Triple-dipole Potential Research Articles

On the Use of Different Potential Energy Functions in Rare-Gas Cluster Optimization by Genetic Algorithms: Application to Argon Clusters

We study the effect of the potential energy function on the global minimum structures of argon clusters arising in the optimization performed by genetic algorithms (GAs). We propose a robust and efficient GA which allows for the calculation of all of the putative global minima of Ar(N) (N = 3-78) clusters modeled with four different potentials. Both energetic and structural properties of such minima are compared among each other and with those previously obtained for the Lennard-Jones function. In addition, the possibility of obtaining global minima of one potential through local optimization over the corresponding cluster geometry given by other potentials was associated with some structural parameters. The influence of the contribution from the three-body (Axilrod-Teller-Muto) triple-dipole potential (including or not a damping function to describe its correct behavior at smaller interatomic distances) has also been analyzed.

Third virial coefficient of nonpolar gases from accurate binary potentials and ternary forces

An explicit model for the third virial coefficient C(T) is presented, based on accurate binary interactions plus three-body forces; its predictions are compared to experimental data for 15 fluids (argon, krypton, xenon, nitrogen, oxygen, fluorine, carbon monoxide, carbon dioxide, perfluoromethane, methane, ethene, ethane, propane, n-butane, and n-pentane). Three-body interactions are represented by the Axilrod–Teller–Muto (ATM) triple-dipole potential, while the binary potential profile is systematically varied using approximate non-conformal (ANC) potentials. Non-conformality affects significantly both two- and three-body contributions to C(T). A formula for C(T) of ANC+ATM systems is obtained in terms of all interaction parameters; the three-body contribution is proportional to the parameter ν of the ATM potential while its temperature dependence is proportional to that of a reference fluid (argon). The ANC+ATM model reproduces C(T) within experimental error for most of the fluids considered, with a small negative deviation in some cases, which may be ascribed to the need of a complement to the ATM term.

Monte Carlo simulations of nitrogen using an ab initio potential

A new ab initio pair potential for nitrogen has been calculated at CCSD(T) level with aug-cc-pVDZ and -pVTZ correlation consistent basis sets. The results were extrapolated to approximate the basis set limit. This potential was used within Gibbs ensemble Monte Carlo (GEMC) simulations to obtain the densities of the coexisting phases, the vapour pressure and the enthalpy of vaporization from 70 K to close to the critical point. The influence of several 3-body interactions (an approximate anisotropic triple dipole potential derived by Stogryn, the isotropic triple dipole potential by Axilrod and Teller (AT), and a 3-body induction potential on the above mentioned properties was investigated. Satisfactory agreement with experimental data was observed. To determine whether the remaining deviations between experimental and computed data are due to inaccuracies in the 2-body or 3-body potential, the 2-body potential was rescaled to reproduce experimental 2nd virial coefficients accurately, and some of the calculations were repeated with the new potential. It turns out that an accurate 2-body potential only in connection with the AT potential yields accurate results for the thermodynamic properties phase equilibria.

Complete ab initio three-body nonadditive potential in Monte Carlo simulations of vapor–liquid equilibria and pure phases of argon

Complete ab initio three-body nonadditivity, including the short-range contributions, has been used for the first time in Monte Carlo simulations of liquid, gas, and liquid–vapor equilibrium of argon. Very good overall agreement with experiment has been achieved for phase equilibrium parameters and pV data of pure phases. The importance of various three-body interaction components has been assessed and their relationship to the liquid structure has been investigated. Although the short-range contributions to the nonadditivity are quite large, cancellations occurring between these and other components at the interatomic distances typical for the liquid make the total three-body nonadditive effect very similar to that given by a simple triple-dipole potential. Performance of the three-term virial equation of state is assessed based on the results of simulations.

Many-body interaction effects on the low-k structure of liquid Kr.

Neutron diffraction measurements and theoretical calculations of the structure factor S(k) of liquid Kr are extended to small k values (k<4 nm(-1)). The results show that many-body interaction contributions have an increasing effect on S(k) as k-->0, reaching at least 40% of the measured intensity. Both the phase diagram and the low-k structural data of dense Kr turn out to be closely reproduced by the hierarchical reference theory if additional many-body forces are taken into account by an augmented strength of the Axilrod-Teller triple-dipole potential. The experimental density derivative of S(k) is also used for a very sensitive test of the theories and interaction models considered here.

Effects of dispersion forces on the structure and thermodynamics of fluid krypton

Semianalytical and numerical calculations are performed to predict the structural and thermodynamic properties of low-density Kr fluid. Assuming that the interatomic forces can be modelled by a pairwise potential plus the three-body Axilrod-Teller potential, two different routes are explored. The first one is based on the hybridized mean spherical approximation integral equation of the theory of liquids and the second one uses large-scale molecular dynamics (MD). Algorithms for MD simulation are constructed on parallel machines to reduce the amount of computer time induced by the calculations of the three-body forces and the pair-correlation function. Our results obtained with the two methods mentioned above are in quite good agreement with the recent small-angle neutron-scattering experiments [Formisano et al., Phys. Rev. Lett. 79, 221 (1997); Benmore et al., J. Phys.: Condens. Matter 11, 3091 (1999)]. Moreover, the reliability of the asymptotic form of the integral equation is assessed for the specific case of dispersion forces including the three-body contributions, by an analysis at low wave vector and low density. It is seen that the effects of the Axilrod-Teller triple-dipole potential cannot be ignored to describe the structure and the thermodynamic properties of fluid krypton even at low density.

Structure of fluid krypton using the integral-equation theory for three-body forces

The formalism of the hybridized mean spherical approximation integral equation is extended to calculate the pair correlation function for krypton, beyond the critical isotherm, by using the two- and three-body contributions to the interatomic potential. The calculations are intended to examine the change of the pair correlation function caused by the inclusion of the Axilrod-Teller triple-dipole potential. It is seen that the three-body interaction cannot be ignored and that their effects become more significant for intermediate densities. In addition, a good agreement is found with the recent small-angle neutron-scattering experiments of Formisano et al. [Phys. Rev. Lett. 79, 221 (1997)].

Model intermolecular potentials and virial coefficients determined from the speed of sound

A simple procedure is given for determining model two- and three-body intermolecular potential energy functions from precise measurements of the speed of sound in the gas phase. The method is applied to the pure gases argon, methane and nitrogen and results are considered for propane and the mixture (methane + propane) obtained recently by similar methods. For the pair potential, the four-parameter model proposed by Maitland and Smith is used while the consequences of three-body forces were generally assumed to be represented adequately by the triple-dipole dispersion potential of Axilrod and Teller. The effect of including additional dispersion and exchange terms in the three-body potential was investigated for argon; each of these terms is significant but their effect on the third virial coefficient may be absorbed accurately in an effective triple-dipole potential. Three or, in some cases, all four of the parameters in the pair potential were optimized, together with the triple-dipole dispersion coefficient, in fits to the acoustic data. Ordinary second and third virial coefficients have been calculated from the model potential and the former are shown to be in excellent agreement with directly measured values. In the case of the third virial coefficient, some small discrepancies are noted; the resolution of these differences could have important implications for our understanding of many-body forces. The values of the dilute-gas viscosity predicted by the model pair potentials are examined also and, in the case of argon, found to be in good agreement with experimental values; for the other systems differences of several per cent were noted.

Influence of three-body forces on the gas-liquid coexistence of simple fluids: The phase equilibrium of argon

The effect of three-body classical dispersion forces on the gas-liquid coexistence of argon is explored by means of a reference hypernetted-chain integral equation and Gibbs ensemble Monte Carlo simulations. We find that the combination of Aziz's [J. Chem. Phys. 99, 4518 (1993)] pair interaction with the standard Axilrod-Teller [J. Chem. Phys. 11, 299 (1943)] triple dipole potential leads to the best prediction of the experimental coexistence curve. By contrast, when the Lennard-Jones potential is used in conjunction with the Axilrod-Teller interaction the calculated curve deviates appreciably from the experimental data, far from improving the reasonable plain Lennard-Jones results. Additionally, the integral equation approach with an effective state-dependent pair potential to account for the three-body interactions leads to very accurate results for the thermodynamic and coexistence properties of the fluid. This feature comes to prove that the theory can be an invaluable tool for studying systems with three-body interactions of the Axilrod-Teller type.

Exploring the influence of three-body classical dispersion forces on phase equilibria of simple fluids: An integral-equation approach.

We investigate a recently introduced integral equation which takes into account three-body interactions via an effective pair potential. The scheme proposed here essentially reduces to solving a reference hypernetted-chain equation with a state-dependent effective potential, and a hard-sphere reference bridge function that minimizes the free energy per particle. Our computational algorithm is shown to be stable and rapidly convergent. As a whole, the proposed procedure yields thermodynamic properties in accordance with simulation results for systems with Axilrod-Teller triple-dipole potential plus a Lennard-Jones interaction, and improves upon previous integral-equation calculations. Predictions obtained for the gas-liquid coexistence of argon are in remarkable agreement with experimental results, and show unequivocally that the influence of the three-body classical dispersion forces (and not only quantum effects) must be explicitly incorporated to account for the deviations between pure Lennard-Jones systems and real fluids. Moreover, the integral equation approach as introduced here proves to be a reliable tool and an inexpensive probe to assess the influence of three-body interactions in a consistent way. For completeness, the no-solution line of the integral equation is also presented. A study of the behavior of the isothermal compressibility in the vicinity of the no-solution boundary shows the presence of a divergence that deviates from a power law at high densities, and the appearance of a singularity with the characteristics of square-root branch point at low density (a feature also found in the hypernetted-chain approximation in a variety of systems). The no-solution line, unfortunately, hides the coexistence curve near the critical point, and this constitutes the only severe drawback in our approach. An alternative for bypassing this shortcoming is explored.

Simulation results for a fluid with the Axilrod-Teller triple dipole potential.

Simulations are performed for a fluid whose atoms interact with the Lennard-Jones pair potential and the Axilrod-Teller three-body potential, and results are tabulated for the internal energy and for the virial pressure. The triple dipole potential is effectively repulsive, as is well known, making the internal energy less negative and the pressure more positive than for the underlying Lennard-Jones fluid. For the relatively low polarizabilities examined, \ensuremath{\nu}/3\ensuremath{\epsilon}${\mathrm{\ensuremath{\sigma}}}^{9}$\ensuremath{\approxeq}0.03, the perturbation approximation of Barker, Fisher, and Watts [Mol. Phys. 21, 657 (1971)] is shown to be essentially exact, and the hypernetted-chain approximation of Attard [Phys. Rev. A 45, 3659 (1992)] is quantitatively accurate at low densities, but only qualitatively correct for the liquid state.

Pair-hypernetted-chain closure for three-body potentials: Results for argon with the Axilrod-Teller triple-dipole potential.

The hypernetted-chain closure is modified to include three-body potentials. The latter is reduced to a state-dependent effective pair potential. Explicit expressions for the Helmholtz free energy and for the chemical potential are derived for the modified closure. The approximation is shown to be relatively self-consistent and in qualitative agreement with perturbation results. An argonlike fluid is discussed using the Lennard-Jones pair potential and the Axilrod-Teller triple-dipole potential. In agreement with earlier work, it is found that the effective three-body contribution is positive, increasing the pressure and the internal energy by about 10% in the liquid state.

Three-body contributions to the thermodynamic properties of supercritical argon

Numerical calculations of the thermodynamic properties of argon were performed using Zwanzig perturbation theory in which the reference radial distribution functions were derived from the Percus–Yevick equation in conjunction with an accurate pair potential (HFD–C). Three-body contributions were approximated by an Axilrod–Teller induced triple dipole potential and the Kirkwood superposition approximation to the triple radial distribution function. Agreement with experimental data is very good over a range of densities up to twice the critical density and to a temperature approaching the critical. Deviation of the model from experiment is evident near the critical point, due to onset of macroscopic fluctuations.

Structure of krypton gas: Monte Carlo results, virial expansions, and real experimental data

We have made Monte Carlo calculations of the pair correlation function $g(r)$ of a dense gas along the 297 K isotherm using a published pair potential for krypton. Eleven states were simulated, and then, using the pair potential plus the Axilrod-Teller triple-dipole potential, seven states were simulated. The effect of the triplet potential was very small except near the principal peak of $g(r)$. We compare, in real and Fourier space, these results to the virial expansion of $g(r)$ at low densities to test its range of validity. This work provided the background for the interpretation of the experimental data of Teitsma and Egelstaff on real krypton gas, and examples are given involving the extraction of the pair- and triplet-potential terms.

Threshold Condition for Forming an Atomic4He Trimer

The Rayleigh-Ritz variational principle is used to formulate a rigorous bound on the threshold conditions under which three helium four atoms will form a bound state when their pair interaction is represented by the Lennard-Jones (n, 6) model and their nonadditive interaction is taken to be the Muto-Axilrod-Teller triple-dipole potential. The accepted parameter values of the model with n equal to 12 predict a bound-state trimer. The oversight, that of failing to use the (12, 6) parameter values of Kihara, Midzuno and Shizume in the extensive work on liquid helium, is pointed out.

Surface energy and surface tension of liquid Ar near the triple point

The effect of the transition zone on the surface energy and the surface tension near the triple point is reported. Both quantities are calculated from the extended Kirkwood-Buff formulas for the cases of a linear, a cubic, and a Fermi density profile. The realistic Barker-Fisher-Watts (BFW) potential and the Parson, Siska, and Lee (MSVIII) potential are used. The experimental radial distribution function of Yarnell et al. is used for the bulk liquid and as an approximation for the transition zone. As a function of the zone width both realistic potentials lead to the same rapid monotonic increase for the surface energy, but a different mild monotonic decrease for the surface tension. The transition zone width can be determined approximately from the surface energy. If the three-body nonadditivity effects on both quantities are ignored, the experimental value 34.9 erg/${\mathrm{cm}}^{2}$ of the surface energy suggests a zone width of 6.2 to 8.5 \AA{}, thus predicting a value of the tension of 12.1 dyn/cm for the BFW potential and 12.8 dyn/cm for the MSVIII potential as compared to the experimental value of 13.1 dyn/cm. Including the three-body nonadditivity effects estimated from the Axilrod-Teller triple-dipole potential, the zone width needed for a correct value of the surface energy is slightly larger (7.2 to 9.5 \AA{}). However, the surface tension thus predicted is too small for all potentials.

Perturbation theory for the radial distribution function in the presence of three-body forces

Perturbation theory is used to consider the effect of the three-body potential on the radial distribution function g (r). We find that the effective pair potential, ueff(1,2) =u (1,2)−kTρℱ g0(1,3) g0(2,3) f (1,2,3) d3, where g0(i,j) is the true pair potential radial distribution function, is an appropriate model for the study of the effect of the triplet potential on the g (r). Dividing the effective pair potential into repulsive and attractive regions, the radial distribution function of the reference potential is evaluated. It is shown, by calculating the total g (r) for the effective pair potential that, to first order, the effect of the triplet potential on the g (r) is accounted for by its effect on the reference g (r). We find that the effect of the triple-dipole potential on the structure of liquids at low temperatures especially near the triple point is substantial.

Extended time dependent Hartree theory of van der Waals interaction: Inclusion of electric quadrupoles

Expressions are derived for the van der Waals interaction free energy, to infinite order, which include dipole and quadrupole interactions. The treatment is an extension of the time-dependent Hartree approximation developed previously for dipolar interactions. The results are obtained via linear response theory. Just as in the dipolar case, the results are expressed in terms of the eigen frequencies of a normal mode matrix. In the present formulation, however, the normal modes are determined not only by electric dipolar coupling but also by coupling of electric quadrupoles with each other and with dipoles. The method is also useful in generating the perturbation terms of the free energy shift. It is found that the second order term, when applied to isotropic oscillators at zero degrees, reduces to the result derived earlier by Margenau. The nonadditive three-body potential, derived here, reduces to the Axilrod–Teller triple dipole potential in the absence of quadrupoles. Numerical results are given for the interaction between two and between three isotropic oscillators.

Third Virial Coefficients of Ar, Kr, and Xe

In view of serious discrepancies in polynomial fitting of PVT measurements over different density ranges, we have reanalyzed the data of the Amsterdam group to obtain improved third virial coefficients Cexpt1(T) for Ar, Kr, and Xe at low temperatures. The third virial coefficients assuming additivity CA(T) and the nonadditivity corrections ΔC(T), assuming the triple-dipole potential to be valid at arbitrarily small distances rij, have been calculated for some of the better pair potentials, and in all cases it is found that CA(T)+Δ C(T) agrees well with the reanalyzed experimental value Cexpt1(T). We consider the values of ΔC when a cutoff at r* is made such that the three-body interaction vanishes when any rij&amp;lt;r*. This shows that the major contribution to the ΔC integral comes from triangles with one or more sides rij smaller than rm, the separation at the pair-potential minimum. Since one expects both higher multipole and exchange-overlap three-body interactions to be appreciable for these small triangles, it is difficult to understand the consistently good agreement between CA+Δ C and the experimental values. The large discrepancies between CA(T) and Cexpt1(T) cannot be reasonably attributed to uncertainties in the pair potential or in the revised polynomial fitting of the PVT data.

Three-Body Quantum Correction to the Free Energy of a Simple Liquid and the Role of Pair and Triplet Potentials in Isotopic Distillation

The lowest-order quantum correction to the partition function and free energy of a simple dense fluid is generalized to take account of many-body forces. The three-body quantum correction is calculated using the triple-dipole potential for the nonadditive interaction, assuming the superposition approximation for the triplet correlation function, and taking the pair correlation functions from x ray and neutron diffraction experiments. In the case of liquid Ar near the triple point, the three-body correction is small enough to be neglected in connection with isotopic distillation experiments. These experiments provide values for 〈 ∇2U 〉, where U is the total interaction energy of the molecules in the liquid, and the results can be used in conjunction with experimental radial distribution functions to test empirical pair potentials u(r) for the Ar—Ar interaction. Several forms of u(r) are investigated and the Barker—Pompe, Kingston, and (exp-6) potentials are found to give the best results for 〈 ∇2U〉. The values of 〈 ∇2U 〉 test the repulsive region of the potential and are much less sensitive to the form of u(r) than is the vapor pressure of the liquid near the triple point. The vapor-pressure and isotopic-distillation tests are compared.