Florian A. Bischoff

The MP2‐F12 method in the TURBOMOLE program package

A detailed description of the explicitly correlated second‐order Møller–Plesset perturbation theory (MP2‐F12) method, as implemented in the TURBOMOLE program package, is presented. The TURBOMOLE implementation makes use of density fitting, which greatly reduces the prefactor for integral evaluation. Methods are available for the treatment of ground states of open‐ and closed‐shell species, using unrestricted as well as restricted (open‐shell) Hartree–Fock reference determinants. Various methodological choices and approximations are discussed. The performance of the TURBOMOLE implementation is illustrated by example calculations of the molecules leflunomide, prednisone, methotrexate, ethylenedioxytetrafulvalene, and a cluster model for the adsorption of methanol on the zeolite H‐ZSM‐5. Various basis sets are used, including the correlation‐consistent basis sets specially optimized for explicitly correlated calculations (cc‐pVXZ‐F12).

Scope and limitations of the SCS-MP2 method for stacking and hydrogen bonding interactions

Fluorobenzenes are pi-acceptor synthons that form pi-stacked structures in molecular crystals as well as in artificial DNAs. We investigate the competition between hydrogen bonding and pi-stacking in dimers consisting of the nucleobase mimic 2-pyridone (2PY) and all fluorobenzenes from 1-fluorobenzene to hexafluorobenzene (n-FB, with n = 1-6). We contrast the results of high level ab initio calculations with those obtained using ultraviolet (UV) and infrared (IR) laser spectroscopy of isolated and supersonically cooled dimers. The 2PY.n-FB complexes with n = 1-5 prefer double hydrogen bonding over pi-stacking, as diagnosed from the UV absorption and IR laser depletion spectra, which both show features characteristic of doubly H-bonded complexes. The 2-pyridone.hexafluorobenzene dimer is the only pi-stacked dimer, exhibiting a homogeneously broadened UV spectrum and no IR bands characteristic for H-bonded species. MP2 (second-order Møller-Plesset perturbation theory) calculations overestimate the pi-stacked dimer binding energies by about 10 kJ/mol and disagree with the experimental observations. In contrast, the MP2 treatment of the H-bonded dimers appears to be quite accurate. Grimmes spin-component-scaled MP2 approach (SCS-MP2) is an improvement over MP2 for the pi-stacked dimers, reducing the binding energy by approximately 10 kJ/mol. When applied to explicitly correlated MP2 theory (SCS-MP2-R12 approach), agreement with the corresponding coupled-cluster binding energies [at the CCSD(T) level] is very good for the pi-stacked dimers, within +/- 1 kJ/mol for the 2PY complexes with 1-fluorobenzene, 1,2-difluorobenzene, 1,2,4,5-tetrafluorobenzene, pentafluorobenzene and hexafluorobenzene. Unfortunately, the SCS-MP2 approach also reduces the binding energy of the H-bonded species, leading to disagreement with both coupled-cluster theory and experiment. The SCS-MP2-R12 binding energies follow the SCS-MP2 binding energies closely, being about 0.5 and 0.7 kJ/mol larger for the H-bonded and pi-stacked forms, respectively, in an augmented correlation-consistent polarized valence quadruple-zeta basis. It seems that the SCS-MP2 and SCS-MP2-R12 methods cannot provide sufficient accuracy to replace the CCSD(T) method for intermolecular interactions where H-bonding and pi-stacking are competitive.

Assessment of basis sets for F12 explicitly-correlated molecular electronic-structure methods

One-electron basis sets for F12 explicitly-correlated molecular electronic-structure methods are assessed by analysing the accuracy of Hartree–Fock energies and valence-only second-order correlation energies of a test set of 106 small molecules containing the atoms H, C, N, O and F. For these molecules, near Hartree–Fock-limit energies and benchmark second-order correlation energies (accurate to within 99.95% of the basis-set limit) are provided. Absolute energies are analysed as well as the Hartree–Fock and second-order correlation contributions to the atomisation energies of the molecules. Standard basis sets such as the Karlsruhe def2-TZVPP and def2-QZVPP sets and the augmented correlation-consistent polarised valence X-tuple zeta (aug-cc-p VXZ, X = D, T, Q, 5) sets are compared with the specialised cc-pVXZ-F12 (X = D, T, Q) sets that were recently optimised by Peterson and co-workers [J. Chem. Phys. 128, 084102 (2008)] for use in F12 theory. The results obtained from F12 explicitly-correlated molecular electronic-structure calculations are compared with those that are obtained by standard electronic-structure calculations followed by basis-set extrapolation based on the X −3 convergence behaviour of the aug-cc-pVXZ basis sets. The most important conclusions are that the cc-pVXZ-F12 sets are the preferred basis sets for F12 theory and that the X −3 extrapolation from the aug-cc-pVQZ and aug-cc-pV5Z is slightly more accurate than F12 theory in the cc-pVTZ-F12 basis but less accurate than F12 theory in the cc-pVQZ-F12 basis.

Low-order tensor approximations for electronic wave functions: Hartree–Fock method with guaranteed precision

Here we report a formulation of the Hartree-Fock method in an adaptive multiresolution basis set of spectral element type. A key feature of our approach is the use of low-order tensor approximations for operators and wave functions to reduce the steep rise of storage and computational costs with the number of degrees of freedom that plague finite element computations. As a proof of principle we implemented Hartree-Fock method without explicit storage of the full-dimensional wave function and with guaranteed precision (microhartree precision for up to 14 electron systems is demonstrated). Even for the one-electron method the use of low-order tensor approximation reduces storage relative to the full representation, albeit with modest increase in cost. Preliminary tests for explicitly-correlated two-electron (six-dimensional) wave function suggest a factor of 50 savings in storage. At least correlated two-electron methods should be feasible with our approach on modern workstations with guaranteed precision.

Second-order electron-correlation and self-consistent spin-orbit treatment of heavy molecules at the basis-set limit.

Second-order perturbation theory using explicitly correlated wave functions has been introduced into a quasirelativistic two-component formalism. The convergence of the correlation energy is as much improved as for the nonrelativistic Hamiltonian, achieving basis-set-limit results in a moderate-size basis set. Equilibrium distances and vibrational frequencies of small molecules of the 6th period of the periodic system of the elements have been calculated, demonstrating the improved behavior of the explicitly correlated wave functions. Taking advantage of density-fitting techniques, the explicitly correlated approach is an economical and appealing alternative to conventional two-component second-order perturbation theory in a large one-particle basis.

Computing many-body wave functions with guaranteed precision: The first-order Møller-Plesset wave function for the ground state of helium atom

We present an approach to compute accurate correlation energies for atoms and molecules using an adaptive discontinuous spectral-element multiresolution representation for the two-electron wave function. Because of the exponential storage complexity of the spectral-element representation with the number of dimensions, a brute-force computation of two-electron (six-dimensional) wave functions with high precision was not practical. To overcome the key storage bottlenecks we utilized (1) a low-rank tensor approximation (specifically, the singular value decomposition) to compress the wave function, and (2) explicitly correlated R12-type terms in the wave function to regularize the Coulomb electron-electron singularities of the Hamiltonian. All operations necessary to solve the Schrödinger equation were expressed so that the reconstruction of the full-rank form of the wave function is never necessary. Numerical performance of the method was highlighted by computing the first-order Møller-Plesset wave function of a helium atom. The computed second-order Møller-Plesset energy is precise to ~2 microhartrees, which is at the precision limit of the existing general atomic-orbital-based approaches. Our approach does not assume special geometric symmetries, hence application to molecules is straightforward.

Computing molecular correlation energies with guaranteed precision

We present an approach to compute accurate correlation energies for atoms and molecules in the framework of multiresolution analysis (MRA), using an adaptive discontinuous multiresolution spectral-element representation for the six-dimensional (two-electron) pair function. The key features of our approach that make it feasible, namely (1) low-rank tensor approximations of functions and operators and (2) analytic elimination of operator singularities via explicit correlation, were retained from the previous work [F. A. Bischoff, R. J. Harrison, and E. F. Valeev, J. Chem. Phys. 137, 104103 (2012)]. Here we generalized the working equations to handle general (non-symmetric) many-electron systems at the MP2 level. The numerical performance is shown for the beryllium atom and the water molecule where literature data for the basis set limits could be reproduced to a few tens of μE(h). The key advantages of molecular MRA-MP2 are the absence of bias and arbitrariness in the choice of the basis set, high accuracy, and low scaling with respect to the system size.

MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation

MADNESS (multiresolution adaptive numerical environment for scientific simulation) is a high-level software environment for solving integral and differential equations in many dimensions that uses adaptive and fast harmonic analysis methods with guaranteed precision that are based on multiresolution analysis and separated representations. Underpinning the numerical capabilities is a powerful petascale parallel programming environment that aims to increase both programmer productivity and code scalability. This paper describes the features and capabilities of MADNESS and briefly discusses some current applications in chemistry and several areas of physics.

Scalar relativistic explicitly correlated R12 methods.

Combinations of explicitly correlated R12 wave functions with relativistic Douglas-Kroll-Hess (DKH) Hamiltonians are discussed. We considered several ways to incorporate the relativistic terms into the second-order Moller-Plesset R12 method and applied them to the helium isoelectronic series to investigate their accuracy and numerical stability. Among the approaches are the evaluation of the relativistic terms via double resolution-of-the-identity and the explicit evaluation of all terms up to O(c(-4)) using the Pauli Hamiltonian. Numerical collapse of the latter can be avoided if the R12 amplitudes are determined by Katos cusp condition. Closed formulas for new two-electron integrals that include the mass-velocity term have been derived and implemented into the LIBINT2 integral library. The proposed approaches are not restricted to DKH and can be combined with other one- and two-component relativistic Hamiltonians.

Regularizing the molecular potential in electronic structure calculations. I. SCF methods

We present a method to remove the singular nuclear potential in a molecule and replace it with a regularized potential that is more amenable to be represented numerically. The singular nuclear potential is canceled by the similarity-transformed kinetic energy operator giving rise to an effective nuclear potential that contains derivative operators acting on the wave function. The method is fully equivalent to the non-similarity-transformed version. We give numerical examples within the framework of multi-resolution analysis for medium-sized molecules.