Central Pair Potential Research Articles

Multiscale coarse-graining and structural correlations: connections to liquid-state theory.

A statistical mechanical framework elucidates the significance of structural correlations between coarse-grained (CG) sites in the multiscale coarse-graining (MS-CG) method (Izvekov, S.; Voth, G. A. J. Phys. Chem. B 2005, 109, 2469; J. Chem. Phys. 2005, 123, 134105). If no approximations are made, then the MS-CG method yields a many-body multidimensional potential of mean force describing the interactions between CG sites. However, numerical applications of the MS-CG method typically employ a set of pair potentials to describe nonbonded interactions. The analogy between coarse-graining and the inverse problem of liquid-state theory clarifies the general significance of three-particle correlations for the development of such CG pair potentials. It is demonstrated that the MS-CG methodology incorporates critical three-body correlation effects and that, for isotropic homogeneous systems evolving under a central pair potential, the MS-CG equations are a discretized representation of the well-known Yvon-Born-Green equation. Numerical calculations validate the theory and illustrate the role of these structural correlations in the MS-CG method.

Molecular dynamics simulations and quantum mechanical studies of the hydrogen bond in water cluster systems

The formation and persistence of hydrogen bonds are inherently dynamic processes. In this research the technique of molecular dynamics is used to examine the evolution of hydrogen bonds as a function of initial conditions such as temperature, internal energy, and geometry. Clusters of water molecules serve as the prototypical hydrogen-bonded system. Inherent in the molecular dynamics simulations is the choice of the potential energy function. The results of molecular dynamics studies on a semi-empirical quantum mechanical potential surface are compared with simulations using a central force pair potential. In addition to the molecular dynamics calculations, ab initio quantum mechanical calculations of cluster geometry and stability were made for comparison with the predictions of the semi-empirical and empirical potentials. For both potentials used in the molecular dynamics simulations, the hydrogen bond was found to be quite flexible. It is found that the lifetime of a specific hydrogen bond can be substantially less than the lifetime of the cluster. This can be the case even when the average number of hydrogen bonds in the cluster remains constant. The flexibility of the hydrogen bond opens dynamically accessible reaction paths for molecular rearrangement or dissociation. Cooperative motions serve effectively to reduce energy barriers for such reactions. Hence the dynamical simulation can provide more realistic insight into the role of the hydrogen bonds in the process of interest than can examination of the static energy surface. The sensitivity of the results to the details of the potential is discussed. Preparation of the initial state of the molecule or cluster is seen to affect directly the evolution of the system.

Bond particle model for semiconductor melts and its application to the liquid structure of germanium

A simple type of liquid-state model is proposed to describe on a primitive level the melt of an elemental group IV semiconductor as a mixture of atoms and bond particles. The latter, on increase of a coupling strength parameter become increasingly localised between pairs of atoms up to local tetrahedral coordination of atoms by bond particles. Angular interatomic correlations are built into the model as bond particle localisation grows, even though the bare interactions between the components of the liquid are formally described solely in terms of central pair potentials. The model is solved for liquid structure by standard integral equation techniques of liquid-state theory and by Monte Carlo simulation, for values of the parameters which are appropriate to liquid germanium down to strongly supercooled states. The calculated liquid structure is compared with the results of diffraction experiments on liquid germanium near freezing and discussed in relation to diffraction data on amorphous germanium. The model suggests simple melting criteria for elemental and polar semiconductors, which are empirically verified.

The ground‐state energy of a system of identical bosons

A system of N identical bosons is studied, each having mass m, which interact in R3 via attractive central pair potentials and obey nonrelativistic quantum mechanics. A lower energy bound is found by the equivalent two‐body method. An upper energy bound established previously on the basis of field theory is now derived by variational methods within conventional quantum mechanics. In the case of the linear potential Vij=γ‖ri−rj‖ the bounds imply that the ground‐state energy is given by E=C(N)(N−1)(ℏ2/m)1/3(γN/2)2/3, where 2.3381<C(N)<2.343 52. The energy is therefore determined in this case with error <0.116% for all N≥2. Similar results are given for other power‐law potentials.

Cauchy relations for central-force random networks.

For many years it has been known that the elastic moduli of networks whose only interatomic forces are central pair potentials and whose atoms occupy centers of inversion at equilibrium obey the Cauchy relations. Recently it has been shown analytically that such networks with one bond missing (thus eliminating all centers of inversion at atomic sites) still obey the Cauchy relations exactly, while similar networks with one site missing do not. The usual hypothesis that must be satisfied in order for the Cauchy relations to hold must be generalized. This paper presents a possible hypothesis and explores its implications via computer simulation of three different two-dimensional random networks.

Low-energy scattering parameters for van der Waals perturbation theory.

The low-energy two-body scattering S- and P-wave parameters, together with a shape-dependent S-wave quantity, which appear in the well-known low-density expansions for the ground-state energy of many-boson and many-fermion systems, are accurately calculated for several central pair potentials for spin-polarized atoms, helium atoms, and nucleons. An integral-equation method allowing accurate determination of the coefficients of the above-mentioned parameters expanded in powers of the attraction strength of the pair potential is presented and applied. The results are needed for perturbation studies of ground-state energies of various quantum many-body systems.

Interactions, local order, and atomic-rearrangement kinetics in amorphous nickel-phosphorous alloys.

Molecular-dynamics computer simulations have been carried out to study the liquid and amorphous solid states for a system comprising 120 Ni and 30 P atoms. This study utilized additive central pair potentials to model the interactions. Emphasis has been placed on the geometry of the resulting potential-energy hypersurface for the system as a whole to explain temperature dependence of short-range order, and to characterize bistable degrees of freedom that dominate low-temperature properties in this, and other, amorphous solids. A small collection of transition states and associated reaction coordinates for such degrees of freedom has been numerically constructed. The corresponding atomic motions tend to be localized mainly on a small subset of the atoms, but are diverse insofar as which chemical species are involved, in barrier height and asymmetry, and in the nonlinearity of the collective reaction coordinate.

A theoretical estimation of orientational quantum effects in polar fluids

Thermodynamic properties of classical polar fluids can be estimated from those of a simple fluid wherein the central pair potential has been chosen to account for pairwise orientational correlations. This mapping of polar to simple fluids has been extended to account for the quantization of rotational motion and the corresponding damping of orientational correlations. Choosing the model parameters to reflect HCl, HBr and HI we find large rotational quantum effects on the phase diagram and second virial coefficients if dispersion forces are neglected but only very minor such effects when the dominant dispersion forces are included.

Local order and structural transitions in amorphous metal-metalloid alloys.

The NiBoPUl alloy system has been investigated in both liquid and amorphous solid forms using molecular-dynamics computer simulation. Atomic interactions were modeled by central pair potentials selected to represent roughly the atomic sizes and relative bond strengths. Self-diffusion constants and pair-correlation functions have been determined as a function of temperature. By means of a mass-weighted steepest descent on the potential-energy hypersurface, dynamical configurations at various temperatures have been mapped onto nearby potential-energy minima (stable atomic packings). This establishes that the liquid phase for the alloy has a temperature-independent inherent structure. Comparison with diffraction data on real Ni-P alloys suggests (and we verify) that improved molecular-dynamics modeling is possible.

Energy inequalities({N}{2})−1EN≤({K}{2})−1EK,2≤K&lt;N, relating two systems of identical bosons

Suppose ${E}_{N}$ is the lowest energy of a translation-invariant system of $N$ identical bosons which interact by attractive central pair potentials and obey nonrelativistic quantum mechanics. Sufficient conditions are prescribed on the potential to prove that ${E}_{N}$ is bounded above by $({N}{2}){({K}{2})}^{\ensuremath{-}1}{E}_{K}$, where $N\ensuremath{\ge}\ensuremath{\mu}K$. The $\ensuremath{\mu}$ numbers are specified for various classes of potential shape. In cases where $\ensuremath{\mu}\ensuremath{\le}\frac{3}{2}$, it follows that ${E}_{N}\ensuremath{\le}\frac{1}{2}N(N\ensuremath{-}1){E}_{2}$ for $N\ensuremath{\ge}3$.

The computation of the vibrational spectrum of hydrogen in aluminium by the recursion method

The vibrational density of states of hydrogen in aluminium was calculated by the recursion method using central pair potentials determined from pseudopotential theory. The coupling between hydrogen and aluminium atoms is fairly weak, and the low-frequency resonance appears in the spectral density due to weak transverse coupling.

Energy and Wave Functions of Interstitial Hydrogen in FCC and BCC Metals

Calculations have been performed of the energy and wave functions of H(D) atoms self-trapped on interstitial sites in fcc and bcc metals. The Schrodinger equation for a H(D) atom is solved in the field of interaction potential with surrounding metal atoms approximated by a sum of central pair potentials. Both ground and excited states have been studied for the purpose of making comparison with observed optical mode spectra of inelastic neutron scattering. It is shown from these calculations that a consistent understanding can be obtained of the excitation energy of a H atom on the T-site in V, Nb, Ta (bcc), the O-site in β-VH x (bct) and the O-site in Pd (fcc). Comparison of the calculated wave functions with Fourier-reconstructed density maps from neutron diffraction experiments is also made.

On the optical-mode excitation energy interstitial hydrogen in metals

An interpretation is proposed of a strong correlation between the optical-mode excitation energy and hydrogen-metal distance in a series of fluorite-type metal hydrides. Discussion is based on central pair potentials between hydrogen and metal atoms, and is shown to give consistent results on hydrogen on T sites in BCC metals as well.

Energy trajectories for theN-boson problem by the method of potential envelopes

This paper concerns the ground-state energy ${E}_{N}$ of a system of $N$ identical bosons interacting via the attractive central pair potential $V({r}_{\mathrm{ij}})=\ensuremath{-}{V}_{0}f(\frac{{r}_{\mathrm{ij}}}{a})$ and obeying nonrelativistic quantum mechanics. It is assumed that the potential shape $f$ is decreasing and can be represented as the envelope of each of two complementary families of power-law potentials $\ensuremath{\alpha}+\ensuremath{\beta}{r}^{p}$ (one family is above $f$ and the other below) for suitable fixed $p={p}_{1}$ and $p={p}_{2}$. If $\ensuremath{\epsilon}=\ensuremath{-}\frac{m{a}^{2}{E}_{N}}{(N\ensuremath{-}1){\ensuremath{\hbar}}^{2}}$ and $v=\frac{\mathrm{Nm}{V}_{0}{a}^{2}}{2{\ensuremath{\hbar}}^{2}}$, then it is proved that the entire collection of nonintersecting energy trajectories $\ensuremath{\epsilon}={F}_{N}(v)$, $N=2, 3, 4, \dots{}$, is bounded between the fixed curves $(v, \ensuremath{\epsilon})=(\ensuremath{-}\ensuremath{\gamma}(p){[{s}^{3}{f}^{\ensuremath{'}}(s)]}^{\ensuremath{-}1}, (\frac{v}{2})[2f(s)+s{f}^{\ensuremath{'}}(s)])$, where the curve parameter $s&gt;0$, and $p={p}_{1}, {p}_{2}$. Potentials, for example, with shapes $f(r)=\frac{{\ensuremath{\alpha}}_{1}}{r}+\frac{{\ensuremath{\alpha}}_{2}}{(r+{\ensuremath{\alpha}}_{3})}\ensuremath{-}{\ensuremath{\alpha}}_{4}\mathrm{ln}r\ensuremath{-}{\ensuremath{\alpha}}_{5}\mathrm{sgn}(q){r}^{q}$, where ${\ensuremath{\alpha}}_{i}\ensuremath{\ge}0$ and $|q|\ensuremath{\le}1$, have the $\ensuremath{\gamma}$ numbers $\ensuremath{\gamma}(\ensuremath{-}1)=2$ and $\ensuremath{\gamma}(1)=\frac{12}{\ensuremath{\pi}}$. The appropriate $\ensuremath{\gamma}$ numbers are provided for other classes of potential shape including perturbed harmonic oscillators, and also for problems in one spatial dimension. The method yields in effect a recipe for the way ${E}_{N}$ depends on $N$ and all the parameters of the pair potential.

Comparison of the major force constant models for cubic systems using a self-consistency condition

Previously noted similarities between the central pair potential, CPP, the DeLaunay angular force, DAF, and the Lehman, Wolfram, and De Wames axially symmetric, AS, models for lattice dynamics coupled with criticisms of the DAF model, not applicable to the CPP model, has resulted in a puzzling and confusing picture. New physical arguments are used to develop a self-consistency condition, essentially unrecognized previously, which must be included in the use of all these force constant models. In this event, all of the above three models are proven to be identical. We then go on to examine the Clark–Gazis–Wallis, CGW, model. A systematic procedure is presented allowing a practical extension of the expression for the force constants from the current first and second nearest neighbour interactions to include third nearest neighbour interactions. For fcc systems we show that the previous expressions are incomplete and, consequently, overly restrictive and misleading and, hence, causing a current controversy. Also, we point out that the CGW model reproduces the general tensor force, GTF, model, at least to third nearest neighbour interactions. It is important to note that the application of the above mentioned self-consistency condition restores the Cauchy relation in all the models. Finally, numerical results are presented for the phonon spectra of the bcc metals Na and Nb and the fcc metals Cu and Au in order to demonstrate the practical effect of the self-consistency condition. This effect is seen to be larger for metals having larger Cauchy discrepancies. It is inferred that the force constant models, and hence the self-consistency condition, must be modified to yield the Cauchy discrepancy.

Internal equilibrium and crystal dynamics of tantalum

A lattice dynamical model for bcc metals, which satisfies the internal force equilibrium condition of the lattice is proposed. The present model combines a linearized Thomas-Fermi Theory for the electron-ion interaction and the central pair potential for ion-ion interaction. as an application, the computed phonon dispersion, vibrational spectrum, and lattice specific heat of tantalum show fairly good agreement with experiments.

Overlap interaction between spheroidal ions and the lattice dynamics of zinc and beryllium

A phenomenological model of lattice dynamics of hcp metals has been developed by assuming the ions to have a spheroidal shape with the axis of revolution parallel to the $c$ axis. Overlap of spheroidal ions causes a noncentral interaction between the first neighbors (as in beryllium). The model is derived from a force system consisting of this noncentral overlap interaction, as six-neighbor central pair potential, and an electron-ion interaction term. Application of this model to zinc and beryllium shows that the force parameters coupling neighbors beyond the second are rather small as compared to the parameters for the first two neighbors. The dispersion curves derived from the model present a good agreement with the experimental curves.

Crystal equilibrium and phonon dispersion in some bcc transition metals

A lattice-dynamical model has been developed for bcc metals which considers ion-ion interaction through second-neighbor central pair potential and electron-ion interaction based on the Sharma-Joshi theory. In the present model the ionic lattice is in equilibrium in a medium of electrons. The theory has been applied to calculate the dispersion curves in a number of transition metals. The results when compared to experimental neutron data show a reasonably good agreement in the [100] and [111] directions, while sufficient deviations are present in the [110] direction.

Lattice dynamics of hcp metals

A lattice dynamical model for hcp metals has been developed which considers ion-ion interaction as a five-neighbour central pair potential and electron-ion interaction in a modified form of Bhatia's model. Complete account of the equilibrium of the hcp lattice including the effect of electron pressure has been considered and as a consequence two equilibrium conditions, necessary for the hcp structure, have been obtained and used in the study. Phonon dispersion curves for four hcp metals, Be, Mg, Y and Ho have been computed for two symmetry directions and a good agreement has been obtained with the neutron data. The results have also been compared with the use of the same model but with a single condition, reported earlier and show in general a significant improvement.

Crystal equilibrium and phonon dispersion in rare‐earth H.C.P. metals

AbstractThe phonon dispersionin two h.c.p. rare‐earth metals Tb and Ho are investigated on the basis of a model which uses a 5‐neighbour−central pair potential for close‐range forces and a screened Coulomb interaction for long‐range forces. The model considers completely the equilibrium of the ionic lattice in a medium of an uniform electron gas. Consequently two equilibrium conditions, necessary for the h.c.p. structure, are deduced and used in the study.