Implementation of empirical dispersion corrections to density functional theory for periodic systems

A recently developed empirical dispersion correction (Grimme et al., J. Chem. Phys. 2010, 132, 154104) to standard density functional theory (DFT‐D3) is implemented in the plane‐wave program package VASP. The DFT‐D3 implementation is compared with an implementation of the earlier DFT‐D2 version (Grimme, J. Comput. Chem. 2004, 25, 1463; Grimme, J. Comput. Chem. 2006, 27, 1787). Summation of empirical pair potential terms is performed over all atom pairs in the reference cell and over atoms in shells of neighboring cells until convergence of the dispersion energy is obtained. For DFT‐D3, the definition of coordination numbers has to be modified with respect to the molecular version to ensure convergence. The effect of three‐center terms as implemented in the original molecular DFT‐D3 version is investigated. The empirical parameters are taken from the original DFT‐D3 version where they had been optimized for a reference set of small molecules. As the coordination numbers of atoms in bulk and surfaces are much larger than in the reference compounds, this effect has to be discussed. The results of test calculations for bulk properties of metals, metal oxides, benzene, and graphite indicate that the original parameters are also suitable for solid‐state systems. In particular, the interlayer distance in bulk graphite and lattice constants of molecular crystals is considerably improved over standard functionals. With the molecular standard parameters (Grimme et al., J. Chem. Phys. 2010, 132, 154104; Grimme, J. Comput. Chem. 2006, 27, 1787) a slight overbinding is observed for ionic oxides where dispersion should not contribute to the bond. For simple adsorbate systems, such as Xe atoms and benzene on Ag(111), the DFT‐D implementations reproduce experimental results with a similar accuracy as more sophisticated approaches based on perturbation theory (Rohlfing and Bredow, Phys. Rev. Lett. 2008, 101, 266106). © 2012 Wiley Periodicals, Inc.     

Influence of collision energy on the dynamics of the reaction H (2S) + NH (X3Σ−) → N (4S) + H2 (X1Σ g + ) by the state-to-state quantum mechanical study

The conversion of cholesterol to pregnenolone is a physiologically essential process which initiates with two sequential hydroxylation processes catalyzed by cytochrome P450 side-chain cleavage enzyme (P450_SCC). Extensive efforts have been exerted; however, the mechanistic details remain obscure. In this work, we employed the dispersion-corrected density functional theoretical (DFT-D) calculations to investigate the mechanistic details of such hydroxylation processes. Calculated results reveal that the active intermediate Compound I (CpdI) of P450_SCC hydroxylates cholesterol efficiently, which coincides with previous spectrometric observations. The hydrogen bond effect of water molecule within the active site lowers the energy barrier significantly. Intriguingly, the adjacent hydrogen bond (H-bond) between the hydroxyl group of the substrate and the oxo group of CpdI in the second hydroxylation affects the H-abstraction significantly. Such H-bond was weakened during the C–H bond activation process, increasing the energy barriers by approximately 2 kcal/mol, which is different to the intermolecular H-bond effect of water₉₀₃ found by Shaik et al. that decreases the barrier by about 4 kcal/mol. Such adjacent H-bond also affects the transition state by bending the alignment of the C–H–O moiety, and consequently lowering the kinetic isotope effect values. Besides, a series of DFT-D calculations (Grimme’s D2, D3-zero, and D3-BJ methods) were performed and accessed to find out an appropriate protocol for H-bond containing hydroxylation process. Our results show that DFT-D single-point energies (SPE) based on geometries optimized with non-dispersion-corrected DFT varies drastically and sometime presents unreasonable results. DFT-D SPE calculations on DFT-D optimized geometries present stable and reasonable results.     

System-dependent dispersion coefficients for the DFT-D3 treatment of adsorption processes on ionic surfaces

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.     

Role and effective treatment of dispersive forces in materials: Polyethylene and graphite crystals as test cases

A semiempirical addition of dispersive forces to conventional density functionals (DFT-D) has been implemented into a pseudopotential plane-wave code. Test calculations on the benzene dimer reproduced the results obtained by using localized basis set, provided that the latter are corrected for the basis set superposition error. By applying the DFT-D/plane-wave approach a substantial agreement with experiments is found for the structure and energetics of polyethylene and graphite, two typical solids that are badly described by standard local and semilocal density functionals.     

Electronic structure and chemical bond in naphthalene and anthracene

We investigated the electronic structure of crystalline naphthalene and anthracene within the framework of density functional theory including van der Waals interactions (DFT-D). It is established that for better agreement with experimental values it is necessary to use the increased values of the van der Waals radii, which is caused by an overestimated value of the van der Waals interactions in crystalline linear oligoacenes. Utilization of the DFT-D leads to a correct account of the dispersion forces, which results in a high precision of the computed lattice parameters and cohesive energy. Based on the relaxed crystal structures, we have computed the total and deformation electron density and determined the mechanism of chemical bonds formation in crystals of naphthalene and anthracene. It has been established that the chemical bond in molecular crystals is formed under the influence of not only intramolecular but also intermolecular interactions. On the basis of the Mulliken population analysis it was revealed that two C(3) atoms in naphthalene (or C(3) and C(4) in anthracene) have a positive charge and the population of the rest of the carbon atoms increased, as compared with isolated molecule.     

Dispersion-corrected density functional theory calculations of the molecular binding of n-alkanes on Pd(111) and PdO(101)

We investigated the molecular binding of n-alkanes on Pd(111) and PdO(101) using conventional density functional theory (DFT) and the dispersion-corrected DFT-D3 method. In agreement with experimental findings, DFT-D3 predicts that the n-alkane desorption energies scale linearly with the molecule chain length on both surfaces, and that n-alkanes bind more strongly on PdO(101) than on Pd(111). The desorption energies computed using DFT-D3 are slightly higher than the measured values for n-alkanes on Pd(111), though the agreement between computation and experiment is a significant improvement over conventional DFT. The measured desorption energies of n-alkanes on PdO(101) and the energies computed using DFT-D3 agree to within better than 2.5 kJ/mol (< 5%) for chain lengths up to n-butane. The DFT-D3 calculations predict that the molecule-surface dispersion energy for a given n-alkane is similar in magnitude on Pd(111) and PdO(101), and that dative bonding between the alkanes and coordinatively unsaturated Pd atoms is primarily responsible for the enhanced binding of n-alkanes on PdO(101). From analysis of the DFT-D3 results, we estimate that the strength of an alkane η(2)(H, H) interaction on PdO(101) is ~16 kJ/mol, while a single η(1) H-Pd dative bond is worth about 10 kJ/mol.     

Protein-ligand interaction energies with dispersion corrected density functional theory and high-level wave function based methods

With dispersion-corrected density functional theory (DFT-D3) intermolecular interaction energies for a diverse set of noncovalently bound protein-ligand complexes from the Protein Data Bank are calculated. The focus is on major contacts occurring between the drug molecule and the binding site. Generalized gradient approximation (GGA), meta-GGA, and hybrid functionals are used. DFT-D3 interaction energies are benchmarked against the best available wave function based results that are provided by the estimated complete basis set (CBS) limit of the local pair natural orbital coupled-electron pair approximation (LPNO-CEPA/1) and compared to MP2 and semiempirical data. The size of the complexes and their interaction energies (ΔE(PL)) varies between 50 and 300 atoms and from -1 to -65 kcal/mol, respectively. Basis set effects are considered by applying extended sets of triple- to quadruple-ζ quality. Computed total ΔE(PL) values show a good correlation with the dispersion contribution despite the fact that the protein-ligand complexes contain many hydrogen bonds. It is concluded that an adequate, for example, asymptotically correct, treatment of dispersion interactions is necessary for the realistic modeling of protein-ligand binding. Inclusion of the dispersion correction drastically reduces the dependence of the computed interaction energies on the density functional compared to uncorrected DFT results. DFT-D3 methods provide results that are consistent with LPNO-CEPA/1 and MP2, the differences of about 1-2 kcal/mol on average (<5% of ΔE(PL)) being on the order of their accuracy, while dispersion-corrected semiempirical AM1 and PM3 approaches show a deviating behavior. The DFT-D3 results are found to depend insignificantly on the choice of the short-range damping model. We propose to use DFT-D3 as an essential ingredient in a QM/MM approach for advanced virtual screening approaches of protein-ligand interactions to be combined with similarly "first-principle" accounts for the estimation of solvation and entropic effects.     

A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions

A thorough energy benchmark study of various density functionals (DFs) is carried out with the new GMTKN30 database for general main group thermochemistry, kinetics and noncovalent interactions [Goerigk and Grimme, J. Chem. Theor. Comput., 2010, 6, 107; Goerigk and Grimme, J. Chem. Theor. Comput., 2011, 7, 291]. In total, 47 DFs are investigated: two LDAs, 14 GGAs, three meta-GGAs, 23 hybrids and five double-hybrids. Besides the double-hybrids, also other modern approaches, i.e., the M05 and M06 classes of functionals and range-separated hybrids, are tested. For almost all functionals, the new DFT-D3 correction is applied in order to consistently test the performance also for important noncovalent interactions; the parameters are taken from previous works or determined for the present study. Basis set and quadrature grid issues are also considered. The general aim of the study is to work out which functionals are generally well applicable and robust to describe the energies of molecules. In summary, we recommend on the GGA level the B97-D3 and revPBE-D3 functionals. The best meta-GGA is oTPSS-D3 although meta-GGAs represent in general no clear improvement compared to numerically simpler GGAs. Notably, the widely used B3LYP functional performs worse than the average of all tested hybrids and is also very sensitive to the application of dispersion corrections. We discourage its usage as a standard method without closer inspection of the results, as it still seems to be often done nowadays. Surprisingly, long-range corrected exchange functionals do in general not perform better than the corresponding standard hybrids. However, the ωB97X-D functional seems to be a promising method. The most robust hybrid is Zhao and Truhlar's PW6B95 functional in combination with DFT-D3. If higher accuracy is required, double-hybrids should be applied. The corresponding DSD-BLYP-D3 and PWPB95-D3 variants are the most accurate and robust functionals of the entire study. Additional calculations with MP2 and and its spin-scaled variants SCS-MP2, S2-MP2 and SOS-MP2 revealed that double-hybrids in general outperform those. Only SCS-MP2 can be recommended, particularly for reaction energies. We suggest its usage when a large self-interaction error is expected that prohibits usage of double-hybrids. Perdews' metaphoric picture of Jacob's Ladder for the classification of density functionals' performance could unbiasedly be confirmed with GMTKN30. We also show that there is no statistical correlation between a functional's accuracy for atomization energies and the performance for chemically more relevant reaction energies.     

Optical charge transfer transitions in supramolecular fullerene and porphyrin compounds

By the DFT method with the planar wave basis set and in the PAW approximation the geometric and electronic structures of four supramolecular compounds of porphyrin and fullerene molecules in the crystalline state are performed: H₂TPP·C₆₀·3 toluene, H₂T_pivPP·C₆₀, H₂T_3,5-dimethylPP·C₇₀·4 toluene, and NiT_4-methylPP·2C₇₀·2 toluene. The geometry is optimized using the PBE functional and the Grimme DFT-D2 dispersion interaction correction. The electronic structure and absorbance spectra are calculated using the HSE functional. It is shown that the H₂TPP·C₆₀·3 toluene structure having a sufficiently wide absorbance wavelength range, which results in a photoinduced electron transition from the higher occupied states formed by porphyrin molecules to the lower unoccupied states formed by fullerene molecules, is most promising for the design of photogalvanic elements.     

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.     

甲烷晶体的晶格能和弹性性质: 不同方法及泛函的评估

通过对甲烷晶体进行结构、晶格能和弹性特性的研究, 评估了不包含和包含色散能量修正的密度泛函理论的性能. 我们分别利用不包含色散能量修正的密度泛函理论(DFT) (包含不同的标准泛函和杂化泛函)和包含色散能量修正的密度泛函理论(DFT-D)计算了甲烷晶体特性, 并与实验作对比. 尽管DFT-D 与传统密度泛函理论及杂化密度泛函理论相比, 修正了甲烷晶体中的范德华(vdW)相互作用, 但是一些修正方案过分修正了这种相互作用. 因此, 人们在使用DFT-D方法时务必谨慎.     

Benchmark database on isolated small peptides containing an aromatic side chain: comparison between wave function and density functional theory methods and empirical force field

A detailed quantum chemical study on five peptides (WG, WGG, FGG, GGF and GFA) containing the residues phenylalanyl (F), glycyl (G), tryptophyl (W) and alanyl (A) -- where F and W are of aromatic character -- is presented. When investigating isolated small peptides, the dispersion interaction is the dominant attractive force in the peptide backbone-aromatic side chain intramolecular interaction. Consequently, an accurate theoretical study of these systems requires the use of a methodology covering properly the London dispersion forces. For this reason we have assessed the performance of the MP2, SCS-MP2, MP3, TPSS-D, PBE-D, M06-2X, BH&H, TPSS, B3LYP, tight-binding DFT-D methods and ff99 empirical force field compared to CCSD(T)/complete basis set (CBS) limit benchmark data. All the DFT techniques with a '-D' symbol have been augmented by empirical dispersion energy while the M06-2X functional was parameterized to cover the London dispersion energy. For the systems here studied we have concluded that the use of the ff99 force field is not recommended mainly due to problems concerning the assignment of reliable atomic charges. Tight-binding DFT-D is efficient as a screening tool providing reliable geometries. Among the DFT functionals, the M06-2X and TPSS-D show the best performance what is explained by the fact that both procedures cover the dispersion energy. The B3LYP and TPSS functionals-not covering this energy-fail systematically. Both, electronic energies and geometries obtained by means of the wave-function theory methods compare satisfactorily with the CCSD(T)/CBS benchmark data.     

Coupled perturbed Hartree-Fock for periodic systems: the role of symmetry and related computational aspects

A general and efficient implementation of the coupled perturbed Hartree-Fock (CPHF) scheme in the CRYSTAL06 code that applies to systems periodic in one dimension (polymers), two dimensions (slabs), three dimensions (crystals) and, as a limiting case, zero dimension (molecules) is presented. The dielectric tensor of large unit cell systems such as boehmite (gamma-AlOOH, 8 atoms/cell), calcite (CaCO3, 10 atoms/cell), and pyrope (Mg3Al2Si3O12, 80 atoms/cell) has been computed. Results are well converged with respect to the computational parameters, in particular, to the number of k points in the reciprocal space and tolerances used in the truncation of the Coulomb and exchange series, showing that the same standard computational conditions used for the self-consistent-field (SCF) step can also be used safely in a CPHF calculation. Point symmetry, being so important in determining crystal properties, also reduces dramatically the computational cost both of the preliminary SCF step and the CPHF calculation, so that the dielectric tensor for large unit cell systems such as pyrope can be computed within 2 CPU hours on a single processor PC.     

Density functional theory-based conformational analysis of a phospholipid molecule (dimyristoyl phosphatidylcholine)

The conformational space of the dimyristoyl phosphatidylcholine (DMPC) molecule has been studied using density functional theory (DFT), augmented with a damped empirical dispersion energy term (DFT-D). Fourteen ground-state isomers have been found with total energies within less than 1 kcal/mol. Despite differences in combinations of their torsion angles, all these conformers share a common geometric profile, which includes a balance of attractive, repulsive, and constraint forces between and within specific groups of atoms. The definition of this profile fits with most of the structural characteristics deduced from measured NMR properties of DMPC solutions. The calculated vibrational spectrum of the molecule is in good agreement with experimental data obtained for DMPC bilayers. These results support the idea that DMPC molecules preserve their individual molecular structures in the various assemblies.     

Phase transition and chemical decomposition of shocked CO-N(2) mixture

Using quantum molecular dynamics simulations based on density functional theory including dispersion corrections (DFT-D), we have studied the thermophysical properties of liquid carbon monoxide and nitrogen (CO-N(2)) mixture under extreme conditions. Density functional theory (DFT) method significantly overestimates the pressure as compared to DFT-D. It is demonstrated that the van der Waals (vdW) interaction has a negative contribution to the pressure and tends to reduce the overestimation of the equilibrium volume. We also demonstrate that a negative slope of Hugoniot curve could possibly be caused by both the absorption of dissociation energy and the uncertainties in composition. As density and temperature increase along the Hoguniot curve, the system appears to undergo a continuous transition and provides for a much richer set of dissociation products. The influence of dissociated carbon and oxygen atoms on nitrogen molecules is also discussed.     

Van der Waals interactions in aromatic systems: structure and energetics of dimers and trimers of pyridine.

Full geometry optimizations at the dispersion-corrected DFT-BLYP level of theory were carried out for dimers and trimers of pyridine. The DFT-D interaction energies were checked against results from single-point SCS-MP2/aug-cc-pVTZ calculations. Three stacked structures and a planar H-bonded dimer were found to be very close in energy (interaction energies in the range from -3.4 to -4.0 kcal mol(-1)). Two T-shaped geometries are higher lying, by about 1 kcal mol(-1), which is explained by the more favorable electrostatic interactions in the stacked and H-bonded arrangements. The DFT-D approach has proved to be a reliable and efficient tool to explore the conformational space of aromatic van der Waals complexes and furthermore provides interaction energies with errors of less than 10-20 % of DeltaE. Comparisons with previous results obtained by using only partially optimized model geometries strongly indicate that unconstrained optimizations are mandatory in such weakly bonded low-symmetry systems.     

The role of dispersive forces determining the energetics of adsorption in Ti zeolites

Ti‐zeolites are interesting materials because of their key role in partial oxidation reactions, as well as under a fundamental point of view being regarded as single site catalysts. Both experimental and computational approaches have been widely applied to the characterization of their active sites, reaching a level of knowledge unmatchable by most other important catalysts. However, several questions are still open, being a proper energetic simulation of the adsorption process of simple molecules, fitting with the experimental outcomes, still missing. The present work wants to underline the role of dispersive forces in correctly determining the adsorption energies of H₂O and NH₃ in Ti chabazite: first dispersive contributions have been included through an ONIOM scheme, comparing the results from semiempirical Grimme scheme and fully ab initio MP2. Being the key contribution of dispersion proved, a fully periodic, Grimme dispersions inclusive approach has been applied, coming to results close to the experimental values. © 2016 Wiley Periodicals, Inc.