Harmonic interaction potential

In the case of a linear interaction between neighboring lipid bilayers, Helfrich has demonstrated that the repulsive free energy due to confinement is inversely proportional to (7b2. While this result is strictly valid for a harmonic interaction potential (linear force), we assume that it can be extended to any interaction. We will examine later under what conditions this approximation is accurate. 

Only k, within a single primitive cell of the reciprocal lattice, yields different solutions then, there will be N non-equivalent values of k. This is why they can be chosen to be any primitive cell of the reciprocal lattice adequately selected in the first BriUouin zone. In the case of a 3-dimensional harmonic interaction potential, we can use the following expression [2] ... 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

There are other sources of nonlinearity in the system, such as the intrinsic anharmonicity of the molecular interactions present also in the corresponding crystals. While these issues are of potential importance to other problems, such as the Griineisen parameter, expression (B.l) only considers the lowest order harmonic interactions and thus does not account for this nonlinear effect. We must note that if this nonlinearity is significant, it could contribute to the nonuniversality of the plateau, in addition to the variation in Tg/(do ratio. It would thus be helpful to conduct an experiment comparing the thermal expansion of different glasses and see whether there is any correlation with the plateau s location. 

The low-lying excited states of the hydrogen molecule conhned in the harmonic potential were studied using the configuration interaction method and large basis sets. Axially symmetric harmonic oscillator potentials were used. The effect of the confinement on the geometry and spectroscopic constants was analyzed. Detailed analysis of the effect of confinement on the composition of the wavefunction was performed. 

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

Under the assumptions of a harmonic oscillator and an interaction potential, which is linear in (r — re), only the next lowest vibrational channel n = n — 1 can be populated. The propensity rule n — n — 1 is a strict selection rule. 

The last task is to calculate the area of the independent pieces S0, which is provided by eq 6 only for a harmonic interaction. To accomplish this, we will seek a harmonic potential Hdz) = (l/2)Bdz — zdj2 4- C (with Bo, zo, and C independent of z), which best approximates H(z) and use So = mdBo)m. 

The effect of electrolyte concentration on the transition from common to Newton black films and the stability of both types of films are explained using a model in which the interaction energy for films with planar interfaces is obtained by adding to the classical DLVO forces the hydration force. The theory takes into account the reassociation of the charges of the interface with the counterions as the electrolyte concentration increases and their replacements by ion pairs. This affects both the double layer repulsion, because the charge on the interface is decreased, and the hydration repulsion, because the ion pair density is increased by increasing the ionic strength. The theory also accounts for the thermal fluctuations of the two interfaces. Each of the two interfaces is considered as formed of small planar surfaces with a Boltzmannian distribution of the interdistances across the liquid film. The area of the small planar surfaces is calculated on the basis of a harmonic approximation of the interaction potential. It is shown that the fluctuations decrease the stability of both kinds of black films. 

The above derivation will be extended to a well -behaved interaction potential, U(z), for which (7(0) — oo U oo) —0 and has a minimum at a distance z0. In the vicinity of the minimum of the enthalpy (per unit area) H(z) = U(z) + pz, the potential will be approximated by a harmonic one, with the effective spring constant B = iiAH(z)/dz%=Z(] , where z0 is the solution of <>H(z)/iiz = 0. It will be assumed that the areas of the small independent surfaces into which the interfaces are decomposed are still given by eq 27, but... 

In Figure 6, the pressure (a) and the root mean square fluctuation (b) are plotted as functions of the average thickness ofthe film, for Kc = 10 x ICC19 J and for (1) the anharmonic (eq 32) and (2) the harmonic (eq 23) interaction potentials. The spring constant for the second case was obtained from the harmonic approximation of eq 32 around its minimum, at p = 0. 

The model which has received the greatest theoretical attention is that of a colinear collision between an atom A and a diatomic molecule BC (Figure 3.1). The molecule is usually assumed to be a harmonic oscillator, and the interaction potential VAB an exponential repulsion. The model was first... 

Consider first the one dimensional, classical problem of a diatomic molecule BC colliding with an atom A (Figure 3.1) to yield either vibrational activation or deactivation of BC. Suppose that the molecule is a harmonic oscillator with frequency v. If the interaction potential is... 

It has been found useful to represent the interaction potential for a dimer of homonuclear diatomic molecules [4,5,46,58] as a spherical harmonic expansion, separating radial and angular dependencies. The radial coefficients include different types of contributions to the interaction potential (electrostatic, dispersion, repulsion due to overlap, induction, spin-spin coupling). For the three dimers of atmospheric relevance, we provided compact expansions, where the angular dependence is represented by spherical harmonics and truncating the series to a small number of physically motivated terms. The number of terms in the series are six for the N2-O2 systems, corresponding to the number of configurations of the dimer (for N2-N2 and O2-O2 this number of terms is reduced to five and four, respectively). 

This transformed external potential (239) has the same form as the potential for the undriven (F = 0) harmonic well problem in the rest frame, except for the term c which depends on time but not on r. Furthermore, because the (Coulomb or other) particle-particle interaction is a function only of differences Tj — tj = f — fj, the interaction potential is also invariant under the transformation to the accelerated frame. Thus, both classically and quantum mechanically, any state or motion in the rest frame has a counterpart motion with superimposed translation X(f), provided that (238) is satisfied. In particular... 

These classical interaction potentials must be parameterized, e.g. the magnitude of the partial charges on each atom in the molecule must be assigned, and the equilibrium bond length and size of the harmonic force constant must be attached to each bond. In the early biomolecular MM forcefields, these parameters were developed to produce molecular models that could reproduce known experimental properties of the bulk system. For example, several MM water models have been developed. ° One of the earliest successful models, TIP3P, was parameterized such that simulations of boxes of TIP3P molecules reproduced known thermodynamic properties of water, such as liquid density and heats of vaporisation. Such a parameterisation scheme is to be applauded, as it ties the molecular model closely to experiment. Indeed many of the common MM models of amino acids were developed by comparison to experiment, e.g. OPLS. Indeed it is such a good... 

Fig. 3 A comparison of different coarse grain lipid models. The Shelley model " of DMPC, and Marrink and Essex models of DPPC are compared to their atomistic equivalents (for ease of comparison, hydrogen atoms of the atomistic models are not shown). Solid lines represent harmonic bonds connecting CG particles, and the CG particle types for the Shelley and Marrink models are labelled (the labels are the same as those used in the main text). The point charges (represented by + and —) and point dipoles (represented by arrows) are shown for the Essex model (the charges and dipoles are located at the centre of their associated CG particle). The Shelley and Marrink models use LJ particles (represented by spheres), while the Essex model uses a combination of LJ particles (spheres) and Gay-Berne particles (ellipsoids). Finally, the blob model proposed by Chao et al is also shown for comparison. This model represents groups of atoms as rigid non-spherical blobs that use interaction potentials based on multipole expansions.

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