Harmonic potentials

For a set of Gaussians, it is rather difficult to establish the analytic behavior of Eqs. f55), or of (50), in the t plane. However, with a single Gaussian (in one spatial dimension) and a harmonic potential surface one classically has... 

For certain values of q and a harmonic potential, the distribution pq (F) can have infinite variance and higher moments. This fact has motivated the use of the g-expectation value to compute the average of an observable A... 

Figure 7-9. Variation of the potential energy of the bonded interaction of two atoms with the distance between them. The solid line comes close to the experimental situation by using a Morse function the broken line represents the approximation by a harmonic potential.

For each pair of interacting atoms (/r is their reduced mass), three parameters are needed D, (depth of the potential energy minimum, k (force constant of the par-tictilar bond), and l(, (reference bond length). The Morse ftinction will correctly allow the bond to dissociate, but has the disadvantage that it is computationally very expensive. Moreover, force fields arc normally not parameterized to handle bond dissociation. To circumvent these disadvantages, the Morse function is replaced by a simple harmonic potential, which describes bond stretching by Hooke s law (Eq. (20)). 

Compared with the Morse potential, Hooke s law performs reasonably well in the equilibrium area near If, where the shape of the Morse function is more or less quadratic (see Figure 7-9 in the minimum-energy region). To improve the performance of the harmonic potential for non-equilibrium bond lengths also, higher-order terms can be added to the potential according to Eq. (21). 

As for bond stretching, the simplest description of the energy necessary for a bond angle to deviate firom the reference value is a harmonic potential following Hooke s law, as shown in Eq. (22). 

For every type of angle including three atoms, two parameters (force constant fe and reference value 0q) are needed. Also, as in the bond deformation case, higher-order contributions such as that given by Eq. (23) are necessary to increase accuracy or to account for larger deformations, which no longer follow a simple harmonic potential. 

The Morse function rises more steeply ihan ihe harmonic potential at short bonding distances. This difference can be important especially during molecular dynamics simulations, where thermal energy takes a molecule away from a potential minimum. ... 

Comparison of the simple harmonic potential (Hooke s law) with the Morse curve. 

Various other ways to incorporate the out-of-plane bending contribution are possible. For e3plane bend involves a cakulation of the angle between a bond from the central atom and the plane defined by I he central atom and the other two atoms (Figure 4.10). A value of 0° corresponds to all four atoms being coplanar. A third approach is to calculate the height of the central atom above a plane defined by the other three atoms (Figure 4.10). With these two definitions the deviation of the out-of-plane coordinate (be it an angle or a distance) can be modelled Lt ing a harmonic potential of the form... 

The first molecular dynamics simulations of a lipid bilayer which used an explicit representation of all the molecules was performed by van der Ploeg and Berendsen in 1982 [van dei Ploeg and Berendsen 1982]. Their simulation contained 32 decanoate molecules arranged in two layers of sixteen molecules each. Periodic boundary conditions were employed and a xmited atom force potential was used to model the interactions. The head groups were restrained using a harmonic potential of the form ... 

Fig. 9.24 A restraining potential that does not penalise struetures in which the distance lies between the leaver and upper distances di and and uses harmonie functions outside this range (left). The harmonic potentials may also he replaeed by linear restraints further from this region (right).

Ej(R). In the crudest useful approximation, Ej(R) is taken to be a so-called harmonic potential... 

Truneating the Taylor series at the quadratie terms (assuming these terms dominate beeause only small displaeements from the equilibrium geometry are of interest), one has the so-ealled harmonic potential ... 

Results using this technique are better for force helds made to describe geometries away from equilibrium. For example, it is better to use Morse potentials than harmonic potentials to describe bond stretching. Some researchers have created force helds for a specihc reaction. These are made by htting to the potential energy surface obtained from ah initio calculations. This is useful for examining dynamics on the surface, but it is much more work than simply using ah initio methods to hnd a transition structure. 

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. 

AMorse function best approximates a bond potential. One of the obvious differences between a Morse and harmonic potential is that only the Morse potential can describe a dissociating bond. 

In light of the differences between a Morse and a harmonic potential, why do force fields use the harmonic potential First, the harmonic potential is faster to compute and easier to parameterize than the Morse function. The two functions are similar at the potential minimum, so they provide similar values for equilibrium structures. As computer resources expand and as simulations of thermal motion (See Molecular Dynamics , page 69) become more popular, the Morse function may be used more often. 

You need to specify two parameters the equilibrium value of the internal coordinate and the force constant for the harmonic potential. The equilibrium restraint value depends on the reason you choose a restraint. If, for example, you would like a particular bond length to remain constant during a simulation, then the equilibrium restraint value would probably be the initial length of the bond. If you want to force an internal coordinate to anew value, the equilibrium internal coordinate is the new value. 

In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

The situation changes when moving on to low temperature. Friction affects not only the prefactor but also the instanton action itself, and the rate constant depends strongly on rj. In what follows we restrict ourselves to the action alone, and for the calculation of the prefactor we refer the reader to the original papers cited. For the cusp-shaped harmonic potential... 

In order to find Eq we study first the auxiliary problem of a cusp-shaped harmonic potential with a wall placed at x = Xp (see fig. 7),... 

Bead-spring models without explicit solvent have also been used to simulate bilayers [40,145,146] and Langmuir monolayers [148-152]. The amphi-philes are then forced into sheets by tethering the head groups to two-dimensional surfaces, either via a harmonic potential or via a rigid constraint. 

An elastic interaction between steps can also be approximated by a harmonic potential when the deviation of the steps from a straight line is small [18]. Even though steps fluctuate with a diverging width, Eq. (36), the separation between neighboring steps or the terrace width fluctuates a little... 

Figure S-13 Demonstration of the Hammond postulate with harmonic potential wells.

The harmonic potential is a good starting place for a discussion of vibrating molecules, but analysis of the vibrational spectrum shows that real diatomic... 

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