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Sandbox Physics
Nonadiabatic molecular dynamics

L4 · Chapter 8

What makes a dynamics comparison convincing

Sandbox Physics ·

A smooth curve and a short runtime do not establish accuracy. A convincing comparison freezes the problem, converges its quantum reference, and distinguishes trajectory sampling noise from persistent method differences. Saved preparation and exit definitions make the result reproducible.

Define the exit before comparing probabilities

For lower-surface incidence in the Tully model, take RR positive to the right, reflection window R<−4R<-4 and transmission window R>4R>4 in Bohr. Project each exit onto the same lower and upper electronic states. Report central remaining probability separately.

Quantum exits integrate projected density. Classical methods apply the same spatial windows; Ehrenfest also uses electronic projection weights. FSSH active-surface counts and amplitude projections are separate diagnostics and must be labeled accordingly.

Finite-time windows differ from asymptotic scattering probabilities. Extend propagation until incoming and central contributions have separated. The comparison adds surviving window probability to outgoing probability collected by direction and electronic state at the distant absorber. Both parts are also saved separately. An independent enlarged-box calculation without absorption tests this correction.

Reading: John C. Tully · Cigrang et al.

Boundary flux explains regional probability change

In a fixed diabatic basis the total density is ρ=χ†χ\rho=\boldsymbol\chi^\dagger\boldsymbol\chi. Subtract the complex-conjugate Schrödinger equation from the original: Hermitian potential terms cancel and leave continuity. Current jj depends on the nuclear spatial phase gradient.

Integrating over [a,b][a,b] gives inflow through the left boundary minus outflow through the right. An absorber adds a loss term. Individual adiabatic-channel continuity equations additionally contain transfer terms.

j=ℏMIm⁡ ⁣(χ†∂Rχ),∂tρ+∂Rj=0j=\frac{\hbar}{M}\operatorname{Im}\!\left(\boldsymbol\chi^\dagger\partial_R\boldsymbol\chi\right),\qquad \partial_t\rho+\partial_R j=0
This form applies in the interior without absorption.
ddt∫abρ dR=j(a,t)−j(b,t)\frac{d}{dt}\int_a^b\rho\,dR=j(a,t)-j(b,t)
Integrated current and regional probability provide separate numerical checks. Net flux also differs from first-arrival probability when recrossing occurs.

Conservation and convergence test different things

Without absorption, quantum norm should stay one. With absorption, inspect surviving norm plus accumulated loss. Compare energy before the packet enters the absorber; removal of amplitude also removes energy.

Increase grid resolution at fixed physical interval, then reduce the time step at fixed grid, then enlarge the box or change its absorber. Inspect outgoing probabilities and high-wave-number tails. Each change isolates a different numerical approximation.

Transform initial amplitudes, potential and measurement projectors together in a basis test. A constant unitary rotation preserves total density; a spatially varying one also transforms kinetic energy. Accurate branching also requires resolved spatial structure and converged time evolution.

Worked example of quadrupling trajectory count

Suppose 240240 of N=400N=400 independent trajectories reach an exit, giving p^=0.6\hat p=0.6. Binomial standard error is approximately p^(1−p^)/N≃0.0245\sqrt{\hat p(1-\hat p)/N}\simeq0.0245. At 16001600 trajectories with similar probability it becomes about 0.01220.0122.

A converged quantum value 0.450.45 differs by 0.150.15, much more than this counting error. Before attributing the gap to dynamics, verify quantum error, preparation and exit time. A standard error cannot replace those checks.

Weighted electronic populations require the sample variance of their trajectory weights, not a binomial formula. Repeat seeds to study sampling stability; vary grid and time step to study quantum discretization.

SE⁡(p^)≃p^(1−p^)N,SE⁡(wˉ)≃swN\operatorname{SE}(\hat p)\simeq\sqrt{\frac{\hat p(1-\hat p)}{N}},\qquad \operatorname{SE}(\bar w)\simeq\frac{s_w}{\sqrt N}
The first formula applies to independent binary counts, the second to independent weights with sample standard deviation.

Save the calculation before discussing cost and bias

Save potential version, units, width convention, electronic preparation, grid, box, time step, duration, trajectory count, seed, frustrated-hop rule and exits. Time series and final trajectories allow measurements to be recalculated.

Define computational cost: solver time, total time including initialization, trajectory steps, or page time including rendering and downloads. Milliseconds from different hardware are not algorithmic complexity. Repeated timing requires the same accuracy and task.

The 2025 roadmap treats systems, electronic structure, preparation and observables as shared benchmark choices. A comparison on one fixed low-dimensional model isolates algorithm behavior. A molecular prediction additionally needs electronic-structure error, missing coordinates and experimental signal formation.

Reading: Cigrang et al.

Check width conventions before reading a paper benchmark

Tully’s scattering calculations use the paper’s incident-packet convention. For a selected figure point, convert the amplitude width to position-probability standard deviation and save that conversion. An amplitude e−(R−R0)2/σ2e^{-(R-R_0)^2/\sigma^2} has position standard deviation σ/2\sigma/2.

Reference values read from a scanned plot also have digitization uncertainty. Agreement with that interval checks the selected points. It neither reproduces the entire paper nor replaces grid and time-step convergence.

Try the prediction

Save a reproducible comparison

  1. Fix one preset and record exit probabilities, remaining probability, loss, energy drift and solver time.
  2. Increase only trajectory count and change the seed; inspect the standard error.
  3. Export JSON and CSV and locate the model, preparation, exits, numerical settings and measured solver cost.

Exercises and explanations

  1. Error bars shrink with more trajectories but the mean difference persists. How should you interpret it?

    Read the explanation

    Sampling noise is decreasing. After excluding preparation, exit definitions and quantum discretization, the persistent difference characterizes that method for the stated setting and observable.

  2. Why refine a grid whose norm is conserved extremely well?

    Read the explanation

    A discrete propagator can preserve norm while resolving wavelengths, the potential or reflection incorrectly. Conservation and convergence are independent requirements.

Sources and calculation scope

  1. John C. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys. 93, 1061–1071 (1990)

    Published primary method paper. Sections II–III define fewest switches and its one-dimensional test models. Our three-method comparison uses the simple avoided-crossing Hamiltonian; the nuclear packet, finite windows and numerical settings are stated with each saved run.

  2. Cigrang et al., Roadmap for Molecular Benchmarks in Nonadiabatic Dynamics, arXiv:2502.14569v3

    Fixed 4 July 2025 author version associated with the published Perspective, DOI 10.1021/acs.jpca.5c02171. Sections II A–F discuss systems, dynamics methods, electronic structure, initial conditions, observables and experimental comparisons. This is a benchmark roadmap, not a completed universal ranking.

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