FR14
Nonadiabatic molecular dynamics
A light pulse can send a molecule toward several possible products. The nuclei move across electronic energy surfaces, and near a small gap the electronic state can change. Start with a vibrating bond, learn how a wave packet branches, then compare quantum evolution, mean forces and surface hopping. Interference connects this simple picture to current research.
Start with a bond that can stretch
A potential-energy surface tells the nuclei which way the electronic energy slopes. One surface often suffices near a stable bond. Excited molecules can approach a second surface so closely that the electrons and nuclei must be followed together. The difficulty is then concrete: how much probability leaves in each channel, and what information does a cheaper approximation lose?
The first three chapters build surfaces, moving electronic bases and a single-passage transition law. Chapters four to seven compare dynamics, introduce a conical intersection and follow phase through recombination and loss of coherence. The final two chapters teach a reproducible comparison and a close reading of primary research. Each chapter includes a worked example, a prediction to test and exercises with explanations.
L1 · 1
How electrons create potential-energy surfaces
Use a vibrating bond to understand energy surfaces, nuclear wave packets and reaction exits before adding a second electronic state.
L2 · 2
What changes when the electronic basis moves
Derive derivative coupling from nuclear kinetic energy and compare adiabatic and diabatic representations with a two-state rotation.
L2 · 3
How much probability changes channel in one passage
Define gap slopes and channel labels, calculate the Landau–Zener parameter, and test fast and slow limits with a worked example.
L3 · 4
Three dynamics methods for the same preparation
Compare quantum packets, Ehrenfest mean forces and fewest-switches surface hopping with common initial conditions and exit windows.
L3 · 5
How two surfaces form a conical intersection
Derive the two conditions for a real two-state degeneracy, construct a local cone, and examine how the incident path affects branching.
L3 · 6
Why paths around a conical intersection can cancel
Derive a two-state loop phase and connect the electronic eigenvector sign to a controlled two-port interference distribution.
L3 · 7
When two nuclear packets can still interfere
Connect amplitude addition to reduced electronic coherence and distinguish spatial separation, phase averaging and trajectory decoherence corrections.
L4 · 8
What makes a dynamics comparison convincing
Define exits and flux, check conservation and convergence, and separate sampling error, numerical error and method bias.
L4 · 9
Reading interference and molecular-benchmark papers
Read fixed versions of the 2016 interference paper and the 2025 benchmark roadmap, distinguishing representation, limited reproduction and open problems.
Experiments along the route
- Set a molecular vibration in motion
- Send the same preparation through three dynamics methods
- Follow a two-component packet through a conical intersection
- Change the loop and compare interference
- Measure branching probabilities and numerical error
The experiments use defined low-dimensional Hamiltonians and simulated records. The two-coordinate conical model and the controlled path-interference model answer different questions. Their parameters are not fitted to a named molecule. The research chapter identifies the separate Shin–Metiu calculation used in the 2016 paper.
Sources and calculation scope
- C. A. Mead & D. G. Truhlar, On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei, J. Chem. Phys. 70, 2284–2296 (1979)
Published original connecting conical intersections with nuclear-motion phase. Bibliography and the 1983 erratum are recorded by the authors’ group. This is further reading; the guided Lab derives its phase from Berry’s fixed two-state example.
- 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.
- Tully 1990 original paper, author-text scan
Open copy of the original paper for checking the transition rule and model equations. The displayed Lab comparison is generated by the repository solver, not digitized experimental data.
- M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45–57 (1984)
Published primary derivation: equations (5)–(6) separate dynamical and geometrical phase; section 3 treats a degeneracy. The path experiment is a controlled two-state model, with the reduction and phase comparison specified on its page.
- C. Zener, Non-adiabatic crossing of energy levels, Proc. R. Soc. Lond. A 137, 696–702 (1932)
Published original result for a linear-in-time diabatic gap and constant coupling. The course defines the slope as the slope of the full diabatic energy difference, avoiding a factor-of-two ambiguity.
- Curchod, Agostini & Gross, An Exact Factorization Perspective on Quantum Interferences in Nonadiabatic Dynamics, arXiv:1604.02351v2
Fixed 30 June 2016 author version; published in J. Chem. Phys. 145, 034103 (2016), DOI 10.1063/1.4958637. Reading scope: equations (5)–(6), (14)–(18), and Figures 3–7. Its Shin–Metiu electron–ion calculation differs from the Tully and conical teaching models.
- 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.