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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.

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.

We report a closed formula expressing the McMurchie-Davidson (MD) key intermediates {[r] (0) ; r x + r y + r z ≤ 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, {[0 κ ] (m+m ' ) }↦{[l κ ] (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.

We present efficient algorithms for computing two-center integrals and integral derivatives, with general interaction kernels K ( r 12 ), 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 ( r 12 ) 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 J AB = ( A |1/ r 12 | B ) (183 MiB) can be computed in 50 ms on a Q2'2018 notebook CPU, without any screening or approximations.

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 Ni IV complexes, aryl-CF 3 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 Ni IV 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-CF 3 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 CF 3 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., (© 2019 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.)

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., Competing Interests: The authors declare no conflict of interest., (© 2019 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.)

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., (©2019 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.)

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.

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., (© 2018 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.)

A recent report on the crystal structure of the pentagonal-pyramidal hexamethylbenzene dication C 6 (CH 3 ) 6 2+ 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 CH 3 + group. The different roles of coordination chemistry are discussed., (© 2018 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.)

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.

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.

A detailed analysis of the C(sp 3 )-H activation process by vinylidene Au I 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., (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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, [Fe III (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 [Fe IV (O)(TMC-py)] 2+ (2) with 1,4-cyclohexadiene at -40 °C by H-atom abstraction. Complex 3 exhibits a UV spectrum with a λ max 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)Fe III -O-Cr III (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.

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.

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., (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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, Cr 2 , 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.

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., (© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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., (© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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., (© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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.

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., (© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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., (Copyright © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

Modern quantum chemistry can make quantitative predictions on an immense array of chemical systems. However, the interpretation of those predictions is often complicated by the complex wave function expansions used. Here we show that an exceptionally simple algebraic construction allows for defining atomic core and valence orbitals, polarized by the molecular environment, which can exactly represent self-consistent field wave functions. This construction provides an unbiased and direct connection between quantum chemistry and empirical chemical concepts, and can be used, for example, to calculate the nature of bonding in molecules, in chemical terms, from first principles. In particular, we find consistency with electronegativities (χ), C 1s core-level shifts, resonance substituent parameters (σR), Lewis structures, and oxidation states of transition-metal complexes.

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.

We extend our density matrix embedding theory (DMET) [Phys. Rev. Lett.2012, 109, 186404] from lattice models to the full chemical Hamiltonian. DMET allows the many-body embedding of arbitrary fragments of a quantum system, even when such fragments are open systems and strongly coupled to their environment (e.g., by covalent bonds). In DMET, empirical approaches to strong coupling, such as link atoms or boundary regions, are replaced by a small, rigorous quantum bath designed to reproduce the entanglement between a fragment and its environment. We describe the theory and demonstrate its feasibility in strongly correlated hydrogen ring and grid models; these are not only beyond the scope of traditional embeddings but even challenge conventional quantum chemistry methods themselves. We find that DMET correctly describes the notoriously difficult symmetric dissociation of a 4 × 3 hydrogen atom grid, even when the treated fragments are as small as single hydrogen atoms. We expect that DMET will open up new ways of treating complex strongly coupled, strongly correlated systems in terms of their individual fragments.

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.

We present a simple, broadly applicable method for determining the numerical properties of quantum chemistry algorithms. The method deliberately introduces random numerical noise into computations, which is of the same order of magnitude as the floating point precision. Accordingly, repeated runs of an algorithm give slightly different results, which can be analyzed statistically to obtain precise estimates of its numerical stability. This noise is produced by automatic code injection into regular compiler output, so that no substantial programming effort is required, only a recompilation of the affected program sections. The method is applied to investigate: (i) the numerical stability of the three-center Obara-Saika integral evaluation scheme for high angular momenta, (ii) if coupled cluster perturbative triples can be evaluated with single precision arithmetic, (iii) how to implement the density fitting approximation in Møller-Plesset perturbation theory (MP2) most accurately, and (iv) which parts of density fitted MP2 can be safely evaluated with single precision arithmetic. In the integral case, we find a numerical instability in an equation that is used in almost all integral programs. Due to the results of (ii) and (iv), we conjecture that single precision arithmetic can be applied whenever a calculation is done in an orthogonal basis set and excessively long linear sums are avoided.

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.

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).

Accurate extrapolation to the complete basis set (CBS) limit of valence correlation energies calculated with explicitly correlated MP2-F12 and CCSD(T)-F12b methods have been investigated using a Schwenke-style approach for molecules containing both first and second row atoms. Extrapolation coefficients that are optimal for molecular systems containing first row elements differ from those optimized for second row analogs, hence values optimized for a combined set of first and second row systems are also presented. The new coefficients are shown to produce excellent results in both Schwenke-style and equivalent power-law-based two-point CBS extrapolations, with the MP2-F12/cc-pV(D,T)Z-F12 extrapolations producing an average error of just 0.17 mE(h) with a maximum error of 0.49 for a collection of 23 small molecules. The use of larger basis sets, i.e., cc-pV(T,Q)Z-F12 and aug-cc-pV(Q,5)Z, in extrapolations of the MP2-F12 correlation energy leads to average errors that are smaller than the degree of confidence in the reference data (approximately 0.1 mE(h)). The latter were obtained through use of very large basis sets in MP2-F12 calculations on small molecules containing both first and second row elements. CBS limits obtained from optimized coefficients for conventional MP2 are only comparable to the accuracy of the MP2-F12/cc-pV(D,T)Z-F12 extrapolation when the aug-cc-pV(5+d)Z and aug-cc-pV(6+d)Z basis sets are used. The CCSD(T)-F12b correlation energy is extrapolated as two distinct parts: CCSD-F12b and (T). While the CCSD-F12b extrapolations with smaller basis sets are statistically less accurate than those of the MP2-F12 correlation energies, this is presumably due to the slower basis set convergence of the CCSD-F12b method compared to MP2-F12. The use of larger basis sets in the CCSD-F12b extrapolations produces correlation energies with accuracies exceeding the confidence in the reference data (also obtained in large basis set F12 calculations). It is demonstrated that the use of the 3C(D) Ansatz is preferred for MP2-F12 CBS extrapolations. Optimal values of the geminal Slater exponent are presented for the diagonal, fixed amplitude Ansatz in MP2-F12 calculations, and these are also recommended for CCSD-F12b calculations.

The simple and efficient CCSD(T)-F12x approximations (x = a,b) we proposed in a recent communication [T. B. Adler, G. Knizia, and H.-J. Werner, J. Chem. Phys. 127, 221106 (2007)] are explained in more detail and extended to open-shell systems. Extensive benchmark calculations are presented, which demonstrate great improvements in basis set convergence for a wide variety of applications. These include reaction energies of both open- and closed-shell reactions, atomization energies, electron affinities, ionization potentials, equilibrium geometries, and harmonic vibrational frequencies. For all these quantities, results better than the AV5Z quality are obtained already with AVTZ basis sets, and usually AVDZ treatments reach at least the conventional AVQZ quality. For larger molecules, the additional cost for these improvements is only a few percent of the time for a standard CCSD(T) calculation. For the first time ever, total reaction energies with chemical accuracy are obtained using valence-double-zeta basis sets.

The recently proposed explicitly correlated CCSD(T)-F12x (x = a,b) approximations [T. B. Adler, G. Knizia, and H.-J. Werner, J. Chem. Phys. 127, 221106 (2007)] are applied to compute equilibrium structures and harmonic as well as anharmonic vibrational frequencies for H(2)O, HCN, CO(2), CH(2)O, H(2)O(2), C(2)H(2), CH(2)NH, C(2)H(2)O, and the trans-isomer of 1,2-C(2)H(2)F(2). Using aug-cc-pVTZ basis sets, the CCSD(T)-F12a equilibrium geometries and harmonic vibrational frequencies are in very close agreement with CCSD(T)/aug-cc-pV5Z values. The anharmonic frequencies are evaluated using vibrational self-consistent field and vibrational configuration interaction methods based on automatically generated potential energy surfaces. The mean absolute deviation of the CCSD(T)-F12a/aug-cc-pVTZ anharmonic frequencies from experimental values amounts to only 4.0 cm(-1).

We present an explicitly correlated version of the high-spin open-shell RMP2 method. The theory is derived in a unitarily invariant form, which is suitable for the insertion of local approximations. It is demonstrated that the rapid basis set convergence of closed-shell MP2-F12 is also achieved in RMP2-F12, and similar Ansatze and approximations can be employed. All integrals are computed using efficient density fitting approximations, and many-electron integrals are avoided using resolution of the identity approximations. The performance of the method is demonstrated by benchmark calculations on a large set of ionization potentials, electron affinities and atomization energies. Using triple-zeta basis sets RMP2-F12 yields results that are closer to the basis set limit than standard RMP2 with augmented quintuple-zeta basis sets for all properties. Different variants of perturbative corrections for the open-shell Hartree-Fock treatment are described and tested.

A new explicitly correlated CCSD(T)-F12 approximation is presented and tested for 23 molecules and 15 chemical reactions. The F12 correction strongly improves the basis set convergence of correlation and reaction energies. Errors of the Hartree-Fock contributions are effectively removed by including MP2 single excitations into the auxiliary basis set. Using aug-cc-pVTZ basis sets the CCSD(T)-F12 calculations are more accurate and two orders of magnitude faster than standard CCSD(T)/aug-cc-pV5Z calculations.