2. Potential models, forces and transferability
Position in the course: Lesson 2 of 11. Complete the preceding derivation and use the explained exercises to check understanding.
1. Decide what the particles represent
A trajectory requires an energy function before it requires an integrator. In an all-atom force field, bonded terms encode stretching, bending and torsion; nonbonded terms describe repulsion, dispersion and electrostatics. A coarse-grained particle may represent several atoms, so its effective interactions depend on the state and on the degrees of freedom that were eliminated. A machine-learned potential approximates a reference energy surface within its training domain. None of these names guarantees transferability to new temperatures, charges or reactions.
Use positions \(R\), energy \(U(R)\) and forces \(F_i=-\nabla_iU\). This lesson assumes differentiable pair interactions and fixed particle identities. Reactive, polarizable and many-body models require additional variables or derivatives. Matching energies without matching their derivatives is insufficient for dynamics.
2. Differentiate the Lennard–Jones model
For a pair at separation \(r\), write the repulsive and attractive terms separately. The radial force is positive outward. Differentiation gives
Set the bracket to zero: \(2\sigma^{12}/r^{12}=\sigma^6/r^6\), hence \(r_m=2^{1/6}\sigma\). Substitution yields \(U(r_m)=-\epsilon\). The zero of energy is at \(r=\sigma\), which is distinct from the minimum. At the minimum the force vanishes, but the curvature is positive:
The reduced mass occurs because the separation is a relative coordinate. This connects a potential parameter to the fastest local vibration and therefore to timestep selection. It is an analytic two-particle model, not a calculated liquid spectrum.
3. A worked dimensionless check
At \(r=\sigma\), the energy is zero and \(F_r=24\epsilon/\sigma\): the particles still repel. At \(r=2\sigma\), \(U/\epsilon=4(2^{-12}-2^{-6})=-0.0615234375\) and \(F_r\sigma/\epsilon=12(2\,2^{-12}-2^{-6})=-0.181640625\). Negative force means attraction. A useful implementation test compares the analytic force to a centered difference, \(-[U(r+\delta)-U(r-\delta)]/(2\delta)\). Sweep \(\delta\): truncation error decreases initially, then floating-point cancellation dominates. One arbitrary displacement does not establish consistency.
4. Cutoffs change the model
Subtracting \(U(r_c)\) makes energy continuous but leaves a force jump. A shifted-force construction subtracts both the energy and the slope at the cutoff:
Its derivative vanishes at \(r_c\). A switching polynomial can smooth more derivatives. These choices alter the interaction near the cutoff, so compare the same convention when reporting properties. Electrostatic \(1/r\) interactions cannot generally be truncated as if they were a short-range Lennard–Jones tail.
5. Validation and misconceptions
Fit quality and physical validity answer different questions. Check forces, elastic response, phase stability and representative configurations held out from fitting. A low global force error can hide rare configurations controlling a barrier. Increasing trajectory length reduces sampling error; it cannot repair the reference surface or an incorrect molecular topology. For reactive conditions, verify whether bonds can change at all in the model.
6. Exercises with reasoning
If \(\epsilon\) doubles while masses and \(\sigma\) remain fixed, how does the harmonic period change? Since \(\omega^2\propto\epsilon\), frequency increases by \(\sqrt2\) and period decreases by \(1/\sqrt2\). The timestep may need reduction even though the equilibrium separation is unchanged.
Does shifting energy by a constant change force? A global constant does not. A piecewise shift at a cutoff can remove an energy jump while retaining a derivative jump; that is why energy continuity alone is inadequate.
7. References
Original analytic teaching illustration under the stated assumptions; no simulation data.
8. Related theory and practice
Quantum mechanics · Monte Carlo · Quantum Monte Carlo · CP2K · Quantum-Espresso