# How are the parameters of a force field obtained for the bond angle HCH?

In my head, I can't see molecules that I could vary the HCH bond angle without varying other bond lengths, angles and dihedrals to build an energy vs. HCH bond angle curve.

If I vary a single HCH angle of a simple methane molecule, for example, I would automatically be varying the other HCH angles of that same molecule, which would affect the construction of the actual energy vs. HCH bond angle curve.

I may be talking nonsense.

You don't necessarily need a potential energy curve to fit a force field. I answered a related question about fitting a force field from quantum calculations.

Let's assume you have a harmonic angle term. You could either do this in terms of the angle bending or the distance between the end atoms, but let's take the angle for our example:

$$E_{angle} = k (\theta - \theta_0)^2$$

The first parameter you need is the optimal angle $$\theta_0$$ which you can get from a geometry optimization, experiment, etc.

Your question stems from the second parameter, $$k$$ - the force constant for whatever this angle type happens to be (e.g. H-C-H in methane for your example).

There are a few ways to get the force constant, but it comes from the second derivative, right? So you can get that from a Hessian or vibrational calculation. My favorite discussions of this center around the "Badger's rule for angle force constants, e.g."

Unfortunately, most such force fields are defined in well-determined sets of internal coordinates, whereas empirical potentials use larger sets of dependent coordinates. This paper illustrates a unique “localized” representation of the angle-deformation potential in dependent coordinates which is exactly diagonal for in-plane bending at trigonal-planar centers and is nearly diagonal for angle bending at tetra coordinate centers.

Modern force field fitting methods typically use scripts that minimize the differences between an in-development parameter set and a set of Hessians and/or experimental data.

As I mentioned in the other question, there's a huge pile of such papers, many with code to derive force fields from quantum chemical data.

• Thank you very much for your comments, Geoff Hutchison! I am very grateful for your help. As this area is new to me, it has been difficult for me to understand some things for now. The first idea I had was to derive the parameters of the force field through potential energy curves. Some things are explicit and I am managing to solve it in Gaussian, for example. Others don't. – Emerson P L Feb 26 at 18:28
• Hutcison I was thinking about one thing ... Every scan that we perform to obtain the energy curve (varying bond length, bond angle and dihedral), indirectly we would be changing the non-bonded interaction parameters. That would be a limitation, right? It would be practically impossible to obtain the isolated parameters of bond length, bond angle and dihedral. Did I say any nonsense? – Emerson P L Feb 26 at 19:42
• @EmersonPL - please read the 1990 Halgren paper. It goes into the issues of 'separability' between QM / experiment and the typical design of force fields. – Geoff Hutchison Feb 26 at 19:47