance channel complexes are formed and can interconvert over a barrier. The main product is >90% TiO⁺ + N₂, and the remainder is TiN⁺ + NO. Both channels need to undergo ISC to form ground-state products but TiO⁺ can be formed in an excited state exothermically. Therefore, kinetic information is obtained only for the TiN⁺ channel, where ISC occurs with a lifetime on the order of 10^-9 s. Statistical modeling indicates that the dipole-preferred Ti⁺ON₂ complex is formed in ∼80% of collisions, and this value is reproduced using a capture model based on the generic ion-dipole + quadrupole long-range potential.

Insights into the effect of distal histidine and water hydrogen bonding on NO ligation to ferrous and ferric heme: a DFT study

The effect of distal histidine on ligation of NO to ferrous and ferric-heme, has been investigated with the high-level density functional theoretical (DFT) method. It has been predicted that the distal histidine significantly stabilizes the interaction of NO ferrous-heme (by −2.70 kcal mol⁻¹). Also, water hydrogen bonding is quite effective in strengthening the Fe–NO bond in ferrous heme. In contrast in ferric heme, due to the large distance between the H₂O and O(NO) and lack of hydrogen bonding, the distal histidine exhibits only a slight effect on the binding of NO to the ferric analogue. Concerning the bond nature of Fe^II–NO and Fe^III–NO in heme, a QTAIM analysis predicts a partially covalent and ionic bond nature in both systems.

The effect of distal histidine on ligation of NO to ferrous and ferric-heme, has been investigated with the high-level density functional theoretical (DFT) method.

Cyclotrimerization of Acetylene under Thermal Conditions: Gas-Phase Kinetics of V+ and Fe+ + C2H2

The kinetics of successive reactions of acetylene (C₂H₂) initiated on either vanadium or iron atomic cations have been investigated under thermal conditions using the variable-ion source and temperature-adjustable selected-ion flow tube apparatus. Consistent with the literature results, the reaction of Fe⁺ + C₂H₂ primarily yields Fe⁺(m/z = (C₂H₂)₃); however, analysis via quantum chemical calculations and statistical modeling shows that the mechanism does not form benzene upon the third acetylene addition. The kinetics are more consistent with successive addition of three acetylene molecules, yielding Fe⁺(C₂H₂)₃, followed by an addition of a fourth acetylene molecule, initiating cyclotrimerization, yielding either Fe⁺(C₂H₂) + neutral benzene or Fe⁺(Bz) + acetylene, where Bz is a benzene ligand. In contrast, the reaction of V⁺ + C₂H₂ yields products via successive associations V⁺(m/z = (C₂H₂)_n) either with or without a bimolecular step involving loss of one H₂ and V⁺C₂(m/z = (C₂H₂)_m), where n and m extend at least up to 11 under conditions of 0.32 Torr at 300 K. Stabilized V⁺(Bz) is not a significant intermediate in the association mechanism. We propose a plausible mechanism for the generation of neutral benzene in this reaction and compare with the Fe⁺ results. The reaction steps that produce benzene result in turnover of the single-atom catalyst, and the large hydrocarbons produced that remain associated to the catalyst are proposed to be polycyclic aromatic hydrocarbons.

Local hybrid functionals augmented by a strong-correlation model

The strong-correlation factor of the recent KP16/B13 exchange-correlation functional has been adapted and applied to the framework of local hybrid (LH) functionals. The expression identifiable as nondynamical (NDC) and dynamical (DC) correlations in LHs is modified by inserting a position-dependent KP16/B13-style strong-correlation factor q_AC(r) based on a local version of the adiabatic connection. Different ways of deriving this factor are evaluated for a simple one-parameter LH based on the local density approximation. While the direct derivation from the LH NDC term fails due to known deficiencies, hybrid approaches, where the factor is determined from the B13 NDC term as in KP16/B13 itself, provide remarkable improvements. In particular, a modified B13 NDC expression using Patra's exchange-hole curvature showed promising results. When applied to the simple LH as a first attempt, it reduces atomic fractional-spin errors and deficiencies of spin-restricted bond dissociation curves to a similar extent as the KP16/B13 functional itself while maintaining the good accuracy of the underlying LH for atomization energies and reaction barriers in weakly correlated situations. The performance of different NDC expressions in deriving strong-correlation corrections is analyzed, and areas for further improvements of strong-correlation corrected LHs and related approaches are identified. All the approaches evaluated in this work have been implemented self-consistently into a developers' version of the Turbomole program.

Replacing hybrid density functional theory: motivation and recent advances

Density functional theory (DFT) is the most widely-used electronic structure approximation across chemistry, physics, and materials science. Every year, thousands of papers report hybrid DFT simulations of chemical structures, mechanisms, and spectra. Unfortunately, hybrid DFT's accuracy is ultimately limited by tradeoffs between over-delocalization and under-binding. This review summarizes these tradeoffs, and introduces six modern attempts to go beyond them while maintaining hybrid DFT's relatively low computational cost: DFT+U, self-interaction corrections, localized orbital scaling corrections, local hybrid functionals, real-space nondynamical correlation, and our rung-3.5 approach. The review concludes with practical suggestions for DFT users to identify and mitigate these tradeoffs' impact on their simulations.

Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition

In this work, we provide a nuanced view of electron correlation in the context of transition metal complexes, reconciling computational characterization via spin and spatial symmetry breaking in single-reference methods with qualitative concepts from ligand-field and molecular orbital theories. These insights provide the tools to reliably diagnose the multi-reference character, and our analysis reveals that while strong (i.e., static) correlation can be found in linear molecules (e.g., diatomics) and weakly bound and antiferromagnetically coupled (monometal-noninnocent ligand or multi-metal) complexes, it is rarely found in the ground-states of mono-transition-metal complexes. This leads to a picture of static correlation that is no more complex for transition metals than it is, e.g., for organic biradicaloids. In contrast, the ability of organometallic species to form more complex interactions, involving both ligand-to-metal σ-donation and metal-to-ligand π-backdonation, places a larger burden on a theory's treatment of dynamic correlation. We hypothesize that chemical bonds in which inter-electron pair correlation is non-negligible cannot be adequately described by theories using MP2 correlation energies and indeed find large errors vs experiment for carbonyl-dissociation energies from double-hybrid density functionals. A theory's description of dynamic correlation (and to a less important extent, delocalization error), which affects relative spin-state energetics and thus spin symmetry breaking, is found to govern the efficacy of its use to diagnose static correlation.

Dioxygen Binding to all 3d, 4d, and 5d Transition Metals from Coupled-Cluster Theory

Understanding how transition metals bind and activate dioxygen (O₂ ) is limited by experimental and theoretical uncertainties, making accurate quantum mechanical descriptors of interest. Here we report coupled-cluster CCSD(T) energies with large basis sets and vibrational and relativistic corrections for 160 3d, 4d, and 5d metal-O₂ systems. We define four reaction energies (120 in total for the 30 metals) that quantify O-O activation and reveal linear relationships between metal-oxygen and O-O binding energies. The CCSD(T) data can be combined with thermochemical cycles to estimate chemisorption and physisorption energies for each metal from metal oxide embedding energies, in good correlation with atomization enthalpies (R² =0.75). Spin-geometry variations can break the linearities, of interest to circumventing the Sabatier principle. Pt, Pd, Co, and Fe form a distinct group with the weakest O₂ binding. R² up to 0.84 between surface adsorption energies and our energies for MO₂ systems indicate relevance also to real catalytic systems.

Strain effects on the electronic, optical and electrical properties of Cu2ZnSnS4: DFT study

Based on the density functional theory and Boltzmann transport theory, we investigated electronic, electrical and optical properties of Kesterite CZTS under different strain conditions. Our results indicate that both biaxial compressive and tensile strain effects lead to change in the band gap of this structure with different strain values. Furthermore, the edge of absorption, under the influence of an increasing compression, moves towards the short wavelengths. Electrical conductivity for pure CZTS and under dilatation and compression shows that with the increase of dilatation the conductivity of the material also increases, this physical property could be exploited to improve the performance of CZTS a suitable absorbent material in solar cells.

Methane Adducts of Gold Dimer Cations: Thermochemistry and Structure from Collision-Induced Dissociation and Association Kinetics

The bond dissociation energies at 0 K (BDE) of Au₂⁺-CH₄ and Au₂CH₄⁺-CH₄ have been determined using two separate experimental methods. Analyses of collision-induced dissociation cross sections for Au₂CH₄⁺ + Xe and Au₂(CH₄)₂⁺ + Xe measured using a guided ion beam tandem mass spectrometer (GIBMS) yield BDEs of 0.71 ± 0.05 and 0.57 ± 0.07 eV, respectively. Statistical modeling of association kinetics of Au₂(CH₄)_0-2⁺ + CH₄ + He measured from 200 to 400 K and at 0.3-0.9 Torr using a selected-ion flow tube (SIFT) apparatus yields slightly higher values of 0.81 ± 0.21 and 0.75 ± 0.25 eV. The SIFT data also place a lower limit on the BDE of Au₂C₂H₈⁺-CH₄ of 0.35 eV, likely an activated isomer, not Au₂(CH₄)₂⁺-CH₄. Particular emphasis is placed on determining the uncertainty in the derivation from association kinetics measurements, including uncertainties in collisional energy transfer, calculated harmonic frequencies, and possible contribution of isomerization of the association complexes. This evaluation indicates that an uncertainty of ±0.2 eV should be expected and that an uncertainty of better than ±0.1 eV is unlikely to be reasonable.

A study of the reactions of Ni+ and NiO+ ions relevant to planetary upper atmospheres

The reactions between Ni+(2D) and O3, O2, N2, CO2 and H2O were studied at 294 K using the pulsed laser ablation at 532 nm of a nickel metal target in a fast flow tube, with mass spectrometric detection of Ni+ and NiO+. The rate coefficient for the reaction of Ni+ with O3 is k(294 K) = (9.7 ± 2.1) × 10-10 cm3 molecule-1 s-1; the reaction proceeds at the ion-permanent dipole enhanced Langevin capture rate with a predicted T-0.16 dependence. Electronic structure theory calculations were combined with Rice-Ramsperger-Kassel-Markus theory to extrapolate the measured recombination rate coefficients to the temperature and pressure conditions of planetary upper atmospheres. The following low-pressure limiting rate coefficients were obtained for T = 120-400 K and He bath gas (in cm6 molecule-2 s-1, uncertainty ±σ at 180 K): log10(k, Ni+ + N2) = -27.5009 + 1.0667log10(T) - 0.74741(log10(T))2, σ = 29%; log10(k, Ni+ + O2) = -27.8098 + 1.3065log10(T) - 0.81136(log10(T))2, σ = 32%; log10(k, Ni+ + CO2) = -29.805 + 4.2282log10(T) - 1.4303(log10(T))2, σ = 28%; log10(k, Ni+ + H2O) = -24.318 + 0.20448log10(T) - 0.66676(log10(T))2, σ = 28%). Other rate coefficients measured (at 294 K, in cm3 molecule-1 s-1) were: k(NiO+ + O) = (1.7 ± 1.2) × 10-10; k(NiO+ + CO) = (7.4 ± 1.3) × 10-11; k(NiO+ + O3) = (2.7 ± 1.0) × 10-10 with (29 ± 21)% forming Ni+ as opposed to NiO2+; k(NiO2+ + O3) = (2.9 ± 1.4) × 10-10, with (16 ± 9)% forming NiO+ as opposed to ONiO2+; and k(Ni+·N2 + O) = (7 ± 4) × 10-12. The chemistry of Ni+ and NiO+ in the upper atmospheres of Earth and Mars is then discussed.

An efficient method for strongly correlated electrons in two-dimensions

This work deals with the problem of strongly correlated electrons in two-dimensions. We give a reduced density matrix (RDM) based tool through which the ground-state energy is given as a functional of the natural orbitals and their occupation numbers. Specifically, the Piris Natural Orbital Functional 7 (PNOF7) is used for studying the 2D Hubbard model and hydrogen square lattices. The singlet ground-state is studied, as well as the doublet mixed quantum state obtained by extracting an electron from the system. Our method satisfies two-index necessary N-representability conditions of the two-particle RDM (2RDM) and guarantees the conservation of the total spin. We show the ability of PNOF7 to describe strong correlation effects in two-dimensional (2D) systems by comparing our results with the exact diagonalization, density matrix renormalization group (DMRG), and auxiliary-field quantum Monte Carlo calculations. PNOF7 overcomes variational 2RDM methods with two- and three-index positivity N-representability conditions, reducing computational cost to mean-field scaling. Consistent results are obtained for small and large systems up to 144 electrons, weak and strong correlation regimes, and many filling situations. Unlike other methods, there is no dependence on dimensionality in the results obtained with PNOF7 and no particular difficulties have been observed to converge PNOF7 away from half-filling. Smooth double occupancy of sites is obtained, regardless of the filling. Symmetric dissociation of 2D hydrogen lattices shows that long-range nondynamic correlation dramatically affects electron detachment energies. PNOF7 compares well with DMRG along the dissociation curve.

A comparative study of CunX (X = Sc, Y; n = 1–10) clusters based on the structures, and electronic and aromatic properties

The MO diagrams and orbital contributions of the HOMO and LUMO for the Cu₇Sc and Cu₇Y clusters.

Au2+ cannot catalyze conversion of methane to ethene at low temperature

The previously reported conversion of methane to ethene catalyzed by Au₂⁺ at thermal energies is investigated through a combination of experiment and theory. The conversion is found not to occur, in-line with well-established thermodynamics.

Temperature and Isotope Dependent Kinetics of Nickel-Catalyzed Oxidation of Methane by Ozone

The temperature dependent kinetics of Ni⁺ + O₃ and of NiO⁺ + CH₄/CD₄ are measured from 300 to 600 K using a selected-ion flow tube apparatus. Together, these reactions comprise a catalytic cycle converting CH₄ to CH₃OH. The reaction of Ni⁺ + O₃ proceeds at the collisional limit, faster than previously reported at 300 K. The NiO⁺ product reacts further with O₃, also at the collisional limit, yielding both higher oxides (up to NiO₅⁺ is observed) as well as undergoing an apparent reduction back to Ni⁺. This apparent reduction channel is due to the oxidation channel yielding NiO₂⁺* with sufficient energy to dissociate. ⁴NiO⁺ + CH₄ (CD₄) (whereas ⁴NiO⁺ refers to the quartet state of NiO⁺) proceeds with a rate constant of (2.6 ± 0.4) × 10^-10 cm³ s^-1 [(1.8 ± 0.5) × 10^-10 cm³ s^-1] at 300 K and a temperature dependence of ∼ T^-0.7±0.3 (∼ T^-1.1±0.4), producing only the ²Ni⁺ + ¹CH₃OH channel up to 600 K. Statistical modeling of the reaction based on calculated stationary points along the reaction coordinate reproduces the experimental rate constant as a function of temperature but underpredicts the kinetic isotope shift. The modeling was found to better represent the data when the crossing from quartet to doublet surface was incomplete, suggesting a possible kinetic effect in crossing from quartet to doublet surfaces. Additionally, the modeling predicts a competing ³NiOH⁺ + ²CH₃ channel to become increasingly important at higher temperatures.

Comparative Study of Nonhybrid Density Functional Approximations for the Prediction of 3d Transition Metal Thermochemistry

The utility of several nonhybrid density functional approximations (DFAs) is considered for the prediction of gas phase enthalpies of formation for a large set of 3d transition metal-containing molecules. Nonhybrid DFAs can model thermochemical values for 3d transition metal-containing molecules with accuracy comparable to that of hybrid functionals. The GAM-generalized gradient approximation (GGA); the TPSS, M06-L, and MN15-L meta-GGAs; and the Rung 3.5 PBE+ΠLDA(s) DFAs all give root-mean-square deviations below that of the widely used B3LYP hybrid. Modern nonhybrid DFAs continue to show utility for transition metal thermochemistry.