Harmonic potential function

The discussion of the previous section amounts to a qualitative treatment of harmonic vibrational motion. The harmonic potential function on which the molecule vibrates has been described in terms of displacement of bond stretches from the equilibrium configuration for the diatomic molecule for water, displacement of... 

Figure 2 Energy and vibrational frequency of a classical harmonic oscillator. The sohd fine is the harmonic potential function. The dashed line represents the motion of the mass-and-spring system, displaced by Ar = 5 units then released...

FIGURE 3. The solid line is a harmonic potential function, and the dashed line is a typical anharmonic function. The horizontal line represents the zero-point vibrational energy level, and the average value of r is shown. It is larger than the equilibrium value of r (vertical dashed line)... 

It is beyond the scope of this short review to list every available molecular mechanics program. Only a selected few programs are mentioned here, without descriptive details of the potential functions, minimization algorithms, or comparative evaluations. Both the CHARMM and AMBER force fields use harmonic potential functions to calculate protein structures. They were developed in the laboratories of Karplus and Kollman, respectively, and work remarkably well. The CFF and force fields use more complex potential functions. Both force fields were developed in commercial settings and based extensively or exclusively on results obtained from quantum mechanics. Unlike the other molecular mechanics methods, the OPLS force field was parameterized by Jorgensen to simulate solution phase phenomena. 

Fig. 1.26. The equilibrium bond lengths for the v 0 and v = 1 states for HCl and DCl. The solid curve is the anharmonic potential function and the dotted curve is the harmonic potential function for both molecules.

Harmonic zero order wavenumbers (vj are used because the diatomic formula was derived assuming a harmonic potential function. For H Cl and the deuterated isotope H Cl the wavenumber ratio calculated from the above equation is... 

M-L bond deformations are often described with a harmonic potential function, equation (1). This description is suitable for small distortions from equilibrium and it has been demonstrated to provide an accurate and transferable description of M-L bond lengths in many transition metal complexes. When large distortions of M-L bond length occur, a more accurate description of the potential energy surface can be obtained with anharmonic potential functions such as that used in MM3, equation (2). 

Figure 1 Energy surfaces for K-0 bond length distortion computed with an ionic potential function (line), a harmonic potential function (triangles), and the default MM3 anharmonic potential function (circles)...

Elastic energy can be approximated by the harmonic potential function (Evans and Hochmuth, 1978)... 

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