Binding energy curves from nonempirical density functionals II. van der Waals bonds in rare-gas and alkaline-earth diatomics

Binding energy curves have been calculated for the ground-state rare-gas diatomics Ne(2) and Ar(2) and for the alkaline-earth diatomic Be(2) using the nonempirical density functionals from the first three rungs of a ladder of approximations: the local spin density (LSD) approximation, the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and the Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA. Binding energy curves in reasonable agreement with those constructed from experiment are found from PBE and TPSS, which incorporate inhomogeneity corrections that satisfy the Lieb-Oxford bound and so describe the short-range part of the van der Waals interaction. At large internuclear separation, these functionals produce an exponentially decaying attraction in place of the correct long-range -C(6)/R(6). Basis-set and exchange-only effects are also discussed.     

Comparative DFT study of van der Waals complexes: rare-gas dimers, alkaline-earth dimers, zinc dimer, and zinc-rare-gas dimers

Recent interest in the application of density functional theory prompted us to test various functionals for the van der Waals interactions in the rare-gas dimers, the alkaline-earth metal dimers, zinc dimer, and zinc-rare-gas dimers. In the present study, we report such tests for 18 DFT functionals, including both some very recent functionals and some well-established older ones. We draw the following conclusions based on the mean errors in binding energies and complex geometries: (1) B97-1 gives the best performance for predicting the geometry of rare-gas dimers, whereas M05-2X and B97-1 give the best energetics for rare-gas dimers. (2) PWB6K gives the best performance for the prediction of the geometry of the alkaline-earth metal dimers, zinc dimers, and zinc-rare-gas dimers. M05-2X gives the best energetics for the metal dimers, whereas B97-1 gives the best energetics for the zinc-rare-gas dimers. (3) The M05 functional is unique in providing good accuracy for both covalent transition-metal dimers and van der Waals metal dimers. (4) The combined mean percentage unsigned error in geometries and energetics shows that M05-2X and MPWB1K are the overall best methods for the prediction of van der Waals interactions in metal and rare-gas van der Waals dimers.     

A density functional theory study of van der Waals interaction in carbon nanotubes

In this article, we investigate the effect of van der Waals force in zigzag carbon nanotubes (CNTs) including single-wall CNT (SWCNT) and double-walled CNT (DWCNT) structures with several interaction configurations. The solid-state density functional theory is employed to calculate the geometric optimization, normal mode frequencies, and IR and Raman spectra with the periodic boundary condition. For SWCNTs, we find that the Raman intensity is not affected by the tube diameter or the electronic structure. The IR absorption, however, increases with the tube diameter. We find that the close metallicity of the electronic structure has a significant impact on the IR simulations. When the van der Waals force is applied outside the CNTs at a distance longer than 3.0, the effect on Raman spectra is minimal but some effects can still be confirmed by IR absorption. When the van der Waals force acts inside the CNTs, the effect on the spectrum can be observed, especially at a distance of 2.8 Å, both IR and Raman can be significantly enhanced in many modes.     

Nonlocal van der Waals density functional: the simpler the better

We devise a nonlocal correlation energy functional that describes the entire range of dispersion interactions in a seamless fashion using only the electron density as input. The new functional is considerably simpler than its predecessors of a similar type. The functional has a tractable and robust analytic form that lends itself to efficient self-consistent implementation. When paired with an appropriate exchange functional, our nonlocal correlation model yields accurate interaction energies of weakly-bound complexes, not only near the energy minima but also far from equilibrium. Our model exhibits an outstanding precision at predicting equilibrium intermonomer separations in van der Waals complexes. It also gives accurate covalent bond lengths and atomization energies. Hence the functional proposed in this work is a computationally inexpensive electronic structure tool of broad applicability.     

Water monomer interaction with gold nanoclusters from van der Waals density functional theory

We investigate the interaction between water molecules and gold nanoclusters Au(n) through a systematic density functional theory study within both the generalized gradient approximation and the nonlocal van der Waals (vdW) density functional theory. Both planar (n = 6-12) and three-dimensional (3D) clusters (n = 17-20) are studied. We find that applying vdW density functional theory leads to an increase in the Au-Au bond length and a decrease in the cohesive energy for all clusters studied. We classify water adsorption on nanoclusters according to the corner, edge, and surface adsorption geometries. In both corner and edge adsorptions, water molecule approaches the cluster through the O atom. For planar clusters, surface adsorption occurs in a O-up/H-down geometry with water plane oriented nearly perpendicular to the cluster. For 3D clusters, water instead favors a near-flat surface adsorption geometry with the water O atom sitting nearly atop a surface Au atom, in agreement with previous study on bulk surfaces. Including vdW interaction increases the adsorption energy for the weak surface adsorption but reduces the adsorption energy for the strong corner adsorption due to increased water-cluster bond length. By analyzing the adsorption induced charge rearrangement through Bader's charge partitioning and electron density difference and the orbital interaction through the projected density of states, we conclude that the bonding between water and gold nanocluster is determined by an interplay between electrostatic interaction and covalent interaction involving both the water lone-pair and in-plane orbitals and the gold 5d and 6s orbitals. Including vdW interaction does not change qualitatively the physical picture but does change quantitatively the adsorption structure due to the fluxionality of gold nanoclusters.     

A van der Waals density functional study of ice Ih

Density functional theory with the van der Waals density functional (vdW-DF) is used to calculate equilibrium crystal structure, binding energy, and bulk modulus of ice Ih. It is found that although it overestimates the equilibrium volume, vdW-DF predicts accurate binding energy of ice Ih, as compared with high level quantum chemistry calculations and experiment. Inclusion of the nonlocal correlation, i.e., van der Waals interaction, leads to an overall improvement over the standard generalized gradient approximation in describing water ice.     

An accurate analytical formula for the van der Waals potentials of homonuclear rare-gas dimers with one adjustable parameter

In the present article, the Tang–Toennies–Yiu (TTY) potential model is modified by introducing one adjustable parameter. Then, the van der Waals potentials of He₂, Ne₂, Ar₂, Kr₂, and Xe₂ are calculated by this model with the adjustable parameter being determined by the well determined well depth D_e of these systems. Based on the derived potentials, the vibrational energy spacings of these systems are also calculated. It is shown that the present derived potentials and vibrational energy spacings agree well with experiment and other theoretical calculations. Finally, the normalization constant A in the asymptotic wave function of rare-gas atoms is estimated. The present derived normalization constant A is very close to the one by calculating the ratio between the Hartree–Fock function and the asymptotic wave function. The results confirm that absorbing the first-order polarization energy into the exchange energy expression is a well approximation for the present systems.     

van der Waals interactions of polycyclic aromatic hydrocarbon dimers

Density functional theory is in principle exact and includes also long-range interactions, such as the van der Waals interactions. These are, however, part of the exchange-correlation energy functional that needs to be approximated, and are absent in the local and semilocal standard implementations. Recently a density functional which includes van der Waals interactions for planar systems has been developed, which we show can be extended to provide a treatment of planar molecules. We use this functional to calculate binding distances and energies for dimers of three of the smallest polycyclic aromatic hydrocarbons (PAHs)--naphthalene, anthracene, and pyrene.     

Linear response time-dependent density functional theory for van der Waals coefficients

A linear response time-dependent density functional theory is described and used to calculate the dynamic polarizabilities and van der Waals C(6) coefficients of complex atom pairs. We present values of C(6) for dimers of main group atoms and the first row of transition metal atoms.     

Self-consistent implementation of a nonlocal van der Waals density functional with a Gaussian basis set

Nearly all common density functional approximations fail to properly describe dispersion interactions responsible for binding in van der Waals complexes. Empirical corrections can fix some of the failures but cannot fully grasp the complex physics and may not be reliable for systems dissimilar to the fitting set. In contrast, the recently proposed nonlocal van der Waals density functional (vdW-DF) was derived from first principles, describes dispersion interactions in a seamless fashion, and yields the correct asymptotics. Implementation of this functional is somewhat cumbersome: Nonlocal dependence on the electron density requires numerical double integration over the space variables and functional derivatives are nontrivial. This paper shows how vdW-DF can be implemented self-consistently with Gaussian basis functions. The gradients of the energy with respect to nuclear displacements have also been derived and coded, enabling efficient geometry optimizations. We test the vdW-DF correlation functional in combination with several exchange approximations. We also study the sensitivity of the method to the basis set size and to the quality of the numerical quadrature grid. For weakly interacting systems, acceptable accuracy in semilocal exchange is achieved only with fine grids, whereas for nonlocal vdW-DF correlation even rather coarse grids are sufficient. The current version of vdW-DF is not well suited for pairing with Hartree-Fock exchange, leading to considerable overbinding.     

Time-dependent density functional theory calculation of van der Waals coefficient of sodium clusters

In this paper we employ all-electron ab initio time-dependent density functional theory based method to calculate the long range dipole-dipole dispersion coefficient (van der Waals coefficient) C(6) of sodium atom clusters containing even number of atoms ranging from 2 to 20 atoms. The dispersion coefficients are obtained via Casimir-Polder relation [Phys. Rev. 3, 360 (1948)]. The calculations are carried out with two different exchange-correlation potentials: (i) the asymptotically correct statistical average of orbital potential (SAOP) and (ii) Vosko-Wilk-Nusair representation [Can. J. Phys. 58, 1200 (1980)] of exchange-correlation potential within local density approximation. A comparison with the other theoretical results has been performed. We also present the results for the static polarizabilities of sodium clusters and also compare them with other theoretical and experimental results. These comparisons reveal that the SAOP results for C(6) and static polarizability are quite accurate and very close to the experimental results. We examine the relationship between volume of the cluster and van der Waals coefficient, and find that to a very high degree of correlation C(6) scales as the square of the volume. We also present the results for van der Waals coefficient corresponding to cluster-Ar atom and cluster-N(2) molecule interactions.     

Rank-1 approximation to the van der Waals interaction

We present and discuss a variational single-product approximation to the van der Waals dispersion interaction leading to a simple formula for C ₆ that seems capable to give more than 99% of the ‘exact’ value. The formula is derived from Hylleraas’ variational principle in the tensor product space of the interacting molecules and therefore enjoys bounding properties. The formula has been tested by computing the C ₆ dispersion constants of H–H, and, at Full CI level, of the following systems: He–He, He–Li, Li–Li, LiH–LiH, HF–HF. Connections with the London formula are discussed. Contribution to the Fernando Bernardi Memorial Issue.     

The “hydrophobic effect”: Essentially a van der Waals interaction

Summary It has been shown thermodynamically, and illustrated by means of a typical example, that the preferential attraction between hydrophobic determinants immersed in water, commonly alluded to as the hydrophobic effect, can be entirely ascribed to van der Waals interactions. Quantitatively, in water, the attraction between two hydrophobic determinants is stronger than the attraction between a hydrophobic and a hydrophilic determinant, although the latter attraction is not so small as to be negligible. The interaction between hydrophilic determinants in water is attractive but small and may be easily overwhelmed by the electrostatic repulsion which occurs between such entities.There is no repulsion by the solvent. On the contrary, the attraction between water and a hydrophobic material as well as between water and a hydrophilic material is strong. This interaction does play a role in determining the overall strength of the interaction between hydrophobic and hydrophilic determinants but cannot render the attraction between such determinants negligible. The attraction between hydrophobic and hydrophilic determinants in an aqueous medium can be made exeedingly small and may indeed (after lowering the surface tension of the liquid) be changed into a repulsion. The latter phenomenon is used in the elution step of the protein separation method called hydrophobic chromatography.

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Zusammenfassung Es wird auf thermodynamischer Grundlage gezeigt und an Hand eines typischen Beispiels erläutert, daß die Anziehung zwischen hydrophoben Partikel in Wasser vollständig durch van der Waals-Wechselwirkungen erklärt werden kann. Die Anziehung zwischen solchen Teilchen oder Molekülen ist stärker als die Anziehung zwischen hydrophoben und hydrophilen Partikeln in Wasser, die jedoch nicht vernachlässigbar klein ist. Die Wechselwirkung zwischen hydrophilen Partikeln ist so klein, daß sie leicht von elektrostatischen Wechselwirkungen überdeckt werden kann.Eine Abstoßung durch das Lösungsmittel findet nicht statt. Im Gegenteil, die Anziehung zwischen Wasser und einem hydrophoben Material ebenso wie diejenige zwischen Wasser und einem hydrophilen Material ist beträchtlich. Diese Wechselwirkung spielt zwar eine Rolle in der gesamten Wechselwirkung zwischen hydrophoben und hydrophilen Partikeln, kann aber die Anziehung zwischen hydrophob und hydrophil in Wasser nicht zum Verschwinden bringen. Die Wechselwirkung zwischen hydrophob und hydrophil kann jedoch durch Erniedrigung der Oberflächenspannung des Wassers abstoßend gemacht werden. Dieser Umstand wird im Elutionsschritt des als hydrophobe Chromatographie bekannten Verfahrens zur Trennung von Proteinen benutzt.

     

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.     

On the angular shape of van der Waals dimers of small polar molecules

Ab initio calculations on twenty van der Waals dimers of small polar molecules at the experimentally observed intermolecular separation R, using Hartree-Fock molecular moments, show that the minima in the electrostatic interaction expanded up to R⁻⁶ converges to angular structures which are close to those observed experimentally for such complexes.     

An application of the van der Waals density functional: Hydrogen bonding and stacking interactions between nucleobases

We apply the van der Waals density functional (vdW-DF) to study hydrogen bonding and stacking interactions between nucleobases. The excellent agreement of our results with high level quantum chemical calculations highlights the value of the vdW-DF for first-principles investigations of biologically important molecules. Our results suggest that, in the case of hydrogen-bonded nucleobase pairs, dispersion interactions reduce the cost of propeller twists while having a negligible effect on buckling. Furthermore, the efficient scaling of DFT methods allowed for the easy optimization of separation distance between nucleobase stacks, indicating enhancements in the interaction energy of up to 3 kcalmol over previous fixed distance calculations. We anticipate that these results are significant for extending the vdW-DF method to model larger vdW complexes and biological molecules.     

The validity of electrostatic predictions of the shapes of van der Waals dimers

The interaction energies of the van der Waals dimers H₂O-H₂O, Cl₂-HF, ClF-HF and N₂O-HF have been calculated for a range of geometries using ab initio SCF techniques. The SCF binding energies have been decomposed into electrostatic, exchange, polarisation and charge-transfer contributions and the intermolecular angles optimised with respect to various combinations of the above components. The effects of exchange, polarisation and charge transfer on the shape of a given dimer are found approximately to cancel, so that in each case a purely electrostatic model is capable of predicting intermolecular angles that agree well with those of a full SCF treatment, as well as with experiment. These findings are consistent with the proposals and earlier calculations of Buckingham and Fowler.     

van der Waals interaction of simple, parallel polymers

We study the mutual interactions of simple parallel polymers within the framework of density-functional theory (DFT). As the conventional implementations of DFT do not treat the long-range dispersion [van der Waals (vdW)] interactions, we develop a systematic correction scheme for the nonlocal energy contribution of the polymer interaction at the intermediate to the asymptotic separations. We primarily focus on the three polymers, polyethylene, isotactic polypropylene, and isotactic polyvinylchloride, but the scheme presented applies also more generally to other simple polymers. From first-principle calculations we extract the geometrical and electronic structures of the polymers and the local part of their interaction energy, as well as the static electric response. The dynamic electrodynamic response is modeled on the basis of these static calculations, from which the nonlocal vdW interaction of the polymers is extracted.     

H-SAPO-18催化甲醇制烯烃反应的芳烃烃池机理：基于范德华校正的密度泛函理论研究

由于可以从非石油资源如煤、天然气、生物质等出发制备低碳烯烃,分子筛催化甲醇制烯烃(MTO)反应在学术界和工业界引起了广泛的研究兴趣. H-SAPO-34是目前表现优异性能的分子筛催化剂之一,其双烯(乙烯+丙烯)的选择性在80%以上,已经实现了工业化应用.为了提升MTO反应的选择性,以及调控乙烯丙烯的选择性之比,非常有必要从反应机理出发来优化设计新的催化剂.然而,由于MTO催化反应产物复杂多样,对MTO反应机理的认识还存在很大的争议.目前基本能够接受的是MTO催化反应沿着烃池机理进行.在此反应机理中,无机分子筛和有机烃池活性中心形成共催化剂,甲醇进攻有机活性中心生成烷基链,此烷基链断裂得到烯烃产物.目前提出的烃池活性中心主要包括多甲基苯和烯烃自身,它们分别沿着各自的循环反应网络(芳烃循环和烯烃循环)生成烯烃产物.有文献指出在H-ZSM-5分子筛中芳烃循环主要生成乙烯,而烯烃循环主要生成丙烯等产物.因此,系统研究分子筛结构对两条循环网络相对贡献程度的影响规律,从而阐述分子筛结构和MTO催化性能之间的关系具有重要的意义. H-SAPO-18是一类结构上与H-SAPO-34相类似的分子筛,其笼由八元环孔道互联.实验研究指出,其也具有优异的MTO催化性能.在本工作中,我们利用包含范德华相互作用校正的交换相关泛函(BEEF-vdW),系统研究了H-SAPO-18分子筛中的芳烃循环反应机理.所有计算用VASP程序包完成, H-SAPO-18用48T周期性结构模型表示.利用静态吸附和相互转化的自由能变化情况,我们首先确认了反应条件下H-SAPO-18中最稳定的多甲基苯的结构.计算结果指出,1,2,4,5-四甲基苯的吸附能最强,而六甲基苯是主要存在的多甲基苯组分.多甲基苯在分子筛孔道内的稳定性主要由两个相反的作用共同影响：范德华相互作用引起的吸引,以及分子筛孔道结构引起的排斥.在芳烃循环路线中,乙基侧链的增长是反应的关键基元步.吉布斯自由能分析指出芳烃循环路线中,在反应温度673 K下H-SAPO-18中的六甲基苯并不比五甲基苯,四甲基苯的活性高,这与H-SAPO-34分子筛中的结果相一致. H-SAPO-18中的四甲基苯、五甲基苯和六甲基苯的总吉布斯自由能垒分别是208,215,239 kJ/mol.六甲基苯循环路线所表现出的高反应能垒的一个原因,是由于分子筛几何限域效应引起的熵增加所致.通过与烯烃循环路线的动力学进行比较,本文芳烃循环路线动力学的工作可以为MTO催化反应机理的研究提供一些启示.