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

Question

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.

Answer 1

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."

• "Maximally diagonal force constants in dependent angle-bending coordinates. II. Implications for the design of empirical force fields" J. Am. Chem. Soc. 1990, 112, 12, 4710-4723

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.

• QubeKit J. Chem. Inf. Model. 2019, 59, 4, 1366-1381 - code at GitHub
• QuickFF J. Comput. Chem. 2015, 36, 1015– 1027 - code at GitHub
• ForceBalance J. Chem. Theory Compute. 2013, 9, 1, 452-460 and J. Phys. Chem. Lett. 2014, 5, 11, 1885-1891 - code at GitHub
• ForceFit J. Comput. Chem., 2010 31: 2307-2316
• Parfit J. Chem. Inf. Model. 2017, 57, 3, 391-396 - code at GitHub