Potential energy as a function

Figure C2.1.4. Potential energy as a function of the rotation about the central C-C bond in butane. The sketches show the projection of the molecule along the central C-C bond.

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

Figure 4-3 Potential Energy as a Function of Compression or Stretching of a One-Dimensional Harmonic Oscillator.

Figure 1.8 Hindered rotation around a carbon-carbon bond, (a) The definition of (p (from 0 = 0) in terms of the ethane molecule, (b) The potential energy as a function of (p. (c) Here (p is shown (from (p = 0) for a carbon-carbon bond along a polyethylene backbone, (d) The potential energy for case (c) shown as a function of (p. [Panels (b) and (d) reprinted with permission from W. J. Taylor, J.Chem.Phys. 16 257 (1948).]...

Figure 2.7 Potential energy as a function of location along the reaction coordinate. The solid line describes an undisturbed liquid the broken line applies to liquids subjected to shearing force.

Figure 6.35 Contour of potential energy as a function of two C-0 bond lengths, r- and r2, in CO2...

Sketch the form of the potential energy as a function of torsional angle cj) for the torsional vibration in (a) ethane, (b) CH3NO2, (c) 2-fluorophenol, (d) CH2FOH, and... 

Fig. 3.1. Potential energy as a function of torsion angle for ethane.

The potential energy surface consists of two valleys separated by a col or saddle. The reacting system will tend to follow a path of minimum potential energy in its progress from the initial state of reactants (A + BC) to the final state of products (AB -F C). This path is indicated by the dashed line from reactants to products in Fig. 5-2. This path is called the reaction coordinate, and a plot of potential energy as a function of the reaction coordinate is called a reaction coordinate diagram. 

Exercise 3.2. Write a computer program that solves Exercise 3.1 for any given time and evaluate the kinetic energy and potential energy as a function of time. 

Figure 5. Potential energy as a function of time for the octa-peptide simulation at 300K starting from (a) an a-helix, and (b) from a structure provided by ECEPP.

Equation describes electrical potential energy as a function of charges q and qi) and their separation distance... 

The first two tenns in Eq. (2) represent the kinetic energy of the nuclei and the electrons, respectively. The remaining three terms specify the potential energy as a function of the interaction between the particles. Equation (3) expresses the potential function for the interaction of each pair of nuclei. In general, this sum is composed of terms that are given by Coulomb s law for the repulsion between particles of like charge. Similarly, Eq. (4) corresponds to the electron-electron repulsion. Finally, Eq. (5) is the potential function for the attraction between a given electron (<) and a nucleus (j). 

During initialization and final analysis of the QCT calculations, the numerical values of the Morse potential parameters that we have used are given as De = 4.580 eV, re = 0.7416 A, and (3 = 1.974 A-1. Moreover, the potential energy as a function of internuclear distances obtained from the analytical expression (with the above parameters) and the LSTH [75,76] surface asymptotically agreed very well. 

For polymers with achiral substituents, molecular mechanics calculations (Discover 3, ver. 4.00 simple minimization) by us of potential energy as a function of dihedral angle afforded symmetrical multi-well-potential curves, with nonminimum 180° values, as shown in Figures 4.37a and 4.37b for H-(Ph2Si)3o-H and H-[(/7-n-BuPh)2Si]30-H, respectively. 

Figure 38 Force field-calculated potential energy as a function of backbone dihedral angle for (a) H-(Ar2Si)3o-H and (b) syndiotactic H-(ArAr Si)30-H (where Ar = 4-n-butylphenyl, and Ar = 4-(S,)-2-methylbutylphenyl).248Reprinted with permission from Koe, J. R. Fujiki, M. Motonaga, M. Nakashima, H. Macromolecules 2001,34, 1082-1089, 2001 American Chemical Society.

The results suggest that, if A... B repulsions dominate, then the spinel form is stable with respect to the olivine form and that if B... B repulsions dominate, the reverse is true. (The relative hardness of A and B is important, i.e. the slopes of their curves of potential energy as a function of separation.)... 

Using the relationship between force and potential energy discussed earlier, we can represent the nuclear force in terms of a simple plot of the nuclear potential energy as a function of distance to the center (Fig. 5.1). Since low-energy particles... 

Fig. 1. Potential energy as a function of internuclear distance for states of the O2 molecule. Curve (a) after Wilkinson and Mulliken, (105). Curve (b) after Vander-slice et al. (86). Curve (c) after Carroll (21). Portions of this figure are taken from ref. (105) with the permission of The Journal of Chemical Physics.

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