THE CONSTRUCTION AND INTERPRETATION OF MCSCF WAVEFUNCTIONS

Chelate Complexes of 3d Transition Metal Ions─A Challenge for Electronic-Structure Methods?

Different electronic-structure methods were assessed for their ability to predict two important properties of the industrially relevant chelating agent nitrilotriacetic acid (NTA): its selectivity with respect to six different first-row transition metal ions and the spin-state energetics of its complex with Fe(III). The investigated methods encompassed density functional theory (DFT), the random phase approximation (RPA), coupled cluster (CC) theory, and the auxiliary-field quantum Monte Carlo (AFQMC) method, as well as the complete active space self-consistent field (CASSCF) method and the respective on-top methods: second-order N-electron valence state perturbation theory (NEVPT2) and multiconfiguration pair-density functional theory (MC-PDFT). Different strategies for selecting active spaces were explored, and the density matrix renormalization group (DMRG) approach was used to solve the largest active spaces. Despite somewhat ambiguous multi-reference diagnostics, most methods gave relatively good agreement with experimental data for the chemical reactions connected to the selectivity, which only involved transition-metal complexes in their high-spin state. CC methods yielded the highest accuracy followed by range-separated DFT and AFQMC. We discussed in detail that even higher accuracies can be obtained with NEVPT2, under the prerequisite that consistent active spaces along the entire chemical reaction can be selected, which was not the case for reactions involving Fe(III). A bigger challenge for electronic-structure methods was the prediction of the spin-state energetics, which additionally involved lower spin states that exhibited larger multi-reference diagnostics. Conceptually different, typically accurate methods ranging from CC theory via DMRG-NEVPT2 in combination with large active spaces to AFQMC agreed well that the high-spin state is energetically significantly favored over the other spin states. This was in contrast to most DFT functionals and RPA which yielded a smaller stabilization and some common DFT functionals and MC-PDFT even predicting the low-spin state to be energetically most favorable.

New physical insights into the supporting subspace factorization of XMS-CASPT2 and generalization to multiple spin states via spin-free formulation

This paper introduces a spin-free formulation of the supporting subspace factorization [C. Song and T. J. Martínez, J. Chem. Phys. 149, 044108 (2018)], enabling a reduction in the computational scaling of the extended multi-state complete active space second-order perturbation (XMS-CASPT2) method for arbitrary spins. Compared to the original formulation that is defined in the spin orbitals and is limited to singlet states, the spin-free formulation in this work treats different spin states equivalently, thus naturally generalizing the idea beyond singlet states. In addition, we will present a new way of deriving the supporting subspace factorization with the purpose of understanding its physical interpretation. In this new derivation, we separate the sources that make CASPT2 difficult into the "same-site interactions" and "inter-site interactions." We will first show how the Kronecker sum can be used to remove the same-site interactions in the absence of inter-site interactions, leading to MP2 energy in dressed orbitals. We will then show how the inter-site interactions can be exactly recovered using Löwdin partition, where the supporting subspace concept will naturally arise. The new spin-free formulation maintains the main advantage of the supporting subspace factorization, i.e., allowing XMS-CASPT2 energies to be computed using highly optimized MP2 energy codes and Fock build codes, thus reducing the scaling of XMS-CASPT2 to the same scaling as MP2. We will present and discuss results that benchmark the accuracy and performance of the new method. To demonstrate how the new method can be useful in studying real photochemical systems, the supporting subspace XMS-CASPT2 is applied to a photoreaction sensitive to magnetic field effects. The new spin-free formulation makes it possible to calculate the doublet and quartet states required in this particular photoreaction mechanism.

Variational Active Space Selection with Multiconfiguration Pair-Density Functional Theory

The selection of an adequate set of active orbitals for modeling strongly correlated electronic states is difficult to automate because it is highly dependent on the states and molecule of interest. Although many approaches have shown some success, no single approach has worked well in all cases. In light of this, we present the "discrete variational selection" (DVS) approach to active space selection, in which one generates multiple trial wave functions from a diverse set of systematically constructed active spaces and then selects between these wave functions variationally. We apply this DVS approach to 207 vertical excitations of small-to-medium-sized organic and inorganic molecules (with 3 to 18 atoms) in the QUESTDB database by (i) constructing various sets of active space orbitals through diagonalization of parametrized operators and (ii) choosing the result with the lowest average energy among the states of interest. This approach proves ineffective when variationally selecting between wave functions using the density matrix renormalization group (DMRG) or complete active space self-consistent field (CASSCF) energy but is able to provide good results when variationally selecting between wave functions using the energy of the translated PBE (tPBE) functional from multiconfiguration pair-density functional theory (MC-PDFT). Applying this DVS-tPBE approach to selection among state-averaged DMRG wave functions, we obtain a mean unsigned error of only 0.17 eV using hybrid MC-PDFT. This result matches that of our previous benchmark without the need to filter out poor active spaces and with no further orbital optimization following active space selection of the SA-DMRG wave functions. Furthermore, we find that DVS-tPBE is able to robustly and effectively select between the new SA-DMRG wave functions and our previous SA-CASSCF results.

Spin current in the early stage of radical reactions and its mechanisms

We study the electronic spin flux (atomic-scale flow of the spin density in molecules) by a perturbation analysis and ab initio nonadiabatic calculations. We derive a general perturbative expression of the charge and spin fluxes and identify the driving perturbation of the fluxes to be the time derivative of the electron-nucleus interaction term in the Hamiltonian. We then expand the expression in molecular orbitals so as to identify relevant components of the fluxes. Our perturbation theory describes the electronic fluxes in the early stage of reactions in an intuitively clear manner. The perturbation theory is then applied to an analysis of the spin flux obtained in ab initio calculations of the radical reaction of O2 and CH3· starting from three distinct spin configurations; (a) CH3· and triplet O2 with total spin of the system set Stot=1/2 (b) CH3· and singlet O2, Stot=1/2, and (c) CH3· and triplet O2, Stot=3/2. Further analysis of the time-dependent behaviors of the spin flux in these numerical simulations reveals (i) the spin flux induces rearrangement of the local spin structure, such as reduction of the spin polarization arising from the triplet O2 and (ii) the spin flux flows from O2 to CH3· in the reaction starting from spin configuration (a) and from CH3· to O2 in that starting from configuration (b), whereas no major intermolecular spin flux was observed in that starting from configuration (c). Our study thus establishes the mechanism of the spin flux that rearranges the local spin structures associated with chemical bonds.

An Accurate Density Coherence Functional for Hybrid Multiconfiguration Density Coherence Functional Theory

We present hybrid multiconfiguration density coherence functional theory (HMC-DCFT), and we optimize a density coherence functional by parametrization against a diverse data set of 59 bond energies and 60 barrier heights. We compare the results to calculations on the same data set by CASSCF, CASPT2, six Kohn-Sham and hybrid Kohn-Sham exchange-correlation functionals, and three on-top functionals for pair-density functional theory (PDFT) and hybrid PDFT. The new functional has better accuracy than all compared methods.

Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory: Multireference Advantages with the Practicality of Linear Response Theory

The density functional theory (DFT) and linear response (LR) time-dependent (TD)-DFT are of the utmost importance for routine computations. However, the single reference formulation of DFT suffers in the description of open-shell singlet systems such as diradicals and bond-breaking. LR-TDDFT, on the other hand, finds difficulties in the modeling of conical intersections, doubly excited states, and core-level excitations. In this Perspective, we demonstrate that many of these limitations can be overcome by recently developed mixed-reference (MR) spin-flip (SF)-TDDFT, providing an alternative yet accurate route for such challenging situations. Empowered by the practicality of the LR formalism, it is anticipated that MRSF-TDDFT can become one of the major workhorses for general routine tasks.

Algebraic Diagrammatic Construction Theory for Simulating Charged Excited States and Photoelectron Spectra

Charged excitations are electronic transitions that involve a change in the total charge of a molecule or material. Understanding the properties and reactivity of charged species requires insights from theoretical calculations that can accurately describe orbital relaxation and electron correlation effects in open-shell electronic states. In this Review, we describe the current state of algebraic diagrammatic construction (ADC) theory for simulating charged excitations and its recent developments. We start with a short overview of ADC formalism for the one-particle Green's function, including its single- and multireference formulations and extension to periodic systems. Next, we focus on the capabilities of ADC methods and discuss recent findings about their accuracy for calculating a wide range of excited-state properties. We conclude our Review by outlining possible directions for future developments of this theoretical approach.

Size-dependent errors in real-time electron density propagation

Real-time (RT) electron density propagation with time-dependent density functional theory (TDDFT) or Hartree-Fock (TDHF) is one of the most popular methods to model the charge transfer in molecules and materials. However, both RT-TDHF and RT-TDDFT within the adiabatic approximation are known to produce inaccurate evolution of the electron density away from the ground state in model systems, leading to large errors in charge transfer and erroneous shifting of peaks in absorption spectra. Given the poor performance of these methods with small model systems and the widespread use of the methods with larger molecular and material systems, here we bridge the gap in our understanding of these methods and examine the size-dependence of errors in RT density propagation. We analyze the performance of RT density propagation for systems of increasing size during the application of a continuous resonant field to induce Rabi-like oscillations, during charge-transfer dynamics, and for peak shifting in simulated absorption spectra. We find that the errors in the electron dynamics are indeed size dependent for these phenomena, with the largest system producing the results most aligned with those expected from linear response theory. The results suggest that although the RT-TDHF and RT-TDDFT methods may produce severe errors for model systems, the errors in charge transfer and resonantly driven electron dynamics may be much less significant for more realistic, large-scale molecules and materials.

A study of core-excited states of organic molecules computed with the generalized active space driven similarity renormalization group

This work examines the accuracy and precision of x-ray absorption spectra computed with a multireference approach that combines generalized active space (GAS) references with the driven similarity renormalization group (DSRG). We employ the x-ray absorption benchmark of organic molecule (XABOOM) set, consisting of 116 transitions from mostly organic molecules [Fransson et al., J. Chem. Theory Comput. 17, 1618 (2021)]. Several approximations to a full-valence active space are examined and benchmarked. Absolute excitation energies and intensities computed with the GAS-DSRG truncated to second-order in perturbation theory are found to systematically underestimate experimental and reference theoretical values. Third-order perturbative corrections significantly improve the accuracy of GAS-DSRG absolute excitation energies, bringing the mean absolute deviation from experimental values down to 0.32 eV. The ozone molecule and glyoxylic acid are particularly challenging for second-order perturbation theory and are examined in detail to assess the importance of active space truncation and intruder states.

Mg(I)-Fe(-II) and Mg(0)-Mg(I) covalent bonding in the MgnFe(CO)4- (n = 1, 2) anion complexes: an infrared photodissociation spectroscopic and theoretical study

Heteronuclear magnesium-iron carbonyl anion complexes MgFe(CO)₄^- and Mg₂Fe(CO)₄^- are produced in the gas phase and are detected by mass-selected infrared photodissociation spectroscopy in the carbonyl stretching frequency region. The geometric structures and the metal-metal bonding are discussed with the aid of quantum chemical calculations. Both complexes are characterized to have a doublet electronic ground state with C_3v symmetry containing a Mg-Fe bond or a Mg-Mg-Fe bonding unit. Bonding analyses indicate that each complex involves an electron-sharing Mg(I)-Fe(-II) σ bond. The Mg₂Fe(CO)₄^- complex involves a relatively weak covalent Mg(0)-Mg(I) σ bond.

The Role of the Redox Non-Innocent Hydroxyl Ligand in the Activation of O2 Performed by [Ni(H)(OH)]. Chemistry 2023;29:e202203128. [PMID: 36447369 DOI: 10.1002/chem.202203128] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

The cationic complex [Ni(H)(OH)]⁺ was previously found to activate dioxygen and methane in gas phase under single collision conditions. These remarkable reactivities were thought to originate from a non-classical electronic structure, where the Ni-center adopts a Ni(II), instead of the classically expected Ni(III) oxidation state by formally accepting an electron from the hydroxo ligand, which formally becomes a hydroxyl radical in the process. Such radicaloid oxygen moieties are envisioned to easily react with otherwise inert substrates, mimicking familiar reactivities of free radicals. In this study, the reductive activation of dioxygen by [Ni(H)(OH)]⁺ to afford the hydroperoxo species was investigated using coupled cluster, multireference ab initio and density functional theory calculations. Orbital and wave function analyses indicate that O₂ binding tranforms the aforementioned non-classical electronic structure to a classical Ni(III)-hydroxyl system, before O₂ reduction takes place. Remarkably, we found no evidence for a direct involvement of the radicaloid hydroxyl in the reaction with O₂ , as is often assumed. The function of the redox non-innocent character of the activator complex is to protect the reactive electronic structure until the complex engages O₂ , upon which a dramatic electronic reorganization releases internal energy and drives the chemical reaction to completion.

Quantum self-consistent equation-of-motion method for computing molecular excitation energies, ionization potentials, and electron affinities on a quantum computer

Near-term quantum computers are expected to facilitate material and chemical research through accurate molecular simulations. Several developments have already shown that accurate ground-state energies for small molecules can be evaluated on present-day quantum devices. Although electronically excited states play a vital role in chemical processes and applications, the search for a reliable and practical approach for routine excited-state calculations on near-term quantum devices is ongoing. Inspired by excited-state methods developed for the unitary coupled-cluster theory in quantum chemistry, we present an equation-of-motion-based method to compute excitation energies following the variational quantum eigensolver algorithm for ground-state calculations on a quantum computer. We perform numerical simulations on H2, H4, H2O, and LiH molecules to test our quantum self-consistent equation-of-motion (q-sc-EOM) method and compare it to other current state-of-the-art methods. q-sc-EOM makes use of self-consistent operators to satisfy the vacuum annihilation condition, a critical property for accurate calculations. It provides real and size-intensive energy differences corresponding to vertical excitation energies, ionization potentials and electron affinities. We also find that q-sc-EOM is more suitable for implementation on NISQ devices as it is expected to be more resilient to noise compared with the currently available methods.

Algorithm for analytic nuclear energy gradients of state averaged DMRG-CASSCF theory with newly derived coupled-perturbed equations

We present an algorithm for evaluating analytic nuclear energy gradients of the state-averaged density matrix renormalization group complete-active-space self-consistent field (SA-DMRG-CASSCF) theory based on the newly derived coupled-perturbed (CP) DMRG-CASSCF equations. The Lagrangian for the conventional SA-CASSCF analytic gradient theory is extended to the SA-DMRG-CASSCF variant that can fully consider a whole set of constraints on the parameters of multi-root canonical matrix product states formed at all the DMRG block configurations. An efficient algorithm to solve the CP-DMRG-CASSCF equations for determining the multipliers was developed. The complexity of the resultant analytic gradient algorithm is overall the same as that of the unperturbed SA-DMRG-CASSCF algorithm. In addition, a reduced-scaling approach was developed to directly compute the SA reduced density matrices (SA-RDMs) and their perturbed ones without calculating separate state-specific RDMs. As part of our implementation scheme, we neglect the term associated with the constraint on the active orbitals in terms of the active-active rotation in the Lagrangian. Thus, errors from the true analytic gradients may be caused in this scheme. The proposed gradient algorithm was tested with the spin-adapted implementation by checking how accurately the computed analytic energy gradients reproduce numerical gradients of the SA-DMRG-CASSCF energies using a common number of renormalized bases. The illustrative applications show that the errors are sufficiently small when using a typical number of the renormalized bases, which is required to attain adequate accuracy in DMRG's total energies.

C-O Bond Activation in Mononuclear Lanthanide Oxocarbonyl Complexes OLn(η2-CO) (Ln = La, Ce, Pr, and Nd)

Fundamental investigation of metal-CO interactions is of great importance for the development of high-performance catalysts to CO activation. Herein, a series of side-on bonded mononuclear lanthanide (Ln) oxocarbonyl complexes OLn(η²-CO) (Ln = La, Ce, Pr, and Nd) have been prepared and identified in solid argon matrices. The complexes exhibit uncommonly low C-O stretching bands near 1630 cm^-1, indicating remarkable C-O bond activation in these Ln analogues. The η²-CO ligand in OLn(η²-CO) can be claimed as an anion on the basis of the experimental observations and quantum chemistry investigations, although the CO anion is commonly considered to be unstable with electron auto-detachment. The CO activation in OLn(η²-CO) is attributed to the photoinduced intramolecular charge transfer from LnO to CO rather than the generally accepted metal → CO π back-donation, which conforms to the traditional Dewar-Chatt-Duncanson motif. Energy decomposition analysis combined with natural orbitals for chemical valence calculations demonstrates that the bonding between LnO and η²-CO arises from the combination of dominant ionic forces (>76%) and normal Lewis "acid-base" interactions. The fundamental findings provide guidelines for the catalyst design of CO activation.

State averaged CASSCF in AMOEBA polarizable water model for simulating nonadiabatic molecular dynamics with nonequilibrium solvation effects

This paper presents a state-averaged complete active space self-consistent field (SA-CASSCF) in the atomic multipole optimized energetics for biomolecular application (AMOEBA) polarizable water model, which enables rigorous simulation of non-adiabatic molecular dynamics with nonequilibrium solvation effects. The molecular orbital and configuration interaction coefficients of the solute wavefunction, and the induced dipoles on solvent atoms, are solved by minimizing the state averaged energy variationally. In particular, by formulating AMOEBA water models and the polarizable continuum model (PCM) in a unified way, the algorithms developed for computing SA-CASSCF/PCM energies, analytical gradients, and non-adiabatic couplings in our previous work can be generalized to SA-CASSCF/AMOEBA by properly substituting a specific list of variables. Implementation of this method will be discussed with the emphasis on how the calculations of different terms are partitioned between the quantum chemistry and molecular mechanics codes. We will present and discuss results that demonstrate the accuracy and performance of the implementation. Next, we will discuss results that compare three solvent models that work with SA-CASSCF, i.e., PCM, fixed-charge force fields, and the newly implemented AMOEBA. Finally, the new SA-CASSCF/AMOEBA method has been interfaced with the ab initio multiple spawning method to carry out non-adiabatic molecular dynamics simulations. This method is demonstrated by simulating the photodynamics of the model retinal protonated Schiff base molecule in water.

Relativistic Kramers-Unrestricted Exact-Two-Component Density Matrix Renormalization Group

In this work we develop a variational relativistic density matrix renormalization group (DMRG) approach within the exact-two-component (X2C) framework (X2C-DMRG), using spinor orbitals optimized with the two-component relativistic complete active space self-consistent field. We investigate fine-structure splittings of p- (Ga, In, Tl) and d-block (Sc, Y, La) atoms and excitation energies of monohydride molecules (GeH, SnH, and TlH) with X2C-DMRG calculations using an all-electron relativistic Hamiltonian in a Kramers-unrestricted basis. We find that X2C-DMRG yields accurate ²P and ²D splittings compared to multireference configuration interaction with singles and doubles (MRCISD). We also investigated the degree of symmetry breaking in the atomic multiplets and convergence of electron correlation in the total energies. Symmetry breaking can be large in some cases (∼30 meV); however, increasing the number of renormalized block states m for the DMRG optimization recovers the symmetry breaking by several orders of magnitude. Encouragingly, we find the convergence of electron correlation to be close to MRCISDTQ5 quality. Relativistic X2C-DMRG approaches are important for cases where spin-orbit coupling is significant and the underlying reference wave function requires a large determinantal space. We are able to obtain quantitatively correct fine-structure splittings for systems up to 10¹⁹ number of determinants with traditional CI approaches, which are currently unfeasible to converge for the field.

Investigation of negative-ion resonances using a subspace-projected multiconfigurational electron propagator perturbed with a complex absorbing potential

The transient negative-ion resonances found in scattering experiments are important intermediates in many chemical processes. These metastable states correspond to the continuum part of the Hamiltonian of the projectile-target composite system. Usual bound-state electronic structure methods are not applicable for these. In this work, we develop a subspace-projection method in connection with an electron propagator (EP) defined in terms of a complete-active-space self-consistent-field initial state. The target Hamiltonian (Ĥ) is perturbed by a complex absorbing potential (CAP) for the analytical continuation of the spectrum of Ĥ to complex eigenvalues associated with the continuum states. The resonance is identified as a pole of the EP, which is stable with respect to variations in the strength of the CAP. The projection into a small subspace reduces the size of the complex matrices to be diagonalized, minimizes the computational cost, and affords some insight into the orbitals that are likely to play some role in the capture of the projectile. Two molecular (Πg2N₂ ^- and ²Π CO^-) and an atomic shaperesonance (²P Be^-) are investigated using this method. The position and width of the resonances are in good agreement with the previously reported values.

State-averaged CASSCF with polarizable continuum model for studying photoreactions in solvents: Energies, analytical nuclear gradients, and non-adiabatic couplings

This paper presents state-averaged complete active space self-consistent field in polarizable continuum model (PCM) for studies of photoreactions in solvents. The wavefunctions of the solute and the PCM surface charges of the solvent are optimized simultaneously such that the state-averaged free energy is variationally minimized. The method supports both fixed weights and dynamic weights where the weights are automatically adjusted based on the energy gaps. The corresponding analytical nuclear gradients and non-adiabatic couplings are also derived. Furthermore, we show how the new method can be entirely formulated in terms of seven basic operations, which allows the implementation to benefit from existing high-performance libraries on graphical processing units. Results demonstrating the accuracy and performance of the implementation are presented and discussed. We also apply the new method to the study of minimal conical intersection search and photoreaction energy pathways in solvents. Effects from the polarity of the solvents and different formulas of dynamic weights are compared and discussed.

Characterizing the E⊗e Jahn–Teller Potential Energy Surfaces by Differential Geometry Tools

The term ‘mathematical chemistry’ is mostly associated with applications of graph theory in topological issues of 3D chemical structures, thought of as a collection of atoms as dots and bonds as lines. We propose here new directions in this field, coming from the side of theoretical chemistry approached with modern computational tools. Possible challenges are proposed in using ancillary tools of differential geometry for examining the potential energy surfaces of certain specific structural prototypes. Concretely, we describe here the geodesics on the surfaces related to the potential energy functions of the so-called E⊗e Jahn–Teller effect, a spontaneous symmetry-breaking phenomenon also known as a case of conical intersection. To illustrate the case, first-principles (ab initio) quantum chemical calculations are performed on the cyclo-propenyl molecular radical C3H3.

Insights into the Metal-CO Bond in O2M(η1-CO) (M = Cr, Mo, W, Nd, and U) Complexes

Investigations on the structures and bonding properties of metal carbonyl compounds provide fundamental understandings on the origin of small-molecule activations. Herein, the geometry and bonding trends of a series of isovalent metal oxocarbonyl complexes O₂M(η¹-CO) (M = Cr, Mo, W, Nd, and U) were studied by combined matrix-isolation infrared spectroscopy and advanced quantum chemical calculations. The title complexes present red shift of C-O stretching bands in the range from 122 to 244 cm^-1, indicating the difference of CO activation ability for the series of isovalent metal dioxides. Density functional theory calculations predict T-shaped structures with a C_2v symmetry for all the title molecules. O₂Nd(η¹-CO) bears little resemblance to the other complexes in bonding characters because of the weak interactions between the NdO₂ and CO moiety. For the other complexes, natural localized molecular orbital analysis reveals a gradual increase of covalent character in M-CO bonds along the metal series Cr → Mo → W→ U. Energy decomposition analysis with natural orbitals for chemical valence calculations demonstrates that the M-CO bonding patterns conform to the conventional Dewar-Chatt-Duncanson motif. The contributions from orbital interactions in total attractions increase from Cr (41.7%) to U (52.7%). The breakdown of the orbital term into pairwise interactions shows that contributions of the M ← CO σ donation decrease from Cr (59.2%) to U (28.4%), while the M → CO π* backdonation increases significantly from Cr (23.8%) to U (67.3%). The more effective overlap and the better energy matching of U 5f and U 6d valence orbitals with CO π* orbitals result in much stronger U → CO π backdonation than the other metal elements.

Simulating X-ray photoelectron spectra with strong electron correlation using multireference algebraic diagrammatic construction theory

A new theoretical approach for the simulations of X-ray photoelectron spectra of strongly correlated molecular systems that combines multireference algebraic diagrammatic construction theory (MR-ADC) with a core–valence separation (CVS) technique.

Theoretical Design of Blue-Color Phosphorescent Complexes for Organic Light-Emitting Diodes: Emission Intensities and Nonradiative Transition Rate Constants in Ir(ppy)2(acac) Derivatives

Theoretical calculations of phosphorescent spectra and nonradiative transition (NRT) rate constants for S₁ ⇝ T₁, T₁ ⇝ S₀, and S₁ ⇝ S₀ were carried out to determine the best candidate for a blue-color phosphorescent complex among several derivatives of bis(2-phenylpyridine)(acetylacetonate)iridium(III). The geometries of the ground state (S₀), the lowest triplet state (T₁), and the lowest excited singlet state (S₁) were optimized at the levels of density functional theory, in which B3LYP functionals and SBKJC+p basis sets were used. The NRT rate constants were derived by using a generating function method within the displaced harmonic oscillator model. The results of the calculation for phosphorescence showed that the introduction of F and/or CN substituents at the 4'/6'-th and 5'-th sites in 2-phenylpyridinate (ppy) ligands, respectively, causes a blue shift of the emission spectra. They also suggest that Ir(5-CN,6-F-ppy)₂(acac), denoted 3(56) in the text, is a good candidate for a blue-color phosphorescent complex because a blue shift of emission spectra and a moderate intensity are obtained for phosphorescence and, furthermore, this complex is calculated to have a large rate constant for S₁ ⇝ T₁ and relatively smaller rate constants for T₁ ⇝ S₀ and S₁ ⇝ S₀ based on the calculations of spin-orbit coupling and nonadiabatic coupling constants.

iCISCF: An Iterative Configuration Interaction-Based Multiconfigurational Self-Consistent Field Theory for Large Active Spaces

An iterative configuration interaction (iCI)-based multiconfigurational self-consistent field (SCF) theory, iCISCF, is proposed to handle systems that require large active spaces. The success of iCISCF stems from three ingredients: (1) efficient selection of individual configuration state functions spanning the active space while maintaining full spin symmetry; (2) the use of Jacobi rotation for optimization of the active orbitals in conjunction with a quasi-Newton algorithm for the core/active-virtual and core-active orbital rotations; (3) a second-order perturbative treatment of the residual space left over by the selection procedure (i.e., iCISCF(2)). Several examples that go beyond the capability of CASSCF are taken as showcases to reveal the efficacy of iCISCF and iCISCF(2), facilitated by iCAS for imposed automatic selection and localization of active orbitals.

Molecular Dynamics Driven by the Many-Body Expansion (MBE-MD)

We present a protocol for classical and nuclear quantum dynamics, in which the energies and forces are generated by the many-body expansion (MBE), and apply it to water clusters using the TTM2.1-F and MB-Pol interaction potentials at various temperatures. We carry out MBE-molecular dynamics (MD) classical and nuclear quantum dynamical simulations, in which the energies and forces of the full system are approximated by the two-, three-, and four-body terms of the MBE, and compare the average potential and the vibrational density of states with the full simulation, i.e., the one for which no MBE is used. Our results indicate that the thermally averaged potential energy from the MBE up to the four-body term converges with near-identical behavior to the one from the full simulation. The three-body makes a substantial contribution (∼20%) to the energy, whereas the four-body is necessary for obtaining quantitatively accurate energetics and forces, albeit making a small contribution to each (∼2%). We further show that the harmonic frequencies are reproduced to within a few wavenumbers (cm^-1) at the four-body level and that the slowest modes to converge with the MBE rank are those involving the strongest hydrogen bonds. Anharmonicity exacerbates this effect, so that a four-body description of the energies and forces is needed to achieve accurate anharmonic vibrational frequencies in the hydrogen-bonded OH-stretching region. We also discuss the asymptotic scaling of the MBE-MD protocol with respect to the cost of the underlying potential energy evaluation, suggesting that electronic structure methods that scale at least as N⁴, N being the size of the system, are needed to result in savings over the traditional full MD simulation. We anticipate that the MBE-MD protocol can evolve into a powerful and practical method, which will allow for highly accurate ab initio MD simulations on a much broader range of molecular systems than can be currently handled.

The Static-Dynamic-Static Family of Methods for Strongly Correlated Electrons: Methodology and Benchmarking

A series of methods (SDSCI, SDSPT2, iCI, iCIPT2, iCISCF(2), iVI, and iCAS) is introduced to accurately describe strongly correlated systems of electrons. Born from the (restricted) static-dynamic-static (SDS) framework for designing many-electron wave functions, SDSCI is a minimal multireference (MR) configuration interaction (CI) approach that constructs and diagonalizes a [Formula: see text] matrix for [Formula: see text] states, regardless of the numbers of orbitals and electrons to be correlated. If the full molecular Hamiltonian H in the QHQ block (which describes couplings between functions of the first-order interaction space Q) of the SDSCI CI matrix is replaced with a zeroth-order Hamiltonian [Formula: see text] before the diagonalization is taken, we obtain SDSPT2, a CI-like second-order perturbation theory (PT2). Unlike most variants of MRPT2, SDSPT2 treats single and multiple states in the same way and is particularly advantageous in the presence of near degeneracy. On the other hand, if the SDSCI procedure is repeated until convergence, we will have iterative CI (iCI), which can converge quickly from the above to the exact solutions (full CI) even when starting with a poor guess. When further combined with the selection of important configurations followed by a PT2 treatment of dynamic correlation, iCI becomes iCIPT2, which is a near-exact theory for medium-sized systems. The microiterations of iCI for relaxing the coefficients of contracted many-electron functions can be generalized to an iterative vector interaction (iVI) approach for finding exterior or interior roots of a given matrix, in which the dimension of the search subspace is fixed by either the number of target roots or the user-specified energy window. Naturally, iCIPT2 can be employed as the active space solver of the complete active space (CAS) self-consistent field, leading to iCISCF(2), which can further be combined with iCAS for automated selection of active orbitals and assurance of the same CAS for all states and all geometries. The methods are calibrated by taking the Thiel set of benchmark systems as examples. Results for the corresponding cations, a new set of benchmark systems, are also reported.

Reaction of the N Atom with Electronically Excited O2 Revisited: A Theoretical Study

The kinetics of the reaction of N with electronically excited O₂ (singlet a¹Δ_g and b¹Σ_g⁺ states), potentially relevant for NO_x formation in nonthermal air plasma, is theoretically studied using the multireference second-order perturbation theory. The corresponding thermodynamically and kinetically favored reaction pathways together with possible intersystem crossings are identified. It has been revealed that the energy barrier for the N + O₂(a¹Δ_g) → NO + O reaction is approximately twice the barrier height for the counterpart process with O₂(X³Σ_g^-). The molecular oxygen in the b¹Σ_g⁺ state, in turn, proved to be even less reactive to atomic nitrogen than O₂(a¹Δ_g). Appropriate thermal rate constants for specified reaction channels are calculated by the variational transition-state theory incorporating corrections for the tunneling effect, nonadiabatic transitions, and anharmonicity of vibrations for transition states and reactants. The corresponding three-parameter Arrhenius expressions for the broad temperature range (T = 300-4000 K) are reported. At last, post-transition-state molecular dynamics simulations indicate that the N + O₂(a¹Δ_g) reaction produces vibrationally much colder NO molecules than the N + O₂(X³Σ_g^-) process.

C-H Bond Activation by the Excited Zinc Atom: Gas-Phase Formation of Methylzinc Hydride (HZnCH3) Based on Multireference Second-Order Perturbation Theory and Coupled Cluster Calculations

The pioneering spectroscopic observations of the methylzinc hydride [HZnCH₃(X¹A₁)] molecule were reported previously by the Ziurys group [J. Am. Chem. Soc. 2010, 132, 17186-17192], and the possible formation mechanisms were suggested therein, including those with the participation of excited zinc atoms in reaction with methane. Herein, the ground singlet state and the lowest excited triplet state potential energy surfaces of the Zn + CH₄ reaction have been explored using high-level electronic structure calculations with multireference second-order perturbation theory and coupled cluster singles and doubles with perturbative triples (CCSD(T)) methods in conjunction with all-electron basis sets (up to aug-cc-pV5Z) and scalar relativistic effects incorporated via the second-order Douglas-Kroll-Hess (DK) method. Based on the ab initio results, a plausible scenario for the formation of HZnCH₃(X¹A₁) is proposed involving the activation of the C-H bond of methane by the lowest excited ³P state atomic zinc. Calculations also highlight the importance of an agostic-like Zn···H-C interactions in the pre-activation complex and good agreement between the structure of the HZnCH₃(X¹A₁) molecule predicted at the DK-CCSD(T)/aug-cc-pVQZ-DK level of theory and that derived from rotational spectroscopy, as well as the discrepancies between the ab initio and density functional theory predictions.

Potential Energy Curves of Molecular Nitrogen for Singly and Doubly Ionized States with Core and Valence Holes

Theoretical description of potential energy curves (PECs) of molecular ions is essential for interpretation and prediction of coupled electron-nuclear dynamics following ionization of parent molecule. However, an accurate representation of these PECs for core or inner valence ionized state is nontrivial, especially at stretched geometries for double- or triple-bonded systems. In this work, we report PECs of singly and doubly ionized states of molecular nitrogen using state-of-the-art quantum chemical methods. The valence, inner valence, and core ionized states have been computed. A double-loop optimization scheme that separates the treatment of the core and the valence orbitals during the orbital optimization step of the multiconfiguration self-consistent field method has been implemented. This technique allows the energy to be converged to any desired ionized state with any number of core or inner-shell holes. The present work also compares the PECs obtained using both delocalized and localized sets of orbitals for the core hole states. The PECs of a number of singly and doubly ionized valence states have also been computed and compared with previous studies. The computed PECs reported here are expected to be of importance for future studies to understand the interplay between photoionization and Auger spectra during the breakup of molecular nitrogen when interacting with intense free electron lasers.

Boron nitride nanotube precursor formation during high-temperature synthesis: kinetic and thermodynamic modelling

We performed integrated modelling of the chemical pathways of formation for boron nitride nanotube (BNNT) precursors during high-temperature synthesis in a B/N₂mixture. Integrated modelling includes quantum chemistry, Quantum-classical molecular dynamics, thermodynamic modelling, and kinetic approaches. We demonstrate that BN compounds are formed via the interaction of molecular nitrogen with small boron clusters, rather than through interactions with less reactive liquid boron. (This process can also be described as N₂molecule fixation.) Liquid boron evaporates to produce these boron clusters (B_mwithm≤ 5), which are subsequently converted into B_mN_nchains. The production of such chains is crucial to the growth of BNNTs because these chains form the building blocks of bigger and longer BN chains and rings, which are in turn the building blocks of fullborenes and BNNTs. Additionally, kinetic modelling revealed that B₄N₄and B₅N₄species in particular play a major role in the N₂molecule fixation process. The formation of these species via reactions with B₄and B₅clusters is not adequately described under the assumption of thermodynamic equilibrium, as is demonstrated in our kinetic modelling. Thus, the accumulation of both B₄N₄and B₅N₄depends on the background gas pressure and the gas cooling rate. Long BN chains and rings, which are precursors of the fullborene and BNNT growth, form via self-assembly of components B₄N₄and B₅N₄. Our modelling results-particularly the increased densities of B₄N₄and B₅N₄species at higher gas pressures-explain the experimentally observed effect of gas pressure on the yield of high-quality BNNTs. The catalytic role of hydrogen was also studied; it is shown that HBNH molecules can be the main precursor of BNNT synthesis in the presence of hydrogen.

iCAS: Imposed Automatic Selection and Localization of Complete Active Spaces

It is shown that in the spirit of "from fragments to molecule" for localizing molecular orbitals [J. Chem. Theory Comput. 2011, 7, 3643], a prechosen set of occupied/virtual valence/core atomic/fragmental orbitals can be transformed to an equivalent set of localized occupied/virtual pre-localized molecular orbitals (pre-LMO), which can then be taken as probes to select the same number of maximally matching localized occupied/virtual Hartree-Fock (HF) or restricted open-shell HF (ROHF) molecular orbitals as the initial local orbitals spanning the desired complete active space (CAS). In each cycle of the self-consistent field (SCF) calculation, the CASSCF orbitals can be localized by means of the noniterative "top-down least-change" algorithm for localizing ROHF orbitals [J. Chem. Phys. 2017, 146, 104104] such that the maximum matching between the orbitals of two adjacent iterations can readily be monitored, leading finally to converged localized CASSCF orbitals that overlap most the guess orbitals. Such an approach is to be dubbed as "imposed CASSCF" (iCASSCF or simply iCAS in short) for good reasons: (1) it has been assumed that only those electronic states that have largest projections onto the active space defined by the prechosen atomic/fragmental orbitals are to be targeted. This is certainly an imposed constraint but has wide applications in organic and transition metal chemistry where valence (or core) atomic/fragmental orbitals can readily be identified. (2) The selection of both initial and optimized local active orbitals is imposed from the very beginning by the pre-LMOs (which span the same space as the prechosen atomic/fragmental orbitals). Apart from the (imposed) automation and localization, iCAS has two additional merits: (a) the guess orbitals are guaranteed to be the same for all geometries, for the pre-LMOs do not change in character with geometry and (b) the use of localized orbitals facilitates the SCF convergence, particularly for large active spaces. Both organic molecules and transition-metal complexes are taken as showcases to reveal the efficacy of iCAS.

Vibronic transitions and spin-orbit coupling of three-membered metallacycles formed by lanthanide-mediated dehydrogenation of dimethylamine

Metal-mediated N-H and C-H bond activation of aliphatic amines is an effective strategy for synthesizing biologically important molecules. Ln (Ln = La and Ce) atom reactions with dimethylamine are carried out in a pulsed-laser vaporization supersonic molecular beam source. A series of dehydrogenation species are observed with time-of-flight mass spectrometry, and the dehydrogenated Ln-containing species in the formula Ln(CH₂NCH₃) are characterized by single-photon mass-analyzed threshold ionization (MATI) spectroscopy and quantum chemical calculations. The theoretical calculations include density functional theory for both Ln species and multiconfiguration self-consistent field and quasi-degenerate perturbation theory for the Ce species. The MATI spectrum of La(CH₂NCH₃) consists of a single vibronic band system, which is assigned to the ionization of the doublet ground state of N-methyl-lanthanaaziridine. The MATI spectrum of Ce(CH₂NCH₃) displays two vibronic band systems, which are attributed to the ionization of two-pair lowest-energy spin-orbit coupling states of N-methyl-ceraaziridine. Both metallaaziridines are three-membered metallacycles and formed by the thermodynamically and kinetically favorable concerted dehydrogenation of the amino group and one of the methyl groups.

Chembot: A Machine Learning Approach to Selective Configuration Interaction

We introduce a machine learning-based approach to selective configuration interaction, dubbed Chembot, that utilizes many novel choices for its model design and training. These choices include the use of a support vector machine to select important configurations, the use of the charge density matrix and configuration energy as features, and heuristics to improve the quality of training data. We test Chembot's ability to obtain near full configuration interaction quality energies and find that it definitively outperforms its purely Stochastic cousin Monte Carlo configuration interaction by requiring fewer iterations to converge, fewer determinants in the variational space, and fewer important configurations to achieve the same energy. In addition, Chembot at times requires fewer determinants in its variational space than the heat-bath configuration interaction method to achieve the same energy. We demystify Chembot's innards and then showcase our claims on the set of small but challenging systems: the hydrogen ring (H₄), stretched methylene (H₂C), and stretched water (H₂O).

Intermediates of Carbon Monoxide Oxidation on Praseodymium Monoxide Molecules: Insights from Matrix-Isolation IR Spectroscopy and Quantum-Chemical Calculations

Identifying reaction intermediates in gas-phase investigations will provide understanding for the related catalysts in fundamental aspects including bonding interactions of the reaction species, oxidation states (OSs) of the anchored atoms, and reaction mechanisms. Herein, carbon monoxide (CO) oxidation by praseodymium monoxide (PrO) molecules has been investigated as a model reaction in solid argon using matrix-isolation IR spectroscopy and quantum-chemical calculations. Two reaction intermediates, OPr(η¹-CO) and OPr(η²-CO), have been trapped and characterized in argon matrixes. The intermediate OPr(η²-CO) shows an extremely low C-O stretching band at 1624.5 cm^-1. Quantum-chemistry studies indicate that the bonding in OPr(η¹-CO) is described as "donor-acceptor" interactions conforming to the Dewar-Chatt-Duncanson motif. However, the bonding in OPr(η²-CO) results evidently from a combination of dominant ionic forces and normal Lewis "acid-base" interactions. The electron density of the singly occupied bonding orbital is strongly polarized to the CO fragment in OPr(η²-CO). Electronic structure analysis suggests that the two captured species exhibit Pr(III) OSs. Besides, the pathways of CO oxidation have been discussed.

Stationary Points on Potential Energy Surface of Cyclic C3H3 with Coupled-Cluster Approaches and Density Functional Theory

Cyclic C₃H₃ is the simplest cyclic hydrocarbon, but its configuration is complicated. The ground ²E″ state at the equilateral triangle geometry with D_3h symmetry undergoes both Jahn-Teller (JT) distortion and pseudo-Jahn-Teller (PJT) distortion to structures with C_3v, C_2v, C_s, or C₂ symmetry. Previous works using complete active space self-consistent field (CASSCF) or multiconfiguration SCF (MCSCF) differ on the characteristics of these structures. To clarify the characteristics of these stationary points on the potential energy surface (PES) of this radical, coupled-cluster methods CCSD, CCSD(T), and EOMEA/IP-CCSD as well as density functional theory are employed to calculate their geometries, harmonic frequencies, and relative energies. Deformations between these stationary points due to JT and PJT effects are analyzed in detail. Most of the results with these methods are consistent with each other, except for the b₂ mode of the ²B₁ state with C_2v symmetry. CCSD and CCSD(T) provide an imaginary frequency for this vibrational mode, while it is calculated to be real with the other methods as well as with EOM-CCSDT-3. This may be related to unstable reference in CCSD and CCSD(T) calculations. On the other hand, most of the employed exchange-correlation functionals provide reliable results on the characteristics of these stationary points. Our results show that the ²A' state with C_s symmetry is the only minimum structure on the PES of cyclic C₃H₃. The ²A₂ and ²B₁ states of the C_2v structure are second-order saddle points, while both the ²A state of the C₂ structure and the ²A″ state of the C_s structure are transition states connecting the global minimum ²A' state.

Modeling Spin-Crossover Dynamics

In this article, we review nonadiabatic molecular dynamics (NAMD) methods for modeling spin-crossover transitions. First, we discuss different representations of electronic states employed in the grid-based and direct NAMD simulations. The nature of interstate couplings in different representations is highlighted, with the main focus on nonadiabatic and spin-orbit couplings. Second, we describe three NAMD methods that have been used to simulate spin-crossover dynamics, including trajectory surface hopping, ab initio multiple spawning, and multiconfiguration time-dependent Hartree. Some aspects of employing different electronic structure methods to obtain information about potential energy surfaces and interstate couplings for NAMD simulations are also discussed. Third, representative applications of NAMD to spin crossovers in molecular systems of different sizes and complexities are highlighted. Finally, we pose several fundamental questions related to spin-dependent processes. These questions should be possible to address with future methodological developments in NAMD.

Diffusion Monte Carlo method on small boron clusters using single- and multi- determinant-Jastrow trial wavefunctions

Multireference character in some small boron clusters could be significant, and a previous all-electron fixed-node diffusion quantum Monte Carlo (FN-DMC) calculation with the single-determinant-Jastrow (SDJ) trial wavefunction shows that the atomization energy (AE) of B₄ ⁺ is overestimated by about 1.4 eV compared with the coupled cluster method with single, doubles, and perturbative triples [CCSD(T)] results. All-electron FN-DMC calculations and those with the pseudopotential (PP) using SDJ and multi-determinant-Jastrow (MDJ) trial wavefunctions with B3LYP orbitals as well as CC calculations at different levels are carried out on B_n ^Q (n = 1-5, Q = -1, 0, 1) clusters. The obtained FN-DMC energies indicate that the node error of the employed SDJ trial wavefunction in all-electron calculations is different from that with the PP for some clusters. The error of AEs and dissociation energies (DEs) from all-electron FN-DMC calculations is larger than that with the PP when the SDJ trial wavefunction is employed, while errors of CC methods do not depend on whether the PP is used. AEs and DEs of the boron clusters are improved significantly when MDJ trial wavefunctions are used in both all-electron calculations and those with the PP, and their error is similar to that of CCSD(T) compared with CCSDT(Q) results. On the other hand, reasonable adiabatic electron detachment energies (ADEs) and ionization potentials (AIPs) are achieved with FN-DMC using SDJ trial wavefunctions and MDJ is less effective on ADEs and AIPs. Furthermore, the relative energy between two structures of B₉ ^- is predicted reliably with FN-DMC using the SDJ trial wavefunction and the effect of MDJ is negligible, while density functional theory results using different exchange-correlation functionals differ significantly.

Quantum Information and Algorithms for Correlated Quantum Matter

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.

Bootstrap embedding with an unrestricted mean-field bath

A suite of quantum embedding methods have recently been developed where the Schmidt decomposition is applied to the full system wavefunction to derive basis states that preserve the entanglement between the fragment and the bath. The quality of these methods can depend heavily on the quality of the initial full system wavefunction. Most of these methods, including bootstrap embedding (BE) [M. Welborn et al; J. Chem. Phys. 145, 074102 (2016)], start from a spin-restricted mean-field wavefunction [call this restricted BE (RBE)]. Given that spin-unrestricted wavefunctions can capture a significant amount of strong correlation at the mean-field level, we suspect that starting from a spin-unrestricted mean-field wavefunction will improve these embedding methods for strongly correlated systems. In this work, BE is generalized to an unrestricted Hartree-Fock bath [call this unrestricted BE (UBE)], and UBE is applied to model hydrogen ring systems. UBE's improved versatility over RBE is utilized to calculate high spin symmetry states that were previously unattainable with RBE. Ionization potentials, electron affinities, and spin-splittings are computed using UBE with accuracy on par with spin-unrestricted coupled cluster singles and doubles. Even for cases where RBE is viable, UBE converges more reliably. We discuss the limitations or weaknesses of each calculation and how improvements to RBE and density matrix embedding theory these past few years can also improve UBE.

Exploring the Reaction Paths on the Potential Energy Surfaces of the S1 and T1 States in Methylenecyclopropane

The reaction paths of methylenecyclopropane 1 on the potential energy surfaces (PESs) of the lowest triplet (T₁ ) state and the lowest excited singlet (S₁ ) state, as well as that of the ground state (S₀ ), were explored by using the nudged elastic band method at the MRMP2//MCSCF/6-31++G(d,p) and DFT(B3LYP)/6-31++G(d,p) levels of theory. After vertical excitation of 1, three transition states on the PES of the lowest triplet state and one transition state on the S₁ PES were found along the reaction path to produce a carbene, cyclobutylidene 2. All of these transition states are lower in energy than the S₁ state produced by vertical excitation at the S₀ energy minimum in 1. Fast transition is predicted to occur from the T₁ state or from the S₁ state to the S₀ state due to strong spin-orbit coupling or nonadiabatic coupling in the geometrical vicinity of 2. On the MRMP2 S₀ PES, the energy barriers of 5.0, 10.3 and 13.5 kcal mol^-1 were obtained for C migration reaction (backward reaction), 1,2-H migration reaction to cyclobutene 3, and 1,3-H migration reaction to bicyclopropane 4, respectively, started at 2. The introduction of phenyl groups makes the energy barriers smaller due to the π conjugation between the carbene center and phenyl groups.

A fully diastereoselective oxidation of a mesoionic dipole with triplet molecular oxygen

Oxidations with molecular oxygen are ubiquitous processes in biological systems where cofactor-dependent enzymes activate either oxygen or hydrogen peroxide to induce multichannel pathways. In stark contrast, such slow atmospheric oxidations are seldom harnessed in chemical synthesis and analysis. The present study unveils an unusual aerobic oxidation of a mesoionic dipole leading easily to a more functionalized skeleton. Although the synthetic scope has not been explored, two key considerations emerge from this transformation, as it proceeds with complete diastereoselection and could be successfully extrapolated to structurally related mesoionic chirons without racemization. How this oxidation actually occurs proved to be puzzling from the onset and only high-level computation reveals a cascade transformation, whose results reconcile theory and experiment. Hopefully, the mechanistic insights should help us to understand better the autoxidative reactions of organic molecules.