Nonadiabatic Wave Packet Dynamics

• Nonadiabatic wave packet dynamics is the quantum evolution where coupled electronic and nuclear motions induce rapid state transitions at avoided crossings.
• Advanced methods like grid-based propagation, variational Gaussian techniques, and surface hopping enable accurate treatment of nonadiabatic coupling and interference.
• Applications span ultrafast photochemistry, molecular scattering, and quantum device simulations, demonstrating its significance in energy transfer and coherence phenomena.

Nonadiabatic wave packet dynamics encompasses the quantum evolution of molecular or condensed-matter systems in which the Born–Oppenheimer separation of electronic and nuclear (or, more generally, fast and slow) degrees of freedom breaks down. These dynamics are dominated by rapid transitions between electronic states induced by nuclear motion near avoided crossings or conical intersections, and are essential for describing ultrafast photochemistry, inelastic scattering, energy flow, and quantum transport in systems where the adiabatic approximation fails. The field brings together exact quantum techniques, semiclassical surface-hopping, Gaussian-based variational propagators, and geometric frameworks to model, analyze, and simulate the full range of nuclear–electronic correlation and interference phenomena observed experimentally and in ab initio simulation.

1. Theoretical Foundations and Representations

Nonadiabatic wave packet dynamics is governed by the time-dependent molecular Schrödinger equation,

$$i\hbar\frac{\partial}{\partial t}\Psi(R, r, t) = \left( \hat{T}_N + \hat{H}_e(R) \right)\Psi(R, r, t),$$

where $R$ and $r$ denote nuclear and electronic coordinates, respectively, $\hat{T}_N$ is the nuclear kinetic energy, and $\hat{H}_e(R)$ is the electronic Hamiltonian at fixed nuclear geometry. The key representations used in nonadiabatic dynamics are:

• Adiabatic Representation: Expands $\Psi(R, r, t)$ in the eigenstates $|\psi_j(R)\rangle$ of $\hat{H}_e(R)$. Nonadiabatic coupling arises via derivative terms $$d_{ij}(R) = \langle \psi_i(R)| \nabla_R | \psi_j(R)\rangle$$ that diverge at conical intersections. The nuclear wave packet is a multicomponent vector $\vec{\chi}(R, t)$ propagated with couplings from $d_{ij}(R)$ (Roncero et al., 13 Dec 2024, Gu, 2023, Joubert-Doriol et al., 2018, Lam et al., 15 Aug 2024).
• Diabatic Representation: Uses a nuclear-coordinate independent electronic basis, yielding a matrix potential $V^d_{ij}(R)$ that can be smooth across crossings (Roncero et al., 13 Dec 2024, Issa et al., 23 May 2025).
• Local Diabatic and Moving Crude Adiabatic Representations: Project electronic structure onto local/varying nuclear frames or the centers of moving Gaussians, effectively regularizing derivative couplings at crossings and enabling exact or nearly exact split-operator propagation and variational treatments (Gu, 2023, Joubert-Doriol et al., 2018, Joubert-Doriol et al., 2018).

The dynamics in all representations ultimately require high-dimensional nonadiabatic coupling vectors, which engender population transfer, interference, and geometric phase effects central to quantum molecular and solid-state dynamics.

2. Quantum and Semiclassical Propagation Methods

Grid-Based and Split-Operator Techniques

Exact nonadiabatic dynamics for low-dimensional systems use full wave function propagation on spatial and electronic grids. In the local diabatic representation, the Strang-splitting scheme

$$U(\Delta t) \approx e^{-i V_d \Delta t/2\hbar}\, e^{-i T_N \Delta t/\hbar}\, e^{-i V_d \Delta t/2\hbar}$$

with a Fourier/sinc discrete-variable representation ensures conservation of the norm and manifestly regular treatment of geometric phase and nonadiabatic population transfer even at conical intersections (Gu, 2023).

Gaussian Wave Packets and Variational Methods

Gaussian-based methods balance accuracy and scalability by expanding the nuclear wavefunction as a sum over coherent or thawed Gaussians: $$\Psi(R, t) = \sum_{k, s} C_{k s}(t)\, g_k(R; q_k, p_k, \alpha_k, z_k) \, \phi_s(r|q_k)$$ with quantum or hybrid-classical equations of motion for the parameters, derived from the time-dependent variational principle (TDVP). Key formulations include:

• Frozen-width Gaussians: Used in surface-hopping and variational multiple cloning (spawning/cloning) schemes to adaptively represent branching and decoherence (Joubert-Doriol et al., 2018, Humeniuk et al., 2016).
• Moving Crude Adiabatic (MCA): Eliminates problematic derivative couplings and permits on-the-fly calculation of electronic matrix elements (no global PES fit), with natural inclusion of geometric phases (Joubert-Doriol et al., 2018, Humeniuk et al., 2016).
• Hagedorn Wave Packets: Generalized time-dependent bases (with variational or direct-update width, center, and phase parameters) enable systematic convergence and accurate dynamics even for anharmonic or coupled-surface problems (Issa et al., 23 May 2025).

Semiclassical and Hybrid Quantum-Classical Approaches

• Surface Hopping Methods: Tully's fewest-switches surface hopping (FSSH) and Landau–Zener single-switch variants represent the nuclear dynamics as classical trajectories, each stochastically “hopping” between adiabatic surfaces according to instantaneous quantum amplitudes and couplings (Schmidt et al., 2019).
• Semiclassical Initial Value Representations (IVR): Herman–Kluk propagators and nonadiabatic variants use semiclassical phase-space integrals over classical paths to construct the quantum propagator; nonadiabatic corrections (“metaplectic hops”) are incorporated at crossings (Kammerer et al., 2020).
• Coupled Wavepackets for Non-Adiabatic Molecular Dynamics (CW-NAMD): Extends thawed Gaussian dynamics by allowing spawning, merging, and full coherent superposition on multiple PESs, thereby capturing quantum interference and decoherence without empirical parameters (White et al., 2016).
• Quantum–Classical Hybrid Schemes: For example, combining variational quantum eigensolvers (VQE/SSVQE) for PES and NAC computation with classical grid-based nuclear propagation, as demonstrated for H₂O⁺ with NISQ-compatible circuits (Hirai et al., 2021).

3. Nonadiabatic Phenomena: Population Transfer, Branching, and Coherence

Wave packet branching, quantum yield, and the creation of electronic coherence are driven by the structure of the potential energy landscapes and the nonadiabatic couplings:

• Conical Intersections (CIs): At points where two adiabatic surfaces become degenerate, the nonadiabatic coupling vectors diverge and the nuclear wave packet naturally splits (branches). This branching ultimately sets the quantum yield (fraction transferred to product channel) (Wang et al., 2022, Lam et al., 15 Aug 2024). The probability and timescale for branching can be estimated via Landau–Zener (for 1D), surface-hopping, or dephasing-induced surface hopping formulas (Wang et al., 2022, Betz et al., 2010, Schmidt et al., 2019).
• Wave Packet Proliferation and Quantum Chaos: In periodic nonadiabatic crossing scenarios, e.g., a two-level atom in a standing-wave field, repeated splitting leads to an exponential growth in the number of branches—“nonadiabatic quantum chaos”—quantified by a dimensionless Landau–Zener parameter (Prants, 2012).
• Electronic/Nuclear Coherences and Interference: Multi-Gaussian and grid-based methods naturally capture Stückelberg oscillations, geometric-phase nodal lines, and complex population recurrences, all of which are hallmarks of coherent nonadiabatic dynamics (Joubert-Doriol et al., 2018, White et al., 2016, Kammerer et al., 2020).

A table summarizing commonly used methods and their capabilities is given below.

Method/Framework	Dimensionality	Transition regime	Decoherence	Geometric phase	On-the-fly PES/NAC	Scalability
Full grid + split-op	low (2–4D)	all	exact	exact	yes (with ab initio)	low
Gaussian (FWG/MCA)	moderate/high	all	variational	variational	yes	moderate
Surface hopping	high	perturbative	classical	not included	yes	high
Hagedorn/IVR	moderate	semiclassical	partial	partial	yes	moderate
CW-NAMD	moderate	all	full	full	yes	moderate

4. Geometric and Analogue-Gravity Structures in Nonadiabatic Dynamics

Recent developments connect nonadiabatic corrections and geometric structures to metrics and curvature in both nuclear configuration space and momentum (Brillouin zone) space (Ren, 6 Jun 2025, Ren et al., 29 Aug 2025). The key insight is that interband admixtures at leading nonadiabatic order induce a Riemannian metric tensor (the “nonadiabatic metric”) governing wave packet evolution: $$G_{ij}(\mathbf{q}) = 2\,\Re\sum_{m \neq 0} \frac{ \langle u_0| i\partial_{q^i} |u_m \rangle\, \langle u_m| i\partial_{q^j} |u_0 \rangle}{E_m - E_0}$$ for Bloch electrons (or an analogous metric in real-space for nuclear wave packets at CIs) (Ren, 6 Jun 2025, Ren et al., 29 Aug 2025). This metric gives rise to geodesic corrections in the equations of motion, extending the concept of Berry curvature (Lorentz force-like) to genuine momentum- and coordinate-space "gravity." In flat-band systems, this metric reduces to an effective mass, and in nonuniform scenarios, creates curvature that can bend or focus packets, alter quantum conductance, and generate robust geometric phase effects.

5. Practical Applications and Benchmark Studies

Nonadiabatic wave packet dynamics methods are routinely applied to:

• Ultrafast photochemical reactions: e.g., monitoring bending and stretching wave packets and their electronic population transfer in SO₂⁺ by time-resolved Coulomb explosion imaging and quantum simulation, extracting transfer timescales (~20 fs) and coupling strengths (~0.1 eV) associated with conical intersections (Lam et al., 15 Aug 2024).
• Quantum simulation of CIs: Engineered qubit–oscillator QED systems have been used to directly emulate and image wave packet branching and "electronic dephasing" at CI, including analytic estimation of final quantum yields by measuring time-resolved observables (Wang et al., 2022).
• Inelastic/reactive molecular scattering: Programs such as MADWAVE3 enable large-scale quantum nonadiabatic simulations in triatomic molecules, with efficient parallelization and full state-to-state S-matrix computation (Roncero et al., 13 Dec 2024).
• Quantum-classical device demonstrations: Hybrid workflows combining variational quantum eigensolvers (VQE, SSVQE) for the computation of nuclear PES and nonadiabatic coupling vectors, and classical grid-based wave packet propagation for femtosecond nonadiabatic transfer (as in H₂O⁺ de-excitation) (Hirai et al., 2021).
• Benchmark model systems: The three-step Hagedorn propagation scheme and surface-hopping Gaussian methods are validated by comparison to Taylor-propagated exact results on 2D Henon–Heiles potentials and reduced models of retinal isomerization (Issa et al., 23 May 2025).

6. Algorithmic Trends and Future Directions

Contemporary progress in nonadiabatic wave packet dynamics is characterized by:

• Unified quantum geometry frameworks: Nonadiabatic corrections appear as quantum metrics in phase space, leading to geometric/curvature-driven anomalous velocities and enforced by geodesic equations in both real and momentum spaces (Ren et al., 29 Aug 2025, Ren, 6 Jun 2025).
• Exact and robust grid-based propagation schemes: The use of local diabatic and overlap-based “connection” representations circumvents phase/gauge singularities at CIs, enabling time-reversible, stable dynamics with large time steps (Gu, 2023).
• Systematic variational improvements: Adaptive spawning/cloning criteria, rigorous TDVP-based equations, and flexible, generalized bases (Hagedorn, Fourier, Gauss–Hermite hybrid) allow convergence to numerically exact results for nonadiabatic dynamics (Issa et al., 23 May 2025, Joubert-Doriol et al., 2018, Joubert-Doriol et al., 2018).
• Scalability via quantum–classical and parallel methods: Mixed quantum–classical and parallel grid-based codes make large, multidimensional, and polyatomic calculations tractable (Roncero et al., 13 Dec 2024, Schmidt et al., 2019).
• Quantum device integration: Proof-of-principle demonstrations show the applicability of near-term quantum hardware for extracting fundamental nonadiabatic observables, with limitations determined by circuit depth and quantum noise (Hirai et al., 2021).

A plausible implication is that as NISQ and beyond-NISQ quantum devices mature, hybrid classical–quantum workflows will be increasingly adopted for high-fidelity nonadiabatic quantum dynamics in complex systems, leveraging advances in both electronic-structure optimization and real-time propagation. The geometric and metric perspective is expected to further unify disparate phenomena—from transport in topological solids to ultrafast photochemical branching—under a single dynamical and conceptual framework.