5. Constrained motion and thermal degrees of freedom
Position in the course: Lesson 5 of 11. Complete the preceding derivation and use the explained exercises to check understanding.
1. A constraint removes a motion
A rigid bond restricts coordinates to a manifold, \(g_a(R)=0\). It is a physical approximation that removes a high-frequency vibration; it is not merely a way to accelerate exactly the same flexible model. Constraints permit larger timesteps only when the remaining modes are resolved. They also change temperature bookkeeping and contribute to the virial. Assume independent holonomic constraints, fixed cell and no time dependence in \(g_a\).
2. Derive the constraint force
Add Lagrange multipliers to Newton's equations. For one bond choose \(g=|r_1-r_2|^2-d^2\). Gradients have equal magnitude and opposite signs, so its internal constraint force preserves total momentum:
Differentiating \(g=0\) once imposes a tangent-velocity condition and twice imposes an acceleration condition:
Position correction alone therefore does not guarantee valid velocities. SHAKE corrects positions iteratively; RATTLE includes the velocity constraint. The allowed constraint tolerance is a numerical control, separate from the choice to freeze the bond.
3. Work a velocity projection
Let \(e=(r_1-r_2)/d\) be the unit bond direction and \(w=(v_1-v_2)\cdot e\) the forbidden relative velocity. Apply equal opposite impulses \(-Je\) and \(+Je\) to particles 1 and 2. The corrected relative component is
For equal masses, \(v_1\cdot e=2\) and \(v_2\cdot e=0\) become \(1\) and \(1\). Momentum is preserved; forbidden kinetic energy \(\mu w^2/2\) is removed. This projection is a diagnostic construction, not a complete production constrained integrator.
4. Count independent modes
With \(N\) particles, \(c\) independent constraints and removed center-of-mass motion, a common bulk count is \(f=3N-c-3\). A free rigid nonlinear molecule has three translational and three rotational modes; a linear molecule has two rotational modes. Redundant constraints cannot be subtracted as though they were independent. For a rigid water molecule, three internal constraints leave six modes; removing the global center-of-mass translation for a collection of molecules removes three modes from the whole collection, not three per molecule.
Incorrect \(f\) can make a valid kinetic distribution look too cold or too hot. Statistical weights on a constrained surface involve geometry and mass metrics; simply deleting coordinates in a flexible partition function can give the wrong equilibrium measure. This is a reason to use a validated constrained thermostat and integrator combination.
5. Limits and checks
Inspect maximum bond deviation, velocity tangent residual and NVE energy drift as tolerance and timestep are tightened independently. Include constraint forces in pressure. For spectroscopy, a removed stretch cannot reappear in a dipole correlation spectrum. Do not constrain a reaction coordinate whose change is the target process.
6. Exercises
Ten rigid nonlinear triatomic molecules in 3D, with global momentum removed: what is \(f\)? There are \(90\) Cartesian coordinates and \(30\) independent internal constraints. Thus \(f=90-30-3=57\), before any other constraints.
Why does a small position residual not establish a correct velocity distribution? A velocity can point off the constraint manifold while its starting position lies on it. The first derivative of \(g\) must vanish as well.
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