Overview of molecular forces: Justification of harmonic form for bond potential

Oliver Smart
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## Justification of harmonic form for bond potential

In general it is found (both experimentally and using quantum chemistry) that bonds between chemically similar atoms in a wide variety of molecules have similar lengths (e.g. bonds from carbon atoms to hydrogens are around 1 angstroms). Because of this it is reasonable to suppose that the potential energy cost for changing the bond length varies so that it has a minimum at the equilibrium value:

From high school mathematics any function minimum can be expanded as a Taylor power series:

Consider using this expansion for the potential energy for bond deformation around the equilibrium value. The term in will be zero because the derivative of the potential must be zero at an equilibrium value. Close to the equilibrium (at low values of ) the term in will increasingly tend to dominate the term in and higher terms. The practical upshot of this is that any function which varies around a minimum will behave as a quadratic sufficiently close to the minimum. i.e., in the case above the function will be:

so it will behave as a harmonic function: with A and B as constants. This approach is similar to the theoretical basis of many of the methods used in energy minimization.

In deriving a potential energy function for a molecule we are not interested in the absolute value of the potential energy so that the first term is unimportant. Therefore a Hooke's Law expression is normally used for the potential energy cost of bond deformation:

The simple expression means that only one constant has to be found for each different type of bond. However, it is only a first order approximation and if more accuracy is required then higher moments can be invoked or the Morse potential can be used.

The same reasoning also holds true for other terms which represent the energy cost for the change in a coordinate which only varies by a small amount around an equilibrium value ( e.g., bond angles). However, if the coordinate can take a variety of values then such an approach is unsuitable (eg., dihedral angles have more than one stable value and must be included in a different way).

As a practical example consider a bond whose potential is modelled by the Morse potential.

(Click on icon to proper sized image).

The Morse potential is given by:

the minimum of the function is at i.e. the equilibrium bond length (see graph above). It can be shown that (a little exercise to keep that high school maths fresh!) the second differential of the potential at the equilibrium position is:

plugging this into the harmonic equation above we achieve the red line shown in the graph. The parabola is a good approximation to Morse potential close to the equilibrium but begins to deviate away from this position. In particular the potential energy of parabola tends to infinity at large distances whereas the Morse potential can provide a more realistic representation of the formation energy of the bond.

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