Refactor angle force constant formula and add test suite

[Eashan-H]

Summary
This PR refactors the get_angle_force_constant method in ffprime/bond.py to strictly adhere to the standard Seminario (1996) method. The previous
implementation produced physically unrealistic values (~2500 kcal/mol/rad²) due to squared eigenvalue analysis and incorrect unit scaling. The corrected
code now yields standard molecular mechanics values (~10–25 kcal/mol/rad²) that match established literature results.

Physical & Mathematical Rationale
The derivation follows the Seminario method for calculating harmonic force constants directly from the QM Hessian. The core logic was updated to address
three critical issues:

1. Eigenanalysis: Replaced the use of $H H^T$ (which effectively squared the force constants) with a direct eigenanalysis of the off-diagonal $3 \times
   3$ Hessian blocks ($H_{ij}$ and $H_{kj}$) connecting the central atom $j$ to terminal atoms $i$ and $k$.
2. Directional Projections: Updated the projection of eigenvalues onto the bending directions ($u_{PA}$ and $u_{PC}$) using absolute weights rather than
   squared weights, as specified in the original paper.
3. Harmonic Combination: Implemented the correct combination rule for the two "arms" of the angle. The angle force constant $k_{\theta}$ is derived via
   the harmonic mean of the projected Cartesian stiffness:
   $$\frac{1}{k_{\theta}} = \frac{r_{ji}^2}{k_{PA}} + \frac{r_{jk}^2}{k_{PC}}$$
   where $r$ is the equilibrium bond length and $k_{P}$ is the projected force constant.

Changes

• ffprime/bond.py:
  • Refactored get_angle_force_constant with the corrected Seminario logic.
  • Updated get_force_constant and get_angle_force_constant to default the hess argument to self.hess_new, improving usability.
  • Improved docstrings with LaTeX equations and clear unit documentation ($kcal/mol/\text{\AA}^2$ to $kcal/mol/rad^2$).
  • Wrapped example execution code in if name == "main": to prevent side effects during library imports.
• tests/test_bond.py:
  • Added a new pytest suite to verify the Bonded class.
  • Included accuracy tests that match the calculated results against the standard Seminario reference for the lig.log example.

Verification Results
The implementation was verified using the provided examples/lig.log and lig.fchk files:

• Angle A1 (2-1-13):
  • Previous value: ~2498.5 kcal/mol/rad²
  • Corrected value: 14.70 kcal/mol/rad² (Matches reference)
• Comprehensive Check: All 26 angles in the example system now fall within the physically expected range of 10.2 to 24.4 kcal/mol/rad².
• Automated Tests: All 4 new test cases in tests/test_bond.py pass successfully.

Closes : Bug: Physically unrealistic angle force constants in get_angle_force_constant #6

[Eashan-H]

Hi @QC-Devs team,

I’m Eashan, a student at IIT Bhilai, and I’ve really enjoyed diving into the ffprime codebase to implement this fix for the Seminario angle force
constants.

Beyond this contribution, I am writing to express my strong interest in participating in Google Summer of Code (GSoC) 2026 with the QC-Devs organization.
I am very interested in the intersection of computational chemistry and software development, and I believe my background aligns well with the technical
challenges of this project.

I would love to learn more about your GSoC project ideas or any specific areas where I can contribute further to demonstrate my skills. Please let me know
if there are specific mentors I should reach out to or a community channel (like Slack or Discord) where I can join the discussion.

Looking forward to your feedback on the PR!

Best regards,
Eashan