Morse functions

To calculate the bonded interaction of two atoms, a Morse function is often used. It has the form described by Eq, (19). 

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). 

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. ... 

The analytic PES function is usually a summation of two- and three-body terms. Spline functions have also been used. Three-body terms are often polynomials. Some of the two-body terms used are Morse functions, Rydberg... 

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. 

A larger value for the stretch force constant Kj. leads to a greater tendency for the bond to remain at its equilibrium distance rg Higher powers of r - rg, giving cubic, quartic, or higher terms are also common. A Morse function might also be employed. 

The Morse function which is given above was obtained from a study of bonding in gaseous systems, and dris part of Swalin s derivation should probably be replaced with a Lennard-Jones potential as a better approximation. The general idea of a variable diffusion step in liquids which is more nearly akin to diffusion in gases than the earlier treatment, which was based on the notion of vacant sites as in solids, remains as a valuable suggestion. 

The a constant is the same as that appearing in the Morse function, but is usually taken as a fitting parameter. 

A third functional form, which has an exponential dependence and the correct general shape, is the Morse potential, eq. (2.5). It does not have the dependence at long range, but as mentioned above, in reality there are also R, /f etc. terms. The D and a parameters of a Morse function describing vdw will of course be much smaller than for str, and Ro will be longer. 

For small systems, where accurate interaction energy profiles are available, it has been shown that the Morse function actually gives a slightly better description than an Exp.-6, which again performs significantly better than a Lennard-Jones 12-6 potential. This is illustrated for the H2-He interaction in Figure 2.9. 

In the absence of stress, the potential energy for the bond is usually approximated with the Morse function (Fig. 18) [87] ... 

Fig. 19. Derivative of the Morse function. The shaded area corresponds to the bond energy under stress...

The VB potential surfaces 1E and 3 7s can be approximated by analytical Morse and anti-Morse functions ... 

Vibrational levels are often analyzed in terms of an harmonic model, but in practice this is not a good approximation for real chemical bonds. An empirical expression that provides a more realistic description of a stretched bond, is the Morse function... 

The three parameters in the Morse function D, B, re are positive and are usually chosen to fit the bond dissociation energy, the harmonic vibrational frequency and the equilibrium bond length. At r = re, the Morse function V = 0. As r — D, V approaches D. For r re, V is large and positive, corresponding to short range repulsion. Although the Morse function has been used extensively, its representation of the potential away from re is not satisfactory. Several modifications have been proposed in Morse function. 

It is better than Morse function, because multiplying the exponential term, bypolynomial convergence can be checked by adding more terms by polynomial. This function can be linearised. A plot of In V versus r will give the value of D and o. ... 

Blais and Bunker developed an analytic function for A + BC — AB + C reactions by using Morse function and a switching function. 

This specific problem was addressed by Burgi and Dunitz (1987), using a Morse function modified to treat fractional bonds. The observed ground state structure of a given molecular fragment, e.g. the O-C-O fragment of the acetals [96], was then treated as a distorted version of a standard structure of known molecular dimensions. Using experimental values of the... 

The potentiul energy for the nuclear motion of a diatomic molecule is closely approximated hy the Morse function... 

---