[3 + 2] Cycloadditions and Retrocycloadditions of Niobium Imido Complexes: An Experimental and Computational Mechanistic Study.

We demonstrate reactivity between a ß-diketiminate-supported niobium(III) imido complex and alkyl azides to form niobatetrazene complexes (BDI)Nb(NtBu)(RNNNNR) (BDI = N,N-bis(2,6-diisopropylphenyl)-3,5-dimethyl-ß-diketiminate; R = cyclohexyl (1), benzyl (2)). Intriguingly, niobatetrazene complexes 1 and 2 can be interconverted via addition of an appropriate alkyl azide, likely through a series of concerted [3 + 2] cycloaddition and retrocycloaddition reactions in which π-loaded bis(imido) intermediates are formed. The bis(imido) intermediates were trapped upon addition of alkyl isocyanides to yield five-coordinate bis(imido) complexes (BDI)Nb(NtBu)(NCy)(CNR) (R = tert-butyl (4a), cyclohexyl (4b)). Two computational methods─density functional theory and density functional tight binding (DFTB)─were employed to calculate the lowest energy pathway across the potential energy surface for this multistep transformation. Reaction path calculations for individual cycloaddition or retrocycloaddition processes along the multistep reaction pathway showed that these transformations occur via a concerted, yet highly asynchronous mechanism, in which the two bond-breaking or -making events do not occur simultaneously. The use of the DFTB method in this work highlights its advantages and utility for studying transition metal systems.

Characterization of a Molecule Partially Confined at the Pore Mouth of a Zeotype.

We investigate the interaction between a molecule and a pore mouth-a critical step in adsorption processes-by characterizing the conformation of a macrocyclic calix[4]arene-TiIV complex, which is grafted on the external surface of a zeotype (*-SVY). X-ray absorption and 13 C{1 H} CPMAS NMR spectroscopies independently detect a unique conformation of this complex when it is grafted at crystallographically equivalent locations that lie at the interface of 7 Šhemispherical microporous cavities and the external surface. Electronic structure calculations support the presence of this unique conformation, and suggest that it is brought about by a specific orientation of the macrocycle that maximizes non-covalent interactions between calix[4]arene upper-rim tert-butyl substituents and the microporous-cavity walls. Our comparative study provides a rare "snapshot" of a molecule partially confined at a pore mouth, an essential intermediate for adsorption into micropores, and demonstrates how surrounding environment controls this confinement in a sensitive fashion.

Electronic Structures of Rhenium(II) ß-Diketiminates Probed by EPR Spectroscopy: Direct Comparison of an Acceptor-Free Complex to Its Dinitrogen, Isocyanide, and Carbon Monoxide Adducts.

Electron paramagnetic resonance (EPR) studies of the rhenium(II) complex Re(η5-Cp)(BDI) (1; BDI = N,N'-bis(2,6-diisopropylphenyl)-3,5-dimethyl-ß-diketiminate) have revealed that this species reversibly binds N2 in solution: flash frozen toluene solutions of 1 disclose entirely different EPR spectra at 10 K when prepared under N2 versus Ar atmospheres. This observation was additionally verified by the synthesis of stable CO and 2,6-xylylisocyanide (XylNC) adducts of 1, which display EPR features akin to those observed in the putative N2 complex. While we found that 1 displays an extremely large gmax value of 3.99, the binding of an additional ligand leads to substantial decreases in this value, displaying gmax values of ca. 2.4. Following the generation of isotopically enriched 15N2 and 13CO adducts of 1, HYSCORE experiments allowed for the measurement of the corresponding hyperfine couplings associated with spin delocalization onto the electron-accepting ligands in these species, which proved to be small. A cumulative assessment of the EPR data, when combined with insights provided by near-infrared (NIR) spectroscopy and time-dependent density functional theory (TDDFT) calculations, indicated that while the binding of electron acceptors to 1 does lead to decreases in gmax in relative accord with the field strength (i.e., π-acidity) of the variable ligand, the magnitude of these decreases is primarily due to the changes in electronic structure at the Re center.

Dynamic Reorganization and Confinement of TiIV Active Sites Controls Olefin Epoxidation Catalysis on Two-Dimensional Zeotypes.

The effect of dynamic reorganization and confinement of isolated TiIV catalytic centers supported on silicates is investigated for olefin epoxidation. Active sites consist of grafted single-site calix[4]arene-TiIV centers or their calcined counterparts. Their location is synthetically controlled to be either unconfined at terminal T-atom positions (denoted as type-(i)) or within confining 12-MR pockets (denoted as type-(ii); diameter ∼7 Å, volume ∼185 Å3) composed of hemispherical cavities on the external surface of zeotypes with *-SVY topology. Electronic structure calculations (density functional theory) indicate that active sites consist of cooperative assemblies of TiIV centers and silanols. When active sites are located at unconfined type-(i) environments, the rate constants for cyclohexene epoxidation (323 K, 0.05 mM TiIV, 160 mM cyclohexene, 24 mM tert-butyl hydroperoxide) are 9 ± 2 M-2 s-1; whereas within confining type-(ii) 12-MR pockets, there is a ∼5-fold enhancement to 48 ± 8 M-2 s-1. When a mixture of both environments is initially present in the catalyst resting state, the rate constants reflect confining environments exclusively (40 ± 11 M-2 s-1), indicating that dynamic reorganization processes lead to the preferential location of active sites within 12-MR pockets. While activation enthalpies are Δ H‡app = 43 ± 1 kJ mol-1 irrespective of active site location, confining environments exhibit diminished entropic barriers (Δ S‡app = -68 J mol-1 K-1 for unconfined type-(i) vs -56 J mol-1 K-1 for confining type-(ii)), indicating that confinement leads to more facile association of reactants at active sites to form transition state structures (volume ∼ 225 Å3). These results open new opportunities for controlling reactivity on surfaces through partial confinement on shallow external-surface pockets, which are accessible to molecules that are too bulky to benefit from traditional confinement within micropores.

Kohn-Sham Density Functional Theory with Complex, Spin-Restricted Orbitals: Accessing a New Class of Densities without the Symmetry Dilemma.

We show that using complex, spin-restricted orbitals in Kohn-Sham (KS) density functional theory allows one to access a new class of densities that is not accessible by either spin-restricted (RKS) or spin-unrestricted (UKS) orbitals. We further show that the real part of a complex RKS (CRKS) density matrix can be nonidempotent when the imaginary part of the density matrix is not zero. Using CRKS orbitals shows significant improvements in the triplet-singlet gaps of a benchmark set, called TS12, for well-established, widely used density functionals. Moreover, it was shown that RKS and UKS yield qualitatively wrong charge densities and spin densities, respectively, leading to worse energetics. We demonstrate that representative modern density functionals show surprisingly no improvement even with a qualitatively more accurate density from CRKS orbitals. To this end, our work not only provides a way to escape the symmetry dilemma whenever there exists a CRKS solution, but also suggests a new route to design better approximate density functionals.

Excited states via coupled cluster theory without equation-of-motion methods: Seeking higher roots with application to doubly excited states and double core hole states.

In this work, we revisited the idea of using the coupled-cluster (CC) ground state formalism to target excited states. Our main focus was targeting doubly excited states and double core hole states. Typical equation-of-motion (EOM) approaches for obtaining these states struggle without higher-order excitations than doubles. We showed that by using a non-Aufbau determinant optimized via the maximum overlap method, the CC ground state solver can target higher energy states. Furthermore, just with singles and doubles (i.e., CCSD), we demonstrated that the accuracy of ΔCCSD and ΔCCSD(T) (triples) far surpasses that of EOM-CCSD for doubly excited states. The accuracy of ΔCCSD(T) is nearly exact for doubly excited states considered in this work. For double core hole states, we used an improved ansatz for greater numerical stability by freezing core hole orbitals. The improved methods, core valence separation (CVS)-ΔCCSD and CVS-ΔCCSD(T), were applied to the calculation of the double ionization potential of small molecules. Even without relativistic corrections, we observed qualitatively accurate results with CVS-ΔCCSD and CVS-ΔCCSD(T). Remaining challenges in ΔCC include the description of open-shell singlet excited states with the single-reference CC ground state formalism as well as excited states with genuine multireference character. The tools and intuition developed in this work may serve as a stepping stone toward directly targeting arbitrary excited states using ground state CC methods.

Independent amplitude approximations in coupled cluster valence bond theory: Incorporation of 3-electron-pair correlation and application to spin frustration in the low-lying excited states of a ferredoxin-type tetrametallic iron-sulfur cluster.

Coupled cluster valence bond (CCVB) is a simple electronic structure method based on a perfect pairing (PP) reference with 2-pair recouplings for strong electron correlation problems. CCVB is spin-pure, size-consistent, and can exactly (in its active space) separate any molecule into atoms for which unrestricted Hartree-Fock (UHF) at dissociation is the sum of the ground state UHF energies of the atoms. However CCVB is far from a complete description of strong correlations. Its first failure to exactly describe spin-recouplings arises at the level of 3 electron pairs, such as the recoupling of 3 triplet oxygen atoms in the dissociation of singlet ozone. Such situations are often associated with spin frustration. To address this limitation, an extension of CCVB, termed CCVB+i3, is reported here that includes an independent (i) amplitude approximation to the 3-pair recouplings. CCVB+i3 thereby has the same basic computational requirements as those of CCVB, which has previously been shown to be an efficient method. CCVB+i3 correctly separates molecules that CCVB cannot. As a by-product, an independent 2-pair amplitude approximation to CCVB, called PP+i2, is also defined. Remarkably, PP+i2 can also correctly separate systems that CCVB cannot. CCVB+i3 is validated on the symmetric dissociation of D3h ozone. CCVB+i3 is then used to explore the role of 3-pair recouplings in an [Fe4S4(SCH3)4]2- cluster that has been used to model the iron-sulfur core of [Fe4S4] ferredoxins. Using localized PP orbitals, such recouplings are demonstrated to be large in some low-lying singlet excited states of the cluster. Significant 3 pair recoupling amplitudes include the usual triangular motif associated with spin frustration and other geometric arrangements of the 3 entangled pairs across the 4 iron centers.

Open-shell coupled-cluster valence-bond theory augmented with an independent amplitude approximation for three-pair correlations: Application to a model oxygen-evolving complex and single molecular magnet.

We report the failure of coupled-cluster valence-bond (CCVB) theory with two-pair configurations [D. W. Small and M. Head-Gordon, J. Chem. Phys. 130, 084103 (2009)] for open-shell (OS) spin-frustrated systems where including three-pair configurations is necessary to properly describe strong spin-correlations. We extend OS-CCVB by augmenting the model with three-pair configurations within the independent amplitude approximation. The resulting new electronic structure model, OS-CCVB+i3, involves only a quadratic number of independent wavefunction parameters. It includes the recently reported closed-shell CCVB+i3 as a special case. Its cost is dominated by integral transformations, and it is capable of breaking multiple bonds exactly for all systems examined so far. The strength of OS-CCVB+i3 is highlighted in realistic systems including the [CaMn3O4] cubane subunit of the oxygen-evolving complex and a molecular magnet with the [Cr9] core unit as well as model systems such as N3, V3O3, and P5. We show that OS-CCVB+i3 is only slightly dependent on the underlying perfect-pairing reference, while OS-CCVB shows a stronger dependence. We also emphasize the compactness of the OS-CCVB+i3 wavefunction compared to the heat-bath configuration interaction wavefunction, a recently introduced soft exponential-scaling approach.

Coupled cluster valence bond theory for open-shell systems with application to very long range strong correlation in a polycarbene dimer.

The Coupled Cluster Valence Bond (CCVB) method, previously presented for closed-shell (CS) systems, is extended to open-shell (OS) systems. The theoretical development is based on embedding the basic OS CCVB wavefunction in a fictitious singlet super-system. This approach reveals that the OS CCVB amplitude equations are quite similar to those of CS CCVB, and thus that OS CCVB requires the same level of computational effort as CS CCVB, which is an inexpensive method. We present qualitatively correct CCVB potential energy curves for all low-lying spin states of P2 and Mn2+. CCVB is successfully applied to the low-lying spin states of some model linear polycarbenes, systems that appear to be a hindrance to standard density functionals. We examine an octa-carbene dimer in a side-by-side orientation, which, in the monomer dissociation limit, exhibits maximal strong correlation over the length of the polycarbene.

A simple way to test for collinearity in spin symmetry broken wave functions: general theory and application to generalized Hartree Fock.

We introduce a necessary and sufficient condition for an arbitrary wavefunction to be collinear, i.e., its spin is quantized along some axis. It may be used to obtain a cheap and simple computational procedure to test for collinearity in electronic structure theory calculations. We adapt the procedure for Generalized Hartree Fock (GHF), and use it to study two dissociation pathways in CO2. For these dissociation processes, the GHF wave functions transform from low-spin Unrestricted Hartree Fock (UHF) type states to noncollinear GHF states and on to high-spin UHF type states, phenomena that are succinctly illustrated by the constituents of the collinearity test. This complements earlier GHF work on this molecule.

Restricted Hartree Fock using complex-valued orbitals: a long-known but neglected tool in electronic structure theory.

Restricted Hartree Fock using complex-valued orbitals (cRHF) is studied. We introduce an orbital pairing theorem, with which we obtain a concise connection between cRHF and real-valued RHF, and use it to uncover the close relationship between cRHF, unrestricted Hartree Fock, and generalized valence bond perfect pairing. This enables an intuition for cRHF, contrasting with the generally unintuitive nature of complex orbitals. We also describe an efficient computer implementation of cRHF and its corresponding stability analysis. By applying cRHF to the Be + H2 insertion reaction, a Woodward-Hoffmann violating reaction, and a symmetry-driven conical intersection, we demonstrate in genuine molecular systems that cRHF is capable of removing certain potential energy surface singularities that plague real-valued RHF and related methods. This complements earlier work that showed this capability in a model system. We also describe how cRHF is the preferred RHF method for certain radicaloid systems like singlet oxygen and antiaromatic molecules. For singlet O2, we show that standard methods fail even at the equilibrium geometry. An implication of this work is that, regardless of their individual efficacies, cRHF solutions to the HF equations are fairly commonplace.

A fusion of the closed-shell coupled cluster singles and doubles method and valence-bond theory for bond breaking.

Closed-shell coupled cluster singles and doubles (CCSD) is among the most important of electronic-structure methods. However, it fails qualitatively when applied to molecular systems with more than two strongly correlated electrons, such as those with stretched or broken covalent bonds. We show that it is possible to modify the doubles amplitudes to obtain a closed-shell CCSD method that retains the computational cost and desirable features of standard closed-shell CCSD, e.g., correct spin symmetry, size extensivity, orbital invariance, etc., but produces greatly improved energies upon bond dissociation of multiple electron pairs; indeed, under certain conditions the dissociation energies are exact.

Post-modern valence bond theory for strongly correlated electron spins.

We give a pedagogical overview of our recently introduced electronic-structure method, Coupled Cluster Valence Bond (CCVB). We show that CCVB can be viewed as an approximation to the accurate, yet very expensive, Spin Coupled Valence Bond model (SCVB). Both of these models are intended for use on strongly correlated molecular systems, especially when the strong correlations are due to electron spin coupling. Using familiar ideas from electronic-structure theory, we provide definitions for these strong-correlation concepts. We show that CCVB and SCVB generally produce similar results, with more substantial discrepancies occurring for systems displaying electronic resonance. We conclude that CCVB is a useful, inexpensive alternative to SCVB.

Orbitals that are unrestricted in active pairs for generalized valence bond coupled cluster methods.

Spin unrestriction is typically defined as a free variation of the molecular orbitals of different spins in order to lower the molecular energy. When applied to approximate active space electron correlation methods, such as generalized valence bond coupled cluster (GVB-CC) models, this approach often leads to undesirable artifacts. We therefore present an alternative unrestricted in active pairs (UAP) procedure for spin polarization in GVB-CC methods, which resembles the corresponding orbitals of unrestricted Hartree-Fock theory. The UAP procedure permits spin polarization only within the two orbital subspaces describing each electron pair (consisting of one nominally occupied and one virtual orbital). This great reduction to just a linear number of degrees of freedom associated with spin polarization eliminates many of the undesirable artifacts associated with unconstrained spin unrestriction. The UAP procedure is tested on a variety of potential curves for bond breaking and the properties of small radicals as well as larger polyenyl radicals, allyl, pentadienyl, and heptatrienyl.

Remarkable Accuracy of an O(N6) Perturbative Correction to Opposite-Spin CCSD: Are Triples Necessary for Chemical Accuracy in Coupled Cluster?

The focus of this work is OS-CCSD-SPT(2), which is a second-order similarity transformed perturbation theory correction to opposite spin coupled cluster singles doubles, where in the latter the same-spin amplitudes are removed and the opposite-spin ones are solved self-consistently. OS-CCSD-SPT(2) is free of empirical parameters, has an instrinsic scaling of O(N6), and makes no use of triples. We demonstrate that, for non-multireference molecules, OS-CCSD-SPT(2) produces relative energies whose accuracy is significantly higher than what is generally expected of a triples-free model. For example, using PBE0 orbitals in the reference, OS-CCSD-SPT(2) exhibits a mean absolute deviation (MAD) of 1.13 kcal/mol with respect to CCSD(2F) benchmark values for the non-multireference subset of W4-08 atomization energies (cf. a MAD > 6.5 kcal/mol for CCSD) and a MAD of 0.68 kcal/mol for the energies of reactions generated from the W4-08 molecules. These MADs are reduced to 0.61 and 0.63 kcal/mol, respectively, by a simple one-parameter spin-component scaling of the OS-CCSD-SPT(2) same-spin correlation energy. OS-CCSD is also naturally amenable to higher order corrections: the associated third-order correction, OS-CCSD-SPT(3), which does involve connected triples and quadruples, exhibits a MAD of 0.44 kcal/mol for the same atomization-energy benchmark.

Incorporating Electronic Information into Machine Learning Potential Energy Surfaces via Approaching the Ground-State Electronic Energy as a Function of Atom-Based Electronic Populations.

Machine learning (ML) approximations to density functional theory (DFT) potential energy surfaces (PESs) are showing great promise for reducing the computational cost of accurate molecular simulations, but at present, they are not applicable to varying electronic states, and in particular, they are not well suited for molecular systems in which the local electronic structure is sensitive to the medium to long-range electronic environment. With this issue as the focal point, we present a new machine learning approach called "BpopNN" for obtaining efficient approximations to DFT PESs. Conceptually, the methodology is based on approaching the true DFT energy as a function of electron populations on atoms; in practice, this is realized with available density functionals and constrained DFT (CDFT). The new approach creates approximations to this function with neural networks. These approximations thereby incorporate electronic information naturally into a ML approach, and optimizing the model energy with respect to populations allows the electronic terms to self-consistently adapt to the environment, as in DFT. We confirm the effectiveness of this approach with a variety of calculations on LinHn clusters.

Nickel-catalysed anti-Markovnikov hydroarylation of unactivated alkenes with unactivated arenes facilitated by non-covalent interactions.

Anti-Markovnikov additions to alkenes have been a longstanding goal of catalysis, and anti-Markovnikov addition of arenes to alkenes would produce alkylarenes that are distinct from those formed by acid-catalysed processes. Existing hydroarylations are either directed or occur with low reactivity and low regioselectivity for the n-alkylarene. Herein, we report the first undirected hydroarylation of unactivated alkenes with unactivated arenes that occurs with high regioselectivity for the anti-Markovnikov product. The reaction occurs with a nickel catalyst ligated by a highly sterically hindered N-heterocyclic carbene. Catalytically relevant arene- and alkene-bound nickel complexes have been characterized, and the rate-limiting step was shown to be reductive elimination to form the C-C bond. Density functional theory calculations, combined with second-generation absolutely localized molecular orbital energy decomposition analysis, suggest that the difference in activity between catalysts containing large and small carbenes results more from stabilizing intramolecular non-covalent interactions in the secondary coordination sphere than from steric hindrance.

Isomerism and dynamic behavior of bridging phosphaalkynes bound to a dicopper complex.

A dicopper complex featuring a symmetrically bridging nitrile ligand and supported by a binucleating naphthyridine-based ligand, [Cu2(µ-η 1 :η 1 -MeCN)DPFN](NTf2)2, was treated with phosphaalkynes (RC[triple bond, length as m-dash]P, isoelectronic analogues of nitriles) to yield dicopper complexes that exhibit phosphaalkynes in rare µ-η 2:η 2 binding coordination modes. X-ray crystallography revealed that these unusual "tilted" structures exist in two isomeric forms (R "up" vs. R "sideways"), depending on the steric profile of the phosphaalkyne's alkyl group (R = Me, Ad, or t Bu). Only one isomer is observed in both solution and the solid state for R = Me (sideways) and t Bu (up). With intermediate steric bulk (R = Ad), the energy difference between the two geometries is small enough that both are observed in solution, and NMR spectroscopy and computations indicate that the solid-state structure corresponds to the minor isomer observed in solution. Meanwhile, treatment of [Cu2(µ-η 1:η 1-MeCN)DPFN](NTf2)2 with 2-butyne affords [Cu2(µ-η 2:η 2-(MeC[triple bond, length as m-dash]CMe))DPFN](NTf2)2: its similar ligand geometry demonstrates that the tilted µ-η 2:η 2 binding mode is not limited to phosphaalkynes but reflects a more general trend, which can be rationalized via an NBO analysis showing maximization of π-backbonding.

Tractable spin-pure methods for bond breaking: Local many-electron spin-vector sets and an approximate valence bond model.

For a given number of electrons, total spin, and matching spin z-component, we construct a set that spans the many-electron spin subspace associated with these spin values. Each vector in the set is tensorially related to spin-pure vectors of six electrons or less. We show that in the limit of separated atoms coupled to any allowed overall spin, the corresponding spin vector has a simple form relative to the introduced sets. From this, we set up a model that is computationally simple, spin pure, size consistent, and able to properly treat molecules as they dissociate into atoms or fragments.

Coupled-Cluster Valence-Bond Singles and Doubles for Strongly Correlated Systems: Block-Tensor Based Implementation and Application to Oligoacenes.

We demonstrate a block-tensor based implementation of coupled-cluster valence-bond singles and doubles (CCVB-SD) [Small, D. W.; Head-Gordon M. J. Chem. Phys. 2012, 137, 114103] which is a simple modification to restricted CCSD (RCCSD) that provides a qualitatively correct description of valence correlations even in strongly correlated systems. We derive the Λ-equation of CCVB-SD and the corresponding unrelaxed density matrices. The resulting production-level implementation is applied to oligoacenes, correlating up to 318 electrons in 318 orbitals. CCVB-SD shows a qualitative agreement with exact methods for short acenes and reaches the bulk limit of oligoacenes in terms of natural orbital occupation numbers, whereas RCCSD shows nonvariational behavior even for relatively short acenes. A significant reduction in polyradicaloid character is found when correlating all valence electrons instead of only the π-electrons.