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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, { [ 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 + H2 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). [ABSTRACT FROM AUTHOR]

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 mEh 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 (∼0.1 mEh). 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. [ABSTRACT FROM AUTHOR]

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-ζ basis sets. [ABSTRACT FROM AUTHOR]

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 H2O, HCN, CO2, CH2O, H2O2, C2H2, CH2NH, C2H2O, and the trans-isomer of 1,2-C2H2F2. 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. [ABSTRACT FROM AUTHOR]

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 Ansätze 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]