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.
- 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