Stefan Grimme

Semiempirical GGA‐type density functional constructed with a long‐range dispersion correction

A new density functional (DF) of the generalized gradient approximation (GGA) type for general chemistry applications termed B97‐D is proposed. It is based on Beckes power‐series ansatz from 1997 and is explicitly parameterized by including damped atom‐pairwise dispersion corrections of the form C6 · R−6. A general computational scheme for the parameters used in this correction has been established and parameters for elements up to xenon and a scaling factor for the dispersion part for several common density functionals (BLYP, PBE, TPSS, B3LYP) are reported. The new functional is tested in comparison with other GGAs and the B3LYP hybrid functional on standard thermochemical benchmark sets, for 40 noncovalently bound complexes, including large stacked aromatic molecules and group II element clusters, and for the computation of molecular geometries. Further cross‐validation tests were performed for organometallic reactions and other difficult problems for standard functionals. In summary, it is found that B97‐D belongs to one of the most accurate general purpose GGAs, reaching, for example for the G97/2 set of heat of formations, a mean absolute deviation of only 3.8 kcal mol−1. The performance for noncovalently bound systems including many pure van der Waals complexes is exceptionally good, reaching on the average CCSD(T) accuracy. The basic strategy in the development to restrict the density functional description to shorter electron correlation lengths scales and to describe situations with medium to large interatomic distances by damped C6 · R−6 terms seems to be very successful, as demonstrated for some notoriously difficult reactions. As an example, for the isomerization of larger branched to linear alkanes, B97‐D is the only DF available that yields the right sign for the energy difference. From a practical point of view, the new functional seems to be quite robust and it is thus suggested as an efficient and accurate quantum chemical method for large systems where dispersion forces are of general importance.

A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu

The method of dispersion correction as an add-on to standard Kohn-Sham density functional theory (DFT-D) has been refined regarding higher accuracy, broader range of applicability, and less empiricism. The main new ingredients are atom-pairwise specific dispersion coefficients and cutoff radii that are both computed from first principles. The coefficients for new eighth-order dispersion terms are computed using established recursion relations. System (geometry) dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination numbers (CN). They are used to interpolate between dispersion coefficients of atoms in different chemical environments. The method only requires adjustment of two global parameters for each density functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of atomic forces. Three-body nonadditivity terms are considered. The method has been assessed on standard benchmark sets for inter- and intramolecular noncovalent interactions with a particular emphasis on a consistent description of light and heavy element systems. The mean absolute deviations for the S22 benchmark set of noncovalent interactions for 11 standard density functionals decrease by 15%-40% compared to the previous (already accurate) DFT-D version. Spectacular improvements are found for a tripeptide-folding model and all tested metallic systems. The rectification of the long-range behavior and the use of more accurate C(6) coefficients also lead to a much better description of large (infinite) systems as shown for graphene sheets and the adsorption of benzene on an Ag(111) surface. For graphene it is found that the inclusion of three-body terms substantially (by about 10%) weakens the interlayer binding. We propose the revised DFT-D method as a general tool for the computation of the dispersion energy in molecules and solids of any kind with DFT and related (low-cost) electronic structure methods for large systems.

Effect of the damping function in dispersion corrected density functional theory

It is shown by an extensive benchmark on molecular energy data that the mathematical form of the damping function in DFT‐D methods has only a minor impact on the quality of the results. For 12 different functionals, a standard “zero‐damping” formula and rational damping to finite values for small interatomic distances according to Becke and Johnson (BJ‐damping) has been tested. The same (DFT‐D3) scheme for the computation of the dispersion coefficients is used. The BJ‐damping requires one fit parameter more for each functional (three instead of two) but has the advantage of avoiding repulsive interatomic forces at shorter distances. With BJ‐damping better results for nonbonded distances and more clear effects of intramolecular dispersion in four representative molecular structures are found. For the noncovalently‐bonded structures in the S22 set, both schemes lead to very similar intermolecular distances. For noncovalent interaction energies BJ‐damping performs slightly better but both variants can be recommended in general. The exception to this is Hartree‐Fock that can be recommended only in the BJ‐variant and which is then close to the accuracy of corrected GGAs for non‐covalent interactions. According to the thermodynamic benchmarks BJ‐damping is more accurate especially for medium‐range electron correlation problems and only small and practically insignificant double‐counting effects are observed. It seems to provide a physically correct short‐range behavior of correlation/dispersion even with unmodified standard functionals. In any case, the differences between the two methods are much smaller than the overall dispersion effect and often also smaller than the influence of the underlying density functional.

Accurate description of van der Waals complexes by density functional theory including empirical corrections

An empirical method to account for van der Waals interactions in practical calculations with the density functional theory (termed DFT‐D) is tested for a wide variety of molecular complexes. As in previous schemes, the dispersive energy is described by damped interatomic potentials of the form C6R−6. The use of pure, gradient‐corrected density functionals (BLYP and PBE), together with the resolution‐of‐the‐identity (RI) approximation for the Coulomb operator, allows very efficient computations for large systems. Opposed to previous work, extended AO basis sets of polarized TZV or QZV quality are employed, which reduces the basis set superposition error to a negligible extend. By using a global scaling factor for the atomic C6 coefficients, the functional dependence of the results could be strongly reduced. The “double counting” of correlation effects for strongly bound complexes is found to be insignificant if steep damping functions are employed. The method is applied to a total of 29 complexes of atoms and small molecules (Ne, CH4, NH3, H2O, CH3F, N2, F2, formic acid, ethene, and ethine) with each other and with benzene, to benzene, naphthalene, pyrene, and coronene dimers, the naphthalene trimer, coronene · H2O and four H‐bonded and stacked DNA base pairs (AT and GC). In almost all cases, very good agreement with reliable theoretical or experimental results for binding energies and intermolecular distances is obtained. For stacked aromatic systems and the important base pairs, the DFT‐D‐BLYP model seems to be even superior to standard MP2 treatments that systematically overbind. The good results obtained suggest the approach as a practical tool to describe the properties of many important van der Waals systems in chemistry. Furthermore, the DFT‐D data may either be used to calibrate much simpler (e.g., force‐field) potentials or the optimized structures can be used as input for more accurate ab initio calculations of the interaction energies.

Semiempirical hybrid density functional with perturbative second-order correlation

A new hybrid density functional for general chemistry applications is proposed. It is based on a mixing of standard generalized gradient approximations (GGAs) for exchange by Becke (B) and for correlation by Lee, Yang, and Parr (LYP) with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional contains only two global parameters that describe the mixture of HF and GGA exchange (a(x)) and of the PT2 and GGA correlation (c), respectively. The parameters are obtained in a least-squares-fit procedure to the G297 set of heat of formations. Opposed to conventional hybrid functionals, the optimum a(x) is found to be quite large (53% with c=27%) which at least in part explains the success for many problematic molecular systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G297 standard benchmark set, a second test suite of atoms, molecules, and reactions that are considered as electronically very difficult (including transition-metal compounds, weakly bonded complexes, and reaction barriers) and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose density functional for molecules (e.g., a mean absolute deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, respectively, for the best other density functionals). Very importantly, also the maximum and minimum errors (outliers) are strongly reduced (by about 10-20 kcal/mol). Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equilibrium properties. Preliminary calculations of the equilibrium bond lengths and harmonic vibrational frequencies for diatomic molecules and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chemical systems emphasizes the need of (virtual) orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chemical method of general purpose.

Improved second-order Møller–Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies

A simple modification of second-order Moller–Plesset perturbation theory (MP2) to improve the description of molecular ground state energies is proposed. The total MP2 correlation energy is partitioned into parallel- and antiparallel-spin components which are separately scaled. The two parameters (scaling factors), whose values can be justified by basic theoretical arguments, have been optimized on a benchmark set of 51 reaction energies composed of 74 first-row molecules. It is found, that the new method performs significantly better than standard MP2: the rms [mean absolute error (MAE)] deviation drops from 4.6 (3.3) to 2.3 (1.8) kcal/mol. The maximum error is reduced from 13.3 to 5.1 kcal/mol. Significant improvements are especially observed for cases which are usually known as MP2 pitfalls while cases already described well with MP2 remain almost unchanged. Even for 11 atomization energies not considered in the fit, uniform improvements [MAE: 8.1 kcal/mol (MP2) versus 3.2 kcal/mol (new)] are found. Th...

Density functional theory with London dispersion corrections

Dispersion corrections to standard Kohn–Sham density functional theory (DFT) are reviewed. The focus is on computationally efficient methods for large systems that do not depend on virtual orbitals or rely on separated fragments. The recommended approaches (van der Waals density functional and DFT‐D) are asymptotically correct and can be used in combination with standard or slightly modified (short‐range) exchange–correlation functionals. The importance of the dispersion energy in intramolecular cases (conformational problems and thermochemistry) is highlighted.

Double-hybrid density functionals with long-range dispersion corrections: higher accuracy and extended applicability

The objective of this work is the further systematic improvement of the accuracy of Double-Hybrid Density Functionals (DHDF) that add non-local electron correlation effects to a standard hybrid functional by second-order perturbation theory (S. Grimme, J. Chem. Phys., 2006, 124, 034108). The only known shortcoming of these generally highly accurate functionals is an underestimation of the long-range dispersion (van der Waals) interactions. To correct this deficiency, we add a previously developed empirical dispersion term (DFT-D) to the energy expression but leave the electronic part of the functional untouched. Results are presented for the S22 set of non-covalent interaction energies, the G3/99 set of heat of formations and conformational energies of a phenylalanyl-glycyl-glycine peptide model. We furthermore propose seven hydrocarbon reactions with strong intramolecular dispersion contributions as a benchmark set for newly developed density functionals. In general, the proposed composite approach is for many chemically relevant properties of similar quality as high-level coupled-cluster treatments. A significant increase of the accuracy for non-covalent interactions is obtained and the corrected B2PLYP DHDF provides one of the lowest ever obtained Mean Absolute Deviations (MAD) for the S22 set (0.2-0.3 kcal mol(-1)). Unprecedented high accuracy is also obtained for the relative energies of peptide conformations that turn out to be very difficult. The significant improvements found for the G3/99 set (reduction of the MAD from 2.4 to 1.7 kcal mol(-1)) underline the importance of intramolecular dispersion effects in large molecules. In all tested cases the results from the standard B3LYP approach are also significantly improved, and we recommend the general use of dispersion corrections in DFT treatments.

Do Special Noncovalent π–π Stacking Interactions Really Exist?†

Noncovalent interactions play an increasingly important role in modern chemical research, and are nowadays considered as cornerstones in supramolecular chemistry, materials science, and even biochemistry. When unsaturated organic groups are involved in noncovalent interactions, the terms “p–p stacking”, or more generally “p–p interactions” are often used. As noted recently, this classification has a quite mysterious flavor. For larger structures, p–p stacking is a phenomenon that is theoretically not well understood, although some progress has been made. From many studies of the benzene dimer and other complexes involving phenyl rings, it can be concluded that the p orbitals do not function as in conventional overlapdriven covalent bonding, although this is not common knowledge. The prototypical benzene dimer is nowadays considered a typical van der Waals complex in which the long-range dispersion interactions (dominant R 6 dependence of the interaction energy on interfragment distance) play the major role. As a consequence, the dimer is unbound at uncorrelated Hartree–Fock and many density functional theory (DFT) levels. This more sophisticated view is increasingly replacing Hunter6s model of p–p interactions, which (over)emphasises the mainly quadrupole–quadrupole electrostatic component of the interaction in benzene-type systems (see Ref. [13] for recent theoretical work on polar psystems). Because van der Waals complexes are formed by almost all neutral, closed-shell molecules, which are considered exclusively herein, what should be so special about the interaction between stacked aromatic units compared to, for example, saturated (hydrogenated) rings of about the same size. This mainly energetic difference is termed herein the p–p stacking effect (PSE). For example, benzene and cyclohexane both exist as fluids at room temperature, which indicates similar intermolecular interactions. According to accurate CCSD(T) computations, the stacked (parallel-displaced, PD) benzene dimer has an even smaller binding energy than the pentane dimer ( 2.8 vs. 3.9 kcalmol ), 14] which has the same number of electrons. These observations seem to be incompatible with the assumption of special p–p interactions. On the other hand, it is known that larger polycyclic aromatic hydrocarbons (PAHs) behave differently to large alkanes; for example, PAHs become increasingly insoluble in common organic solvents with increasing size. Thus the magnitude of the intermolecular interactions and possibly also their fundamental character is more strongly size-dependent in aromatic systems than in saturated systems. The clarification of this matter, and the question as to whether the term “p–p interaction” makes sense from a theoretical point of view, is the central topic of the work presented herein. The linear condensed acenes, from benzene (number of rings n= 1) to tetracene (n= 4), and the corresponding perhydrogenated ring systems (all trans–all anti stereoisomers) were used as models. Homo-dimers of stacked (aromatic with Ci, except for the PD benzene dimer, which has C2h symmetry, and saturated with C2h symmetry) and T-shaped orientation (aromatic only, C2v) are investigated. The Tshaped forms are important in the crystal packing of aromatic molecules, as analyzed in detail by Desiraju and Gavezzotti. For saturated dimers, no well-defined T-shaped structures could be found. Energy-minimized dimer structures for n= 1 and n= 4 are shown as an example in Figure 1.

Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals-Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions.

We present an extended and improved version of our recently published database for general main group thermochemistry, kinetics, and noncovalent interactions [J. Chem. Theory Comput. 2010, 6, 107], which is dubbed GMTKN30. Furthermore, we suggest and investigate two new double-hybrid-meta-GGA density functionals called PTPSS-D3 and PWPB95-D3. PTPSS-D3 is based on reparameterized TPSS exchange and correlation contributions; PWPB95-D3 contains reparameterized PW exchange and B95 parts. Both functionals contain fixed amounts of 50% Fock-exchange. Furthermore, they include a spin-opposite scaled perturbative contribution and are combined with our latest atom-pairwise London-dispersion correction [J. Chem. Phys. 2010, 132, 154104]. When evaluated with the help of the Laplace transformation algorithm, both methods scale as N(4) with system size. The functionals are compared with the double hybrids B2PLYP-D3, B2GPPLYP-D3, DSD-BLYP-D3, and XYG3 for GMTKN30 with a quadruple-ζ basis set. PWPB95-D3 and DSD-BLYP-D3 are the best functionals in our study and turned out to be more robust than B2PLYP-D3 and XYG3. Furthermore, PWPB95-D3 is the least basis set dependent and the best functional at the triple-ζ level. For the example of transition metal carbonyls, it is shown that, mainly due to the lower amount of Fock-exchange, PWPB95-D3 and PTPSS-D3 are better applicable than the other double hybrids. Finally, we discuss in some detail the XYG3 functional [Proc. Nat. Acad. Sci. U.S.A. 2009, 106, 4963], which makes use of B3LYP orbitals and electron densities. We show that it is basically a highly nonlocal variant of B2PLYP and that its partially good performance is mainly due to a larger effective amount of perturbative correlation compared to other double hybrids. We finally recommend the PWPB95-D3 functional in general chemistry applications.