--- title: "Benchmark Nonadiabatic Reaction Methods Interactive Lab | Sandbox Physics" description: "Can more trajectories remove an approximation’s error in a reaction probability?" canonical_url: "https://sandboxphysics.com/labs/nonadiabatic-method-benchmark/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/nonadiabatic-method-benchmark/" --- Count trajectories and measure the error # Benchmark Nonadiabatic Reaction Methods **Can more trajectories remove an approximation’s error in a reaction probability?** Compare all three methods with the same Hamiltonian, initial packet and exit windows. Finer grids test the quantum reference, and more trajectories reduce sampling noise. A stable difference after these checks measures approximation error for that observable. Compare channel probabilities, sampling error and computation time for quantum, Ehrenfest and fewest-switches dynamics. Save each calculation, change the ensemble size and see which differences survive. > Interactive simulation: open this page in a web browser to change the parameters and run the model. Model assumptions The reference is a converged numerical solution of a fixed two-state Tully model, not experimental chemistry data. Sampling errors describe independent Wigner preparations and hopping draws. Time-step and grid errors are separate. Propagation costs are measured on your device and exclude snapshots and rendering; they are not a universal ranking of the algorithms. $t=0$ One nuclear coordinate. Height shows electronic energy; ribbon thickness shows density on a separate scale. Cyan is lower, coral is upper. At the common final time ## Separate sampling scatter from method bias The exit windows begin four bohr from the crossing. Channel estimates include probability absorbed far beyond its exit; surviving and absorbed parts are saved separately. Probability between the windows is unresolved. Quantum upper exit $0$ Hopping minus quantum $0$ Hopping standard error $0$ Quantum unresolved $0$ ### Four exit probabilities Bars share the same probability scale. Thin whiskers show one sampling standard error for the trajectory estimates. ### Saved estimates and their sampling error Save runs with the same physical preparation to compare sample sizes. The line is the quantum value; each gold point is a saved hopping estimate. Norm, energy, flux and measured cost Probability still on the grid $0$ Probability absorbed outside the exits $0$ Norm plus absorbed $0$ Quantum energy drift $0$ Largest trajectory energy drift $0$ Net outward window flux $0$ Quantum / mean force / hopping milliseconds $0$ Net outward flux subtracts incoming flux through the left window. Initially the packet is outside the windows: final exterior probability plus absorbed loss equals initial exterior probability plus net outward flux. Trigonometric interpolation evaluates current at the exact window planes; fractional-cell quadrature integrates their densities. Costs measure propagation on this device, excluding snapshots, figures and downloads. Quantum energy drift is meaningful before absorption; the absorbing layer removes probability beyond nineteen bohr. Model and numerical limits The quantum solver uses 512 periodic Fourier points on a forty-eight-bohr box and a Strang time step no larger than 0.8 atomic time units. It applies the exact local two-state exponential and an outer absorbing mask. No stepwise normalization is applied. The offline benchmark halves the step and doubles the grid. Ehrenfest trajectories follow the electronic mean force. Fewest-switches trajectories follow one adiabatic surface, propagate electronic amplitudes coherently, rescale momentum at accepted hops and reject energetically forbidden hops without velocity reversal. No decoherence correction is added. [Read the benchmark method](https://sandboxphysics.com/stories/nonadiabatic-dynamics/reproducible-benchmarks/) Physics tutorial ## Benchmark Nonadiabatic Reaction Methods **Background** Compare channel probabilities, sampling error and computation time for quantum, Ehrenfest and fewest-switches dynamics. Save each calculation, change the ensemble size and see which differences survive. **Why it matters** A change in electronic state changes the force on the nuclei. Approximations differ in how they keep those two changes together. Start with the essentials - **Focus question**: Can more trajectories remove an approximation’s error in a reaction probability? - **One-sentence intuition**: More trajectories reduce sampling scatter. A systematic difference from the quantum solution can remain. ### Core mathematical model #### A nuclear coordinate coupled to two electronic states $$ H=-\frac{1}{2M}\partial_x^2 I+\begin{pmatrix}z(x)&w(x)\\w(x)&-z(x)\end{pmatrix},\quad M=2000 $$ Atomic units set the reduced Planck constant to one. Position is in bohr, energy in hartree and time in atomic time units. The matrix eigenvalues are the two adiabatic energies. #### The original simple avoided crossing $$ z(x)=0.01\,\operatorname{sgn}(x)(1-e^{-1.6|x|}),\qquad w(x)=0.005c\,e^{-x^2} $$ Electronic mixing multiplies the original off-diagonal coupling. The original Tully model has mixing equal to one. The initial packet is on the lower adiabatic surface. #### A shared Gaussian preparation $$ \chi(x)\propto e^{-(x+8)^2/(4\sigma_x^2)+ip_0x},\quad \sigma_p=\frac{1}{2\sigma_x} $$ The classical methods independently sample the positive Wigner distribution of this same minimum-uncertainty packet. Their initial electronic state is the local lower eigenvector. ### Common difficulties #### A population is not an exit probability **Typical misconception** Electronic population alone gives the reaction yield. **Better mental model** An exit also requires the nuclear packet to reach its chosen spatial window. During the animation the incoming packet is still on the left; the table uses the common final time and reports unresolved weight separately. ### Run the experiment 1. 01 #### Watch the split Choose Two outgoing branches and play the evolution. Compare the three rows at the same time. **What to observe:** The quantum density spreads onto both energy curves. Surface-hopping trajectories separate by their active surface; mean-force trajectories can travel between the two curves. 2. 02 #### Close an exit Choose An upper exit closes and inspect the late-time channel probabilities. **What to observe:** The upper asymptote requires enough total energy. A finite-width quantum packet contains an energy distribution; it can leave a small high-energy tail. 3. 03 #### Separate scatter and bias Open Ensemble and seed, increase Trajectories and save several independent seeds. Compare the final estimates with their standard errors. **What to observe:** Repeated estimates fluctuate less when more trajectories are used. The gap between each method and the quantum result need not shrink. Physical conclusion The reference is a converged numerical solution of a fixed two-state Tully model, not experimental chemistry data. Sampling errors describe independent Wigner preparations and hopping draws. Time-step and grid errors are separate. Propagation costs are measured on your device and exclude snapshots and rendering; they are not a universal ranking of the algorithms. ## Related experiments and articles Nonadiabatic molecular dynamics · Reading path ### From energy surfaces to reaction branches Articles on this path that use this experiment, followed by the other experiments they suggest. 1. [Article **What makes a dynamics comparison convincing**](https://sandboxphysics.com/stories/nonadiabatic-dynamics/reproducible-benchmarks/) 2. [Article **Reading interference and molecular-benchmark papers**](https://sandboxphysics.com/stories/nonadiabatic-dynamics/reading-nonadiabatic-research/) 3. [Lab **Geometric Phase Around a Conical Intersection** Check the Berry loop sign step by step](https://sandboxphysics.com/labs/conical-geometric-phase/) Quantum & fields · Starter path ## Start from a single measurement Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/) 1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/) 2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/) 3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/) 4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/) 5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/) 6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/) [View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)