We investigate the performance of the dispersion correction D3 with and without an explicit three-body dispersion term for the energetic and structural properties of rare gas and molecular crystals. Therefore, the two- and three-body gradient of the dispersion energy is implemented in the periodic plane-wave program VASP. It is combined with different density functionals at the level of the general gradient approximation (GGA) and hybrid functionals. Cohesive energies and lattice parameters for the rare gas crystals Ar, Kr, and Xe and a set of 23 molecular crystals are calculated and compared to experimental reference values. In general, all tested methods yield very good results. For the molecular crystals the mean absolute deviation of lattice energies from reference data (about 1-2 kcal/mol) is close to or below their uncertainties. The influence of the three-body Axilrod-Teller-Muto dispersion term on energy and structure is found to be rather small. While on a GGA level cohesive energies become slightly worse, for hybrid functionals the three-body term improves the results.

Aromatic interactions play a key role in many chemical and biological systems. However, even if very simple models are chosen, the systems of interest are often too large to be handled with standard wave function theory (WFT). Although density functional theory (DFT) can easily treat systems of more than 200 atoms, standard semilocal (hybrid) density functional approximations fail to describe the London dispersion energy, a factor that is essential for accurate predictions of inter- and intramolecular noncovalent interactions. Therefore dispersion-corrected DFT provides a unique tool for the investigation and analysis of a wide range of complex aromatic systems. In this Account, we start with an analysis of the noncovalent interactions in simple model dimers of hexafluorobenzene (HFB) and benzene, with a focus on electrostatic and dispersion interactions. The minima for the parallel-displaced dimers of HFB/HFB and HFB/benzene can only be explained when taking into account all contributions to the interaction energy and not by electrostatics alone. By comparison of saturated and aromatic model complexes, we show that increased dispersion coefficients for sp(2)-hybridized carbon atoms play a major role in aromatic stacking. Modern dispersion-corrected DFT yields accurate results (about 5-10% error for the dimerization energy) for the relatively large porphyrin and coronene dimers, systems for which WFT can provide accurate reference data only with huge computational effort. In this example, it is also demonstrated that new nonlocal, density-dependent dispersion corrections and atom pairwise schemes mutually agree with each other. The dispersion energy is also important for the complex inter- and intramolecular interactions that arise in the molecular crystals of aromatic molecules. In studies of hexahelicene, dispersion-corrected DFT yields "the right answer for the right reason". By comparison, standard DFT calculations reproduce intramolecular distances quite accurately in single-molecule calculations while inter- and intramolecular distances become too large when dispersion-uncorrected solid-state calculations are carried out. Dispersion-corrected DFT can fix this problem, and these results are in excellent agreement with experimental structure and energetic (sublimation) data. Uncorrected treatments do not even yield a bound crystal state. Finally, we present calculations for the formation of a cationic, quadruply charged dimer of a porphyrin derivative, a case where dispersion is required in order to overcome strong electrostatic repulsion. A combination of dispersion-corrected DFT with an adequate continuum solvation model can accurately reproduce experimental free association enthalpies in solution. As in the previous examples, consideration of the electrostatic interactions alone does not provide a qualitatively or quantitatively correct picture of the interactions of this complex.

The chemically relevant intramolecular bond distance between a η3-allylic system and the metal center in a ruthenium complex has been reinvestigated by state-of-the-art dispersion-corrected DFT calculations. Previous works revealed significant deviations between the experimental X-ray crystal structure and results of molecular DFT gas-phase calculations. It was stated that this complex represents a general failure of DFT for transition-metal complex geometries. Our work shows that periodic DFT-D3 calculations for the crystal excellently reproduce all experimental data. The previously found deviations are attributed to crystal packing, the counterions, and improperly accounted dispersion interactions in the theoretical treatment.

The structures and relative energies of the three naturally occurring modifications of titanium dioxide (rutile, brookite and anatase) were investigated. For an accurate description, atom-pairwise dispersion-corrected density functional theory (DFT-D) was applied. The DFT-D3 scheme was extended non-empirically to improve the description of Ti atoms in bulk systems. New dispersion coefficients were derived from TDDFT calculations for electrostatically embedded TiO2 clusters. The dispersion coefficient C-6(TiTi) is reduced by a factor of 18 compared to the free atom. The three TiO2 modifications were optimized in periodic plane-wave calculations with dispersion-corrected GGA (PBE, revPBE) and hybrid density functionals (PBE0, revPBE0). The calculated lattice parameters are in good agreement with experimental data, in particular the dispersion-corrected PBE0 and revPBE0 hybrid functionals. Although the observed relative stabilities could not be reproduced in all cases, dispersion corrections improve the results. For an accurate description of bulk metal oxides, London dispersion is a prominent force that should not be neglected when energies and structures are computed with DFT. Additionally, the influence of dispersion interactions on the relaxation of the TiO2(110) surface is investigated.

Microporous organic molecular crystals (MOMCs) are materials composed of discrete organic molecules interacting noncovalently. The gas uptake of CO2, CH4, N-2, and Xe in one of the most widely investigated MOMCs, trispiro(benzodioxole[2'.2:2 ''.4:2 '''.6]1,3,5,2,4,6-triazatriphosphinine) (1), was computed by grand canonical Monte Carlo (GCMC) simulations based on our lately developed force field method vdW3, which is fitted to accurate ab initio potentials (double hybrid functional B2PLYP with a D3 dispersion correction using def2-TZVPP basis sets). The B2PLYP-D3 results compare very well with CCSD(T)/CBS interaction energies for benzene-gas model complexes. Our multiscale modeling approach to accurate interaction potentials is found to be essential in order to obtain reasonable adsorption isotherms and heats of adsorption. The good agreement between the simulation results and experimental data in all cases gives us the opportunity to design novel complex materials for gas adsorption based solely on first-principles calculations.

Dispersion-corrected density functional theory calculations (DFT-D3) were performed for the adsorption of CO on MgO and C(2) H(2) on NaCl surfaces. An extension of our non-empirical scheme for the computation of atom-in-molecules dispersion coefficients is proposed. It is based on electrostatically embedded M(4)X(4) (M=Na, Mg) clusters that are used in TDDFT calculations of dynamic dipole polarizabilities. We find that the C(MM)(6) dispersion coefficients for bulk NaCl and MgO are reduced by factors of about 100 and 35 for Na and Mg, respectively, compared to the values of the free atoms. These are used in periodic DFT calculations with the revPBE semi-local density functional. As demonstrated by calculations of adsorption potential energy curves, the new C(6) coefficients lead to much more accurate energies (E(ads)) and molecule-surface distances than with previous DFT-D schemes. For NaCl/C(2) H(2) we obtained at the revPBE-D3(BJ) level a value of E(ads) =-7.4 kcal mol(-1) in good agreement with experimental data (-5.7 to -7.1 kcal mol(-1)). Dispersion-uncorrected DFT yields an unbound surface state. For the MgO/CO system, the computed revPBE-D3(BJ) value of E(ads) =-4.1 kcal mol(-1) is also in reasonable agreement with experimental results (-3.0 kcal mol(-1)) when thermal corrections are taken into account. Our new dispersion correction also improves computed lattice constants of the bulk systems significantly compared to plain DFT or previous DFT-D results. The extended DFT-D3 scheme also provides accurate non-covalent interactions for ionic systems without empirical adjustments and is suggested as a general tool in surface science.

A benchmark set of 24 isomerization reactions of large organic molecules (consisting of 24 to 81 atoms) is presented (termed ISOL). The molecules are much larger than what is typically considered in thermochemical tests. To obtain reference isomerization energies, complete basis set (CBS) extrapolations at the (SCS)-MP2 level have been computed that are augmented by perturbative third-order corrections (SCS-MP3 and MP2.5 methods). Based on these carefully examined reference data, a diverse set of common density functionals varying from GGA to double-hybrid functional level with and without dispersion correction (DFT-D) is tested. Double-hybrid and the PBE0 hybrid functionals are found to be the methods of choice for the type of main group thermochemistry examined here. For all isomerizations with an average reaction energy of 22.7 kcal mol(-1) (in a range between 0.5 and 74.5 kcal mol(-1)), PBE0-D, B2PLYP-D and B2GP-PLYP-D yield mean absolute deviations of 2.5, 4.1 and 2.9 kcal mol(-1). Most importantly it is found that the use of a dispersion correction is essential if such large molecules are considered. For all DFT methods the MAD is lowered very significantly by 1.4-5.0 kcal mol(-1) when DFT-D is used. Intramolecular (mainly medium-range) London dispersion interactions account in some cases for more than 50% (41 kcal mol(-1)) of the isomerization energy even though the size of the systems remains unchanged. This study also demonstrates for the first time clearly that typical DFT errors are larger than expected (about 5 kcal mol(-1)) and that chemical accuracy (about 1 kcal mol(-1)) even for these electronically well-behaved molecules is currently not reached by DFT. We propose this new test set as a difficult challenge for electronic structure methods that claim to be routinely applicable to large molecules. We also suggest to use a distance range resolved dispersion energy as a diagnostic for problematic cases in DFT.

The influence of London dispersion effects in density functional theory for the computation of molecular crystals is investigated. The structure of dithienobicyclo[4.4.1]-undeca-3,8-dien-11-one ethylene glycol ketal (Resvan) that has been studied previously under isolated molecule conditions is considered as an example. The intramolecular stacking of the two thiophene rings is strongly influenced by non-covalent interactions. Previous investigations revealed significant deviations between computed molecular and experimentally observed crystal sulfur-sulfur distances. The influence of the crystal environment on this distance is estimated by calculating potential energy curves for the "gas phase" and in the crystal using the periodic plane-wave code VASP. With three common dispersion corrected (DFT-D3) density functionals (PBE, revPBE and PBEsol), an average shortening of the sulfur-sulfur distance by 0.10 A ("packing" effect) is computed which leads to a corrected "experimental" gas-phase value of R(S-S) = 4.19 A. This value is in good agreement with the results of reliable quantum chemical methods and serves as a challenging benchmark for electronic structure methods. Overall the crystal environment leads to a widening of the molecular structure due to intermolecular dispersion interaction. This is opposite to uncorrected DFT that suggest a compression due to Pauli exchange repulsion. In the crystal the intra- and inter-molecular dispersion effects are large but of opposite sign and their effect almost cancels out so that in this example dispersion-uncorrected DFT yields reasonable solid state structures. This is different for the computation of the lattice energy for which reasonable values are obtained only at the DFT-D3 level.

Time-dependent double-hybrid density functional methods are evaluated for the calculation of vertical singlet-singlet valence excitation energies of a wide variety of organic molecules. Beside the already published TD-B2-PLYP method, an analogous approach based on the recently published ground state B2GP-PLYP functional is presented for the first time. Double-hybrid functionals contain a hybrid-GGA-like part for which a conventional TDDFT linear response treatment is carried out. The thus obtained excitation energies are afterwards corrected by adding a non-local correlation portion, which is based on an CIS(D) type excited state perturbative correction. Both, TD-B2-PLYP and TD-B2GP-PLYP, are first applied to the 142 vertical singlet excitation energies in a benchmark set by Schreiber et al., that contains small and medium sized organic molecules. In a second part, a new benchmark set composed of five large organic dyes is proposed. Accurate reference values are derived from experimental 0-0 excitation energies in solution. A back-correction scheme based on TDDFT computations is presented by which solvent, relaxation and vibrational effects are removed, yielding experimental vertical gas phase excitation energies with an estimated accuracy of about +/-0.1 eV. The TD-B2-PLYP, TD-B2GP-PLYP and a variety of conventional TDDFT methods are then applied to this new benchmark set. The results for both considered test sets show that the new double-hybrid approaches yield the smallest mean absolute deviations of 0.22 eV for the first benchmark set and 0.19 eV (TD-B2-PLYP) and 0.16 eV (TD-B2GP-PLYP) for the new organic dye test set. Apart from a break-down of the perturbative correction for very high-lying transitions (larger than 8 eV), it is generally found that the double-hybrid functionals show high robustness and accuracy that cannot be obtained with conventional density functionals (e.g. B3-LYP).