Specific Force Constant Analysis and Computational Observables

Specific Force Constant Analysis and Computational Observables 

The first derivative of the energy with respect to a given bond length, i.e., dU/db, is a well-defined quantity that is a linear combination of Cartesian first 

Proceeding along these lines, if we now take the second derivative d U/dbf, the number of unknowns is even further reduced because now we probe the change in the force on the i bond due to a change in its own length, and all contributions to the molecular potential energy that do not couple — directly or indirectly —the force on the bond to itself are eliminated. Thus, taking the derivative of Eq. [23] we obtain 

We stress that it is specifically the second derivative that extracts the relevant information about the bond-stretching term. As noted above, this is modulated only by derivatives of the nonbonded term and the third-order terms. If, for example, we tried to parameterize v, by, say, changing just one C —H bond length in ethane and fitting the energy profile along this coordinate, then we will find that this profile contains contributions from other coordinates that actually change with the increase in the bond stretch. 

Although in this case the form of the bond potential is well known, this simple example clearly demonstrates the way in which the technique can be used to extract information about the functional form from ab initio calculations. If we did not know how the bond stretch should be adequately represented, this analysis would clearly confirm that a Morse function is more accurate than a harmonic function. 

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