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Propagation of Wave Packets Close to Conical Intersections

Marianne Curely

TL;DR

This work analyzes semiclassical propagation of wave packets for a system of two Schrödinger equations with a matrix-valued potential that exhibits a codimension-2 conical crossing. The authors develop a comprehensive framework that couples classical quantities (trajectories, actions, and adiabatic vectors) with harmonic-profile dynamics, then reduce the passage through the gap to a Landau–Zener problem using a time-rescaled unknown. The main achievement is a precise transfer formula that, after the crossing, decomposes the solution into two outgoing Gaussian wave packets on the two adiabatic surfaces, with explicit phases, drift terms for the classical trajectories, and rigorous remainder bounds of order $O(\varepsilon^{1/14-\beta})$. This work extends prior codimension-1 results to codimension-2 crossings, providing a robust basis for surface-hopping-type descriptions in molecular dynamics and improving understanding of nonadiabatic transitions near conical intersections.

Abstract

In this paper, we study the propagation of wave packets close to conical intersections with respect to a system of two Schr{ö}dinger equations presenting a codimension 2 crossing. We focus on the dynamics that occur when the wave packets pass through an area close to the crossing, and our main results provide an explicit formula for the outgoing wave packet in terms of the incoming one, with a complete description of its phase and of the classical trajectories it follows, including a drift.

Propagation of Wave Packets Close to Conical Intersections

TL;DR

This work analyzes semiclassical propagation of wave packets for a system of two Schrödinger equations with a matrix-valued potential that exhibits a codimension-2 conical crossing. The authors develop a comprehensive framework that couples classical quantities (trajectories, actions, and adiabatic vectors) with harmonic-profile dynamics, then reduce the passage through the gap to a Landau–Zener problem using a time-rescaled unknown. The main achievement is a precise transfer formula that, after the crossing, decomposes the solution into two outgoing Gaussian wave packets on the two adiabatic surfaces, with explicit phases, drift terms for the classical trajectories, and rigorous remainder bounds of order . This work extends prior codimension-1 results to codimension-2 crossings, providing a robust basis for surface-hopping-type descriptions in molecular dynamics and improving understanding of nonadiabatic transitions near conical intersections.

Abstract

In this paper, we study the propagation of wave packets close to conical intersections with respect to a system of two Schr{ö}dinger equations presenting a codimension 2 crossing. We focus on the dynamics that occur when the wave packets pass through an area close to the crossing, and our main results provide an explicit formula for the outgoing wave packet in terms of the incoming one, with a complete description of its phase and of the classical trajectories it follows, including a drift.

Paper Structure

This paper contains 50 sections, 35 theorems, 253 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The equation of molecular dynamics
  3. On the existence of solutions
  4. The Born-Oppenheimer approximation
  5. The problem setting
  6. Wave packets
  7. Classical quantities
  8. Classical trajectories
  9. Actions
  10. Real-valued time-dependent eigenvectors along the trajectory
  11. Real-valued time-dependent eigenvectors along the trajectory
  12. Profile equations
  13. Adiabatic propagation and aim of the study
  14. The framework
  15. Classical quantities after the crossing and main result
  16. ...and 35 more sections

Key Result

Theorem 1.6

Let Assumption hypsurV2 and hypcrossing hold. We consider $\varepsilon \in (0,1)$, $\delta > 0$ and $(t_0,T) \in \mathbb{R}^2$. We assume that $\psi^{\varepsilon}_0$ satisfies Assumption hypsurID at $t_0$ and that the trajectory $t \mapsto \Phi_{-}^{t,t_0}(z_0) \in \lbrace \lvert w \lvert \, > \delt where the vector-valued function $t \mapsto \Vec{Y}_{-}(t)$ is defined in eqVec with $\vec{Y}_0$ as

Figures (1)

  • Figure 1: Evolution of phase and transfer coefficient during wave packet propagation.

Theorems & Definitions (69)

  • Definition 1.3: Wave packets
  • Remark 1.5
  • Theorem 1.6: Adiabatic propagation
  • Remark 1.7
  • Definition 1.8
  • Definition 1.10
  • Theorem 1.11
  • Remark 1.12
  • Lemma 1.13
  • Theorem 1.14: The ingoing wave packet
  • ...and 59 more