Three-body Dispersion Interactions Research Articles

Three-Body Dispersion Corrections to the Spherical Atom Model: The PFD-3B Density Functional.

The performance of the isotropic spherical atom model can be significantly enhanced through combination with anisotropic three-body dispersion interactions to give the new PFD-3B density functional, which reduces the mean absolute deviation (MAD) relative to CCSD(T)/CBS benchmark energies from 0.78 to 0.19 kcal/mol for the S22 test set. Comparison with the extended S22 × 5 test set in the figure indicates that this accuracy is maintained through large variations in geometry. The performance of the PFD-3B functional over the S22 × 5 test set is superior to any of the functionals previously applied to this set. Over the S22 set of examples, the MADs from the CCSD(T)/CBS values for Re, De, and ωe, are 0.032 Å, 0.21 kcal/mol, and 6 cm-1, respectively. Over a comparable set of 26 examples containing second and third row atoms, the MADs from the CCSD(T)/CBS values for Re, De, and ωe, are 0.033 Å, 0.19 kcal/mol, and 5 cm-1, respectively. If used to optimize the geometry of the 48 examples, on average the PFD-3B functional introduces an error of only 0.042 kcal/mol in CCSD(T) single-point energies. This small error combines with the reported analytical first and second derivatives to makes the PFD-3B functional an attractive model for geometry optimization and zero-point energy calculations.

Assessing Electronic Structure Methods for Long-Range Three-Body Dispersion Interactions: Analysis and Calculations on Well-Separated Metal Atom Trimers.

Three-body dispersion interactions are much weaker than their two-body counterpart. However, their importance grows quickly as the number of interacting monomers rises. To explore the numerical performance of correlation methods for long-range three-body dispersion, we performed calculations on eight very simple dispersion-dominated model metal trimers: Na3, Mg3, Zn3, Cd3, Hg3, Cu3, Ag3, and Au3. One encouraging aspect is that relatively small basis sets of augmented triple-ζ size appear to be adequate for three-body dispersion in the long-range. Coupled cluster calculations were performed at high levels to assess MP3, CCSD, CCSD(T), empirical density functional theory dispersion (D3), and the many-body dispersion (MBD) approach. We found that the accuracy of CCSD(T) was generally significantly lower than for two-body interactions, with errors sometimes reaching 20% in the investigated systems, while CCSD and particularly MP3 were generally more erratic. MBD is found to perform better than D3 at large distances, whereas the opposite is true at shorter distances. When computing reference numbers for three-body dispersion, care should be taken to appropriately represent the effect of the connected triple excitations, which are significant in most cases and incompletely approximated by CCSD(T).

Dispersion Interactions between Neutral Atoms and the Quantum Electrodynamical Vacuum

Dispersion interactions are long-range interactions between neutral ground-state atoms or molecules, or polarizable bodies in general, due to their common interaction with the quantum electromagnetic field. They arise from the exchange of virtual photons between the atoms, and, in the case of three or more atoms, are not additive. In this review, after having introduced the relevant coupling schemes and effective Hamiltonians, as well as properties of the vacuum fluctuations, we outline the main properties of dispersion interactions, both in the nonretarded (van der Waals) and retarded (Casimir–Polder) regime. We then discuss their deep relation with the existence of the vacuum fluctuations of the electromagnetic field and vacuum energy. We describe some transparent physical models of two- and three-body dispersion interactions, based on dressed vacuum field energy densities and spatial field correlations, which stress their deep connection with vacuum fluctuations and vacuum energy. These models give a clear insight of the physical origin of dispersion interactions, and also provide useful computational tools for their evaluation. We show that this aspect is particularly relevant in more complicated situations, for example when macroscopic boundaries are present. We also review recent results on dispersion interactions for atoms moving with noninertial motions and the strict relation with the Unruh effect, and on resonance interactions between entangled identical atoms in uniformly accelerated motion.

Density-functional approach to the three-body dispersion interaction based on the exchange dipole moment.

We implement and compute the density functional nonadditive three-body dispersion interaction using a combination of Tang-Karplus formalism and the exchange-dipole moment model of Becke and Johnson. The computation of the C9 dispersion coefficients is done in a non-empirical fashion. The obtained C9 values of a series of noble atom triplets agree well with highly accurate values in the literature. We also calculate the C9 values for a series of benzene trimers and find a good agreement with high-level ab initio values reported recently in the literature. For the question of damping of the three-body dispersion at short distances, we propose two damping schemes and optimize them based on the benzene trimers data, and the fitted analytic potentials of He3 and Ar3 trimers fitted to the results of high-level wavefunction theories available from the literature. Both damping schemes respond well to the optimization of two parameters.

A microscopic approach to Casimir and Casimir–Polder forces between metallic bodies

We consider the Casimir–Polder interaction energy between a metallic nanoparticle and a metallic plate, as well as the Casimir interaction energy between two macroscopic metal plates, in terms of the many-body dispersion interactions between their constituents. Expressions for two- and three-body dispersion interactions between the microscopic parts of a real metal are first obtained, both in the retarded and non-retarded limits. These expressions are then used to evaluate the overall two- and three-body contributions to the macroscopic Casimir–Polder and Casimir force, and to compare them with each other, for the two following geometries: metal nanoparticle/half-space and half-space/half-space, where all the materials are assumed perfect conductors. The above evaluation is obtained by summing up the contributions from the microscopic constituents of the bodies (metal nanoparticles). In the case of nanoparticle/half-space, our results fully agree with those that can be extracted from the corresponding macroscopic results, and explicitly show the non-applicability of the pairwise approximation for the geometry considered. In both cases, we find that, while the overall two-body contribution yields an attractive force, the overall three-body contribution is repulsive. Also, they turn out to be of the same order, consistently with the known non applicability of the pairwise approximation. The issue of the rapidity of convergence of the many-body expansion is also briefly discussed.

First-Principles Many-Body Force Fields from the Gas Phase to Liquid: A “Universal” Approach

We extend our previously developed approach for generating "physically-motivated" force fields from symmetry-adapted perturbation theory by introducing explicit terms to account for nonadditive three-body exchange and dispersion interactions, yielding transferability from the gas- to condensed-phase. These Axilrod-Teller-Muto-type three-body terms require no additional parametrization and can be implemented with high computational efficiency. We demonstrate the accuracy of our force fields for a diverse set of six organic liquids/fluids, examining a wide variety of structural, thermodynamic, and dynamic properties. We find that three-body dispersion and exchange interactions make significant contributions to the internal pressure of condensed phase systems and cannot be neglected in truly ab initio force field development. These resulting force field parameters are extremely transferable over wide ranges in temperature and pressures and across chemical systems, and should be widely applicable in condensed phase simulation.

Higher-order electric multipole contributions to retarded non-additive three-body dispersion interaction energies between atoms: Equilateral triangle and collinear configurations

The theory of molecular quantum electrodynamics (QED) is used to calculate higher electric multipole contributions to the dispersion energy shift between three atoms or molecules arranged in a straight line or in an equilateral triangle configuration. As in two-body potentials, three-body dispersion interactions are viewed in the QED formalism to arise from exchange of virtual photons between coupled pairs of particles. By employing an interaction Hamiltonian that is quadratic in the electric displacement field means that third-order perturbation theory can be used to yield the energy shift for a particular combination of electric multipole polarizable species, with only six time-ordered diagrams needing to be summed over. Specific potentials evaluated include dipole-dipole-quadrupole (DDQ), dipole-quadrupole-quadrupole (DQQ), and dipole-dipole-octupole (DDO) terms. For the geometries of interest, near-zone limiting forms are found to exhibit an R(-11) dependence on separation distance for the DDQ interaction, and an R(-13) behaviour for DQQ and DDO shifts, agreeing with an earlier semi-classical computation. Retardation weakens the potential in each case by R(-1) in the far-zone. It is found that by decomposing the octupole moment into its irreducible components of weights-1 and -3 that the former contribution to the DDO potential may be taken to be a higher-order correction to the leading triple dipole energy shift.

Benchmarking of London Dispersion-Accounting Density Functional Theory Methods on Very Large Molecular Complexes.

A new test set (S12L) containing 12 supramolecular noncovalently bound complexes is presented and used to evaluate seven different methods to account for dispersion in DFT (DFT-D3, DFT-D2, DFT-NL, XDM, dDsC, TS-vdW, M06-L) at different basis set levels against experimental, back-corrected reference energies. This allows conclusions about the performance of each method in an explorative research setting on "real-life" problems. Most DFT methods show satisfactory performance but, due to the largeness of the complexes, almost always require an explicit correction for the nonadditive Axilrod-Teller-Muto three-body dispersion interaction to get accurate results. The necessity of using a method capable of accounting for dispersion is clearly demonstrated in that the two-body dispersion contributions are on the order of 20-150% of the total interaction energy. MP2 and some variants thereof are shown to be insufficient for this while a few tested D3-corrected semiempirical MO methods perform reasonably well. Overall, we suggest the use of this benchmark set as a "sanity check" against overfitting to too small molecular cases.

The long-range non-additive three-body dispersion interactions for the rare gases, alkali, and alkaline-earth atoms

The long-range non-additive three-body dispersion interaction coefficients Z(111), Z(112), Z(113), and Z(122) are computed for many atomic combinations using standard expressions. The atoms considered include hydrogen, the rare gases, the alkali atoms (up to Rb), and the alkaline-earth atoms (up to Sr). The term Z(111) arising from three mutual dipole interactions is known as the Axilrod-Teller-Muto coefficient or the DDD (dipole-dipole-dipole) coefficient. Similarly, the terms Z(112), Z(113), and Z(122) arise from the mutual combinations of dipole (1), quadrupole (2), and octupole (3) interactions between atoms and they are sometimes known, respectively, as dipole-dipole-quadrupole, dipole-dipole-octupole, and dipole-quadrupole-quadrupole coefficients. Results for the four Z coefficients are given for the homonuclear trimers, for the trimers involving two like-rare-gas atoms, and for the trimers with all combinations of the H, He, and Li atoms. An exhaustive compilation of all coefficients between all possible atomic combinations is presented as supplementary data.

Influence of two-body and three-body interatomic forces on gas, liquid, and solid phases

Accurate molecular dynamics simulations are reported which quantify the contributions of two- and three-body interactions in the gas, liquid, and solid phases of argon at both subcritical and supercritical conditions. The calculations use an accurate two-body potential in addition to contributions from three-body dispersion interactions from third-order triple-dipole interactions. The number dependence of three-body interactions is quantified, indicating that a system size of at least five hundred atoms is required for reliable calculations. The results indicate that, although the contribution of three-body interaction to the overall energy is small, three-body interactions significantly affect the pressure at which vapor-liquid and solid-liquid transitions are observed. In particular, three-body interactions substantially increase the pressure of the freezing point. Unlike two-body interactions, which vary with both density and temperature, for a given density, three-body interactions have a near-constant 'background' value irrespective of the temperature. Both two-body interactions and kinetic energy have an important role in vapor-liquid equilibria whereas solid-liquid equilibria are dominated by two-body interactions.

Effect of three-body interactions on the vapor-liquid phase equilibria of binary fluid mixtures

Gibbs-Duhem Monte Carlo simulations are reported for the vapor-liquid phase coexistence of binary argon+krypton mixtures at different temperatures. The calculations employ accurate two-body potentials in addition to contributions from three-body dispersion interactions resulting from third-order triple-dipole interactions. A comparison is made with experiment that illustrates the role of three-body interactions on the phase envelope. In all cases the simulations represent genuine predictions with input parameters obtained independently from sources other than phase equilibria data. Two-body interactions alone are insufficient to adequately describe vapor-liquid coexistence. In contrast, the addition of three-body interactions results in very good agreement with experiment. In addition to the exact calculation of three-body interactions, calculations are reported with an approximate formula for three-body interactions, which also yields good results.

Molecular simulation of the phase behavior of noble gases using accurate two-body and three-body intermolecular potentials

Gibbs ensemble Monte Carlo simulations are reported for the vapor–liquid phase coexistence of argon, krypton, and xenon. The calculations employ accurate two-body potentials in addition to contributions from three-body dispersion interactions resulting from third-order triple-dipole, dipole–dipole–quadrupole, dipole–quadrupole–quadrupole, quadrupole–quadrupole–quadrupole, and fourth-order triple-dipole terms. It is shown that vapor–liquid equilibria are affected substantially by three-body interactions. The addition of three-body interactions results in good overall agreement of theory with experimental data. In particular, the subcritical liquid-phase densities are predicted accurately.

Novel simulation model for many-body multipole dispersion interactions

We present a novel simulation technique, within the framework of a molecular dynamics simulation, which accounts for both two- and three-body dispersion interactions, up to the triple-quadrupole interaction. This technique involves a unification of molecular dynamics and quantum-mechanical variational methods, in the spirit of the Car-Parrinello method. The advantage of this new method compared to existing techniques for simulating three-body dispersion forces, is that it allows for a consistent treatment of both dispersion damping and periodic boundary conditions at the pair and three-body level. The latter means that it would be possible, for the first time, to include many-body dispersion effects in the simulation of bulk properties of materials, without making use of effective pair potentials.

Novel simulation model for many-body multipole dispersion interactions

We present a novel simulation technique, within the framework of a molecular dynamics simulation, which accounts for both two- and three-body dispersion interactions, up to the triple-quadrupole interaction. This technique involves a unification of molecular dynamics and quantum-mechanical variational methods, in the spirit of the Car-Parrinello method. The advantage of this new method compared to existing techniques for simulating three-body dispersion forces, is that it allows for a consistent treatment of both dispersion damping and periodic boundary conditions at the pair and three-body level. The latter means that it would be possible, for the first time, to include many-body dispersion effects in the simulation of bulk properties of materials, without making use of effective pair potentials.

Effect of three-body dispersion interactions on the surface dynamics of Ar(111)

Molecular dynamics simulations were performed for Ar(111) using an accurate two-body potential developed by Barker and co-workers and the Axilrod-Teller three-body dispersion interaction. The decay of surface order parameters accompany surface premelting at approximately 0.78 T m for the two-body potential alone, while inclusion of the three-body potential raises the premelting temperature to approximately 0.85 T m. Prior to surface premelting, a discontinuity is observed in the slope of the Arrhenius plot of the surface layer's mass transfer diffusivity at about 0.67 T m for the two-body potential alone, with inclusion of three-body potential raising this transition temperature to about 0.77 T m. The partial quenching of surface premelting induced by addition of the three-body dispersion interaction arises despite a significant increase in the degree of outward surface relaxation.

A novel simulation model for three-body dispersion interactions

We present a novel molecular dynamics simulation technique, which accounts for both two- and three-body dispersion interactions. This technique is a unified approach of molecular dynamics and quantum mechanical variational methods, in the spirit of the Car - Parrinello method (1985 Phys. Rev. Lett. 55 2471). We use a highly simplified model for the electronic structure of the atoms, which is, nevertheless, sufficient to correctly reproduce the London two-body, and the Axilrod - Teller three-body dispersion forces in an appropriate limit. The advantage of this new method is that it allows for a consistent treatment of both dispersion damping and periodic boundary conditions at the pair and three-body levels.

Monte Carlo simulation of vapour-liquid equilibria in “Lennard-Jones + three-body potential” binary fluid mixtures

Gibbs-ensemble molecular simulations are reported for the vapour-liquid phase coexistence of a binary mixture characterised by an intermolecular potential which accounts for three-body interactions in addition to pairwise interactions. The simulations incorporate the influence of both three-body dispersion and three-body repulsion interactions. The effect of the non-additive Axilrod-Teller dispersion coefficient is investigated. It is concluded that the vapour-liquid coexistence of binary mixtures is unaffected by three-body interactions.

The effects of three-body dispersion interactions on liquid-liquid phase equilibrium

Using perturbation theory, we show that three-body dispersion interactions influence the phase diagrams of partially miscible liquid mixtures. In our model mixtures, the argon-like particles interact through Maitland-Smith pair potentials and Axilrod-Teller three-body potentials. We find that ternary liquid-liquid coexistence curves are sensitive to vABC , the strength of the Axilrod-Teller interaction appearing for the first time in the ternary mixture. Effective pair potentials predict the ternary phase diagrams well if vABC satisfies Tang's rule.

Computation of transport coefficients and the dynamical structure factor of Xe in the presence of long-range three-body interactions.

The importance of the three-body dispersion interaction (Axilrod-Teller potential) in liquid Xe near its triple point is investigated by numerical simulations. Although large for the heavy rare gas Xe, its effect on collective dynamical properties such as the dynamical structure factor and transport coefficients is very small. When used in conjunction with an accurate pair potential, good agreement is obtained with experiment for both the pressure and transport coefficients.

Three-body exchange interaction in dense helium

Self-consistent phonon and Monte Carlo calculations show that the three-body exchange interaction is important in dense helium. Together with a realistic pair potential and the Axilrod-Teller three-body dispersion interaction, it brings calculations into agreement with the experimental equation of state. This interaction stabilizes the hcp structure for pressures greater than about 60 GPa. A speculative phase diagram of high-pressure helium is proposed.