The Force Field Energy Function
The total potential energy is a sum of bonded and non-bonded contributions:
\[E_{total} = \underbrace{\sum_{bonds}k_b(r-r_0)^2 + \sum_{angles}k_\theta(\theta-\theta_0)^2 + \sum_{torsions}V_n[1+\cos(n\phi-\gamma)]}_{bonded} + \underbrace{\sum_{i<j}\left[\frac{A_{ij}}{r_{ij}^{12}}-\frac{B_{ij}}{r_{ij}^6}+\frac{q_iq_j}{4\pi\varepsilon_0 r_{ij}}\right]}_{non\text{-}bonded}\]The Lennard-Jones $r^{-12}$ repulsion models Pauli exclusion while the $r^{-6}$ term captures London dispersion. Partial charges $q_i$ generate the electrostatic interactions.
Parameterisation
Force field parameters are derived from a combination of:
- High-level QM calculations (bond lengths, torsion profiles, ESP charges)
- Experimental thermodynamic data (heats of vaporisation, densities)
- Spectroscopic observables (IR frequencies)
| Force Field | Primary Application | Charge Scheme |
|---|---|---|
| AMBER | Proteins, nucleic acids | RESP |
| CHARMM | Membrane proteins, lipids | CGenFF |
| OPLS-AA | Organic liquids, drug-like | CM5 |
| GROMOS | Carbohydrates | Partial equalisation |
Strengths and Limitations
MM geometry optimisation minimises $E_{total}$ with gradient-based methods such as conjugate gradient or L-BFGS, converging in $O(10^3)$–$O(10^5)$ steps. The computational cost scales as $O(N\log N)$ with particle mesh Ewald for long-range electrostatics.
Key limitations include the inability to model bond breaking or formation, fixed charge models that ignore electronic polarisation, and transferability issues outside the parameterisation set. QM/MM hybrid methods address the first two by treating a reactive centre with DFT or semiempirical quantum mechanics while the environment is described by MM.