Infrared photodissociation spectroscopy of (Al2O3)2-5FeO+: influence of Fe-substitution on small alumina clusters.

The infrared photodissociation spectra of He-tagged (Al2O3)nFeO+ (n = 2-5), are reported in the Al-O and Fe-O stretching and bending spectral region (430-1200 cm-1) and assigned based on calculated harmonic IR spectra from density functional theory (DFT). The substitution of Fe for an Al center occurs preferentially at 3-fold oxygen coordination sites located at the cluster rim and with the Fe atom in the +III oxidation state. The accompanying elongation of metal oxygen bonds leaves the Al-O network structure nearly unperturbed (isomorphous substitution). Contrary to the Al2FeO4+ (n = 1), valence isomerism is not observed, which is attributed to a smaller M:O ratio (M = Al, Fe) and consequently decreasing electron affinities with increasing cluster size.

Valence and Structure Isomerism of Al2FeO4+: Synergy of Spectroscopy and Quantum Chemistry.

We provide spectroscopic and computational evidence for a substantial change in structure and gas phase reactivity of Al3O4+ upon Fe-substitution, which is correctly predicted by multireference (MR) wave function calculations. Al3O4+ exhibits a cone-like structure with a central trivalent O atom (C3v symmetry). The replacement of the Al- by an Fe atom leads to a planar bicyclic frame with a terminal Al-O•- radical site, accompanied by a change from the Fe+III/O-II to the Fe+II/O-I valence state. The gas phase vibrational spectrum of Al2FeO4+ is exclusively reproduced by the latter structure, which MR wave function calculations correctly identify as the most stable isomer. This isomer of Al2FeO4+ is predicted to be highly reactive with respect to C-H bond activation, very similar to Al8O12+ which also features the terminal Al-O•- radical site. Density functional theory, in contrast, predicts a less reactive Al3O4+-like "isomorphous substitution" structure of Al2FeO4+ to be the most stable one, except for functionals with very high admixture of Fock exchange (50%, BHLYP).

Evaluating the Solvent Stark Effect from Temperature-Dependent Solvatochromic Shifts of Anthracene.

The solvent Stark effect on the spectral shifts of anthracene is studied with temperature-dependent solvatochromic measurements. The Stark contribution ΔvStark to the absorption shift Δvp in polar solvents is measured to be ΔvStark =(53±35) cm-1 , in reasonable agreement with dielectric continuum theory estimate of 28 cm-1 , whereas the major shift Δvp ∼300 cm-1 presumably originates from the solute quadrupole. We pay attention to the accurate correction of Δvp for the nonpolar contribution that is crucial when the shifts are modest in magnitude.

Temperature- and Structure-Dependent Optical Properties and Photophysics of BODIPY Dyes.

We report on the temperature- and structural-dependent optical properties and photophysics of a set of boron dipyrromethene (BODIPY) dyes with different substitution patterns of their meso-aryl subunit. Single-crystal X-ray diffraction analysis of the compounds enabled a classification of the dyes into a sterically hindered and a unhindered group. The steric hindrance refers to a blocked rotational motion of the aryl subunit around the bond connecting this moiety to the meso-position of the BODIPY core. The energy barriers related to this rotation were simulated by DFT calculations. As follows from the relatively low rotational barrier calculated to about 17 kcal/mol, a free rotation is only possible for sterically unhindered compounds. Rotational barriers of more than 40 kcal/mol determined for the sterically hindered compounds suggest an effective freezing of the rotational motion in these molecules. With the aid of temperature-dependent spectroscopic measurements, we could show that the ability to rotate directly affects the optical properties of our set of BODIPY dyes. This accounts for the strong temperature dependence of the fluorescence of the sterically unhindered compounds which show a drastic decrease in fluorescence quantum yield and a significant shortening in fluorescence lifetime upon heating. The optical properties of the sterically hindered compounds, however, are barely affected by temperature. Our results suggest a nonradiative deactivation of the first excited singlet state of the sterically unhindered compounds caused by a conical intersection of the potential energy surfaces of the ground and first excited state which is accessible by rotation of the meso-subunit. This is in good agreement with previously reported deactivation mechanisms. In addition, our results suggest the presence of a second nonradiative depopulation pathway of the first excited singlet state which is particularly relevant for the sterically hindered compounds.

Direct determination of optimal pair-natural orbitals in a real-space representation: The second-order Moller-Plesset energy.

An efficient representation of molecular correlated wave functions is proposed, which features regularization of the Coulomb electron-electron singularities via the F12-style explicit correlation and a pair-natural orbital factorization of the correlation components of the wave function expressed in the real space. The pair-natural orbitals are expressed in an adaptive multiresolution basis and computed directly by iterative variational optimization. The approach is demonstrated by computing the second-order Moller-Plesset energies of small- and medium-sized molecules. The resulting MRA-PNO-MP2-F12 method allows for the first time to compute correlated wave functions in a real-space representation for systems with dozens of atoms (as demonstrated here by computations on alkanes as large as C10H22), with precision exceeding what is achievable with the conventional explicitly correlated MP2 approaches based on the atomic orbital representations.

Chemically accurate adsorption energies for methane and ethane monolayers on the MgO(001) surface.

A hybrid QM:QM method that combines MP2 as high-level method on cluster models with density functional theory (PBE+D2) as low-level method on periodic models is applied to adsorption of methane and ethane on the MgO(001) surface for which reliable experimental desorption enthalpies are available. Two coverages are considered, monolayer (every second Mg2+ ion occupied) and one quarter coverage (one of eight Mg2+ ions occupied). Structure optimizations are performed at the hybrid MP2:(PBE+D2) level, with the MP2 energies and forces counterpoise corrected for basis set superposition error and extrapolated to the complete basis set limit. For the MP2 calculations on the adsorbate monolayer a two-body expansion of the lateral molecule-molecule interactions is applied. Higher order correlation effects are evaluated at the hybrid MP2:(PBE+D2) equilibrium structures as coupled cluster [CCSD(T)] - MP2 differences adopting smaller basis sets. The final adsorption energies obtained for monolayer coverage are -14.0 ± 1.0 and -23.3 ± 0.6 kJ mol-1 for CH4·MgO(001) and C2H6·MgO(001), respectively. They agree within 1 kJ mol-1 - well within chemical accuracy limits - with reference energies of -15.0 ± 0.6 and -24.4 ± 0.6 kJ mol-1, respectively. The latter have been derived from measured desorption enthalpy barriers, taking zero-point vibrational energy (ZPVE) and thermal enthalpy contributions into account.

Gas-Phase Vibrational Spectroscopy of the Aluminum Oxide Anions (Al2 O3 )1-6 AlO2.

We use cryogenic ion trap vibrational spectroscopy in combination with density functional theory to probe how the structural variability of alumina manifests itself in the structures of the gas-phase clusters (Al2 O3 )n AlO2- with n=1-6. The infrared photodissociation spectra of the D2 -tagged complexes, measured in the fingerprint spectral range (400-1200 cm-1 ), are rich in spectral features and start approaching the vibrational spectrum of amorphous alumina particles for n>4. Aided by a genetic algorithm, we find a trend towards the formation of irregular structures for larger n, with the exception of n=4, which exhibits a C3v ground-state structure. Locating the global minima of the larger systems proves challenging.

Analytic second nuclear derivatives of Hartree-Fock and DFT using multi-resolution analysis.

We present the formalism, implementation, and numerical results for the computation of second derivatives with respect to nuclear displacements of molecules in the formalism of multi-resolution analysis. The highly singular nuclear potentials are partially regularized to improve the numerical stability. Vibrational frequencies are well reproduced to within an RMS of a few cm-1 compared to large basis set LCAO (linear combination of atomic orbitals) calculations. Intermolecular modes, hindered rotations, and heavy atoms may lead to loss of precision. Tight precision thresholds are therefore necessary to converge to numerically stable results.

Gas phase structures and charge localization in small aluminum oxide anions: Infrared photodissociation spectroscopy and electronic structure calculations.

We use cryogenic ion trap vibrational spectroscopy in combination with quantum chemical calculations to study the structure of mono- and dialuminum oxide anions. The infrared photodissociation spectra of D2-tagged AlO1-4 (-) and Al2O3-6 (-) are measured in the region from 400 to 1200 cm(-1). Structures are assigned based on a comparison to simulated harmonic and anharmonic IR spectra derived from electronic structure calculations. The monoaluminum anions contain an even number of electrons and exhibit an electronic closed-shell ground state. The Al2O3-6 (-) anions are oxygen-centered radicals. As a result of a delicate balance between localization and delocalization of the unpaired electron, only the BHLYP functional is able to qualitatively describe the observed IR spectra of all species with the exception of AlO3 (-). Terminal Al-O stretching modes are found between 1140 and 960 cm(-1). Superoxo and peroxo stretching modes are found at higher (1120-1010 cm(-1)) and lower energies (850-570 cm(-1)), respectively. Four modes in-between 910 and 530 cm(-1) represent the IR fingerprint of the common structural motif of dialuminum oxide anions, an asymmetric four-member Al-(O)2-Al ring.

Numerically accurate linear response-properties in the configuration-interaction singles (CIS) approximation.

In the present work, we report an efficient implementation of configuration interaction singles (CIS) excitation energies and oscillator strengths using the multi-resolution analysis (MRA) framework to address the basis-set convergence of excited state computations. In MRA (ground-state) orbitals, excited states are constructed adaptively guaranteeing an overall precision. Thus not only valence but also, in particular, low-lying Rydberg states can be computed with consistent quality at the basis set limit a priori, or without special treatments, which is demonstrated using a small test set of organic molecules, basis sets, and states. We find that the new implementation of MRA-CIS excitation energy calculations is competitive with conventional LCAO calculations when the basis-set limit of medium-sized molecules is sought, which requires large, diffuse basis sets. This becomes particularly important if accurate calculations of molecular electronic absorption spectra with respect to basis-set incompleteness are required, in which both valence as well as Rydberg excitations can contribute to the molecule's UV/VIS fingerprint.

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.

Regularizing the molecular potential in electronic structure calculations. II. Many-body methods.

In Paper I of this series [F. A. Bischoff, "Regularizing the molecular potential in electronic structure calculations. I. SCF methods," J. Chem. Phys. 141, 184105 (2014)] a regularized molecular Hamilton operator for electronic structure calculations was derived and its properties in SCF calculations were studied. The regularization was achieved using a correlation factor that models the electron-nuclear cusp. In the present study we extend the regularization to correlated methods, in particular the exact solution of the two-electron problem, as well as second-order many body perturbation theory. The nuclear and electronic correlation factors lead to computations with a smaller memory footprint because the singularities are removed from the working equations, which allows coarser grid resolution while maintaining the precision. Numerical examples are given.

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.

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.

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).

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.

Reduction of Hartree-Fock Wavefunctions to Kohn-Sham Effective Potentials Using Multiresolution Analysis.

We present a highly accurate numerical implementation for computing the Kohn-Sham effective potentials for molecules based on a Hartree-Fock wavefunction and density, following the RKS approach of Staroverov and co-workers [ J. Chem. Phys. 2014, 140, 18A535]. Potentials and orbitals are represented in a multiresolution wavelet basis, avoiding basis set incompleteness-related issues. Together with the RKS method, the often occurring problems of oscillating potentials are removed. The MRA implementation of the RKS method allows the generation of molecular Kohn-Sham potentials of benchmark quality. Numerical data for atoms up to Kr and a number of molecules are given, with a special emphasis on the role of nodal planes in the calculations, as showcased in HCN and benzene.

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.

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 Kato's 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.

Thermochemistry of FeOmHnz Species: Assessment of Some DFT Functionals.

Thermochemical data for 20 anionic, cationic, and neutral gas-phase species, including Fe0/+, FeO-/0/+/2+, FeOH0/+/2+, FeO2-/0/+, OFeOH0/+, Fe(OH)20/+, Fe(H2O)+/2+, and Fe(H2O)2+/2+ with oxidation states between +I and +IV for Fe and -I and -II for O, compiled by Schröder [ J. Phys. Chem. A 2008, 112, 13215], are used to assess the performance of the "Jacob's ladder" functionals PBE, TPSS, PBE0, and TPSSh for the SVP, TZVP, and QZVP basis sets. In addition, the BP86 and B3LYP functionals are considered. The TPSSh functional performs best. With the TZVP basis set (recommended), the mean absolute and the maximum errors are 24 and 63 kJ/mol, respectively. With 32 and 78 kJ/mol, respectively, BP86 is second best, better than PBE.