A density-fitting implementation of the density-based basis-set correction method.

This work reports an efficient density-fitting implementation of the density-based basis-set correction (DBBSC) method in the MOLPRO software. This method consists in correcting the energy calculated by a wave-function method with a given basis set by an adapted basis-set correction density functional incorporating the short-range electron correlation effects missing in the basis set, resulting in an accelerated convergence to the complete-basis-set limit. Different basis-set correction density-functional approximations are explored and the complementary-auxiliary-basis-set single-excitation correction is added. The method is tested on a benchmark set of reaction energies at the second-order Møller-Plesset (MP2) level and a comparison with the explicitly correlated MP2-F12 method is provided. The results show that the DBBSC method greatly accelerates the basis convergence of MP2 reaction energies, without reaching the accuracy of the MP2-F12 method but with a lower computational cost.

The p-block challenge: assessing quantum chemistry methods for inorganic heterocycle dimerizations.

The elements of the p-block of the periodic table are of high interest in various chemical and technical applications like frustrated Lewis-pairs (FLP) or opto-electronics. However, high-quality benchmark data to assess approximate density functional theory (DFT) for their theoretical description are sparse. In this work, we present a benchmark set of 604 dimerization energies of 302 "inorganic benzenes" composed of all non-carbon p-block elements of main groups III to VI up to polonium. This so-called IHD302 test set comprises two classes of structures formed by covalent bonding and by weaker donor-acceptor (WDA) interactions, respectively. Generating reliable reference data with ab initio methods is challenging due to large electron correlation contributions, core-valence correlation effects, and especially the slow basis set convergence. To compute reference values for these dimerization reactions, after thorough testing, we applied a computational protocol using state-of-the-art explicitly correlated local coupled cluster theory termed PNO-LCCSD(T)-F12/cc-VTZ-PP-F12(corr.). It includes a basis set correction at the PNO-LMP2-F12/aug-cc-pwCVTZ level. Based on these reference data, we assess 26 DFT methods in combination with three different dispersion corrections and the def2-QZVPP basis set, five composite DFT approaches, and five semi-empirical quantum mechanical methods. For the covalent dimerizations, the r2SCAN-D4 meta-GGA, the r2SCAN0-D4 and ωB97M-V hybrids, and the revDSD-PBEP86-D4 double-hybrid functional are found to be the best-performing methods among the evaluated functionals of the respective class. However, since def2 basis sets for the 4th period are not associated to relativistic pseudo-potentials, we obtained significant errors in the covalent dimerization energies (up to 6 kcal mol-1) for molecules containing p-block elements of the 4th period. Significant improvements were achieved for systems containing 4th row elements by using ECP10MDF pseudopotentials along with re-contracted aug-cc-pVQZ-PP-KS basis sets introduced in this work with the contraction coefficients taken from atomic DFT (PBE0) calculations. Overall, the IHD302 set represents a challenge to contemporary quantum chemical methods. This is due to a large number of spatially close p-element bonds which are underrepresented in other benchmark sets, and the partial covalent bonding character for the WDA interactions. The IHD302 set may be helpful to develop more robust and transferable approximate quantum chemical methods in the future.

Thermochemical evaluation of adaptive and fixed density functional theory quadrature schemes.

A systematic study is made of the accuracy and efficiency of a number of existing quadrature schemes for molecular Kohn-Sham Density-Functional Theory (DFT) using 408 molecules and 254 chemical reactions. Included are the fixed SG-x (x = 0-3) grids of Gill et al., Dasgupta, and Herbert, the 3-zone grids of Treutler and Ahlrichs, a fixed five-zone grid implemented in Molpro, and a new adaptive grid scheme. While all methods provide a systematic reduction of errors upon extension of the grid sizes, significant differences are observed in the accuracies for similar grid sizes with various approaches. For the tests in this work, the SG-x fixed grids are less suitable to achieve high accuracies in the DFT integration, while our new adaptive grid performed best among the schemes studied in this work. The extra computational time to generate the adaptive grid scales linearly with molecular size and is negligible compared with the time needed for the self-consistent field iterations for large molecules. A comparison of the grid accuracies using various density functionals shows that meta-GGA functionals need larger integration grids than GGA functionals to reach the same degree of accuracy, confirming previous investigations of the numerical stability of meta-GGA functionals. On the other hand, the grid integration errors are almost independent of the basis set, and the basis set errors are mostly much larger than the errors caused by the numerical integrations, even when using the smallest grids tested in this work.

A combined first- and second-order optimization method for improving convergence of Hartree-Fock and Kohn-Sham calculations.

In this work, we investigate the optimization of Hartree-Fock (HF) orbitals with our recently proposed combined first- and second-order (SO-SCI) method, which was originally developed for multi-configuration self-consistent field (MCSCF) and complete active space SCF (CASSCF) calculations. In MCSCF/CASSCF, it unites a second-order optimization of the active orbitals with a Fock-based first-order treatment of the remaining closed-virtual orbital rotations. In the case of the single-determinant wavefunctions, the active space is replaced by a preselected "second-order domain," and all rotations involving orbitals in this subspace are treated at second-order. The method has been implemented for spin-restricted and spin-unrestricted Hartree-Fock (RHF, UHF), configuration-averaged Hartree-Fock (CAHF), as well as Kohn-Sham (KS) density functional theory (RKS, UKS). For each of these cases, various choices of the second-order domain have been tested, and appropriate defaults are proposed. The performance of the method is demonstrated for several transition metal complexes. It is shown that the SO-SCI optimization provides faster and more robust convergence than the standard SCF procedure but requires, in many cases, even less computation time. In difficult cases, the SO-SCI method not only speeds up convergence but also avoids convergence to saddle-points. Furthermore, it helps to find spin-symmetry broken solutions in the cases of UHF or UKS. In the case of CAHF, convergence can also be significantly improved as compared to a previous SCF implementation. This is particularly important for multi-center cases with two or more equal heavy atoms. The performance is demonstrated for various two-center complexes with different lanthanide atoms.

MCSCF optimization revisited. II. Combined first- and second-order orbital optimization for large molecules.

A new orbital optimization for the multiconfiguration self-consistent field method is presented. This method combines a second-order (SO) algorithm for the optimization of the active orbitals with the first-order super configuration interaction (SCI) optimization of the remaining closed-virtual rotations and is denoted as the SO-SCI method. The SO-SCI method significantly improves the convergence as compared to the conventional SCI method. In combination with density fitting, the intermediates from the gradient calculation can be reused to evaluate the two-electron integrals required for the active Hessian without introducing a large computational overhead. The orbitals and CI coefficients are optimized alternately, but the CI-orbital coupling is accounted for by the limited memory Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method. This further improves the speed of convergence. The method is applicable to large molecules. The efficiency and robustness of the presented method is demonstrated in benchmark calculations for 21 aromatic molecules as well as for various transition metal complexes with up to 826 electrons and 5154 basis functions.

The Molpro quantum chemistry package.

Molpro is a general purpose quantum chemistry software package with a long development history. It was originally focused on accurate wavefunction calculations for small molecules but now has many additional distinctive capabilities that include, inter alia, local correlation approximations combined with explicit correlation, highly efficient implementations of single-reference correlation methods, robust and efficient multireference methods for large molecules, projection embedding, and anharmonic vibrational spectra. In addition to conventional input-file specification of calculations, Molpro calculations can now be specified and analyzed via a new graphical user interface and through a Python framework.

Toward fast and accurate ab initio calculation of magnetic exchange in polynuclear lanthanide complexes.

Ab initio calculations of the magnetic exchange in polynuclear lanthanide complexes are very challenging and often not feasible, due to large active spaces, the large number of required states or the necessity to include dynamical correlation into the calculations. We present an approach which allows for the computationally efficient calculation of exchange splittings in polynuclear lanthanide complexes including dynamical correlation. This is achieved by extending the local-density-fitted configuration-averaged Hartree-Fock (LDF-CAHF) method to systems with more than one group of open-shell orbitals (e.g. at different metal atoms) and combining it with linear-scaling many-state pair-natural-orbital complete active space perturbation theory of second order (PNO-CASPT2). In order to assess the performance of the method, we apply it to the asymmetric dinuclear complex [hqH2][Yb2(hq)4(NO3)3]·MeOH (hqH = 8-hydroxyquinoline).

Multi-state local complete active space second-order perturbation theory using pair natural orbitals (PNO-MS-CASPT2).

A multistate complete active space second-order perturbation theory (CASPT2) method is presented, which utilizes domains of pair natural orbitals and projected atomic orbitals for the virtual space to achieve linear scaling of the computational effort with the number of inactive orbitals. The method is applied to compute excitation energies of medium size aromatic molecules, and it is shown that the impact of the local approximations on the computed excitation energies is negligible. The applicability and efficiency of the method are demonstrated for two large molecular systems with up to 400 correlated electrons, nearly 3000 basis functions, and 45 electronic states. Furthermore, some approximations in the CASPT2 zeroth-order Hamiltonian, which decouple different configuration spaces, are proposed and tested. These approximations allow us to reuse many integrals and amplitudes from the ground state in the excited states, thereby significantly reducing the computational effort for calculations with many states. Using appropriate correction terms, the impact of these approximations is shown to be small.

Second-order MCSCF optimization revisited. I. Improved algorithms for fast and robust second-order CASSCF convergence.

A new improved implementation of the second-order multiconfiguration self-consistent field optimization method of Werner and Knowles [J. Chem. Phys. 82, 5053 (1985)] is presented. It differs from the original method by more stable and efficient algorithms for minimizing the second-order energy approximation in the so-called microiterations. Conventionally, this proceeds by alternating optimizations of the orbitals and configuration (CI) coefficients and is linearly convergent. The most difficult part is the orbital optimization, which requires solving a system of nonlinear equations that are often strongly coupled. We present a much improved algorithm for solving this problem, using an iterative subspace method that includes part of the orbital Hessian explicitly, and discuss different strategies for performing the uncoupled optimization in a most efficient manner. Second, we present a new solver in which the orbital-CI coupling is treated explicitly. This leads to quadratic convergence of the microiterations but requires many additional evaluations of reduced (transition) density matrices. In difficult optimization problems with a strong coupling of the orbitals and CI coefficients, it leads to much improved convergence of both the macroiterations and the microiterations. Third, the orbital-CI coupling is treated approximately using a quasi-Newton approach with Broyden-Fletcher-Goldfarb-Shanno updates of the orbital Hessian. It is demonstrated that this converges almost as well as the explicitly coupled method but avoids the additional effort for computing many transition density matrices. The performance of the three methods is compared for a set of 21 aromatic molecules, an Fe(ii)-porphine transition metal complex, as well as for the [Cu2O2(NH3) 6]2+, FeCl3, Co2(CO)6C2H2, and Al4O2 complexes. In all cases, faster and more stable convergence than with the original implementation is achieved.

Correspondence on "Core Electron Topologies in Chemical Compounds: Case Study of Carbon versus Silicon".

The conclusions of a recent Communication of Yoshida, Raebiger, Shudo, and Ohno published in this journal, that varying core orbital topologies with minuscule negative tails upon bond formation determine the different chemistries of carbon and silicon and affect ionization energies, excitation energies and bond properties of molecules, are now shown to be based on computational artifacts and oversimplified models. The all-electron wave function uniquely determines the observables, while its representation by one-electron orbital products depends on the details of the chosen approximation and therefore need to be considered with great care.

Analytical energy gradients for explicitly correlated wave functions. II. Explicitly correlated coupled cluster singles and doubles with perturbative triples corrections: CCSD(T)-F12.

We present an implementation of analytical energy gradients for the explicitly correlated coupled cluster singles and doubles method with perturbative triples corrections [CCSD(T)-F12]. The accuracy of the CCSD(T)-F12 analytical gradient technique is demonstrated by computing equilibrium geometries for a set of closed-shell molecules containing first- and second-row elements. Near basis-set limit equilibrium geometries are obtained with triple-zeta correlation consistent basis sets. Various approximations in the F12 treatment are compared, and the effects of these are found to be small.

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.

Communication: Multipole approximations of distant pair energies in local correlation methods with pair natural orbitals.

The accuracy of multipole approximations for distant pair energies in local second-order Møller-Plesset perturbation theory (LMP2) as introduced by Hetzer et al. [Chem. Phys. Lett. 290, 143 (1998)] is investigated for three chemical reactions involving molecules with up to 92 atoms. Various iterative and non-iterative approaches are compared, using different energy thresholds for distant pair selection. It is demonstrated that the simple non-iterative dipole-dipole approximation, which has been used in several recent pair natural orbitals (PNO)-LMP2 and PNO-LCCSD (local coupled-cluster with singles and doubles) methods, may underestimate the distant pair energies by up to 50% and can lead to significant errors in relative energies, unless very tight thresholds are used. The accuracy can be much improved by including higher multipole orders and by optimizing the distant pair amplitudes iteratively along with all other amplitudes. A new approach is presented in which very small special PNO domains for distant pairs are used in the iterative approach. This reduces the number of distant pair amplitudes by 3 orders of magnitude and keeps the additional computational effort for the iterative optimization of distant pair amplitudes minimal.

Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2).

We present a CASPT2 method which exploits local approximations to achieve linear scaling of the computational effort with the molecular size, provided the active space is small and local. The inactive orbitals are localized, and the virtual space for each electron pair is spanned by a domain of pair-natural orbitals (PNOs). The configuration space is internally contracted, and the PNOs are defined for uniquely defined orthogonal pairs. Distant pair energies are obtained by multipole approximations, so that the number of configurations that are explicitly treated in the CASPT2 scales linearly with molecular size (assuming a constant active space). The PNOs are generated using approximate amplitudes obtained in a pair-specific semi-canonical basis of projected atomic orbitals (PAOs). The evaluation and transformation of the two-electron integrals use the same parallel local density fitting techniques as recently described for linear-scaling PNO-LMP2 (local second-order Møller-Plesset perturbation theory). The implementation of the amplitude equations, which are solved iteratively, employs the local integrated tensor framework. The efficiency and accuracy of the method are tested for excitation energies and correlation energies. It is demonstrated that the errors introduced by the local approximations are very small. They can be well controlled by few parameters for the distant pair approximation, initial PAO domains, and the PNO domains.

On the use of Abelian point group symmetry in density-fitted local MP2 using various types of virtual orbitals.

Electron correlation methods based on symmetry-adapted canonical Hartree-Fock orbitals can be speeded up significantly in the well known group theoretical manner, using the fact that integrals vanish unless the integrand is totally symmetric. In contrast to this, local electron correlation methods cannot benefit from such simplifications, since the localized molecular orbitals (LMOs) generally do not transform according to irreducible representations of the underlying point group symmetry. Instead, groups of LMOs become symmetry-equivalent and this can be exploited to accelerate local calculations. We describe an implementation of such a symmetry treatment for density-fitted local Møller-Plesset perturbation theory, using various types of virtual orbitals: Projected atomic orbitals, orbital specific virtuals, and pair natural orbitals. The savings by the symmetry treatment are demonstrated by calculations for several large molecules having different point group symmetries. Benchmarks for the parallel execution efficiency of our method are also presented.

Accurate thermochemistry from explicitly correlated distinguishable cluster approximation.

An explicitly correlated version of the distinguishable-cluster approximation is presented and extensively benchmarked. It is shown that the usual F12-type explicitly correlated approaches are applicable to distinguishable-cluster theory with single and double excitations, and the results show a significant improvement compared to coupled-cluster theory with singles and doubles for closed and open-shell systems. The resulting method can be applied in a black-box manner to systems with single- and multireference character. Most noticeably, optimized geometries are of coupled-cluster singles and doubles with perturbative triples quality or even better.

Communication: Improved pair approximations in local coupled-cluster methods.

In local coupled cluster treatments the electron pairs can be classified according to the magnitude of their energy contributions or distances into strong, close, weak, and distant pairs. Different approximations are introduced for the latter three classes. In this communication, an improved simplified treatment of close and weak pairs is proposed, which is based on long-range cancellations of individually slowly decaying contributions in the amplitude equations. Benchmark calculations for correlation, reaction, and activation energies demonstrate that these approximations work extremely well, while pair approximations based on local second-order Møller-Plesset theory can lead to errors that are 1-2 orders of magnitude larger.

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.

Pyrazine excited states revisited using the extended multi-state complete active space second-order perturbation method.

We demonstrate that the recently developed extended multi-state complete active space second-order perturbation theory (XMS-CASPT2) [Shiozaki et al., J. Chem. Phys., 2011, 135, 081106] provides qualitatively correct potential energy surfaces for low-lying excited singlet states of pyrazine, while the potential energy surfaces of the standard MS-CASPT2 methods are ill-behaved near the crossing point of two reference potential energy surfaces. The XMS-CASPT2 method is based on the extended multi-configuration quasi-degenerate perturbation theory proposed earlier by Granovsky [J. Chem. Phys., 2011, 134, 214113]. We show that the conical intersection at the XMS-CASPT2 level can be described without artifacts if the entire method is invariant with respect to any unitary rotations of the reference functions. The photoabsorption spectra of the 1(1)B(3u) and 1(1)B(2u) states of pyrazine are simulated, based on a vibronic-coupling model Hamiltonian. The XMS-CASPT2 spectrum of the 1(1)B(3u) band is found to be comparable to the one computed by a more expensive multireference configuration interaction (MRCI) method, while the XMS-CASPT2 simulation of the 1(1)B(2u) band is slightly inferior to the MRCI one.

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.