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.

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.

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.

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.

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.

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.

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.

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.

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.

Electron flow in reaction mechanisms--revealed from first principles.

The "curly arrow" of Robinson and Ingold is the primary tool for describing and rationalizing reaction mechanisms. Despite this approach's ubiquity and stellar success, its physical basis has never been clarified and a direct connection to quantum chemistry has never been found. Here we report that the bond rearrangements expressed by curly arrows can be directly observed in ab initio computations, as transformations of intrinsic bond orbitals (IBOs) along the reaction coordinate. Our results clarify that curly arrows are rooted in physical reality-a notion which has been challenged before-and show how quantum chemistry can directly establish reaction mechanisms in intuitive terms and unprecedented detail.

The Stabilizing Effects in Gold Carbene Complexes.

Bonding and stabilizing effects in gold carbene complexes are investigated by using Kohn-Sham density functional theory (DFT) and the intrinsic bond orbital (IBO) approach. The π-stabilizing effects of organic substituents at the carbene carbon atom coordinated to the gold atom are evaluated for a series of recently isolated and characterized complexes, as well as intermediates of prototypical 1,6-enyne cyclization reactions. The results indicate that these effects are of particular importance for gold complexes especially because of the low π-backbonding contribution from the gold atom.

Fe or Fe-NO catalysis? A quantum chemical investigation of the [Fe(CO)3(NO)](-)-catalyzed Cloke-Wilson rearrangement.

A quantum chemical investigation of the Bu4N[Fe(CO)3(NO)]-catalyzed Cloke-Wilson rearrangement of vinyl cyclopropanes is reported. It was found that allylic C-C bond activation can proceed through a SN2' or SN2-type mechanism. The application of the recently reported intrinsic bond orbital (IBO) method for all structures indicated that one Fe-N π bond is directly involved. Further analysis showed that during the reaction oxidation occurs at the NO ligand exclusively.

The electronic ground state of [Fe(CO)3 (NO)](-) : a spectroscopic and theoretical study.

During the past 10 years iron-catalyzed reactions have become established in the field of organic synthesis. For example, the complex anion [Fe(CO)3 (NO)](-) , which was originally described by Hogsed and Hieber, shows catalytic activity in various organic reactions. This anion is commonly regarded as being isoelectronic with [Fe(CO)4 ](2-) , which, however, shows poor catalytic activity. The spectroscopic and quantum chemical investigations presented herein reveal that the complex ferrate [Fe(CO)3 (NO)](-) cannot be regarded as a Fe(-II) species, but rather is predominantly a Fe(0) species, in which the metal is covalently bonded to NO(-) by two π-bonds. A metal-N σ-bond is not observed.

Analytical energy gradients for second-order multireference perturbation theory using density fitting.

We present algorithms for computing analytical energy gradients for multi-configuration self-consistent field methods and partially internally contracted complete active space second-order perturbation theory (CASPT2) using density fitting (DF). Our implementation is applicable to both single-state and multi-state CASPT2 analytical gradients. The accuracy of the new methods is demonstrated for structures and excitation energies of valence and Rydberg states of pyrrole, as well as for structures and adiabatic singlet-triplet energy splittings for the hydro-, the O,O(')-formato-, and the N,N(')-diiminato-copper-dioxygen complexes. It is shown that the effects of density fitting on optimized structures and relative energies are negligible. For cases in which the total cost is dominated by the integral evaluations and transformations, the DF-CASPT2 gradient calculations are found to be faster than the corresponding conventional calculations by typically a factor of three to five using triple-ζ basis sets, and by about a factor of ten using quadruple-ζ basis sets.

Density matrix embedding: a simple alternative to dynamical mean-field theory.

We introduce density matrix embedding theory (DMET), a quantum embedding theory for computing frequency-independent quantities, such as ground-state properties, of infinite systems. Like dynamical mean-field theory, DMET maps the bulk interacting system to a simpler impurity model and is exact in the noninteracting and atomic limits. Unlike dynamical mean-field theory, DMET is formulated in terms of the frequency-independent local density matrix, rather than the local Green's function. In addition, it features a finite, algebraically constructible bath of only one bath site per impurity site, with no bath discretization error. Frequency independence and the minimal bath make DMET a computationally simple and efficient method. We test the theory in the one-dimensional and two-dimensional Hubbard models at and away from half filling, and we find that compared to benchmark data, total energies, correlation functions, and metal-insulator transitions are well reproduced, at a tiny computational cost.

Explicitly correlated multireference configuration interaction: MRCI-F12.

An internally contracted multireference configuration interaction is developed which employs wave functions that explicitly depend on the electron-electron distance (MRCI-F12). This MRCI-F12 method has the same applicability as the MRCI method, while having much improved basis-set convergence with little extra computational cost. The F12b approximation is used to arrive at a computationally efficient implementation. The MRCI-F12 method is applied to the singlet-triplet separation of methylene, the dissociation energy of ozone, properties of diatomic molecules, and the reaction barrier and exothermicity of the F + H(2) reaction. These examples demonstrate that already with basis sets of moderate size the method provides near complete basis set MRCI accuracy, and hence quantitative agreement with the experimental data. As a side product, we have also implemented the explicitly correlated multireference averaged coupled pair functional method (MRACPF-F12).

A new internally contracted multi-reference configuration interaction method.

We present a new internally contracted multi-reference configuration interaction (MRCI) method which, at the same time, efficiently handles large active orbital spaces, long configuration expansions, and many closed-shell orbitals in the reference function. This is achieved by treating the closed-shell orbitals explicitly, so that all required coupling coefficients and density matrices only depend on active orbital labels. As a result, closed-shell orbitals are handled as efficiently as in a closed-shell single-reference program, and this opens up the possibility to perform high-accuracy MRCI calculations for much larger molecules than before. The enormously complex equations are derived using a new domain-specific computer algebra system and semi-automatically implemented using a newly developed integrated tensor framework. The accuracy and efficiency of the MRCI method is demonstrated with applications to dioxygen-copper complexes with different ligands, some of which involve more than 30 atoms, and to spin-state splittings of ferrocene.