Van Der Waals Density Functionals Research Articles

Nonlocal van der Waals functionals: The case of rare-gas dimers and solids

Recently, the nonlocal van der Waals (vdW) density functionals [M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004)] have attracted considerable attention due to their good performance for systems where weak interactions are important. Since the physics of dispersion is included in these functionals, they are usually more accurate and show less erratic behavior than the semilocal and hybrid methods. In this work, several variants of the vdW functionals have been tested on rare-gas dimers (from He2 to Kr2) and solids (Ne, Ar, and Kr) and their accuracy compared to standard semilocal approximations, supplemented or not by an atom-pairwise dispersion correction [S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem. Phys. 132, 154104 (2010)]. An analysis of the results in terms of energy decomposition is also provided.

Structural, elastic, and vibrational properties of layered titanium dichalcogenides: A van der Waals density functional study

Using the van der Waals density functionals (vdW-DFs), we present results of a density functional theory (DFT) study on the structure of layered titanium dichalcogenides TiX(2) (X = S, Se, or Te). By accounting for non-local component in long-ranged electron correlation effects, the vdW-DFs lead to more accurate predictions of the structural parameters relative to conventional DFT. In particular, the vdW-optB86b functional [J. Klimeš et al., Phys. Rev. B 83, 195131 (2011)] gives the most agreement with experimental measurements for the lattice constant involving the vdW interactions. Based on the relaxed structure, the calculated elastic constants and vibrational frequencies are compared to experimental work and overall good agreement is obtained.

Improved description of soft layered materials with van der Waals density functional theory

The accurate description of van der Waals forces within density functional theory is currently one of the most active areas of research in computational physics and chemistry. Here we report results on the structural and energetic properties of graphite and hexagonal boron nitride, two layered materials where interlayer binding is dominated by van der Waals forces. Results from several density functionals are reported, including the optimized Becke88 van der Waals (optB88-vdW) and the optimized PBE van der Waals (optPBE-vdW) (Klimeš et al 2010 J. Phys.: Condens. Matter 22 022201) functionals. Where comparison to experiment and higher-level theory is possible, the results obtained from the two new van der Waals density functionals are in good agreement. An analysis of the physical nature of the interlayer binding in both graphite and hexagonal boron nitride is also reported.

Are we van der Waals ready?

We apply a range of density-functional-theory-based methods capable of describing van der Waals interactions with weakly bonded layered solids in order to investigate their accuracy for extended systems. The methods under investigation are the local-density approximation, semi-empirical force fields, non-local van der Waals density functionals and the random-phase approximation. We investigate the equilibrium geometries, elastic constants and binding energies of a large and diverse set of compounds and arrive at conclusions about the reliability of the different methods. The study also points to some directions of further development for the non-local van der Waals density functionals.

Benchmarking van der Waals density functionals with experimental data: potential-energy curves for H2 molecules on Cu(111), (100) and (110) surfaces

Detailed physisorption data from experiment for the H2 molecule on low-index Cu surfaces challenge theory. Recently, density functional theory (DFT) has been developed to account for nonlocal correlation effects, including van der Waals (dispersion) forces. We show that the functional vdW-DF2 gives a potential-energy curve, potential-well energy levels and difference in lateral corrugation promisingly close to the results obtained by resonant elastic backscattering–diffraction experiments. The backscattering barrier is sensitive to the choice of exchange functional approximation. Further, the DFT-D3 and TS-vdW corrections to traditional DFT formulations are also benchmarked, and deviations are analyzed.

Structure and stability of weakly chemisorbed ethene adsorbed on low-index Cu surfaces: performance of density functionals with van der Waals interactions

We have investigated the performance of popular density functionals that include van der Waals interactions for the experimentally well-characterized problem of ethene (C2H4) adsorbed on the low-index surfaces of copper. This set of functionals does not only include three van der Waals density functionals—vdwDF-PBE, vdwDF-revPBE and optB86b-vdwDF—and two dispersion-corrected functionals—Grimme and TS–but also local and semi-local functionals such as LDA and PBE. The adsorption system of ethene on copper was chosen because it is a weakly chemisorbed system for which the vdW interactions are expected to give a significant contribution to the adsorption energy. Overall the density functionals that include vdW interactions increased substantially the adsorption energies compared to the PBE density functional but predicted the same adsorption sites and very similar C–C bonding distances except for two of the van der Waals functionals. The top adsorption site was predicted almost exclusively for all functionals on the (110), (100) and (111) surfaces, which is in agreement with experiment for the (110) surface but not for the (100) surface. On the (100) surface, all functionals except two van der Waals density functionals singled out the observed cross-hollow site from the calculated C–C bonding distances and adsorption heights. On the top sites on the (110) surface and the cross-hollow site on the Cu(100) surface, the ethene molecule was found to form a weak chemisorption bond. On the (111) surface, all functionals gave a C–C bonding distance and an adsorption height more typical for physisorption, in agreement with experiments.

Comparing van der Waals Density Functionals for CO2 Adsorption in Metal Organic Frameworks

The accuracy of five recently proposed van der Waals (vdW) density functionals (optB86b, optB88, optPBE, revPBE, and rPW86), the semiempirical vdW method of Grimme (DFT-D2), and conventional local (LDA) and gradient-corrected (GGA-PBE) density functionals are assessed with respect to experimental enthalpies (ΔH) for CO2 adsorption in four prototypical metal organic frameworks (MOFs) containing coordinatively unsaturated metal sites: M/DOBDC (M = Mg, Ni, and Co) and Cu-HKUST-1. Although the LDA and GGA functionals partially capture trends, they significantly overbind (LDA) and underbind (GGA) CO2 with respect to the experimental enthalpies. The addition of a semiempirical r–6 dispersion term to the GGA exchange-correlation energy using “off the shelf” DFT-D2 parameters results in a substantial improvement both in trends and in the magnitude of the adsorption enthalpies. However, on average this approach still underbinds CO2 as compared to the experimental data by ∼7 kJ/mol (18%). Better accuracy is obtained with some of the nonempirical vdW density functionals, with the revPBE-based functional of Dion et al. [Phys. Rev. Lett.2004, 92, 246401] yielding an average error of only ∼2 kJ/mol (4%) relative to experiment. This improvement in energetics is accompanied by a slight decrease in the accuracy of predicted structures, as the revPBE functional overestimates the metal–CO2 bond length by about 10%. The identification of an efficient vdW density functional capable of predicting the thermodynamics of CO2 adsorption will facilitate rapid computational screening for optimal CO2 adsorbents.

Insight into the Adsorption of Water on the Clean CeO2(111) Surface with van der Waals and Hybrid Density Functionals

Understanding the interaction between water and ceria surfaces is crucial in many catalytic applications. For the clean CeO2(111) surface, density functional theory (DFT) calculations using different generalized gradient approximations (GGAs) to the exchange-correlation functional and the DFT(GGA)+U method have found that the most stable configuration is on top of a surface cerium atom. However, they disagree on the nature of the adsorption state, with water molecularly adsorbed with one or two Os–H hydrogen bonds (Os indicates a surface oxygen atom) or as a hydroxyl pair (OsHads–OHads), with only one recent report suggesting that these two structures are very close in energy. In this work, we studied the adsorption of water on CeO2(111) employing different approximations to exchange and correlation within DFT, namely, the Perdew–Burke–Ernzerhof (PBE) GGA, DFT(PBE)+U, the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional, and van der Waals (vdW) density functionals [DFT(vdW-DF/vdW-DF2)+U] with optimized exchange functionals (for vdW-DF, optB86b, revPBE, and optPBE; for vdW-DF2, rPW86). All of these methods predict close energies (10–100 meV range) for the two lowest-energy structures, the molecular structure with one Os–H bond and the hydroxyl pair. Our calculations show that these two species should be distinguishable by their infrared (IR) spectra. In particular, a rocking libration at 850 cm–1 could be used as an IR fingerprint to reveal the presence of the molecular structure. We found that the inclusion of vdW interactions increases binding energies by ∼0.18 eV, bringing them closer to the available experimental values.

A simplified implementation of van der Waals density functionals for first-principles molecular dynamics applications

We present a simplified implementation of the non-local van der Waals correlation functional introduced by Dion et al. [Phys. Rev. Lett. 92, 246401 (2004)] and reformulated by Román-Pérez et al. [Phys. Rev. Lett. 103, 096102 (2009)]. The proposed numerical approach removes the logarithmic singularity of the kernel function. Complete expressions of the self-consistent correlation potential and of the stress tensor are given. Combined with various choices of exchange functionals, five versions of van der Waals density functionals are implemented. Applications to the computation of the interaction energy of the benzene-water complex and to the computation of the equilibrium cell parameters of the benzene crystal are presented. As an example of crystal structure calculation involving a mixture of hydrogen bonding and dispersion interactions, we compute the equilibrium structure of two polymorphs of aspirin (2-acetoxybenzoic acid, C(9)H(8)O(4)) in the P2(1)/c monoclinic structure.

Benchmark Assessment of the Accuracy of Several van der Waals Density Functionals.

The nonlocal correlation functional VV10, developed recently in our group, describes the whole range of dispersion interactions in a seamless and general fashion using only the electron density as input. The VV10 functional has a simple analytic form that can be adjusted for pairing with the exchange functional of choice. In this paper, we use several benchmark data sets of weakly interacting molecular complexes to test the accuracy of two VV10 variants, differing in their treatment of the exchange component. For the sake of comparison, several other density functionals suitable for noncovalent interactions were also tested against the same benchmarks. We find that the "default'' version of VV10 with semilocal exchange gives very accurate geometries and binding energies for most van der Waals complexes but systematically overbinds hydrogen-bonded complexes. The alternative variant of VV10 with long-range corrected hybrid exchange performs exceptionally well for all types of weak bonding sampled in this study, including hydrogen bonds.

CO2 Capture by Metal–Organic Frameworks with van der Waals Density Functionals

We use density functional theory calculations with van der Waals corrections to study the role of dispersive interactions on the structure and binding of CO(2) within two distinct metal-organic frameworks (MOFs): Mg-MOF74 and Ca-BTT. For both classes of MOFs, we report calculations with standard gradient-corrected (PBE) and five van der Waals density functionals (vdW-DFs), also comparing with semiempirical pairwise corrections. The vdW-DFs explored here yield a large spread in CO(2)-MOF binding energies, about 50% (around 20 kJ/mol), depending on the choice of exchange functional, which is significantly larger than our computed zero-point energies and thermal contributions (around 5 kJ/mol). However, two specific vdW-DFs result in excellent agreement with experiments within a few kilojoules per mole, at a reduced computational cost compared to quantum chemistry or many-body approaches. For Mg-MOF74, PBE underestimates adsorption enthalpies by about 50%, but enthalpies computed with vdW-DF, PBE+D2, and vdW-DF2 (40.5, 38.5, and 37.4 kJ/mol, respectively) compare extremely well with the experimental value of 40 kJ/mol. vdW-DF and vdW-DF2 CO(2)-MOF bond lengths are in the best agreement with experiments, while vdW-C09(x) results in the best agreement with lattice parameters. On the basis of the similar behavior of the reduced density gradients around CO(2) for the two MOFs studied, comparable results can be expected for CO(2) adsorption in BTT-type MOFs. Our work demonstrates for this broad class of molecular adsorbate-periodic MOF systems that parameter-free and computationally efficient vdW-DF and vdW-DF2 approaches can predict adsorption enthalpies with chemical accuracy.

Al enhances the H2storage capacity of graphene at nanoribbon borders but not at central sites: A study using nonlocal van der Waals density functionals

We performed ab initio density-functional-theory calculations to investigate the adsorption of molecular hydrogen on pristine and Al-doped hydrogen-passivated zigzag graphene nanoribbons (GNRs) using nonlocal van der Waals functionals that have recently been proposed for an accurate description of exchange and correlation effects in weakly bound systems. Our results, which are compared with those obtained using the standard local density and generalized gradient approximations, show that neither pristine GNRs nor substitutionally Al-doped GNRs with an Al atom occupying a central, lateral, or subedge site satisfy the binding-energy criterion specified by the U.S. Department of Energy for novel hydrogen-storage materials. However, the criterion is satisfied by a GNR doped with an Al atom at an edge site and also by zigzag GNRs with adsorbed Al at lateral hole sites.

Performance of dispersion-corrected density functional theory for the interactions in ionic liquids

Potential energy curves for the dissociation of cation-anion associates representing the building units of ionic liquids have been computed with dispersion corrected DFT methods. Non-local van der Waals density functionals (DFT-NL) for the first time as well as an atom pair-wise correction method (DFT-D3) have been tested. Reference data have been computed at the extrapolated MP2/CBS and estimated CCSD(T)/CBS levels of theory. The investigated systems are combined from two cations (1-butyl-3-methylimidazolium and tributyl(methyl)posphonium) and three anions (chloride, dicyanamide, acetate). We find substantial stabilization from London dispersion energy near equilibrium of 5-7 kcal mol(-1) (about 5-6% of the interaction energy). Equilibrium distances are shortened by 0.03-0.09 Å and fundamental (inter-fragment) vibrational frequencies (which are in the range 140-180 cm(-1)) are increased by typically 10 cm(-1) when dispersion corrections are made. The dispersion-corrected hybrid functional potentials are in general in excellent agreement with the corresponding CCSD(T) reference data (typical deviations of about 1-2%). The DFT-D3 method performs unexpectedly well presumably because of cancellation of errors between the dispersion coefficients of the cations and anions. Due to self-interaction error, semi-local density functionals exhibit severe SCF convergence problems, and provide artificial charge-transfer and inaccurate interaction energies for larger inter-fragment distances. Although these problems may be alleviated in condensed phase simulations by effective Coulomb screening, only dispersion-corrected hybrid functionals with larger amounts of Fock-exchange can in general be recommended for such ionic systems.

Water wetting on representative metal surfaces: Improved description from van der Waals density functionals

We present a simple and useful approach based on the van der Waals density functional to investigate water wetting on two representative metal surfaces, Cu(110) and Ru(0001). We found that a mixed van der Waals density functional, by incorporating the Perdew–Burke–Ernzerhof (PBE) exchange for water–metal interactions and the revised PBE (revPBE) exchange of Zhang and Yang for hydrogen bonding, respectively, correctly predicts the wetting of extended water-chains and half-dissociated water layer on Ru(0001), as well as wetting of H-down bilayers on Cu(110), after correcting zero-point energy.

Van Der Waals Density Functionals for Graphene Layers and Graphite

In this communication, we present results of theoretical studies of various systems where Van der Waals interaction plays a considerable role. In the first-principle calculations performed in the density functional theory framework we implement novel functionals accounting for Van der Waals forces and employ to the test cases of graphite and graphene layers. It turns out that this approach provides a solution to the long standing problem of overbinding between graphene layers in bulk graphite, giving the distance between the carbon layers in excellent agreement with experiment. In graphene bilayers, Van der Waals functionals lead to energetic barriers for A–B to A–A ordering of graphene bilayers that are by a factor of two smaller than the barriers obtained with standard functionals. It may be of crucial importance, particularly, if one uses atomistic ab initio methods as a starting point for multi-scale modeling of materials and for determination of effective potentials.

Entropy of Liquid Water from Ab Initio Molecular Dynamics

We have computed the entropy of liquid water using a two-phase thermodynamic model and trajectories generated by ab initio molecular dynamics simulations. We present the results obtained with semilocal, hybrid, and van der Waals density functionals. We show that in all cases, at the experimental equilibrium density and at temperatures in the vicinity of 300 K, the computed entropies are underestimated, with respect to experiment, and the liquid exhibits a degree of tetrahedral order higher than in experiments. The underestimate is more severe for the PBE and PBE0 functionals than for several van der Waals functionals.

Structural and Vibrational Properties of Liquid Water from van der Waals Density Functionals.

We present results for the structural and vibrational properties of the water molecule, water dimer, and liquid water at the experimental equilibrium density, as obtained with several van der Waals density functionals. The functional form originally proposed by Dion et al. [ Phys. Rev. Lett. 2004 , 92 , 246401 ], with an appropriately chosen local exchange functional, yields a description of the liquid superior to that of the semilocal functional PBE. In particular, a specific choice of the local exchange functional (optB88) fitted to quantum chemistry calculations yields the best agreement with experimental results for pair correlation functions although it is slightly inferior to other van der Waals functionals in describing infrared spectra. When using optB88, liquid water displays a hydrogen-bonded network less tightly bound than when using the PBE approximation. The performance of optB88 is definitely inferior to that of the PBE0 hybrid functional for the isolated molecule but only moderately so for the liquid. However, the computational cost of optB88 is much less than that of hybrid functionals; therefore the use of optB88 appears to be a sensible alternative to calculations implying the evaluation of the Fock operator, in cases when simulations of large systems are required.

Comparison of the performance of dispersion-corrected density functional theory for weak hydrogen bonds

Potential energy curves for five complexes with weak to medium strong hydrogen bonds have been computed with dispersion corrected DFT methods. The electronic density based vdW-DF2 and VV10 van der Waals density functionals have been tested, as well as an atom pair-wise correction method (DFT-D3). The short-range exchange-correlation components BLYP and rPW86-PBE together with the extended aug-cc-pVQZ basis sets have been employed. Reference data have been computed at the estimated CCSD(T)/CBS(aQ-a5) level of theory. The investigated systems are CH(4)·NH(3), Cl(3)CH·NH(3), NH(3)·NH(3), CH(3)F·C(2)H(2) and CH(3)F·H(2)O with binding energies ranging from -0.7 kcal mol(-1) to -5.5 kcal mol(-1). We find that all dispersion corrected methods perform reasonably well for these hydrogen bonds, but also observe distinct differences. The BLYP-D3 method provides the best results for three out of five systems. For the fluorinated complexes, the VV10 method gives remarkably good results. The vdW-DF2 method yields good interaction energies similar to the other methods (mean average deviation of 0.2-0.3 kcal mol(-1)), but fails to provide accurate equilibrium separations. Based on these results and previous experience with the computation of non-covalent interactions, for large-scale applications we can recommend DFT-D3 based structure optimizations with subsequent checking of interaction energies by single-point VV10 computations. Comparison of the DFT-D3 and VV10 results leads to the conclusion that the short-range exchange-correlation functional and not the dispersion correction mainly determines the achievable accuracy.

Overcoming systematic DFT errors for hydrocarbon reaction energies

Despite the widespread use and numerous successful applications of density functional theory, descriptions of hydrocarbon reaction energies remain problematic. Illustrative examples include large underestimation of energies associated with alkane bond separation reactions and poor general description of intramolecular dispersion in hydrocarbons (e.g., B3LYP, MAD = 14.1 kcal mol−1). More recent, but not readily availably functionals, along with efficient posteriori corrections, not only show considerable improvement in the energy description of hydrocarbons but also help identify the sources of error in traditional DFT. Interactions in branched alkanes and compact hydrocarbons are adequately mimicked by systems compressed below their typical van der Waals distances. At these distances, standard DFT exchange functionals are overly repulsive for non-bonded density overlaps, and significant improvement is offered by the long-range corrected exchange functionals (e.g., LC-BLYP0.33, MAD = 5.5 kcal mol−1). For those systems, the neglect of long-range dispersion is found to be a critical shortcoming, as well as “overlap dispersion”, for which non-negligible amounts are captured by the correlation functional. Accounting for the missing dispersion interactions is of key importance. Accordingly, most noteworthy improvements over standard functionals are obtained by using non-local van der Waals density functionals (e.g., LC-S-VV09, MAD = 3.6 kcal mol−1, rPW86-VV09, MAD = 5.8 kcal mol−1), a dispersion corrected double hybrid (B2PLYP-D, MAD = 2.5 kcal mol−1), or by the addition of an atom pairwise density-dependent dispersion correction to a standard functional (e.g., PBE-dDXDM, MAD = 0.8 kcal mol−1). To a lesser extent, the reduction of the delocalization error (e.g., MCY3, MAD = 6.3 kcal mol−1) or careful parameter fitting (e.g., M06-2X, MAD = 5.6 kcal mol−1) also lowers the errors.

A first-principles study of weakly bound molecules using exact exchange and the random phase approximation

We present a study of the binding energy (BE) curves of rare gas and alkaline-earth dimers using an energy functional that includes exact exchange (EXX) and correlation energies within the random phase approximation (RPA). Our results for the equilibrium positions and long range behavior of the potential energy curves show great improvements over those obtained at the density functional theory level, within local and semilocal approximations. BEs are improved as well in the case of rare gas dimers. For Ar and Kr, the accuracy of our results is comparable to that of so-called van der Waals density functionals, although EXX/RPA yields BE curves that agree better with experiment for large separation distances, as expected. We also discuss shortcomings of the EXX/RPA perturbative approach and analyze possible sources of error in the description of the potential energy curve of alkaline-earth dimers, in particular, Be(2), exhibiting an unphysical maximum at large separations. We suggest that the lack of self-consistency in current EXX/RPA approaches might be largely responsible for most of the observed shortcomings. Finally, we present a tight-binding approach to obtain the eigenvalues of the dielectric matrix entering the calculation of the RPA correlation energy that greatly improves the efficiency of EXX/RPA calculations.