Code generation in ORCA: progress, efficiency and tight integration

An improved version of ORCA's automated generator environment (ORCA-AGE II) is presented. The algorithmic improvements and the move to C++ as the programming language lead to a performance gain of up to two orders of magnitude compared to the previously developed PYTHON toolchain. Additionally, the restructured modular design allows for far more complex code engines to be implemented readily. Importantly, we have realised an extremely tight integration with the ORCA host program. This allows for a workflow in which only the wavefunction Ansatz is part of the source code repository while all actual high-level code is generated automatically, inserted at the appropriate place in the host program before it is compiled and linked together with the hand written code parts. This construction ensures longevity and uniform code quality. Furthermore the new developments allow ORCA-AGE II to generate parallelised production-level code for highly complex theories, such as fully internally contracted multireference coupled-cluster theory (fic-MRCC) with an enormous number of contributing tensor contractions. We also discuss the automated implementation of nuclear gradients for arbitrary theories. All these improvements enable the implementation of theories that are too complex for the human mind and also reduce development times by orders of magnitude. We hope that this work enables researchers to concentrate on the intellectual content of the theories they develop rather than be concerned with technical details of the implementation.

Local Energy Decomposition Analysis of London Dispersion Effects: From Simple Model Dimers to Complex Biomolecular Assemblies

ConspectusLondon dispersion (LD) forces are ubiquitous in chemistry, playing a pivotal role in a wide range of chemical processes. For example, they influence the structure of molecular crystals, the selectivity of organocatalytic transformations, and the formation of biomolecular assemblies. Harnessing these forces for chemical applications requires consistent quantification of the LD energy across a broad and diverse spectrum of chemical scenarios. Despite the great progress made in recent years in the development of experimental strategies for LD quantification, quantum chemical methods remain one of the most useful tools in the hand of chemists for the study of these weak interactions. Unfortunately, the accurate quantification of LD effects in complex systems poses many challenges for electronic structure theories. One of the problems stems from the fact that LD forces originate from long-range electronic dynamic correlation, and hence, their rigorous description requires the use of complex, highly correlated wave function-based methods. These methods typically feature a steep scaling with the system size, limiting their applicability to small model systems. Another core challenge lies in disentangling short-range from long-range dynamic correlation, which from a rigorous quantum mechanical perspective is not possible.In this Account, we describe our research endeavors in the development of broadly applicable computational methods for LD quantification in molecular chemistry as well as challenging applications of these schemes in various domains of chemical research. Our strategy lies in the use of local correlation theories to reduce the computational cost associated with complex electronic structure methods while providing at the same time a simple means of decomposition of dynamic correlation into its long-range and short-range components. In particular, the local energy decomposition (LED) scheme at the domain-based local pair natural orbital coupled cluster (DLPNO-CCSD(T)) level has emerged as a powerful tool in our research, offering a clear-cut quantitative definition of the LD energy that remains valid across a plethora of different chemical scenarios. Typical applications of this scheme are examined, encompassing protein-ligand interactions and reactivity studies involving many fragments and complex electronic structures. In addition, our research also involves the development of novel cost-effective methodologies, which exploit the LED definition of the LD energy, for LD energy quantification that are, in principle, applicable to systems with thousands of atoms. The Hartree-Fock plus London Dispersion (HFLD) scheme, correcting the HF interaction energy using an approximate CCSD(T)-based LD energy, is a useful, parameter-free electronic structure method for the study of LD effects in systems with hundreds of molecular fragments. However, the usefulness of the LED scheme reaches beyond providing an interpretation of the calculated DLPNO-CCSD(T) or DLPNO-MP2 interaction energies. For example, the dispersion energies obtained from the LED can be fruitfully used in order to parametrize semiempirical dispersion models. We will demonstrate this in the context of an emerging semiempirical method, namely, the Natural Orbital Tied Constructed Hamiltonian (NOTCH) method. NOTCH incorporates LED-derived LD energies and shows promising accuracy at a minimum amount of empiricism. Thus, it holds substantial promise for large and complex systems.

Molybdenum(VI) Nitrido Complexes with Tripodal Silanolate Ligands. Structure and Electronic Character of an Unsymmetrical Dimolybdenum μ-Nitrido Complex Formed by Incomplete Nitrogen Atom Transfer

In contrast to a tungsten nitrido complex endowed with a tripodal silanolate ligand framework, which was reported in the literature to be a dimeric species with a metallacyclic core, the corresponding molybdenum nitrides 3 are monomeric entities comprising a regular terminal nitride unit, as proven by single-crystal X-ray diffraction (SC-XRD). Their electronic character is largely determined by the constraints imposed on the metal center by the podand ligand architecture. 95Mo nuclear magnetic resonance (NMR) and, to a lesser extent, 14N NMR spectroscopy allow these effects to be studied, which become particularly apparent upon comparison with the spectral data of related molybdenum nitrides comprising unrestrained silanolate, alkoxide, or amide ligands. Attempted nitrogen atom transfer from these novel terminal nitrides to [(tBuArN)3Mo] (Ar = 3,5-dimethylphenyl) as the potential acceptor stopped at the stage of unsymmetric dimolybdenum μ-nitrido complex 13a as the first intermediate along the reaction pathway. SC-XRD, NMR, electron paramagnetic resonance, and ultraviolet-visible spectroscopy as well as magnetometry in combination with density functional theory allowed a clear picture of the geometric and electronic structure of this mixed-valent species to be drawn. 13a is formally best described as an adduct of the type [(Mo[O])+III-(μN)-III-(Mo[N])+VI], S = 1/2 complex with (Mo[O])+III in the low-spin configuration, whereas related complexes such as [(AdS)3Mo-(μN)-Mo(NtBuAr)3] (19; Ad = 1-adamantyl) have previously been regarded in the literature as mixed-valent Mo+IV/Mo+V species. The spin population at the two Mo centers is uneven and notably larger at the more reduced Mo[O] atom, whereas the only spin present at the (μN) bridge is derived from spin polarization.

Structural characterization and reactivity of a room-temperature-stable, antiaromatic cyclopentadienyl cation salt

The singlet states of cyclopentadienyl (Cp) cations are considered as true prototypes of an antiaromatic system. Unfortunately, their high intrinsic reactivity inhibited their isolation in the solid state as a salt, and controlled reactions are also scarce. Here we present the synthesis and solid state structure of the room-temperature-stable Cp cation salt [Cp(C6F5)5]+[Sb3F16]-. Although the aromatic triplet state of the [Cp(C6F5)5]+ cation is energetically favoured in the gas phase according to quantum chemical calculations, coordination of the cation by either [Sb3F16]- or C6F6 in the crystal lattice stabilizes the antiaromatic singlet state, which is present in the solid state. The calculated hydride and fluoride ion affinities of the [Cp(C6F5)5]+ cation are higher than those of the perfluorinated tritylium cation [C(C6F5)3]+. Reactions of [Cp(C6F5)5]+[Sb3F16]- with CO, which probably yields the corresponding carbonyl complex, and of radical Cp(C6F5)5∙ with selected model substrates (Cp2Fe, (Ph3C∙)2 and Cp*Al) are also presented.

Oxygen-Donor Metalloligands Induce Slow Magnetization Relaxation in Zero Field for a Cobalt(II) Complex with {CoO4} Motif

Most 3d metal-based single-molecule magnets (SMMs) use N-ligands or ligands with even softer donors to impart a particular coordination geometry and increase the zero-field splitting parameter |D|, while complexes with hard O-donor ligands showing slow magnetization relaxation are rare. Here, we report that a diamagnetic NiII complex of a tetradentate ligand featuring two N-heterocyclic carbene and two alkoxide-O donors, [LO,ONi], can serve as a {O,O'}-chelating metalloligand to give a trinuclear complex [(LO,ONi)Co(LO,ONi)](OTf)2 (2) with an elongated tetrahedral {CoIIO4} core, D = -74.3 cm-1, and a spin reversal barrier Ueff = 86.9 cm-1 in the absence of an external dc field. The influence of diamagnetic NiII on the electronic structure of the {CoO4} unit in comparison to [Co(OPh)4]2- (A) has been probed with multireference ab initio calculations. These reveal a contrapolarizing effect of the NiII, which forms stronger metal-alkoxide bonds than the central CoII, inducing a change in ligand field splitting and a 5-fold increase in the magnetic anisotropy in 2 compared to A, with an easy magnetization axis along the Ni-Co-Ni vector. This demonstrates a strategy to enhance the SMM properties of 3d metal complexes with hard O-donors by modulating the ligand field character via the coordination of diamagnetic ions and the benefit of robust metalloligands in that regard.

A theoretical spectroscopy study of the photoluminescence properties of narrow band Eu2+-doped phosphors containing multiple candidate doping centers. Prediction of an unprecedented narrow band red phosphor

We have previously presented a computational protocol that is based on an embedded cluster model and operates in the framework of TD-DFT in conjunction with the excited state dynamics (ESD) approach. The protocol is able to predict the experimental absorption and emission spectral shapes of Eu2+-doped phosphors. In this work, the applicability domain of the above protocol is expanded to Eu2+-doped phosphors bearing multiple candidate Eu doping centers. It will be demonstrated that this protocol provides full control of the parameter space that describes the emission process. The stability of Eu doping at various centers is explored through local energy decomposition (LED) analysis of DLPNO-CCSD(T) energies. This enables further development of the understanding of the electronic structure of the targeted phosphors, the diverse interactions between Eu and the local environment, and their impact on Eu doping probability, and control of the emission properties. Hence, it can be employed to systematically improve deficiencies of existing phosphor materials, defined by the presence of various intensity emission bands at undesired frequencies, towards classes of candidate Eu2+-doped phosphors with desired narrow band red emission. For this purpose, the chosen study set consists of three UCr4C4-based narrow-band phosphors, namely the known alkali lithosilicates RbNa[Li3SiO4]2:Eu2+ (RNLSO2), RbNa3[Li3SiO4]4:Eu2+ (RNLSO) and their isotypic nitridolithoaluminate phosphors consisting of CaBa[LiAl3N4]2:Eu2+ (CBLA2) and the proposed Ca3Ba[LiAl3N4]4:Eu2+ (CBLA), respectively. The theoretical analysis presented in this work led us to propose a modification of the CBLA2 phosphor that should have improved and unprecedented narrow band red emission properties. Finally, we believe that the analysis presented here is important for the future rational design of novel Eu2+-doped phosphor materials, with a wide range of applications in science and technology.

Nature of S-States in the Oxygen-Evolving Complex Resolved by High-Energy Resolution Fluorescence Detected X-ray Absorption Spectroscopy

Photosystem II, the water splitting enzyme of photosynthesis, utilizes the energy of sunlight to drive the four-electron oxidation of water to dioxygen at the oxygen-evolving complex (OEC). The OEC harbors a Mn4CaO5 cluster that cycles through five oxidation states Si (i = 0-4). The S3 state is the last metastable state before the O2 evolution. Its electronic structure and nature of the S2 → S3 transition are key topics of persisting controversy. Most spectroscopic studies suggest that the S3 state consists of four Mn(IV) ions, compared to the Mn(III)Mn(IV)3 of the S2 state. However, recent crystallographic data have received conflicting interpretations, suggesting either metal- or ligand-based oxidation, the latter leading to an oxyl radical or a peroxo moiety in the S3 state. Herein, we utilize high-energy resolution fluorescence detected (HERFD) X-ray absorption spectroscopy to obtain a highly resolved description of the Mn K pre-edge region for all S-states, paying special attention to use chemically unperturbed S3 state samples. In combination with quantum chemical calculations, we achieve assignment of specific spectroscopic features to geometric and electronic structures for all S-states. These data are used to confidently discriminate between the various suggestions concerning the electronic structure and the nature of oxidation events in all observable catalytic intermediates of the OEC. Our results do not support the presence of either peroxo or oxyl in the active configuration of the S3 state. This establishes Mn-centered storage of oxidative equivalents in all observable catalytic transitions and constrains the onset of the O-O bond formation until after the final light-driven oxidation event.

Cluster-in-Molecule Approach with Explicitly Correlated Methods for Large Molecules

In this article, we present a series of explicitly correlated local correlation methods developed under the cluster-in-molecule (CIM) framework, including explicitly correlated second-order Møller-Plesset perturbation (MP2), coupled-cluster singles and doubles (CCSD), domain-based local pair natural orbital CCSD (DLPNO-CCSD), and DLPNO-CCSD with perturbative triples (DLPNO-CCSD(T)). In these methods, F12 correction is decomposed into contributions from each occupied local molecular orbital and then evaluated independently in a given cluster, which consists of a subset of localized orbitals. These newly developed methods allow F12 calculations of large molecules (up to 145 atoms for quasi-one-dimensional systems) on a single node. We use these methods to investigate the relative stability between extended and folded alkane C30H62, the relative stability of four secondary structures of a polyglycine Ace(Gly)10NH2, and the binding energies of two host-guest complexes. The results demonstrate that the combination of CIM with F12 methods is a promising way to investigate large molecules with small basis set errors.

Assessment of DLPNO-MP2 Approximations in Double-Hybrid DFT

The unfavorable scaling (N5) of the conventional second-order Møller-Plesset theory (MP2) typically prevents the application of double-hybrid (DH) density functionals to large systems with more than 100 atoms. A prominent approach to reduce the computational demand of electron correlation methods is the domain-based local pair natural orbital (DLPNO) approximation that is successfully used in the framework of DLPNO-CCSD(T). Its extension to MP2 [Pinski P.; Riplinger, C.; Valeev, E. F.; Neese, F. J. Chem. Phys. 2015, 143, 034108.] paved the way for DLPNO-based DH (DLPNO-DH) methods. In this work, we assess the accuracy of the DLPNO-DH approximation compared to conventional DHs on a large number of 7925 data points for thermochemistry and 239 data points for structural features, including main-group and transition-metal systems. It is shown that DLPNO-DH-DFT can be applied successfully to perform energy calculations and geometry optimizations for large molecules at a drastically reduced computational cost. Furthermore, PNO space extrapolation is shown to be applicable, similar to its DLPNO-CCSD(T) counterpart, to reduce the remaining error.

Reactive high-spin iron(IV)-oxo sites through dioxygen activation in a metal-organic framework

In nature, nonheme iron enzymes use dioxygen to generate high-spin iron(IV)=O species for a variety of oxygenation reactions. Although synthetic chemists have long sought to mimic this reactivity, the enzyme-like activation of O2 to form high-spin iron(IV) = O species remains an unrealized goal. Here, we report a metal-organic framework featuring iron(II) sites with a local structure similar to that in α-ketoglutarate-dependent dioxygenases. The framework reacts with O2 at low temperatures to form high-spin iron(IV) = O species that are characterized using in situ diffuse reflectance infrared Fourier transform, in situ and variable-field Mössbauer, Fe Kβ x-ray emission, and nuclear resonance vibrational spectroscopies. In the presence of O2, the framework is competent for catalytic oxygenation of cyclohexane and the stoichiometric conversion of ethane to ethanol.

Theoretical Magnetic Relaxation and Spin-Phonon Coupling Study in a Series of Molecular Engineering Designed Bridged Dysprosocenium Analogues

A detailed computational study of hypothetical sandwich dysprosium double-decker complexes, bridged by various numbers of aliphatic linkers, was performed to evaluate the effect of the structural modifications on their ground-state magnetic sublevels and assess their potential as candidates for single-molecule magnets (SMMs). The molecular structures of seven complexes were optimized using the TPSSh functional, and the electronic structure and magnetic properties were investigated using the complete active space self-consistent field method (CASSCF). Estimates of the magnetic moment blocking barrier (Ueff) and blocking temperatures (TB) are reported. In addition, a new method based on computed derivatives of effective demagnetization barriers Ueff with respect to vibrational normal modes was introduced and applied to evaluate the impact of spin-phonon coupling on the SMM properties. On the basis of the computed parameters, we have identified promising candidates with properties superior to those of the existing single-molecule magnets.

Dimerization of confined Brønsted acids in enantioselective organocatalytic reactions

The formation of Brønsted acid aggregates in the course of asymmetric organocatalytic reactions is often overlooked in mechanistic studies, even though it might have a deep impact on the stereo-controlling factors of the transformations. In this work, we shed light on the influence of the catalyst structure and reaction conditions on the spontaneity of the aggregation process for popular chiral organocatalysts derived from phosphoric acids using high-level quantum mechanical calculations. Our study encompasses small and sterically unhindered chiral phosphoric acids as well as large and "confined" imidodiphosphates and imidodiphosphorimidates. These systems have recently proven particularly effective in promoting a large number of highly relevant asymmetric transformations. While cooperative catalytic effects of sterically less hindered chiral phosphoric acid catalysts are well appreciated in literature, it is found that the formation of catalyst dimers in solution is possible for both standard and confined catalysts. The spontaneity of the aggregation process depends on reaction conditions like solvent polarity, polarizability, temperature, the nature of the interaction with the substrate, as well as the catalyst architecture. Finally, it is shown that, at low temperatures (153 K), the aggregation process can profoundly influence the reaction kinetics and selectivity.

Triplet states in the reaction center of Photosystem II

In oxygenic photosynthesis sunlight is harvested and funneled as excitation energy into the reaction center (RC) of Photosystem II (PSII), the site of primary charge separation that initiates the photosynthetic electron transfer chain. The chlorophyll ChlD1 pigment of the RC is the primary electron donor, forming a charge-separated radical pair with the vicinal pheophytin PheoD1 (ChlD1+PheoD1-). To avert charge recombination, the electron is further transferred to plastoquinone QA, whereas the hole relaxes to a central pair of chlorophylls (PD1PD2), subsequently driving water oxidation. Spin-triplet states can form within the RC when forward electron transfer is inhibited or back reactions are favored. This can lead to formation of singlet dioxygen, with potential deleterious effects. Here we investigate the nature and properties of triplet states within the PSII RC using a multiscale quantum-mechanics/molecular-mechanics (QM/MM) approach. The low-energy spectrum of excited singlet and triplet states, of both local and charge-transfer nature, is compared using range-separated time-dependent density functional theory (TD-DFT). We further compute electron paramagnetic resonance properties (zero-field splitting parameters and hyperfine coupling constants) of relaxed triplet states and compare them with available experimental data. Moreover, the electrostatic modulation of excited state energetics and redox properties of RC pigments by the semiquinone QA- is described. The results provide a detailed electronic-level understanding of triplet states within the PSII RC and form a refined basis for discussing primary and secondary electron transfer, charge recombination pathways, and possible photoprotection mechanisms in PSII.

A Conserved Second Sphere Residue Tunes Copper Site Reactivity in Lytic Polysaccharide Monooxygenases

Lytic polysaccharide monooxygenases (LPMOs) are powerful monocopper enzymes that can activate strong C-H bonds through a mechanism that remains largely unknown. Herein, we investigated the role of a conserved glutamine/glutamate in the second coordination sphere. Mutation of the Gln in NcAA9C to Glu, Asp, or Asn showed that the nature and distance of the headgroup to the copper fine-tune LPMO functionality and copper reactivity. The presence of Glu or Asp close to the copper lowered the reduction potential and decreased the ratio between the reduction and reoxidation rates by up to 500-fold. All mutants showed increased enzyme inactivation, likely due to changes in the confinement of radical intermediates, and displayed changes in a protective hole-hopping pathway. Electron paramagnetic resonance (EPR) and X-ray absorption spectroscopic (XAS) studies gave virtually identical results for all NcAA9C variants, showing that the mutations do not directly perturb the Cu(II) ligand field. DFT calculations indicated that the higher experimental reoxidation rate observed for the Glu mutant could be reconciled if this residue is protonated. Further, for the glutamic acid form, we identified a Cu(III)-hydroxide species formed in a single step on the H2O2 splitting path. This is in contrast to the Cu(II)-hydroxide and hydroxyl intermediates, which are predicted for the WT and the unprotonated glutamate variant. These results show that this second sphere residue is a crucial determinant of the catalytic functioning of the copper-binding histidine brace and provide insights that may help in understanding LPMOs and LPMO-inspired synthetic catalysts.

Theoretical spectroscopy for unraveling the intensity mechanism of the optical and photoluminescent spectra of chiral Re(I) transition metal complexes

In this work, we present a computational study that is able to predict the optical absorption and photoluminescent properties of the chiral Re(I) family of complexes [fac-ReX(CO)3L], where X is either Cl or I and L is N-heterocyclic carbene extended with π-conjugated [5]-helicenic unit. The computational strategy is based on carefully calibrated time dependent density functional theory calculations and operates in conjunction with an excited state dynamics approach to treat in addition to absorption (ABS) and photoluminescence (PL), electronic circular dichroism (ECD), and circularly polarized luminescence (CPL) spectroscopies, respectively. The employed computational approach provides, an addition, access to the computation of phosphorescence rates in terms of radiative and non-radiative relaxation processes. The chosen molecules consist of representative examples of non-helicenic (NHC) and helicenic diastereomers. The agreement between theoretical and experimental spectra, including absorption (ABS, ECD) and emission (PL, CPL), is excellent, validating a quantitative interpretation of the spectral features on the basis of natural transition orbitals and TheoDore analyses. It is demonstrated that across the set of studied Re(I) diastereomers, the emission process in the case of NHC diastereomers is metal to ligand charge transfer in nature and is dominated by the easy-axis anisotropy of the emissive excited multiplet. On the contrary, in the cases of the helicenic diastereomers, the emission process is intra ligand charge transfer in nature and is dominated by the respective easy-plane anisotropy of the emissive excited multiplet. This affects remarkably the photoluminescent properties of the molecules in terms of PL and CPL spectral band shapes, spin-vibronic coupling, relaxation times, and the respective quantum yields. Spin-vibronic coupling effects are investigated at the level of the state-average complete active space self-consistent field in conjunction with quasi-degenerate second order perturbation theory. It is in fact demonstrated that a spin-vibronic coupling mechanism controls the observed photophysics of this class of Re(I) complexes.

Dissecting the Molecular Origin of g-Tensor Heterogeneity and Strain in Nitroxide Radicals in Water: Electron Paramagnetic Resonance Experiment versus Theory

Nitroxides are common EPR sensors of microenvironmental properties such as polarity, numbers of H-bonds, pH, and so forth. Their solvation in an aqueous environment is facilitated by their high propensity to form H-bonds with the surrounding water molecules. Their g- and A-tensor elements are key parameters to extracting the properties of their microenvironment. In particular, the gxx value of nitroxides is rich in information. It is known to be characterized by discrete values representing nitroxide populations previously assigned to have different H-bonds with the surrounding waters. Additionally, there is a large g-strain, that is, a broadening of g-values associated with it, which is generally correlated with environmental and structural micro-heterogeneities. The g-strain is responsible for the frequency dependence of the apparent line width of the EPR spectra, which becomes evident at high field/frequency. Here, we address the molecular origin of the gxx heterogeneity and of the g-strain of a nitroxide moiety (HMI: 2,2,3,4,5,5-hexamethylimidazolidin-1-oxyl, C9H19N2O) in water. To treat the solvation effect on the g-strain, we combined a multi-frequency experimental approach with ab initio molecular dynamics simulations for structural sampling and quantum chemical EPR property calculations at the highest realistically affordable level, including an explicitly micro-solvated HMI ensemble and the embedded cluster reference interaction site model. We could clearly identify the distinct populations of the H-bonded nitroxides responsible for the gxx heterogeneity experimentally observed, and we dissected the role of the solvation shell, H-bond formation, and structural deformation of the nitroxide in the creation of the g-strain associated with each nitroxide subensemble. Two contributions to the g-strain were identified in this study. The first contribution depends on the number of hydrogen bonds formed between the nitroxide and the solvent because this has a large and well-understood effect on the gxx-shift. This contribution can only be resolved at high resonance frequencies, where it leads to distinct peaks in the gxx region. The second contribution arises from configurational fluctuations of the nitroxide that necessarily lead to g-shift heterogeneity. These contributions cannot be resolved experimentally as distinct resonances but add to the line broadening. They can be quantitatively analyzed by studying the apparent line width as a function of microwave frequency. Interestingly, both theory and experiment confirm that this contribution is independent of the number of H-bonds. Perhaps even more surprisingly, the theoretical analysis suggests that the configurational fluctuation broadening is not induced by the solvent but is inherently present even in the gas phase. Moreover, the calculations predict that this broadening decreases upon solvation of the nitroxide.

Air-stable four-coordinate cobalt(ii) single-ion magnets: experimental and ab initio ligand field analyses of correlations between dihedral angles and magnetic anisotropy

For single-ion magnets (SIMs), understanding the effects of the local coordination environment and ligand field on magnetic anisotropy is key to controlling their magnetic properties. Here we present a series of tetracoordinate cobalt(ii) complexes of the general formula [^FL₂Co]X₂ (where ^FL is a bidentate diamido ligand) whose electron-withdrawing -C₆F₅ substituents confer stability under ambient conditions. Depending on the cations X, these complexes adopt structures with greatly varying dihedral twist angle δ between the N-Co-N' chelate planes in the solid state (48.0 to 89.2°). AC and DC field magnetic susceptibility measurements show this to translate into very different magnetic properties, the axial zero-field splitting (ZFS) parameter D ranging from -69 cm^-1 to -143 cm^-1 with substantial or negligible rhombic component E, respectively. A close to orthogonal arrangement of the two N,N'-chelating σ- and π-donor ligands at the Co(ii) ion is found to raise the energy barrier for magnetic relaxation to above 400 K. Multireference ab initio methods were employed to describe the complexes' electronic structures, and the results were analyzed within the framework of ab initio ligand field theory to probe the nature of the metal-ligand bonding and spin-orbit coupling. A relationship between the energy gaps of the first few electronic transitions and the ZFS was established, and the ZFS was correlated with the dihedral angle δ as well as with the metal-ligand bonding variations, viz. the two angular overlap parameters e_σ and e_πs. These findings not only give rise to a Co(ii) SIM showing open hysteresis up to 3.5 K at a sweep rate of 30 Oe s^-1, but they also provide design guidelines for Co(ii) complexes with favorable SIM signatures or even switchable magnetic relaxation properties.

Measuring Electron Correlation: The Impact of Symmetry and Orbital Transformations

In this perspective, the various measures of electron correlation used in wave function theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behavior with respect to the choice of the N-electron and the one-electron basis. The impact of symmetry is discussed, and we emphasize that the distinction among determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wave function expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wave functions are discussed and the effect of orbital rotations on the multireference character is reviewed by analyzing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and N-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one.

Synthesis and isolation of a triplet bismuthinidene with a quenched magnetic response

Large Spin-Orbit Coupling (SOC) is an intrinsic property of the heavy-elements that directly affects the electronic structures of the compounds. Herein we report the synthesis and characterization of a mono-coordinate bismuthinidene featuring a rigid and bulky ligand. All magnetic measurements (SQUID, NMR) point to a diamagnetic compound. However, multiconfigurational quantum chemical calculations predict the ground state of the compound to be dominated (76%) by a spin-triplet. The apparent diamagnetism is explained by an extremely large SOC induced positive zero-field-splitting of more than 4500 cm^-1 that leaves the M_S = 0 magnetic sublevel thermally isolated in the electronic ground state.

Development of NOTCH, an all-electron, beyond-NDDO semiempirical method: Application to diatomic molecules

In this work, we develop a new semiempirical method, dubbed NOTCH (Natural Orbital Tied Constructed Hamiltonian). Compared to existing semiempirical methods, NOTCH is less empirical in its functional form as well as parameterization. Specifically, in NOTCH, (1) the core electrons are treated explicitly; (2) the nuclear-nuclear repulsion term is calculated analytically, without any empirical parameterization; (3) the contraction coefficients of the atomic orbital (AO) basis depend on the coordinates of the neighboring atoms, which allows the size of AOs to depend on the molecular environment, despite the fact that a minimal basis set is used; (4) the one-center integrals of free atoms are derived from scalar relativistic multireference equation-of-motion coupled cluster calculations instead of empirical fitting, drastically reducing the number of necessary empirical parameters; (5) the (AA|AB) and (AB|AB)-type two-center integrals are explicitly included, going beyond the neglect of differential diatomic overlap approximation; and (6) the integrals depend on the atomic charges, effectively mimicking the "breathing" of AOs when the atomic charge varies. For this preliminary report, the model has been parameterized for the elements H-Ne, giving only 8 empirical global parameters. Preliminary results on the ionization potentials, electron affinities, and excitation energies of atoms and diatomic molecules, as well as the equilibrium geometries, vibrational frequencies dipole moments, and bond dissociation energies of diatomic molecules, show that the accuracy of NOTCH rivals or exceeds those of popular semiempirical methods (including PM3, PM7, OM2, OM3, GFN-xTB, and GFN2-xTB) as well as the cost-effective ab initio method Hartree-Fock-3c.

Does Serial Femtosecond Crystallography Depict State-Specific Catalytic Intermediates of the Oxygen-Evolving Complex?

Recent advances in serial femtosecond crystallography (SFX) of photosystem II (PSII), enabled by X-ray free electron lasers (XFEL), provided the first geometric models of distinct intermediates in the catalytic S-state cycle of the oxygen-evolving complex (OEC). These models are obtained by flash-advancing the OEC from the dark-stable state (S₁) to more oxidized intermediates (S₂ and S₃), eventually cycling back to the most reduced S₀. However, the interpretation of these models is controversial because geometric parameters within the Mn₄CaO₅ cluster of the OEC do not exactly match those expected from coordination chemistry for the spectroscopically verified manganese oxidation states of the distinct S-state intermediates. Here we focus on the first catalytic transition, S₁ → S₂, which represents a one-electron oxidation of the OEC. Combining geometric and electronic structure criteria, including a novel effective oxidation state approach, we analyze existing 1-flash (1F) SFX-XFEL crystallographic models that should depict the S₂ state of the OEC. We show that the 1F/S₂ equivalence is not obvious, because the Mn oxidation states and total unpaired electron counts encoded in these models are not fully consistent with those of a pure S₂ state and with the nature of the S₁ → S₂ transition. Furthermore, the oxidation state definition in two-flashed (2F) structural models is practically impossible to elucidate. Our results advise caution in the extraction of electronic structure information solely from the literal interpretation of crystallographic models and call for re-evaluation of structural and mechanistic interpretations that presume exact correspondence of such models to specific catalytic intermediates of the OEC.

Including vibrational effects in magnetic circular dichroism spectrum calculations in the framework of excited state dynamics

In this work, we present a computational approach that is able to incorporate vibrational effects in the computations of magnetic circular dichroism (MCD) spectra. The method combines our previous implementations to model absorption as well as fluorescence and phosphorescence spectra in the framework of excited state dynamics with a new technique to calculate MCD intensities, where molecular orientational averages are treated via semi-numerical quadrature. The implementation relies on a path integral approach that is employed to compute nuclear dynamics under the harmonic oscillator approximation (accounting for the nuclear potential energy surface) together with quasi-degenerate perturbative theory (to include the perturbation of an external magnetic field). We evaluate our implementation with a selected molecular set consisting of five aromatic organic molecules, namely, 1,4-benzoquinone, naphthalene, 2-naphthylamine, 2-naphthaldehyde, and benzene; we also included the MnO4- and the [Co(NH3)6]3+ transition metal complexes. This set is used to validate the ability of the approach to compute MCD A- and B-terms in conjunction with time-dependent density functional theory. The computed intensities are discussed in terms of the overall quality of the electronic structure treatments, vibrational modes, and the quality of the nuclear Hessians. It is shown that in the cases in which the potential energy surface is accurately represented, electric dipole-forbidden transitions are vibrationally activated, producing intensities relative to the dipole-allowed transitions in the same order of magnitude as the experimental measurements.

Exchange Interactions and Magnetic Properties of a Molecular Mn18-Ring Complex

[Mn3O(OAc)7(HOAc)]6·xAcOH (x = 6-9) represents a rare example of a compound containing molecular Mn18-rings. These are formed by Mn3(µ3-O) subunits in which the high-spin Mn(III) centers are bridged by three pairs of acetate anions (AcO-). An AcOH molecule coordinates to one of the Mn atoms leading to [Mn3(µ3-O)(µ2-OAc)6(AcOH)]-units, designated in short as Mn3-units, that are interconnected by acetate anions via the other two Mn atoms to form Mn18-rings. Magnetic measurements show weak ferromagnetic interactions between them that are suppressed in strong magnetic field. Quantum-chemical calculations on Mn3 model complexes using independently DFT and ab-initio multi reference methods (CASSCF/NEVPT2) show a correlation between the orientation of the pseudo-Jahn-Teller axes of pairs of Mn(III) magnetic centers and corresponding exchange coupling energies. Weak coupling between Mn3-units within the Mn18-ring allowed to simulate the magnetic susceptibility vs temperature dependence in terms of basically uncoupled magnetic moments of each Mn3-unit within the ring.

Exploring the Accuracy Limits of PNO-Based Local Coupled-Cluster Calculations for Transition-Metal Complexes

While the domain-based local pair natural orbital coupled-cluster method with singles, doubles, and perturbative triples (DLPNO-CCSD(T)) has proven instrumental for computing energies and properties of large and complex systems accurately, calculations on first-row transition metals with a complex electronic structure remain challenging. In this work, we identify and address the two main error sources that influence the DLPNO-CCSD(T) accuracy in this context, namely, (i) correlation effects from the 3s and 3p semicore orbitals and (ii) dynamic correlation-induced orbital relaxation (DCIOR) effects that are not described by the local MP2 guess. We present a computational strategy that allows us to completely eliminate the DLPNO error associated with semicore correlation effects, while increasing, at the same time, the efficiency of the method. As regards the DCIOR effects, we introduce a diagnostic for estimating the deviation between DLPNO-CCSD(T) and canonical CCSD(T) for systems with significant orbital relaxation.

SparseMaps - A systematic infrastructure for reduced-scaling electronic structure methods. VI. Linear-scaling explicitly correlated N-electron valence state perturbation theory with pair natural orbital

In this work, a linear scaling explicitly correlated N-electron valence state perturbation theory (NEVPT2-F12) is presented. By using the idea of domain based local pair natural orbital (DLPNO), the computational scaling of the conventional NEVPT2-F12 is reduced to near linear scaling. For low-lying excited states of organic molecules, the excitation energies predicted by DLPNO-NEVPT2-F12 are as accurate as the exact NEVPT2-F12 results. Some cluster models of rhodopsin are studied by the new algorithm. Our new method is able to study systems with more than 3300 basis functions and an active space containing 12 π-electrons and 12 π-orbitals. However, even larger calculations or active spaces would still be feasible.

On the Single-Molecule Magnetic Behavior of Linear Iron(I) Arylsilylamides

The rational design of 3d-metal-based single-molecule magnets (SMM) requires a fundamental understanding of their intrinsic electronic and structural properties and how they translate into experimentally observable features. Here, we determined the magnetic properties of the linear iron(I) silylamides K{crypt}[FeL₂] and [KFeL₂] (L = -N(Dipp)SiMe₃; crypt = 4,7,13,16,21,24-Hexaoxa-1,10-diazabicyclo[8.8.8]hexacosan). For the former, slow-relaxation of the magnetization with a spin reversal barrier of U_eff = 152 cm^-1 as well as a closed-waist magnetic hysteresis and magnetic blocking below 2.5 K are observed. For the more linear [KFeL₂], in which the potassium cation is encapsulated by the aryl substituents of the amide ligands, the relaxation barrier and the blocking temperature increase to U_eff = 184 cm^-1 and T_B = 4.5 K, respectively. The increase is rationalized by a more pronounced axial anisotropy in [KFeL₂] determined by dc-SQUID magnetometry. The effective relaxation barrier of [KFeL₂] is in agreement with the energy spacing between the ground and first-excited magnetic states, as obtained by field-dependent IR-spectroscopy (178 cm^-1), magnetic measurements (208 cm^-1), as well as theoretical analysis (212 cm^-1). In comparison with the literature, the results show that magnetic coercivity in linear iron(I) silylamides is driven by the degree of linearity in conjunction with steric encumbrance, whereas the ligand symmetry is a marginal factor.

Quantum computing in pharma: A multilayer embedding approach for near future applications

Quantum computers are special purpose machines that are expected to be particularly useful in simulating strongly correlated chemical systems. The quantum computer excels at treating a moderate number of orbitals within an active space in a fully quantum mechanical manner. We present a quantum phase estimation calculation on F₂ in a (2,2) active space on Rigetti's Aspen-11 QPU. While this is a promising start, it also underlines the need for carefully selecting the orbital spaces treated by the quantum computer. In this work, a scheme for selecting such an active space automatically is described and simulated results obtained using both the quantum phase estimation (QPE) and variational quantum eigensolver (VQE) algorithms are presented and combined with a subtractive method to enable accurate description of the environment. The active occupied space is selected from orbitals localized on the chemically relevant fragment of the molecule, while the corresponding virtual space is chosen based on the magnitude of interactions with the occupied space calculated from perturbation theory. This protocol is then applied to two chemical systems of pharmaceutical relevance: the enzyme [Fe] hydrogenase and the photosenzitizer temoporfin. While the sizes of the active spaces currently amenable to a quantum computational treatment are not enough to demonstrate quantum advantage, the procedure outlined here is applicable to any active space size, including those that are outside the reach of classical computation.

The SHARK integral generation and digestion system

In this paper, the SHARK integral generation and digestion engine is described. In essence, SHARK is based on a reformulation of the popular McMurchie/Davidson approach to molecular integrals. This reformulation leads to an efficient algorithm that is driven by BLAS level 3 operations. The algorithm is particularly efficient for high angular momentum basis functions (up to L = 7 is available by default, but the algorithm is programmed for arbitrary angular momenta). SHARK features a significant number of specific programming constructs that are designed to greatly simplify the workflow in quantum chemical program development and avoid undesirable code duplication to the largest possible extent. SHARK can handle segmented, generally and partially generally contracted basis sets. It can be used to generate a host of one- and two-electron integrals over various kernels including, two-, three-, and four-index repulsion integrals, integrals over Gauge Including Atomic Orbitals (GIAOs), relativistic integrals and integrals featuring a finite nucleus model. SHARK provides routines to evaluate Fock like matrices, generate integral transformations and related tasks. SHARK is the essential engine inside the ORCA package that drives essentially all tasks that are related to integrals over basis functions in version ORCA 5.0 and higher. Since the core of SHARK is based on low-level basic linear algebra (BLAS) operations, it is expected to not only perform well on present day but also on future hardware provided that the hardware manufacturer provides a properly optimized BLAS library for matrix and vector operations. Representative timings and comparisons to the Libint library used by ORCA are reported for Intel i9 and Apple M1 max processors.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Calculation of exchange couplings in the electronically excited state of molecular three-spin systems

Photogenerated molecular three-spin systems, composed of a chromophore and a covalently bound stable radical, are promising candidates for applications in the field of molecular spintronics. Through excitation with light, an excited doublet state and a quartet state are generated, whereby their energy difference depends on the exchange interaction J TR between the chromophore triplet state (T) and the stable radical (R). In order to establish design rules for new materials to be used in molecular spintronics devices, it is of great importance to gain knowledge on the magnitude of J TR as well as the factors influencing J TR on a molecular level. Here, we present a robust and reliable computational method to determine excited state exchange couplings in three-electron-three-centre systems based on a CASSCF/QD-NEVPT2 approach. The methodology is benchmarked and then applied to a series of molecules composed of a perylene chromophore covalently linked to various stable radicals. We calculate the phenomenological exchange interaction J TR between chromophore and radical, which can be compared directly to the experiment, but also illustrate how the individual exchange interactions J ij can be extracted using an effective Hamiltonian that corresponds to the Heisenberg-Dirac-Van-Vleck Hamiltonian. The latter procedure enables a more detailed analysis of the contributions to the exchange interaction J TR and yields additional insight that will be invaluable for future design optimisation.

Comprehensive Studies of Magnetic Transitions and Spin-Phonon Couplings in the Tetrahedral Cobalt Complex Co(AsPh3)2I2

A combination of inelastic neutron scattering (INS), far-IR magneto-spectroscopy (FIRMS), and Raman magneto-spectroscopy (RaMS) has been used to comprehensively probe magnetic excitations in Co(AsPh₃)₂I₂ (1), a reported single-molecule magnet (SMM). With applied field, the magnetic zero-field splitting (ZFS) peak (2D') shifts to higher energies in each spectroscopy. INS placed the ZFS peak at 54 cm^-1, as revealed by both variable-temperature (VT) and variable-magnetic-field data, giving results that agree well with those from both far-IR and Raman studies. Both FIRMS and RaMS also reveal the presence of multiple spin-phonon couplings as avoided crossings with neighboring phonons. Here, phonons refer to both intramolecular and lattice vibrations. The results constitute a rare case in which the spin-phonon couplings are observed with both Raman-active (g modes) and far-IR-active phonons (u modes; space group P2₁/c, no. 14, Z = 4 for 1). These couplings are fit using a simple avoided crossing model with coupling constants of ca. 1-2 cm^-1. The combined spectroscopies accurately determine the magnetic excited level and the interaction of the magnetic excitation with phonon modes. Density functional theory (DFT) phonon calculations compare well with INS, allowing for the assignment of the modes and their symmetries. Electronic calculations elucidate the nature of ZFS in the complex. Features of different techniques to determine ZFS and other spin-Hamiltonian parameters in transition-metal complexes are summarized.

Cluster-in-Molecule Method Combined with the Domain-Based Local Pair Natural Orbital Approach for Electron Correlation Calculations of Periodic Systems

The cluster-in-molecule (CIM) method was extended to systems with periodic boundary conditions (PBCs) in a previous work (PBC-CIM) [J. Chem. Theory Comput.2019, 15, 2933], which is able to compute the electronic structures of periodic systems at second-order Møller-Plesset perturbation theory (MP2) and coupled cluster singles and doubles (CCSD) levels. However, the high computational costs of CCSD with respect to the size of clusters limit the usage of PBC-CIM to crystals with small or medium unit cells. In this work, we further develop the PBC-CIM method by employing the domain-based local pair natural orbital (DLPNO) methods for the electron correlation calculations of clusters to reduce the computational costs. The combined approach allows CCSD with perturbative triples, denoted as CCSD(T), to be computationally available for accurate descriptions of periodic systems. The distant-pair correction is also implemented to improve the accuracy of PBC-CIM. As in the molecular cases, the distant pair correction significantly improves the accuracy of various PBC-CIM methods with few additional costs. The PBC-CIM-DLPNO-CCSD(T) approach has been applied to investigate the optimized lattice parameter of the cubic LiCl crystal and two adsorption problems (CO on the NaCl(100) surface and H₂O on the h-BN surface). The results show that the CIM-DLPNO-CCSD(T) method offers accurate and efficient descriptions for the studied systems. Another application to the cohesive energy of the acetic acid crystal reveals that large basis sets are necessary for reliable calculations on the cohesive energies of molecular crystals.

Magnetic exchange and valence delocalization in a mixed valence [Fe2+Fe3+Te2]+ complex: insights from theory and interpretations of magnetic and spectroscopic data

A mixed valence binuclear Fe^2.5+-Fe^2.5+ (Robin-Day Class III) transition metal complex, [Fe^2.5+μTe₂Fe^2.5+]^1-, composed of two FeN₂Te₂ pseudo-tetrahedral units with μ-bridging Te^2- ligands was reported to exist in an unprecedented S = 3/2 ground state (Nature Chemistry, https://doi.org/10.1038/s41557-021-00853-5). For this and the homologous complexes containing Se^2- and S^2-, the Anderson-Hasegawa double exchange spin-Hamiltonian was broadly used to interpret the corresponding structural, spectroscopic and magnetic data. First principles multireference ab initio calculations are used here to simulate magnetic and spectroscopic EPR data; analysis of the results affords a rationale for the stabilization of the S = 3/2 ground state of the Fe₂ pair. Complete Active Space Self-Consistent Field (CASSCF) calculations and dynamical correlation accounted for by means of N-Electron Valence Perturbation Theory to Second Order (NEVPT2) reproduce well the g-factors determined from simulations of X-band EPR spectra. A crucial technical tool to achieve these results is: (i) use of a localized orbital formulation of the many-particle problem at the scalar-relativistic CASSCF step; (ii) choice of state averaging over states of a given spin (at the CASCI/NEVPT2 step); and (iii) accounting for spin-orbit coupling within the non-relativistic Born-Oppenheimer (BO) many-particle basis using Quasi-Degenerate Perturbation Theory (QDPT). The inclusion of the S = 5/2 spin manifold reproduced the observed increase in the magnetic susceptibility (χT) in the high temperature range (T > 100 K), which is explained by thermal population of the S = 5/2 excited state at energy 160 cm^-1 above the S = 3/2 ground state. Theoretical values of χT from experimentally reported data points in the temperature range from 3 to 30 K were further computed and analyzed using a model which takes spin-phonon coupling into account. The model considerations and the computational protocols of this study are generally applicable to any Class I/II mixed valence dimer. The work can potentially stimulate further experimental and theoretical work on bi- and oligonuclear transition metal complexes of importance to bioinorganic chemistry and life sciences.

Going beyond the Electric-Dipole Approximation in the Calculation of Absorption and (Magnetic) Circular Dichroism Spectra including Scalar Relativistic and Spin-Orbit Coupling Effects

In this work, a time-dependent density functional theory (TD-DFT) scheme for computing optical spectroscopic properties in the framework of linearly and circularly polarized light is presented. The scheme is based on a previously formulated theory for predicting optical absorption and magnetic circular dichroism (MCD) spectra. The scheme operates in the framework of the full semi-classical field-matter interaction operator, thus generating a powerful and general computational scheme capable of computing the absorption (ABS), circular dichroism (CD), and MCD spectra. In addition, our implementation includes the treatment of relativistic effects in the framework of quasidegenerate perturbation theory, which accounts for scalar relativistic effects (in the self-consistent field step) and spin-orbit coupling (in the TD-DFT step), as well as external magnetic field perturbations. Hence, this formalism is also able to probe spin-forbidden transitions. The random orientations of molecules are taken into account by a semi-numerical approach involving a Lebedev numerical quadrature alongside analytical integration. It is demonstrated the numerical quadrature requires as few as 14 points for satisfactory converged results thus leading to a highly efficient scheme, while the calculation of the exact transition moments creates no computational bottlenecks. It is demonstrated that at zero magnetic field, the CD spectrum is recovered while the sum of left and right circularly polarized light contributions provides the linear absorption spectrum. The virtues of this efficient and general protocol are demonstrated on a selected set of organic molecules where the various contributions to the spectral intensities have been analyzed in detail.

Can domain-based local pair natural orbitals approaches accurately predict phosphorescence energies?

Since the discovery of the peculiar conducting and optical properties of aromatics, many efforts have been made to characterize and predict their phosphorescence. This physical process is exploited in modern Organic Emitting Light Diodes (OLEDs), and it is also one of the processes decreasing the efficiency of Dye-sensitized solar cells (DSSCs). Herein, we propose a computational strategy for the accurate calculation of singlet-triplet gaps of aromatic compounds, which provides results that are in excellent agreement with available experimental data. Our approach relies on the domain-based local pair natural orbital (DLPNO) variant of the "gold standard" CCSD(T) method. The convergence of our results with respect to the key technical parameters of the calculation, such as the basis set used, the approximations employed in the perturbative triples correction, and the dimension of the PNOs space, was thoroughly discussed.

Electronic and Optical Properties of Eu2+-Activated Narrow-Band Phosphors for Phosphor-Converted Light-Emitting Diode Applications: Insights from a Theoretical Spectroscopy Perspective

In this work, we present a computational protocol that is able to predict the experimental absorption and emission spectral shapes of Eu²⁺-doped phosphors. The protocol is based on time-dependent density functional theory and operates in conjunction with an excited-state dynamics approach. It is demonstrated that across the study set consisting of representative examples of nitride, oxo-nitride, and oxide Eu²⁺-doped phosphors, the energy distribution and the band shape of the emission spectrum are related to the nature of the 4f–5d transitions that are probed in the absorption process. Since the 4f orbitals are very nearly nonbonding, the decisive quantity is the covalency of the 5d acceptor orbitals that become populated in the electronically excited state that leads to emission. The stronger the (anti) bonding interaction between the lanthanide and the ligands is in the excited state, the larger will be the excited state distortion. Consequently, the corresponding emission will get broader due to the vibronic progression that is induced by the structural distortion. In addition, the energy separation of the absorption bands that are dominated by states with valence 4f–5d and a metal to ligand charge transfer character defines a measure for the thermal quenching of the studied Eu²⁺-doped phosphors. Based on this analysis, simple descriptors are identified that show a strong correlation with the energy position and bandwidth of the experimental emission bands without the need for elaborate calculations. Overall, we believe that this study serves as an important reference for designing new Eu²⁺-doped phosphors with desired photoluminescence properties.

Theoretical analysis of the long-distance limit of NMR chemical shieldings

After some years of controversy, it was recently demonstrated how to obtain the correct long-distance limit [point-dipole approximation (PDA)] of pseudo-contact nuclear magnetic resonance chemical shifts from rigorous first-principles quantum mechanics [Lang et al., J. Phys. Chem. Lett. 11, 8735 (2020)]. This result confirmed the classical Kurland-McGarvey theory. In the present contribution, we elaborate on these results. In particular, we provide a detailed derivation of the PDA both from the Van den Heuvel-Soncini equation for the chemical shielding tensor and from a spin Hamiltonian approximation. Furthermore, we discuss in detail the PDA within the approximate density functional theory and Hartree-Fock theories. In our previous work, we assumed a relatively crude effective nuclear charge approximation for the spin-orbit coupling operator. Here, we overcome this assumption by demonstrating that the derivation is also possible within the fully relativistic Dirac equation and even without the assumption of a specific form for the Hamiltonian. Crucial ingredients for the general derivation are a Hamiltonian that respects gauge invariance, the multipolar gauge, and functional derivatives of the Hamiltonian, where it is possible to identify the first functional derivative with the electron number current density operator. The present work forms an important foundation for future extensions of the Kurland-McGarvey theory beyond the PDA, including induced magnetic quadrupole and higher moments to describe the magnetic hyperfine field.

Open-Shell Variant of the London Dispersion-Corrected Hartree-Fock Method (HFLD) for the Quantification and Analysis of Noncovalent Interaction Energies

The London dispersion (LD)-corrected Hartree–Fock (HF) method (HFLD) is an ab initio approach for the quantification and analysis of noncovalent interactions (NCIs) in large systems that is based on the domain-based local pair natural orbital coupled-cluster (DLPNO-CC) theory. In the original HFLD paper, we discussed the implementation, accuracy, and efficiency of its closed-shell variant. Herein, an extension of this method to open-shell molecular systems is presented. Its accuracy is tested on challenging benchmark sets for NCIs, using CCSD(T) energies at the estimated complete basis set limit as reference. The HFLD scheme was found to be as accurate as the best-performing dispersion-corrected exchange-correlation functionals, while being nonempirical and equally efficient. In addition, it can be combined with the well-established local energy decomposition (LED) for the analysis of NCIs, thus yielding additional physical insights.

Ionization Energies and Redox Potentials of Hydrated Transition Metal Ions: Evaluation of Domain-Based Local Pair Natural Orbital Coupled Cluster Approaches

Hydrated transition metal ions are prototypical systems that can be used to model properties of transition metals in complex chemical environments. These seemingly simple systems present challenges for computational chemistry and are thus crucial in evaluations of quantum chemical methods for spin-state and redox energetics. In this work, we explore the applicability of the domain-based pair natural orbital implementation of coupled cluster (DLPNO-CC) theory to the calculation of ionization energies and redox potentials for hydrated ions of all first transition row (3d) metals in the 2+/3+ oxidation states, in connection with various solvation approaches. In terms of model definition, we investigate the construction of a minimally explicitly hydrated quantum cluster with a first and second hydration layer. We report on the convergence with respect to the coupled cluster expansion and the PNO space, as well as on the role of perturbative triple excitations. A recent implementation of the conductor-like polarizable continuum model (CPCM) for the DLPNO-CC approach is employed to determine self-consistent redox potentials at the coupled cluster level. Our results establish conditions for the convergence of DLPNO-CCSD(T) energetics and stress the absolute necessity to explicitly consider the second solvation sphere even when CPCM is used. The achievable accuracy for redox potentials of a practical DLPNO-based approach is, on average, 0.13 V. Furthermore, multilayer approaches that combine a higher-level DLPNO-CCSD(T) description of the first solvation sphere with a lower-level description of the second solvation layer are investigated. The present work establishes optimal and transferable methodological choices for employing DLPNO-based coupled cluster theory, the associated CPCM implementation, and cost-efficient multilayer derivatives of the approach for open-shell transition metal systems in complex environments.

A Combined Spectroscopic and Computational Study on the Mechanism of Iron-Catalyzed Aminofunctionalization of Olefins Using Hydroxylamine Derived N-O Reagent as the "Amino" Source and "Oxidant"

Herein, we study the mechanism of iron-catalyzed direct synthesis of unprotected aminoethers from olefins by a hydroxyl amine derived reagent using a wide range of analytical and spectroscopic techniques (Mössbauer, Electron Paramagnetic Resonance, Ultra-Violet Visible Spectroscopy, X-ray Absorption, Nuclear Resonance Vibrational Spectroscopy, and resonance Raman) along with high-level quantum chemical calculations. The hydroxyl amine derived triflic acid salt acts as the “oxidant” as well as “amino” group donor. It activates the high-spin Fe(II) (S_t = 2) catalyst [Fe(acac)₂(H₂O)₂] (1) to generate a high-spin (S_t = 5/2) intermediate (Int I), which decays to a second intermediate (Int II) with S_t = 2. The analysis of spectroscopic and computational data leads to the formulation of Int I as [Fe(III)(acac)₂-N-acyloxy] (an alkyl-peroxo-Fe(III) analogue). Furthermore, Int II is formed by N–O bond homolysis. However, it does not generate a high-valent Fe(IV)(NH) species (a Fe(IV)(O) analogue), but instead a high-spin Fe(III) center which is strongly antiferromagnetically coupled (J = −524 cm^–1) to an iminyl radical, [Fe(III)(acac)₂-NH·], giving S_t = 2. Though Fe(NH) complexes as isoelectronic surrogates to Fe(O) functionalities are known, detection of a high-spin Fe(III)-N-acyloxy intermediate (Int I), which undergoes N–O bond cleavage to generate the active iron–nitrogen intermediate (Int II), is unprecedented. Relative to Fe(IV)(O) centers, Int II features a weak elongated Fe–N bond which, together with the unpaired electron density along the Fe–N bond vector, helps to rationalize its propensity for N-transfer reactions onto styrenyl olefins, resulting in the overall formation of aminoethers. This study thus demonstrates the potential of utilizing the iron-coordinated nitrogen-centered radicals as powerful reactive intermediates in catalysis.

Electronic structure analysis of electrochemical CO2 reduction by iron-porphyrins reveals basic requirements to design catalysts bearing non-innocent ligands

Electrocatalytic CO2 reduction is a possible solution to the increasing CO2 concentration in the earth atmosphere, because it enables storage of energy while using the harmful CO2 feedstock as starting...

An induced-fit model for asymmetric organocatalytic reactions: a case study of the activation of olefins via chiral Brønsted acid catalysts

We elucidate the stereo-controlling factors of the asymmetric intramolecular hydroalkoxylation of terminal olefins catalyzed by bulky Brønsted acids [Science2018, 359 (6383), 1501–1505] using high-level electronic structure methods.

An efficient implementation of the NEVPT2 and CASPT2 methods avoiding higher-order density matrices

A factorization of the matrix elements of the Dyall Hamiltonian in N-electron valence state perturbation theory allowing their evaluation with a computational effort comparable to the one needed for the construction of the third-order reduced density matrix at the most is presented. Thus, the computational bottleneck arising from explicit evaluation of the fourth-order density matrix is avoided. It is also shown that the residual terms arising in the case of an approximate complete active space configuration interaction solution and containing even the fifth-order density matrix for two excitation classes can be evaluated with little additional effort by choosing again a favorable factorization of the corresponding matrix elements. An analogous argument is also provided for avoiding the fourth-order density matrix in complete active space second-order perturbation theory. Practical calculations indicate that such an approach leads to a considerable gain in computational efficiency without any compromise in numerical accuracy or stability.

First-Principles Study of Optical Absorption Energies, Ligand Field and Spin-Hamiltonian Parameters of Cr3+ Ions in Emeralds

Herein, we study the electronic structure, energies, and vibronic structure of optical d-d transitions of Cr³⁺ ions doped in beryl (Be₃Si₆Al₂O₁₈:Cr³⁺, emerald). A computational protocol is developed that combines periodic density functional theory (for modeling of the bulk crystalline lattice of emerald) and the multireference configuration interaction complete active space self-consistent field method supplemented with n-electron valence second-order perturbation theory (for the calculation of the energy levels, wave functions, and spin-Hamiltonian and ligand-field parameters of the trigonal Cr³⁺ centers in the [CrO₆]^9- clusters embedded in an extended point charge field). Ligand-field parameters were extracted from mapping the effective ligand-field Hamiltonian onto the full many-particle Hamiltonian from one side and from a direct fit to energies of computed d-d transitions on the other side. These have been analyzed using ab initio ligand-field theory. The quality of the theoretical predictions is critically assessed through a detailed comparison with the available experimental data.

Carbon Monoxide Binding to the Iron-Molybdenum Cofactor of Nitrogenase: a Detailed Quantum Mechanics/Molecular Mechanics Investigation

Carbon monoxide (CO) is a well-known inhibitor of nitrogenase activity. Under turnover conditions, CO binds to FeMoco, the active site of Mo nitrogenase. Time-resolved IR measurements suggest an initial terminal CO at 1904 cm^-1 that converts to a bridging CO at 1715 cm^-1, and an X-ray structure shows that CO can displace one of the bridging belt sulfides of FeMoco. However, the CO-binding redox state(s) of FeMoco (E_n) and the role of the protein environment in stabilizing specific CO-bound intermediates remain elusive. In this work, we carry out an in-depth analysis of the CO-FeMoco interaction based on quantum chemical calculations addressing different aspects of the electronic structure. (1) The local electronic structure of the Fe-CO bond is studied through diamagnetically substituted FeMoco. (2) A cluster model of FeMoco within a polarizable continuum illustrates how CO binding may affect the spin-coupling between the metal centers. (3) A QM/MM model incorporates the explicit influence of the amino acid residues surrounding FeMoco in the MoFe protein. The QM/MM model predicts both a terminal and a bridging CO in the E₁ redox state. The scaled calculated CO frequencies (1922 and 1716 cm^-1, respectively) are in good agreement with the experimentally observed IR bands supporting CO binding to the E₁ state. Alternatively, an E₂ state QM/MM model, which has the same atomic structure as the CO-bound X-ray structure, features a semi-bridging CO with a scaled calculated frequency (1718 cm^-1) similar to the bridging CO in the E₁ model.

Effect of Spin-Orbit Coupling on Phonon-Mediated Magnetic Relaxation in a Series of Zero-Valent Vanadium, Niobium, and Tantalum Isocyanide Complexes

Spin-vibronic coupling leads to spin relaxation in paramagnetic molecules, and an understanding of factors that contribute to this phenomenon is essential for designing next-generation spintronics technology, including single-molecule magnets and spin-based qubits, wherein long-lifetime magnetic ground states are desired. We report spectroscopic and magnetic characterization of the isoelectronic and isostructural series of homoleptic zerovalent transition metal triad M(CNDipp)₆ (M = V, Nb, Ta; CNDipp = 2,6-diisopropylphenyl isocyanide) and show experimentally the significant increase in spin relaxation rate upon going from V to Nb to Ta. Correlated electronic calculations and first principle spin-phonon computations support the role of spin-orbit coupling in modulating spin-phonon relaxation. Our results provide experimental evidence that increasing magnetic anisotropy through spin-orbit coupling interactions leads to increased spin-vibronic relaxation, which is detrimental to long spin lifetime in paramagnetic molecules.

Cobalt-Catalyzed Hydrosilylation of Carbon Dioxide to the Formic Acid, Formaldehyde, and Methanol Level-How to Control the Catalytic Network?

The selective hydrosilylation of carbon dioxide (CO₂) to either the formic acid, formaldehyde, or methanol level using a molecular cobalt(II) triazine complex can be controlled based on reaction parameters such as temperature, CO₂ pressure, and concentration. Here, we rationalize the catalytic mechanism that enables the selective arrival at each product platform. Key reactive intermediates were prepared and spectroscopically characterized, while the catalytic mechanism and the energy profile were analyzed with density functional theory (DFT) methods and microkinetic modeling. It transpired that the stepwise reduction of CO₂ involves three consecutive catalytic cycles, including the same cobalt(I) triazine hydride complex as the active species. The increasing kinetic barriers associated with each reduction step and the competing hydride transfer steps in the three cycles corroborate the strong influence of the catalyst environment on the product selectivity. The fundamental mechanistic insights provide a consistent description of the catalytic system and rationalize, in particular, the experimentally verified opportunity to steer the reaction toward the formaldehyde product as the chemically most challenging reduction level.

Addressing the System-Size Dependence of the Local Approximation Error in Coupled-Cluster Calculations

Over the last two decades, the local approximation has been successfully used to extend the range of applicability of the "gold standard" singles and doubles coupled-cluster method with perturbative triples CCSD(T) to systems with hundreds of atoms. The local approximation error grows in absolute value with the increasing system size, i.e., by increasing the number of electron pairs in the system. In this study, we demonstrate that the recently introduced two-point extrapolation scheme for approaching the complete pair natural orbital (PNOs) space limit in domain-based pair natural orbital CCSD(T) calculations drastically reduces the dependence of the error on the system size, thus opening up unprecedented opportunities for the calculation of benchmark quality relative energies for large systems.