Coupled-cluster treatment of complex open-shell systems: the case of single-molecule magnets.

We investigate the reliability of two cost-effective coupled-cluster methods for computing spin-state energetics and spin-related properties of a set of open-shell transition-metal complexes. Specifically, we employ the second-order approximate coupled-cluster singles and doubles (CC2) method and projection-based embedding that combines equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) with density functional theory (DFT). The performance of CC2 and EOM-CCSD-in-DFT is assessed against EOM-CCSD. The chosen test set includes two hexaaqua transition-metal complexes containing Fe(II) and Fe(III), and a large Co(II)-based single-molecule magnet with a non-aufbau ground state. We find that CC2 describes the excited states more accurately, reproducing EOM-CCSD excitation energies within 0.05 eV. However, EOM-CCSD-in-DFT excels in describing transition orbital angular momenta and spin-orbit couplings. Moreover, for the Co(II) molecular magnet, using EOM-CCSD-in-DFT eigenstates and spin-orbit couplings, we compute spin-reversal energy barriers, as well as temperature-dependent and field-dependent magnetizations and magnetic susceptibilities that closely match experimental values within spectroscopic accuracy. These results underscore the efficiency of CC2 in computing state energies of multi-configurational, open-shell systems and highlight the utility of the more cost-efficient EOM-CCSD-in-DFT for computing spin-orbit couplings and magnetic properties of complex and large molecular magnets.

Signatures of s-wave scattering in bound electronic states.

We compute EOM-EA-CCSD and EOM-EA-CCSDT potential energy curves and one-electron properties of several anions at bond lengths close to where these states become unbound. We compare the anions of HCl and pyrrole, which are associated with s-wave scattering, with N2 and H2, which correspond to resonances. For HCl and pyrrole, we observe, on inclusion of diffuse basis functions, a pronounced bending effect in the anionic potential energy curves near the crossing points with their corresponding neutral molecules. Additionally, we observe that the Dyson orbital and second moment of the electron density become extremely large in this region; for HCl, the size of the latter becomes 5 orders of magnitude larger over a range of 5 pm. This behaviour is not observed in H2 or N2. Our work thus shows that bound state electronic-structure methods can distinguish between anions that turn into electronic resonances and those associated with s-wave scattering states.

Influence of Transition Metal Electron Configuration on the Structure of Metal-EDTA Complexes.

The vibrational spectra of cold complexes of ethylenediaminetetraacetic acid (EDTA) with transition metal dications in vacuo show how the electronic structure of the metal provides a geometric template for interaction with the functional groups of the binding pocket. The OCO stretching modes of the carboxylate groups of EDTA serve as structural probes, informing on the spin state of the ion as well as the coordination number in the complex. The results highlight the flexibility of EDTA in accepting a large range of metal cations in its binding site.

Magnetic exchange interactions in binuclear and tetranuclear iron(III) complexes described by spin-flip DFT and Heisenberg effective Hamiltonians.

Low-energy spectra of single-molecule magnets (SMMs) are often described by Heisenberg Hamiltonians. Within this formalism, exchange interactions between magnetic centers determine the ground-state multiplicity and energy separation between the ground and excited states. In this contribution, we extract exchange coupling constants (J) for a set of iron (III) binuclear and tetranuclear complexes from all-electron calculations using non-collinear spin-flip time-dependent density functional theory (NC-SF-TDDFT). For 12 binuclear complexes with J-values ranging from -6 to -132 cm-1 , our benchmark calculations using the short-range hybrid ωPBEh functional and 6-31G(d,p) basis set agree well with the experimentally derived values (mean absolute error of 4.7 cm-1 ). For the tetranuclear SMMs, the computed J constants are within 6 cm-1 from the experimentally derived values. We explore the range of applicability of the Heisenberg model by analyzing bonding patterns in these Fe(III) complexes using natural orbitals (NO), their occupations, and the number of effectively unpaired electrons. The results illustrate the efficiency of the spin-flip protocol for computing the exchange couplings and the utility of the NO analysis in assessing the validity of effective spin Hamiltonians.

Equation-of-Motion Coupled-Cluster Protocol for Calculating Magnetic Properties: Theory and Applications to Single-Molecule Magnets.

We present a new computational protocol for computing macroscopic magnetic properties of transition-metal complexes using the equation-of-motion coupled-cluster (EOM-CC) framework. The approach follows a two-step state-interaction scheme: we first compute zero-order states using nonrelativistic EOM-CC and then use these states to evaluate matrix elements of the spin-orbit and Zeeman operators. Diagonalization of the resulting Hamiltonian yields spin-orbit- and field-perturbed eigenstates. Temperature- and field-dependent magnetization and susceptibility are computed by numerical differentiation of the partition function. To compare with powder-sample experiments, these quantities are numerically averaged over field orientations. We applied this protocol to several single-molecule magnets (SMMs) with Fe(II) and Fe(III) in trigonal pyramidal, linear, and trigonal bipyramidal coordination environments. We described the underlying electronic structure by the electron-attachment (EOM-EA) and spin-flip (EOM-SF) variants of EOM-CC. The computed energy barriers for spin inversion, and macroscopic magnetization and susceptibility agree well with experimental data. Trends in magnetic anisotropy and spin-reversal energy barriers are explained in terms of a molecular orbital picture rigorously distilled from spinless transition density matrices between many-body states. The results illustrate excellent performances of EOM-CC in describing magnetic behavior of mononuclear transition-metal SMMs.

Including dispersion in density functional theory for adsorption on flat oxide surfaces, in metal-organic frameworks and in acidic zeolites.

We examine the performance of nine commonly used methods for including dispersion interactions in density functional theory (DFT): three different parametrizations of damped 1/Rn terms (n = 6, 8, …) added to the DFT energy (Grimme's D2 and D3 parameterizations as well as that of Tkatchenko and Scheffler), three different implementations of the many-body dispersion approach (MBD, MBD/HI and MBD/FI), the density-dependent energy correction, called dDsC, and two "first generation" van der Waals density functionals, revPBE-vdW and optB86b-vdW. As test set we use eight molecule-surface systems for which agreement has been reached between experiment and hybrid QM:QM calculations within chemical accuracy limits (±4.2 kJ mol-1). It includes adsorption of carbon monoxide and dioxide in the Mg2(2,5-dioxido-1,4-benzenedicarboxylate) metal-organic framework (Mg-MOF-74, CPO-27-Mg), adsorption of carbon monoxide as well as of monolayers of methane and ethane on the MgO(001) surface, as well as adsorption of methane, ethane and propane in H-chabazite (H-CHA). D2 with Ne parameters for Mg2+, D2(Ne), MBD/HI and MBD/FI perform best. With the PBE functional, the mean unsigned errors are 6.1, 5.6 and 5.4 kJ mol-1, respectively.

Chemically Accurate Adsorption Energies: CO and H2O on the MgO(001) Surface.

Hybrid MP2:DFT-D structure optimizations are performed at BSSE-free CBS-extrapolated potential energy surfaces for molecule-oxide surface interactions (BSSE, basis set superposition error; CBS, complete basis set limit). Subsequently single point MP2 calculations are performed to estimate the effects of increasing the basis set size in the CBS extrapolation and increasing the cluster model size. The resulting estimates of the periodic MP2 limit agree within 1 kJ/mol with Local MP2 calculations using periodic boundary conditions. Single point CCSD(T) calculations are performed to determine ΔCC = CCSD(T) - MP2 energy differences. The final hybrid MP2:DFT-D+ΔCC estimate for CO on the MgO(001) surface at low coverage, -21.2 ± 0.5 kJ/mol, is in close agreement with the reference energy derived from temperature-programmed desorption experiments, -20.6 ± 2.4 kJ/mol. For H2O on MgO(001), at limiting zero coverage, we predict an adsorption energy of -53.7 ± 4.2 kJ/mol which falls in the range of values, -55.8 ± 12.2 kJ/mol, derived from a high coverage low energy electron diffraction experiments and estimated lateral interactions.

Ab initio study of methanol and ethanol adsorption on Brønsted sites in zeolite H-MFI.

We examine the interaction of methanol and ethanol with a bridging OH group of H-MFI (Al12-O20(H)-Si3 site). The computational standard approach for molecule-surface interaction uses density functional theory with inclusion of dispersion for energies and harmonic vibrational frequencies for entropies and finite temperature effects for enthalpies. At 300 K, this yields -117 and -135 kJ mol-1 for adsorption enthalpies of methanol and ethanol, respectively, and 59 and 61 kJ mol-1, respectively for their entropy terms -T·ΔS. To reach chemical accuracy (±4 kJ mol-1) we go beyond this approach. The energies are calculated using a hybrid QM:QM scheme (QM - quantum mechanics) which combines plane-wave density functional theory accounting for the periodicity of the system with wave function-based methods (Møller-Plesset perturbation and Coupled Cluster theories). Finite temperature and entropy contributions are calculated from anharmonic vibrational partition functions. This yields as final predictions for methanol and ethanol -84 and -104 kJ mol-1, respectively, for the enthalpies of adsorption, 56 and 48 kJ mol-1, respectively, for the -T·ΔS term, and -28 and -56 kJ mol-1, respectively, for the Gibbs free energies at 300 K.

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.

Ab†Initio Calculation of Rate Constants for Molecule-Surface Reactions with Chemical Accuracy.

The ab initio prediction of reaction rate constants for systems with hundreds of atoms with an accuracy that is comparable to experiment is a challenge for computational quantum chemistry. We present a divide-and-conquer strategy that departs from the potential energy surfaces obtained by standard density functional theory with inclusion of dispersion. The energies of the reactant and transition structures are refined by wavefunction-type calculations for the reaction site. Thermal effects and entropies are calculated from vibrational partition functions, and the anharmonic frequencies are calculated separately for each vibrational mode. This method is applied to a key reaction of an industrially relevant catalytic process, the methylation of small alkenes over zeolites. The calculated reaction rate constants (free energies), pre-exponential factors (entropies), and enthalpy barriers show that our computational strategy yields results that agree with experiment within chemical accuracy limits (less than one order of magnitude).

Geometrical correction for the inter- and intramolecular basis set superposition error in periodic density functional theory calculations.

We extend the previously developed geometrical correction for the inter- and intramolecular basis set superposition error (gCP) to periodic density functional theory (DFT) calculations. We report gCP results compared to those from the standard Boys-Bernardi counterpoise correction scheme and large basis set calculations. The applicability of the method to molecular crystals as the main target is tested for the benchmark set X23. It consists of 23 noncovalently bound crystals as introduced by Johnson et al. (J. Chem. Phys. 2012, 137, 054103) and refined by Tkatchenko et al. (J. Chem. Phys. 2013, 139, 024705). In order to accurately describe long-range electron correlation effects, we use the standard atom-pairwise dispersion correction scheme DFT-D3. We show that a combination of DFT energies with small atom-centered basis sets, the D3 dispersion correction, and the gCP correction can accurately describe van der Waals and hydrogen-bonded crystals. Mean absolute deviations of the X23 sublimation energies can be reduced by more than 70% and 80% for the standard functionals PBE and B3LYP, respectively, to small residual mean absolute deviations of about 2 kcal/mol (corresponding to 13% of the average sublimation energy). As a further test, we compute the interlayer interaction of graphite for varying distances and obtain a good equilibrium distance and interaction energy of 6.75 Å and -43.0 meV/atom at the PBE-D3-gCP/SVP level. We fit the gCP scheme for a recently developed pob-TZVP solid-state basis set and obtain reasonable results for the X23 benchmark set and the potential energy curve for water adsorption on a nickel (110) surface.