LESSON NOTES · 11

11. Free energies, umbrella sampling and reaction coordinates

Position in the course: Lesson 11 of 11. Complete the preceding derivation and use the explained exercises to check understanding.

1. A potential-energy barrier is not a free-energy barrier

At finite temperature, many microstates contribute to a macroscopic state. A minimum-energy path describes energy along a selected path; free energy includes the entropic weight of configurations around it. Define a collective variable \(\xi(R)\) before assigning a free-energy profile. If it hides a slow orthogonal variable, apparently converged sampling along \(\xi\) can remain biased.

For a classical canonical system with fixed Hamiltonian and temperature, the marginal density is

\[ p(\xi)=Z^{-1}\int dR\,e^{-\beta U(R)}\delta(\xi-\xi(R)),\qquad F(\xi)=-k_BT\ln p(\xi)+C. \]

The additive constant is unobservable. The measure matters: a radial separation histogram includes a \(4\pi r^2\) geometric factor. Removing that factor changes the definition to a radial potential of mean force. State which profile is plotted.

2. Derive umbrella reweighting

Add a static bias \(W(\xi)\) to obtain easier sampling. Its marginal is \(p_W(\xi)=Z_W^{-1}e^{-\beta W(\xi)}Zp(\xi)\). Rearranging gives

\[ p(\xi)\propto p_W(\xi)e^{\beta W(\xi)},\qquad F(\xi)=-k_BT\ln p_W(\xi)-W(\xi)+C_W. \]

For an observable \(A\), the single-window identity is

\[ \langle A\rangle_0=\frac{\langle Ae^{\beta W}\rangle_W}{\langle e^{\beta W}\rangle_W}. \]

It is exact only when the biased ensemble explores the relevant support and its estimates converge. Exponentially large weights can cause severe variance. Several harmonic umbrellas, \(W_i=\kappa_i(\xi-\xi_i)^2/2\), are combined using relative normalization estimates such as WHAM or MBAR. Overlap links their otherwise arbitrary constants.

3. A worked overlap scale

For a locally flat unbiased profile with a harmonic umbrella, the biased coordinate is Gaussian with variance \(k_BT/\kappa\). If \(k_BT=1\) and \(\kappa=4\) in consistent reduced units, the width is \(0.5\). Window spacing of \(0.25\) should produce appreciable local overlap, whereas spacing of \(2\) suggests weak overlap. This heuristic uses an analytic local model; real profiles may be steep, nonlinear or multimodal, so inspect actual histograms and replica agreement. No biased simulation is reported here.

4. Thermodynamic integration offers another route

For \(U_\lambda=(1-\lambda)U_0+\lambda U_1\), differentiation of \(F_\lambda=-k_BT\ln Z_\lambda\) gives

\[ \partial_\lambda F_\lambda=\left\langle\partial_\lambda U_\lambda\right\rangle_\lambda,\qquad F_1-F_0=\int_0^1d\lambda\,\langle U_1-U_0\rangle_\lambda. \]

Endpoint singularities, phase transitions and poor overlap require suitable paths and sometimes soft-core interactions. A dense integration grid does not fix unequilibrated ensembles. Free-energy perturbation uses \(\Delta F=-k_BT\ln\langle e^{-\beta\Delta U}\rangle_0\) and can be dominated by rare states; inspect overlap before trusting a small standard error.

5. Limitations and design

Equilibrate each window, examine neighboring overlap and slow orthogonal coordinates, compare independent initialization and propagate correlated uncertainty. A constrained mean force is not automatically identical to the derivative of a histogram free energy: coordinate metrics and constraint corrections may matter. Metadynamics uses a time-dependent bias and requires its own reweighting and convergence framework. An equilibrium profile alone does not determine a rate without dynamical information and an adequate dividing surface.

6. Exercises

If \(p(\xi_a)/p(\xi_b)=e^{-3}\), what is \(F(\xi_a)-F(\xi_b)\)? It is \(3k_BT\). Low probability corresponds to high free energy under the same measure.

Can disconnected umbrella windows be normalized uniquely from their separate histograms? No. Without overlap or another normalization constraint, their relative constants are undetermined. More samples within each disconnected window do not create missing overlap.

7. References

Original analytic teaching illustration under the stated assumptions; no simulation data.

Quantum mechanics · Monte Carlo · Quantum Monte Carlo · CP2K · Quantum-Espresso

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