Harmonic potentials, bond stretches

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)). 

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). 

In a Urey-Bradley force field, angle bending is achieved using 1,3 non-bonded interaction rather than an explicit angle-bending potential. The stretch-bond term in such a forci field would be modelled by a harmonic function of the distance between the 1,3 atoms ... 

Bond stretching is most often described by a harmonic oscillator equation. It is sometimes described by a Morse potential. In rare cases, bond stretching will be described by a Leonard-Jones or quartic potential. Cubic equations have been used for describing bond stretching, but suffer from becoming completely repulsive once the bond has been stretched past a certain point. 

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. 

First of all, we have to take account of every bond-stretching motion. We could write a simple harmonic potential for each bond, as discussed above. For a bond A-B, we would therefore write... 

Usually the constants involved in these cross terms are not taken to depend on all the atom types involved in the sequence. For example the stretch/bend constant in principle depends on all three atoms. A, B and C. However, it is usually taken to depend only on the central atom, i.e. = k , or chosen as a universal constant independent of atom type. It should be noted that cross tenns of the above type are inherently unstable if the geometry is far from equilibrium. Stretching a bond to infinity, for example, will make str/bend go towards — oo if 0 is less than If the bond stretch energy itself is harmonic (or quartic) this is not a problem as it approaches +oo faster, however, if a Morse type potential is used, special precautions will have to be made to avoid long bonds in geometry optimizations and simulations. 

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... 

However, as seen in Fig. 3.2, this idealized harmonic oscillator (Fig. 3.2b) is satisfactory only for low vibrational energy levels. For real molecules, the potential energy rises sharply at small values of r, when the atoms approach each other closely and experience significant charge repulsion furthermore, as the atoms move apart to large values of r, the bond stretches until it ultimately breaks and dissociation occurs (Fig. 3.2c). 

These potential energy terms and their attendant empirical parameters define the force field (FF). More complicated FFs which use different and/or more complex functional forms are also possible. For example, the simple harmonic oscillator expression for bond stretching can be replaced by a Morse function, Euorse (3), or additional FF terms may be added such as the stretch-bend cross terms, Estb, (4) used in the Merck molecular force field (MMFF) (7-10) which may be useful for better describing vibrations and conformational energies. 

A harmonic potential is a good approximation of the bond stretching function near the energy minimum (Fig. 2.7). Therefore, most programs use this approximation (see Eq. 2.6) however the limits of the simplification have to be kept in mind, in those cases where the anharmonicity becomes important. Apart from the possibility of including cubic terms to model anharmonicity fsee the second term in Eq. 2.14), which is done in the programs MM2 and MM3[1,2,2 241, the selective inclusion of 1,3-nonbonded interactions can also be used to add anharmonicity to the total potential energy function. 

Abstract The problem of the low-barrier hydrogen bond in protonated naphthalene proton sponges is reviewed. Experimental data related to the infra-red and NMR spectra are presented, and the isotope effects are discussed. An unusual potential for the proton motion that leads to a reverse anharmonicity was shown The potential energy curve becomes much steeper than in the case of the harmonic potential. The isotopic ratio, i.e., vH/VD (v-stretching vibration frequency), reaches values above 2. The MP2 calculations reproduce the potential energy curve and the vibrational H/D levels quite well. A critical review of contemporary theoretical approaches to the barrier height for the proton transfer in the simplest homoconjugated ions is also presented. 

DREIDING [183] is an all-purpose organic or bio-organic molecular force field. It has been most widely used for large biomolecular systems. It uses either harmonic or Morse potential for the bond stretching, and the second power polynomials in... 

---