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.

http://dns2.asia.edu.tw/~ysho/YSHO-English/1000%20WC/PDF/J%20Che%20Phy132,%20154104.pdf

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

We present two new hybrid meta exchange- correlation functionals, called M06 and M06-2X. The M06 functional is parametrized including both transition metals and nonmetals, whereas the M06-2X functional is a high-nonlocality functional with double the amount of nonlocal exchange (2X), and it is parametrized only for nonmetals.The functionals, along with the previously published M06-L local functional and the M06-HF full-Hartree–Fock functionals, constitute the M06 suite of complementary functionals. We assess these four functionals by comparing their performance to that of 12 other functionals and Hartree–Fock theory for 403 energetic data in 29 diverse databases, including ten databases for thermochemistry, four databases for kinetics, eight databases for noncovalent interactions, three databases for transition metal bonding, one database for metal atom excitation energies, and three databases for molecular excitation energies. We also illustrate the performance of these 17 methods for three databases containing 40 bond lengths and for databases containing 38 vibrational frequencies and 15 vibrational zero point energies. We recommend the M06-2X functional for applications involving main-group thermochemistry, kinetics, noncovalent interactions, and electronic excitation energies to valence and Rydberg states. We recommend the M06 functional for application in organometallic and inorganometallic chemistry and for noncovalent interactions.

/pdf/the-m06-suite-of-density-functionals-for-main-group-1748swzp1l.pdf

The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals

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.

Effect of the damping function in dispersion corrected density functional theory

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

We report re-optimization of a recently proposed long-range corrected (LC) hybrid density functional [J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106] to include empirical atom–atom dispersion corrections. The resulting functional, ωB97X-D yields satisfactory accuracy for thermochemistry, kinetics, and non-covalent interactions. Tests show that for non-covalent systems, ωB97X-D shows slight improvement over other empirical dispersion-corrected density functionals, while for covalent systems and kinetics it performs noticeably better. Relative to our previous functionals, such as ωB97X, the new functional is significantly superior for non-bonded interactions, and very similar in performance for bonded interactions.

/pdf/long-range-corrected-hybrid-density-functionals-with-damped-44ti6za9f3.pdf

Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections

MP2 and CCSD(T) complete basis set (CBS) limit interaction energies and geometries for more than 100 DNA base pairs, amino acid pairs and model complexes are for the first time presented together. Extrapolation to the CBS limit is done by using two-point extrapolation methods and different basis sets (aug-cc-pVDZ – aug-cc-pVTZ, aug-cc-pVTZ – aug-cc-pVQZ, cc-pVTZ – cc-pVQZ) are utilized. The CCSD(T) correction term, determined as a difference between CCSD(T) and MP2 interaction energies, is evaluated with smaller basis sets (6-31G** and cc-pVDZ). Two sets of complex geometries were used, optimized or experimental ones. The JSCH-2005 benchmark set, which is now available to the chemical community, can be used for testing lower-level computational methods. For the first screening the smaller training set (S22) containing 22 model complexes can be recommended. In this case larger basis sets were used for extrapolation to the CBS limit and also CCSD(T) and counterpoise-corrected MP2 optimized geometries were sometimes adopted.

Benchmark database of accurate (MP2 and CCSD(T) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs.

Standard density functional theory (DFT) is augmented with a damped empirical dispersion term. The damping function is optimized on a small, well balanced set of 22 van der Waals (vdW) complexes and verified on a validation set of 58 vdW complexes. Both sets contain biologically relevant molecules such as nucleic acid bases. Results are in remarkable agreement with reference high-level wave function data based on the CCSD(T) method. The geometries obtained by full gradient optimization are in very good agreement with the best available theoretical reference. In terms of the standard deviation and average errors, results including the empirical dispersion term are clearly superior to all pure density functionals investigated-B-LYP, B3-LYP, PBE, TPSS, TPSSh, and BH-LYP-and even surpass the MP2/cc-pVTZ method. The combination of empirical dispersion with the TPSS functional performs remarkably well. The most critical part of the empirical dispersion approach is the damping function. The damping parameters should be optimized for each density functional/basis set combination separately. To keep the method simple, we optimized mainly a single factor, s(R), scaling globally the vdW radii. For good results, a basis set of at least triple-zeta quality is required and diffuse functions are recommended, since the basis set superposition error seriously deteriorates the results. On average, the dispersion contribution to the interaction energy missing in the DFT functionals examined here is about 15 and 100% for the hydrogen-bonded and stacked complexes considered, respectively.

Density functional theory augmented with an empirical dispersion term. Interaction energies and geometries of 80 noncovalent complexes compared with ab initio quantum mechanics calculations

Non-covalent interactions play an important role in chemistry, physics and especially in biodisciplines. They determine the structure of biomacromolecules such as DNA and proteins and are responsible for the molecular recognition process. Theoretical evaluation of interaction energies is difficult; however, perturbation as well as variation (supermolecular) methods are briefly described. Accurate interaction energies can be obtained by complete basis set limit calculations providing a large portion of correlation energy is covered (e.g. by performing CCSD(T) calculations). The role of H-bonding and stacking interactions in the stabilisation of DNA, oligopeptides and proteins is described, and the importance of London dispersion energy is shown.

Non-covalent interactions in biomacromolecules

Scaled MP3 interaction energies calculated as a sum of MP2/CBS (complete basis set limit) interaction energies and scaled third-order energy contributions obtained in small or medium size basis sets agree very closely with the estimated CCSD(T)/CBS interaction energies for the 22 H-bonded, dispersion-controlled and mixed non-covalent complexes from the S22 data set. Performance of this so-called MP2.5 (third-order scaling factor of 0.5) method has also been tested for 33 nucleic acid base pairs and two stacked conformers of porphine dimer. In all the test cases, performance of the MP2.5 method was shown to be superior to the scaled spin-component MP2 based methods, e.g. SCS–MP2, SCSN–MP2 and SCS(MI)–MP2. In particular, a very balanced treatment of hydrogen-bonded compared to stacked complexes is achieved with MP2.5. The main advantage of the approach is that it employs only a single empirical parameter and is thus biased by two rigorously defined, asymptotically correct ab-initio methods, MP2 and MP3. The method is proposed as an accurate but computationally feasible alternative to CCSD(T) for the computation of the properties of various kinds of non-covalently bound systems.

Scaled MP3 Non‐Covalent Interaction Energies Agree Closely with Accurate CCSD(T) Benchmark Data

The strengths of noncovalent interactions are generally very sensitive to a number of geometric parameters. Among the most important of these parameters is the separation between the interacting moieties (in the case of an intermolecular interaction, this would be the intermolecular separation). Most works seeking to characterize the properties of intermolecular interactions are mainly concerned with binding energies obtained at the potential energy minimum (as determined at some particular level of theory). In this work, in order to extend our understanding of these types of noncovalent interactions, we investigate the distance dependence of several types of intermolecular interactions, these are hydrogen bonds, stacking interactions, dispersion interactions, and X-H···π interactions. There are several methods that have traditionally been used to treat noncovalent interactions as well as many new methods that have emerged within the past three or four years. Here we obtain reference data using estimated CCSD(T) values at the complete basis set limit (using the CBS(T) method); potential energy curves are also produced using several other methods thought to be accurate for intermolecular interactions, these are MP2/cc-pVTZ, MP2/aug-cc-pVDZ, MP2/6-31G*(0.25), SCS(MI)-MP2/cc-pVTZ, estimated MP2.5/CBS, DFT-SAPT/aug-cc-pVTZ, DFT/M06-2X/6-311+G(2df,2p), and DFT-D/TPSS/6-311++G(3df,3pd). The basis set superposition error is systematically considered throughout the study. It is found that the MP2.5 and DFT-SAPT methods, which are both quite computationally intensive, produce potential energy curves that are in very good agreement to those of the reference method. Among the MP2 techniques, which can be said to be of medium computational expense, the best results are obtained with MP2/cc-pVTZ and SCS(MI)-MP2/cc-pVTZ. DFT-D/TPSS/6-311++G(3df,3pd) is the DFT-based method that can be said to give the most well-balanced description of intermolecular interactions.

Jiří Černý

Papers

Benchmark database of accurate (MP2 and CCSD(T) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs.

Density functional theory augmented with an empirical dispersion term. Interaction energies and geometries of 80 noncovalent complexes compared with ab initio quantum mechanics calculations

Non-covalent interactions in biomacromolecules

Scaled MP3 Non‐Covalent Interaction Energies Agree Closely with Accurate CCSD(T) Benchmark Data

On the Structure and Geometry of Biomolecular Binding Motifs (Hydrogen-Bonding, Stacking, X-H···π): WFT and DFT Calculations.