Generalization of Intrinsic Orbitals to Kramers-Paired Quaternion Spinors, Molecular Fragments, and Valence Virtual Spinors.

Localization of molecular orbitals finds its importance in the representation of chemical bonding (and antibonding) and in the local correlation treatments beyond mean-field approximation. In this paper, we generalize the intrinsic atomic and bonding orbitals [G. Knizia, J. Chem. Theory Comput. 2013, 9, 11, 4834-4843] to relativistic applications using complex and quaternion spinors, as well as to molecular fragments instead of atomic fragments only. By performing a singular value decomposition, we show how localized valence virtual orbitals can be expressed on this intrinsic minimal basis. We demonstrate our method on systems of increasing complexity, starting from simple cases such as benzene, acrylic acid, and ferrocene molecules, and then demonstrate the use of molecular fragments and inclusion of relativistic effects for complexes containing heavy elements such as tellurium, iridium, and astatine. The aforementioned scheme is implemented into a standalone program interfaced with several different quantum chemistry packages.

Regional Embedding Enables High-Level Quantum Chemistry for Surface Science.

Compared to common density functionals, ab initio wave function methods can provide greater reliability and accuracy, which could prove useful when modeling adsorbates or defects of otherwise periodic systems. However, the breaking of translational symmetry necessitates large supercells that are often prohibitive for correlated wave function methods. As an alternative, this paper introduces the regional embedding approach, which enables correlated wave function treatments of only a target fragment of interest through small, fragment-localized orbital spaces constructed using a simple overlap criterion. Applications to the adsorption of water on lithium hydride, hexagonal boron nitride, and graphene substrates show that regional embedding combined with focal-point corrections can provide converged CCSD(T) (coupled-cluster) adsorption energies with very small fragment sizes.

Molecular integrals from Fast Fourier Transforms (FFT) instead of recurrences: The McMurchie-Davidson case.

We report a closed formula expressing the McMurchie-Davidson (MD) key intermediates {[r](0); rx + ry + rz ≤ L} directly in terms of the set of basic integrals {[0](m); m ≤ L}, without any recurrences. This formula can be evaluated at O(L) cost per output [r](0) with dense matrix multiplications and Fast Fourier Transforms (FFT). Key to this is the fact that the transformation that builds Cartesian angular momentum from the basic integrals, {[0κ](m+m')}↦{[lκ](m)} (κ ∈ {x, y, z}), can be phrased as a circulant-matrix/vector product, which is susceptible to FFTs. After simplification, a simple formula yields the final [r](0) in one step, as contraction of four auxiliary vectors over a common Fourier index k-one vector for the [0](m) and one for each Cartesian axis. Similar transformations occur in many integral approaches beside MD, making this idea potentially broadly applicable. The simple resulting code and data structures may make it attractive for novel hardware platforms.

Fast Evaluation of Two-Center Integrals over Gaussian Charge Distributions and Gaussian Orbitals with General Interaction Kernels.

We present efficient algorithms for computing two-center integrals and integral derivatives, with general interaction kernels K(r12), over Gaussian charge distributions of general angular momenta l. While formulated in terms of traditional ab initio integration techniques, full derivations and required secondary information, as well as a reference implementation, are provided to make the content accessible to other fields. Concretely, the presented algorithms are based on an adaption of the McMurchie-Davidson Recurrence Relation (MDRR) combined with analytical properties of the solid harmonic transformation; this obviates all intermediate recurrences except the adapted MDRR itself, and allows it to be applied to fully contracted auxiliary kernel integrals. The technique is particularly well-suited for semiempirical molecular orbital methods, where it can serve as a more general and efficient replacement of Slater-Koster tables, and for first-principles quantum chemistry methods employing density fitting. But the formalism's high efficiency and ability of handling general interaction kernels K(r12) and multipolar Gaussian charge distributions may also be of interest for modeling electrostatic interactions and short-range exchange and charge penetration effects in classical force fields and model potentials. With the presented technique, a 4894 × 4894 univ-JKFIT Coulomb matrix JAB = (A|1/r12|B) (183 MiB) can be computed in 50 ms on a Q2'2018 notebook CPU, without any screening or approximations.

Epoxidation of Alkenes by Peracids: From Textbook Mechanisms to a Quantum Mechanically Derived Curly-Arrow Depiction.

Using the intrinsic bond orbital (IBO) analysis based on accurate quantum mechanical calculations of the reaction path for the epoxidation of propene using peroxyacetic acid, we find that the four commonly used curly arrows for representing this reaction mechanism are insufficient and that seven curly arrows are required as a result of changes to σ and π bonding interactions, which are usually neglected in all textbook curly arrow representations. The IBO method provides a convenient quantitative method for deriving curly arrows in a rational manner rather than the normal ad hoc representations used ubiquitously in teaching organic chemistry.

σ-Noninnocence: Masked Phenyl-Cation Transfer at Formal NiIV.

Reductive elimination is an elementary organometallic reaction step involving a formal oxidation state change of -2 at a transition-metal center. For a series of formal high-valent NiIV complexes, aryl-CF3 bond-forming reductive elimination was reported to occur readily (Bour et al. J. Am. Chem. Soc. 2015, 137, 8034-8037). We report a computational analysis of this reaction and find that, unexpectedly, the formal NiIV centers are better described as approaching a +II oxidation state, originating from highly covalent metal-ligand bonds, a phenomenon attributable to σ-noninnocence. A direct consequence is that the elimination of aryl-CF3 products occurs in an essentially redox-neutral fashion, as opposed to a reductive elimination. This is supported by an electron flow analysis which shows that an anionic CF3 group is transferred to an electrophilic aryl group. The uncovered role of σ-noninnocence in metal-ligand bonding, and of an essentially redox-neutral elimination as an elementary organometallic reaction step, may constitute concepts of broad relevance to organometallic chemistry.

Cationic Gold(I) Diarylallenylidene Complexes: Bonding Features and Ligand Effects.

Using computational approaches, we qualitatively and quantitatively assess the bonding components of a series of experimentally characterized Au(I) diarylallenylidene complexes (N.Kim, R.A.Widenhoefer, Angew. Chem. Int. Ed. 2018, 57, 4722-4726). Our results clearly demonstrate that Au(I) engages only weakly in π-backbonding, which is, however, a tunable bonding component. Computationally identified trends in bonding are clearly correlated with the substitution patterns of the aryl substituents in the Au(I) diarylallenylidene complexes and good agreement is found with the previously reported experimental data, such as IR spectra, 13 C NMR chemical shifts and rates of decomposition together with their corresponding barrier heights, further substantiating the computational findings. The description of the bonding patterns in these complexes allow predictions of their spectroscopic features, their reactivity and stability.

Visualizing Complex-Valued Molecular Orbitals.

We report an implementation of a program for visualizing complex-valued molecular orbitals. The orbital phase information is encoded on each of the vertices of triangle meshes using the standard color wheel. Using this program, we visualized the molecular orbitals for systems with spin-orbit couplings, external magnetic fields, and complex absorbing potentials. Our work has not only created visually attractive pictures but also clearly demonstrated that the phases of the complex-valued molecular orbitals carry rich chemical and physical information on the system, which has often been unnoticed or overlooked.

cPCET versus HAT: A Direct Theoretical Method for Distinguishing X-H Bond-Activation Mechanisms.

Proton-coupled electron transfer (PCET) events play a key role in countless chemical transformations, but they come in many physical variants which are hard to distinguish experimentally. While present theoretical approaches to treat these events are mostly based on physical rate coefficient models of various complexity, it is now argued that it is both feasible and fruitful to directly analyze the electronic N-electron wavefunctions of these processes along their intrinsic reaction coordinate (IRC). In particular, for model systems of lipoxygenase and the high-valent oxoiron(IV) intermediate TauD-J it is shown that by invoking the intrinsic bond orbital (IBO) representation of the wavefunction, the common boundary cases of hydrogen atom transfer (HAT) and concerted PCET (cPCET) can be directly and unambiguously distinguished in a straightforward manner.

The Pentagonal-Pyramidal Hexamethylbenzene Dication: Many Shades of Coordination Chemistry at Carbon.

A recent report on the crystal structure of the pentagonal-pyramidal hexamethylbenzene dication C6 (CH3 )62+ by Malischewski and Seppelt [Angew. Chem. Int. Ed. 2017, 56, 368] confirmed the structural proposal made in the first report of this compound in 1973 by Hogeveen and Kwant [Tetrahedron Lett. 1973, 14, 1665]. The widespread attention that this compound quickly gained led us to reinvestigate its electronic structure. On the basis of intrinsic bond orbital analysis, effective oxidation state analysis, ring current analysis, and comparison with well-established coordination complexes, it is demonstrated that the central carbon atom behaves like a transition metal. The central (apical) carbon atom, although best described as a highly Lewis-acidic carbon atom coordinated with an anionic cyclopentadienyl ligand, is also capable of acting as an electron-pair donor to a formal CH3+ group. The different roles of coordination chemistry are discussed.

Efficient Treatment of Local Meta-generalized Gradient Density Functionals via Auxiliary Density Expansion: The Density Fitting J + X Approximation.

We report an efficient technique to treat density functionals of the meta-generalized gradient approximation (mGGA) class in conjunction with density fitting of Coulomb terms (DF-J) and exchange-correlation terms (DF-X). While the kinetic energy density τ cannot be computed in the context of a DF-JX calculation, we show that the Laplacian of the density υ can be computed with almost no extra cost. With this technique, υ-form mGGAs become only slightly more expensive (10%-20%) than GGAs in DF-JX treatment-and several times faster than regular τ-based mGGA calculations with DF-J and regular treatment of the density functional. We investigate the translation of υ-form mGGAs into τ-form mGGAs by employing a kinetic energy functional but find this insufficiently reliable at this moment. However, υ and τ are believed to carry essentially equivalent information beyond ρ and ||∇ρ|| ( Phys. Rev. B 2007 , 75 , 155109 , DOI: 10.1103/PhysRevB.75.155109 ), so a reparametrization of accurate mGGAs from the τ-form into the υ-form should be possible. Once such functionals become available, we expect the presented technique to become a powerful tool in the computation of reaction paths, intermediates, and transition states of medium sized molecules.

Analytical energy gradients for explicitly correlated wave functions. I. Explicitly correlated second-order Møller-Plesset perturbation theory.

We present the theory and algorithms for computing analytical energy gradients for explicitly correlated second-order Møller-Plesset perturbation theory (MP2-F12). The main difficulty in F12 gradient theory arises from the large number of two-electron integrals for which effective two-body density matrices and integral derivatives need to be calculated. For efficiency, the density fitting approximation is used for evaluating all two-electron integrals and their derivatives. The accuracies of various previously proposed MP2-F12 approximations [3C, 3C(HY1), 3*C(HY1), and 3*A] are demonstrated by computing equilibrium geometries for a set of molecules containing first- and second-row elements, using double-ζ to quintuple-ζ basis sets. Generally, the convergence of the bond lengths and angles with respect to the basis set size is strongly improved by the F12 treatment, and augmented triple-ζ basis sets are sufficient to closely approach the basis set limit. The results obtained with the different approximations differ only very slightly. This paper is the first step towards analytical gradients for coupled-cluster singles and doubles with perturbative treatment of triple excitations, which will be presented in the second part of this series.

C(sp3 )-H Bond Activation by Vinylidene Gold(I) Complexes: A Concerted Asynchronous or Stepwise Process?

A detailed analysis of the C(sp3 )-H activation process by vinylidene AuI complexes is described based on an intrinsic bond orbital analysis. Based on our analysis this event can be divided into three phases: (i) hydride transfer, (ii) C-C bond formation, and (iii) σ to π rearrangement of the lone pair coordinated to Au. Small perturbations of the system lead to either a concerted asynchronous reaction, or a stepwise reaction featuring an intermediate with a C-H-C three-centre two-electron (3c-2e) bond. The role of π-donating substituents is highlighted and provides a way of controlling reactions of this type in future experimental studies.

Characterization of the Fleeting Hydroxoiron(III) Complex of the Pentadentate TMC-py Ligand.

Nonheme mononuclear hydroxoiron(III) species are important intermediates in biological oxidations, but well-characterized examples of synthetic complexes are scarce due to their instability or tendency to form µ-oxodiiron(III) complexes, which are the thermodynamic sink for such chemistry. Herein, we report the successful stabilization and characterization of a mononuclear hydroxoiron(III) complex, [FeIII(OH)(TMC-py)]2+ (3; TMC-py = 1-(pyridyl-2'-methyl)-4,8,11-trimethyl-1,4,8,11-tetrazacyclotetradecane), which is directly generated from the reaction of [FeIV(O)(TMC-py)]2+ (2) with 1,4-cyclohexadiene at -40 °C by H-atom abstraction. Complex 3 exhibits a UV spectrum with a λmax at 335 nm (ε ≈ 3500 M-1 cm-1) and a molecular ion in its electrospray ionization mass spectrum at m/z 555 with an isotope distribution pattern consistent with its formulation. Electron paramagnetic resonance and Mössbauer spectroscopy show 3 to be a high-spin Fe(III) center that is formed in 85% yield. Extended X-ray absorption fine structure analysis reveals an Fe-OH bond distance of 1.84 Å, which is also found in [(TMC-py)FeIII-O-CrIII(OTf)3]+ (4) obtained from the reaction of 2 with Cr(OTf)2. The S = 5/2 spin ground state and the 1.84 Å Fe-OH bond distance are supported computationally. Complex 3 reacts with 1-hydroxy-2,2,6,6-tetramethylpiperidine (TEMPOH) at -40 °C with a second-order rate constant of 7.1 M-1 s-1 and an OH/OD kinetic isotope effect value of 6. On the basis of density functional theory calculations, the reaction between 3 and TEMPOH is classified as a proton-coupled electron transfer as opposed to a hydrogen-atom transfer.

Automated Construction of Molecular Active Spaces from Atomic Valence Orbitals.

We introduce the atomic valence active space (AVAS), a simple and well-defined automated technique for constructing active orbital spaces for use in multiconfiguration and multireference (MR) electronic structure calculations. Concretely, the technique constructs active molecular orbitals capable of describing all relevant electronic configurations emerging from a targeted set of atomic valence orbitals (e.g., the metal d orbitals in a coordination complex). This is achieved via a linear transformation of the occupied and unoccupied orbital spaces from an easily obtainable single-reference wave function (such as from a Hartree-Fock or Kohn-Sham calculations) based on projectors to targeted atomic valence orbitals. We discuss the premises, theory, and implementation of the idea, and several of its variations are tested. To investigate the performance and accuracy, we calculate the excitation energies for various transition-metal complexes in typical application scenarios. Additionally, we follow the homolytic bond breaking process of a Fenton reaction along its reaction coordinate. While the described AVAS technique is not a universal solution to the active space problem, its premises are fulfilled in many application scenarios of transition-metal chemistry and bond dissociation processes. In these cases the technique makes MR calculations easier to execute, easier to reproduce by any user, and simplifies the determination of the appropriate size of the active space required for accurate results.

On the Accessible Reaction Channels of Vinyl Gold(I) Species: π- and σ-Pathways.

The potential of vinyl Au species to react either through a controlled π- or σ-pathway is demonstrated. This nomenclature is directly derived from the orbitals of the vinyl Au species leading to the newly formed bonds. When the π-bond of the vinyl Au intermediate is transformed into a σ-bond, we name it π-pathway, and a σ- to σ- transformation is named σ-pathway. Examples of reactions following these pathways are a Au-catalysed [3,3]-sigmatropic rearrangement and a protodeauration reaction. These reactions have been studied using intrinsic bond orbitals (IBOs) and allow for the clear identification of these pathways. Energies for the reaction path of the Au-catalysed [3,3]-sigmatropic rearrangement were in addition computed using CCSD(T)-F12. Analysis of the intrinsic reaction coordinate (IRC) of the [3,3]-sigmatropic rearrangement using IBOs further allows us to refine the previous mechanistic proposal and identifies a hidden intermediate along the reaction path.

Combining Internally Contracted States and Matrix Product States To Perform Multireference Perturbation Theory.

We present two efficient and intruder-free methods for treating dynamic correlation on top of general multiconfiguration reference wave functions - including such as obtained by the density matrix renormalization group (DMRG) with large active spaces. The new methods are the second order variant of the recently proposed multireference linearized coupled cluster method (MRLCC) [ Sharma, S.; Alavi, A. J. Chem. Phys. 2015 , 143 , 102815 ] and of N-electron valence perturbation theory (NEVPT2), with expected accuracies similar to MRCI+Q and (at least) CASPT2, respectively. Great efficiency gains are realized by representing the first order wave function with a combination of internal contraction (IC) and matrix product state perturbation theory (MPSPT). With this combination, only third order reduced density matrices (RDMs) are required. Thus, we obviate the need for calculating (or estimating) RDMs of fourth or higher order; these had so far posed a severe bottleneck for dynamic correlation treatments involving the large active spaces accessible to DMRG. Using several benchmark systems, including first and second row containing small molecules, Cr2, pentacene, and oxo-Mn(Salen), we show that active spaces containing at least 30 orbitals can be treated using this method. On a single node, MRLCC2 and NEVPT2 calculations can be performed with over 550 and 1100 virtual orbitals, respectively. We also critically examine the errors incurred due to the three sources of errors introduced in the present implementation - calculating second order instead of third order energy corrections, use of internal contraction, and approximations made in the reference wave function due to DMRG.

Gold(I) Vinylidene Complexes as Reactive Intermediates and Their Tendency to π-Backbond.

We herein report a computational study of the bonding in gold(I) vinylidene complexes and compare them to their carbene and CO analogues. The relevance of these intermediates is analysed for the intramolecular cyclisation leading to vinyl sulfonates.

Scalable electron correlation methods I.: PNO-LMP2 with linear scaling in the molecular size and near-inverse-linear scaling in the number of processors.

We propose to construct electron correlation methods that are scalable in both molecule size and aggregated parallel computational power, in the sense that the total elapsed time of a calculation becomes nearly independent of the molecular size when the number of processors grows linearly with the molecular size. This is shown to be possible by exploiting a combination of local approximations and parallel algorithms. The concept is demonstrated with a linear scaling pair natural orbital local second-order Møller-Plesset perturbation theory (PNO-LMP2) method. In this method, both the wave function manifold and the integrals are transformed incrementally from projected atomic orbitals (PAOs) first to orbital-specific virtuals (OSVs) and finally to pair natural orbitals (PNOs), which allow for minimum domain sizes and fine-grained accuracy control using very few parameters. A parallel algorithm design is discussed, which is efficient for both small and large molecules, and numbers of processors, although true inverse-linear scaling with compute power is not yet reached in all cases. Initial applications to reactions involving large molecules reveal surprisingly large effects of dispersion energy contributions as well as large intramolecular basis set superposition errors in canonical MP2 calculations. In order to account for the dispersion effects, the usual selection of PNOs on the basis of natural occupation numbers turns out to be insufficient, and a new energy-based criterion is proposed. If explicitly correlated (F12) terms are included, fast convergence to the MP2 complete basis set (CBS) limit is achieved. For the studied reactions, the PNO-LMP2-F12 results deviate from the canonical MP2/CBS and MP2-F12 values by <1 kJ mol(-1), using triple-ζ (VTZ-F12) basis sets.