Interactions harmonic

The result can be derived by assuming a harmonic interaction between consecutive steps, with the strength of interaction A gff = 7r 7/Lcou. 

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 high-frequency mode is coupled through the anharmonicity coefficient y with the low-frequency mode which is a resonant one due to harmonic interaction with the surface reservoir. For simplicity, the last term is written in the Gaitler-London approximation (compare with more general Eq. (4.1.8) where this restriction is absent). The required GF of the high-frequency mode can be obtained from the... 

The G-reduced Hamiltonians are necessary and sufficient for at least three important classes of Hamiltonians (i) all one-particle Hamiltonians, (ii) bosons or fermions with harmonic interactions [24], and (iii) all Hamiltonians with... 

G. Gidofalvi and D. A. Mazziotti, Boson correlation energies via variational minimization with the two-particle reduced density matrix exact iV-representabihty conditions for harmonic interactions. Phys. Rev. A 69, 042511 (2004). 

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. 

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. 

We already noted that, for a harmonic interaction, o, = constant and (z) = z0 = Zq — (p/B) varies linearly with the applied pressure. However, for an anharmonic interaction, oa is a function of the applied pressure (or, equivalently, ofthe average thickness ofthe film (z), which differs from Zq because of the asymmetry of the distribution). The functional dependence of the pressure on the average thickness differs in the anharmonic and harmonic cases. The pressure is higher in the former case, because ofthe... 

The simplest of these approaches includes Gaussian Network Models (GNM) or Elastic Network Models (ENM) which assume that the native state represents the minimum energy configuration. A structure is represented as a network of beads connected by harmonic springs.12,13 One bead represents one residue and is usually centered on the position of the Ca carbon. Single parameter harmonic interactions are assigned to bead pairs which fall within a certain cutoff distance Rc. In case of proteins, Rc is usually around 8-10 A. The representation of the molecule in the... 

Figure 3-4. Representation of the molecule in the ENM. Each node represents the center of the bead which is the C carbon (CA) in case of one-bead protein models. The lines (pseudobonds) represent the harmonic interactions between beads that are within certain distance apart...

The reaction for stractural diffusion involves replacing the bond-stretching (harmonic) interaction of the 0 -H of the reactant HsO with a non-bonded interaction while the reverse switch is made for the reactant 0(H20) and the H. Then the proton is moved along the 0 0 axis such that the ratio of 0 -H and O-H distances are the same in the product molecules as they were in the reactant molecules. 

The Xj s are variations from equilibrium position of interatomic distances or angles. In this harmonic approximation we have terms in xj, but also crossed terms in XjXj, that represent harmonic interactions between two different vibrations j and / that are often related to nearly lying atoms. The presence of such terms means that one cannot excite the vibration defined by Xj without also exciting that defined by Xj,. One can, however, eliminate such cross terms in the harmonic approximation by performing a linear transform on the x s, which depends on the various coefficients found in the harmonic development of V, which are called force constants. The newly defined coordinates are those of normal modes of the molecule. They are linear combinations of the x s, which implies that the x s may also be expressed in the form of the reciprocal linear combination of these normal modes. The... 

This last interaction term is harmonic becanse it is quadratic in the vibrational coordinates and harmonic terms are by definition those that are of degree less or eqnal to 2. go is defined in eq. (7.A5) of the appendix. The effect of this harmonic interaction term is to shift the centre of the band by a quantity eqnal in wavennmber to -( > (gg WqI c), with c the velocity of light. In the case of a mixed dimer the effect of this last term is, as already mentioned, negligible, because the two vibrations are not resonant. The magnitude of this term can conseqnently be taken equal to 110 cm in the case of (CD3COOH)2 and 80 cm in the case of (CD3COOD)2. These are values that are consistent with eq. (7.A7). The comparison of the bands of these mixed dimers with those of homodimers conseqnently conveys usefnl information on these stretching vibrations of H-bonds, more particnlarly on their harmonic interactions. 

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] ... 

We employ the above formalism and study the conductance of alkane chains of variable length (1-30 units). We have numerically verified that heat conduction in these systems is dominated by harmonic interactions [18]. A recent experiment confirmed this observation, demonstrating ballistic energy transfer in short hydrocarbon mono-layers [24]. 

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