Three-body parameter

Figure 1. The most stable Ain clusters, for n=7, 15, determined using the three-body parameter deduced from the ab initio Alis results.

M. Beminger, A. Zenesini, B. Huang, W. Harm, H.-C. Nagerl, F. Ferlaino, et al., Universality of the three-body parameter for Efimov states in ultracold cesium, Phys. Rev. Lett. 107(2011) 120401. 

The most extensive systematic determination of 4f parameters for ions across the lanthanide series is by Carnall et al. (1988, 1989) for lanthanides in LaFs. The 4f free ion parameters are reduced by a few percent in crystals relative to gaseous ions, but do not change very much from one crystalline environment to another. For this reason, the tabulations of Carnall are still used as the starting point for many of the parametrized calculations within the 4f shell. In most cases, the energy level data are not sufficiently complete to unambiguously determine all of the free ion parameter values, so some of the parameters, such as the three-body parameters, r, are often kept fixed at Carnall s values. The determination of exactly which parameters to keep fixed, and which to vary, is usually decided on a case-by-case basis, based upon how many multiplets are covered by the data, and upon the indeterminacies that arise during the fitting process itself. 

For an overcritical mass ratio M/m > 13.6 we have a well-known phenomenon of the fall of a particle to the center in an attractive l/R potential [26]. In this case the shape of the wavefunction at distances of the order of can significantly influence the large-scale behavior, and a short-range three-body parameter is required to describe the system. The wavefunction of heavy atoms x(R) acquires many nodes at short distances R, which indicates the appearance of three-body bound Efimov states. 

The three-body parameter ro determines the phase of the wavefunction at small distances and, in principle, depends on 1. The wavefunction 10.56 has infinitely many nodes, which means that in the zero-range approximation there are infinitely many trimer states. This is one of the properties of three-body systems with resonant interactions discovered by Efimov [54], We see that the fall to the center is possible in many angular momentum channels, provided the mass ratio is sufficiently large. However, for practical purposes and for simplicity, it is sufficient to consider the case where the Efimov effect occurs only for the angular momentum channel with the lowest possible / for a given symmetry. This implies that when the heavy atoms are fermions and one has odd I, in order to confine ourselves to / = 1 we should have the mass ratio in the range 14 < M/m < 76. For bosonic heavy atoms where / is even, we set I = 0 and consider M/m < 39 to avoid the Efimov effect for / > 2. In both cases we need a single three-body parameter tq. 

Besides the Efimov trimers, one light and two heavy atoms may form universal trimer states well described in the zero-range approximation without introducing the three-body parameter [69]. In particular, they exist for the orbital angular momentum / = 1 and mass ratios below the critical value, where the Efimov effect is absent and short-range physics drops out of consideration. One of such states emerges at Mfmf S and crosses the trimer formation threshold (cn = —2 eo ) at M/m 12.7. The existence of this state is already seen in the Bom-Oppenheimer picture. It appears as a bound state of fermionic heavy atoms in the potential +(/ ) for / = 1. The other state exists at M/m even closer to the critical mass ratio and never becomes sufficiently deeply bound to be formed in cold dimer-dimer collisions. The universal trimer states also exist for / > 1 and M/m > 13.6 [69]. However, trimer formation in dimer-dimer collisions at such mass ratios is dominated by the contribution of Efimov trimers with smaller /. Therefore, below, we focus on the formation of Efimov trimers. 

It is straightforward to extend this theory to account for inelastic processes of the trimer formation and the relaxation of dimers into deeply bound states. Let us first assume that the rate of the relaxation into deep molecular states is negligible and neglect this process. Then the three-body parameter is real, and the trimer formation rate is determined by the imaginary part of the i-wave scattering length. The rate constant is given by [26]... 

Figure 10.12 shows the results for the inelastic collisional rate in the case of bosonic molecules with the mass ratio M/m = 28.5 characteristic of Yb- Li dimers. The solid line corresponds to the case of a real three-body parameter. It is convenient to introduce a related quantity, ao, defined as the value of a at which the energy,... 

The parameters fx and hx clearly depend on all molecular interactions in the mixture, that is, on both those between like molecules, f and h i, and those between unlike molecules, fij and hij. Since intermolecular forces are, at least approximately, pair-wise additive, we do not need to Introduce three-body parameters, fxjk most successful recipe for combining the... 

One additional important reason why nonbonded parameters from quantum chemistry cannot be used directly, even if they could be calculated accurately, is that they have to implicitly account for everything that has been neglected three-body terms, polarization, etc. (One should add that this applies to experimental parameters as well A set of parameters describing a water dimer in vacuum will, in general, not give the correct properties of bulk liquid water.) Hence, in practice, it is much more useful to tune these parameters to reproduce thermodynamic or dynamical properties of bulk systems (fluids, polymers, etc.) [51-53], Recently, it has been shown, how the cumbersome trial-and-error procedure can be automated [54-56A],... 

For this reason, we will restrict our subsequent approach to planar configurations of the two electrons and of the nucleus, with the polarization axis within this plane. This presents the most accurate quantum treatment of the driven three body Coulomb problem to date, valid in the entire nonrelativistic parameter range, without any adjustable parameter, and with no further approximation beyond the confinement of the accessible configuration space to two dimensions. Whilst this latter approximation certainly does restrict the generality of our model, semiclassical scaling arguments suggest that the unperturbed three... 

It should also be noted that ternary and higher order polymer-polymer interactions persist in the theta condition. In fact, the three-parameter theoretical treatment of flexible chains in the theta state shows that in real polymers with finite units, the theta point corresponds to the cancellation of effective binary interactions which include both two body and fundamentally repulsive three body terms [26]. This causes a shift of the theta point and an increase of the chain mean size, with respect to Eq. (2). However, the power-law dependence, Eq. (3), is still valid. The RG calculations in the theta (tricritical) state [26] show that size effect deviations from this law are only manifested in linear chains through logarithmic corrections, in agreement with the previous arguments sketched by de Gennes [16]. The presence of these corrections in the macroscopic properties of experimental samples of linear chains is very difficult to detect. 

The three main parameters of clinical pharmacokinetics are clearance, distribution volume, and bioavailability. Clearance is the rate at which the body eliminates a drug. In order to achieve a steady-state concentration, the drug must be given so that the rate of clearance equals the rate of administration. If the drug is given as quickly as it is eliminated, a consistent level in the body will be maintained. 

Abstract. The physical nature of nonadditivity in many-particle systems and the methods of calculations of many-body forces are discussed. The special attention is devoted to the electron correlation contributions to many-body forces and their role in the Be r and Li r cluster formation. The procedure is described for founding a model potential for metal clusters with parameters fitted to ab initio energetic surfaces. The proposed potential comprises two-body, three-body, and four body interation energies each one consisting of exchange and dispersion terms. Such kind of ab initio model potentials can be used in the molecular dynamics simulation studies and in the cinalysis of binding in small metal clusters. 

For 1, the parameters found at the first step are used as the initial set. In the fit of V3 + Vi, for the better reproduction of the three-body energy it was found more effective to add to the E onadd AgN), the EsiAga) for triangles not presented in the Agjv geometry. 

Two functional forms are checked for the He2-Br2 potential energy function. One is based on the pairwise atom-atom interaction, which has been widely used in all previous calculations on triatomic and tetratomic, Rgn-X2 with n=l,2, complexes. The parameters for the two-body He-He interactions are taken from Ref. ° The second one is given by summing up three-body HeBr2 interactions and the He-He one,... 

Reprint F is an example of analyzing a reaction in formal kinetics. Gray and Scott introduced the autocatalytic A + 2B = 3B as a simple model reaction that proved to have a rich behavior, much richer than the Brusselator for example. However, A + 2B smacks of a three-body interaction, which is a sufficiently rare occurrence as to be avoided. I had done a pseudo-steady-state analysis before I visited Leeds at Gray s invitation, and the chance of working with the fons et origo of this reaction, so to speak, was an opportunity to make sure that the limiting behavior was not lost when certain parameters were small, but not actually zero. For another analysis of autocatalytic behavior, see [107]. 

Now consider an isolated chain of length n n. Parts nt < raj intrinsically should show only weak interaction effects, the effective -parameter zb tnj2 being small (We ignore three-body interactions.) Thus nj-blobs should be random walk-like intrinsically. A long chain should show a swollen configuration, built from Gaussian temperature blobs. This yields... 

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