Effective Three-body Potential Research Articles

Many-Body Potentials for Aqueous Be2+ Derived from ab Initio Calculations.

An effective three-body potential for the aqueous Be2+ ion has been constructed from a large number of high-level ab initio cluster calculations. The new potential was validated in subsequent molecular dynamics simulations of both gas phase ion-water clusters and bulk liquid. The structures of the first and second solvation shells were studied using radial distribution functions and angular distribution functions. The vibrational spectrum of Be2+ and first shell waters was examined by computing power spectra from the molecular dynamics simulations. The observed bands showed reasonable agreement with experimental spectroscopic frequencies. The potential of mean force for water exchange between the first and second solvation shells was calculated and the energy barrier for exchange was found to have improved agreement with experiment relative to two-body force fields. Examination of the solvation structure near the transition state yielded results consistent with an associative mechanism.

First-principles modeling of three-body interactions in highly compressed solid helium

We present a set of three-body interaction models based on the Slater-Kirkwood (SK) potential that are suitable for the study of the energy, structural, and elastic properties of solid $^{4}\mathrm{He}$ at high pressure. Our effective three-body potentials are obtained from the fit to total energies and atomic forces computed with the van der Waals density functional theory method due to Grimme, and represent an improvement with respect to previously reported three-body interaction models. In particular, we show that some of the introduced SK three-body potentials reproduce closely the experimental equation of state and bulk modulus of solid helium up to a pressure of $\ensuremath{\sim}60$ GPa, when used in combination with standard pairwise interaction models in diffusion Monte Carlo simulations. Importantly, we find that recent predictions reporting a surprisingly small variation of the kinetic energy and Lindeman ratio on quantum crystals under increasing pressure are likely to be artifacts deriving from the use of incomplete interaction models. Also, we show that the experimental variation of the shear modulus, ${C}_{44}$, at pressures $0\ensuremath{\le}P\ensuremath{\le}25$ GPa can be quantitatively described by our set of SK three-body potentials. At higher compression, however, the agreement between our ${C}_{44}$ calculations and experiments deteriorates and thus we argue that higher order many-body terms in the expansion of the atomic interactions probably are necessary in order to better describe elasticity in very dense solid $^{4}\mathrm{He}$.

Spectral flow of trimer states of two heavy impurities and one light condensed boson

The spectral flow of three-body (trimer) states consisting of two heavy (impurity) particles sitting in a condensate of light bosons is considered. Assuming that the condensate is weakly interacting and that an impurity and a boson have a resonant zero-range two-body interaction, we use the Born-Oppenheimer approximation to determine the effective three-body potential. We solve the resulting Schrödinger equation numerically and determine the trimer binding energies as a function of the coherence length of the light bosonic condensate particles. The binding energy is found to be suppressed by the presence of the condensate when the energy scale corresponding to the coherence length becomes of order the trimer binding energy in the absence of the condensate. We find that the Efimov scaling property is reflected in the critical values of the condensate coherence length at which the trimers are pushed into the continuum.

Many-body potentials for aqueous Li(+), Na(+), Mg(2+), and Al(3+): comparison of effective three-body potentials and polarizable models.

Many-body potentials for the aqueous Li(+), Na(+), Mg(2+), and Al(3+) ions have been constructed from ab initio cluster calculations. Pure pair, effective pair, effective three-body, and effective polarizable models were created and used in subsequent molecular dynamics simulations. The structures of the first and second solvation shells were studied using radial distribution functions and angular-radial distribution functions. The effective three-body and polarizable potentials yield similar first-shell structures, while the contraction of the O-O distances between the first and second solvation shells is more pronounced with the polarizable potentials. The definition of the tilt angle of the water molecules around the ions is discussed. When a proper definition is used, it is found that for Li(+), Mg(2+), and Al(3+) the water molecules prefer a trigonal orientation, but for Na(+) a tetrahedral orientation (ion in lone-pair direction) is preferred. The self-diffusion coefficients for the water molecules and the ions were calculated; the ionic values follow the order obtained from experiment, although the simulated absolute values are smaller than experiment for Mg(2+) and Al(3+).

The solvation of Li+ and Na+ in acetonitrile from ab initio-derived many-body ion–solvent potentials

Several Li+- and Na+-acetonitrile models were derived from ab initio calculations at the counterpoise-corrected MP2/TZV++(d,p) level for distorted ion-(MeCN)n clusters with n=1, 4 and 6. Two different many-body ion-acetonitrile models were constructed: an effective three-body potential for use with the six-site effective pair model of Böhm et al., and an effective polarizable many-body model. The polarizable acetonitrile model used in the latter model is a new empirical model which was also derived in the present paper. Mainly for comparative purposes, two ion-acetonitrile pair potentials were also constructed from the ab initio cluster calculations: one pure pair potential and one effective pair potential. Using all these potential models, MD simulations in the NPT ensemble were performed for the pure acetonitrile liquid and for Li+(MeCN) and Na+(MeCN) solutions with 1 ion in 512 solvent molecules and with a simulation time of at least 120 ps per system. Thermodynamic properties, solvation-shell structure and the self-diffusion coefficient of the ions and of the solvent molecules were calculated and compared between the different models and with experimental data, where available. The Li+ ion is found to be four-coordinated when the new many-body potentials are used, in contrast to the six-coordinated structure obtained for the pure pair and effective pair potentials. The coordination number of Na+ is close to six for all the models derived here, although the coordination number becomes slightly smaller with the many-body potentials. For both ions, the solvent molecules in the first shell point their nitrogen ends towards the cation, while in the second shell the opposite orientation is the most common.

Effective three-body potentials for Li+(aq) and Mg2+(aq)

A method for the extraction of effective three-body potential parameters from high-level ab initio cluster calculations is presented and compared to effective pair potentials extracted at the same level. Dilute Li+(aq) and Mg2+(aq) solutions are used as test cases and long molecular-dynamics simulations using these newly developed potentials were performed. Resulting thermodynamical, structural, and dynamical properties are compared to experiment as well as to the empirical effective pair potentials of Åqvist. Moreover, a new time-saving method for the correction of cluster energies computed with a relatively cheap ab initio method, to yield expensive, high-level ab initio energies, is presented. The effective pair approach is shown to give inconsistent results when compared to the effective three-body potentials. The performance of three different charge compensation methods (uniform charge plasm, Bogusz net charge correction, and counter ions) is compared for a large number of different system sizes. For most properties studied here, the system-size dependence is found to be small for system sizes with 256 water molecules or more. However, for the self-diffusion coefficients, a 1/L dependence is found, i.e., a very large system-size dependence. A very simple method for correcting for this deficiency is proposed. The results for most properties are found to compare reasonably well to experiment when using the effective three-body potentials.

Computation of 2+ resonance in 6He: bound state in the continuum

Calculation of the 2+ resonance state of 6He in a three-body model (4He + n + n) is done by adopting a novel theoretical technique. The effective three-body potential for the 2+ state of 6He is obtained using hyperspherical harmonics and it presents a shallow well followed by a low and wide barrier. Numerical diffculties present in the calculation of the resonance energy of such a shallow well–barrier combination are overcome by the construction of a one-parameter (λ) isospectral potential having a bound state in the continuum. This potential develops a deep well followed by a high barrier for small positive values of λ. This effectively traps the system in a sharp resonant state and facilitates calculation of the resonance energy accurately. We obtain the first 2+ resonance at 1.814 MeV with a width of 135 keV. We also get a second 2+ resonance at 4.9 MeV.

Three-body potentials originating from cluster distortion.

The distortion of composite particles (clusters) during their mutual interaction leads to effective three-body forces. In a nonrelativistic theory, they originate from two sources. One is the coupling potential which couples a channel to a distortion channel. The second one is the energy denominator (propagator) which arises whenever a distortion channel is formally eliminated and represented by an elimination potential. In the present paper this second source is studied. A three-cluster system with only one distortion state in a two-cluster subsystem is considered. Two possibilities of defining a three-body potential from the three-cluster elimination potential are tested, namely (i) extraction of a two-body potential with ``frozen'' energy dependence and (ii) off-shell transformation and subsequent extraction of a two-body potential. The mathematical formalism is illustrated by a numerical example. In a simplified model of the triton, two \ensuremath{\Delta} particles and a nucleon form the distortion channel. It is seen that the interaction of the third nucleon with the two \ensuremath{\Delta} particles has a remarkable influence on the effective three-body potential. Considering the fact that excited nucleons like \ensuremath{\Delta} particles tend to interact more strongly with nucleons than nucleons interact among themselves, we find the overbinding problem of the triton becoming more serious. From the present microscopic study it also becomes clear that an N-body Schr\"odinger equation with purely phenomenological energy-dependent two-body potentials is in general undefined because of insufficient information.

The use of an effective three-body potential in the nucleus

A discussion is given of the possibility of taking into account the state dependence of the Brueckner t-matrix by using an effective potential containing a three-body force. It is shown that this can be chosen so that the binding energy and its density dependence and also the single particle energies are given correctly. The contribution of the three-body force to the binding energy per particle is shown to be about 17 MeV if the Brueckner and Gammel calculations of the t-matrix are used.