Nonadiabatic Transition

The following sections are included:

• Preface

• Contents

“Adiabaticity” or “adiabatic state” is a very basic well-known concept in natural sciences. This concept implies that there are two sets of variables which describe the system of interest and the system can be well characterized by the eigenstates defined at each fixed value of one set of variables which change slowly compared to the other set. This slowly varying set of variables are called “adiabatic parameters” and the eigenstates are called “adiabatic states.” Good adiabaticity means that a system stays mostly on the same adiabatic state, as the adiabatic parameter changes slowly. So, if we can find such a good adiabatic parameter, it would be very useful and helpful to describe and understand static properties and dynamic behaviour of the system. However, if the good adiabaticity holds all the time, then no state change happens at all and nothing exciting occurs. This world would have been dead. Namely, we expect that in some regions of the adiabatic parameter the adiabaticity breaks down and transitions among the adiabatic states are induced somehow. This transition, i.e. a transition among adiabatic states, is called “nonadiabatic transition.” The adiabaticity breaks down when the adiabatic parameter changes quickly, because the rapidly changing set of variables cannot fully follow the change of the adiabatic parameter and the state of the system changes accordingly. In this world, this kind of transition is occurring everywhere, not only in natural phenomena but also in social phenomena. Whatever rather abrupt changes of states occur, they can be understood as nonadiabatic transitions. So, in a sense, we may say that nonadiabatic transitions represent one of the very basic mechanisms of the mutability of the universe…

The following sections are included:

• Physics

• Chemistry

• Biology

• Economics

The following sections are included:

• Landau–Zener–Stueckelberg Theory

• Rosen–Zener–Demkov Theory

• Nikitin's Exponential Model

• Nonadiabatic Transition Due to Coriolis Coupling and Dynamical State Representation

The following sections are included:

• Wentzel–Kramers–Brillouin Semiclassical Theory

• Stokes Phenomenon

The following sections are included:

• Exact Solutions of the Linear Curve Crossing Problems
  • Landau–Zener type
  • Nonadiabatic tunneling type

• Complete Semiclassical Solutions of General Curve Crossing Problems
  • Landau–Zener (LZ) type
    • E ≥ E_X (b² ≥ 0)
    • E ≤ E_X (b² ≤ 0)
    • Numerical examples
  • Nonadiabatic Tunneling (NT) Type
    • E ≤ E_t (b² ≤ −1)
    • E_t ≤ E ≤ E_b (|b²| ≤ 1)
    • E ≥ E_b (b² ≥ 1)
    • Complete reflection
    • Numerical examples

• Non-Curve-Crossing Case
  • Rosen–Zener–Demkov model
  • Diabatically avoided crossing model

• Exponential Potential Model

• Mathematical Implications
  • Case (i)
  • Case (ii)
  • Case (iii)

The following sections are included:

• Exact Solution of Quadratic Potential Problem

• Semiclassical Solution in General Case
  • Two-crossing case: β ≥ 0 (see Fig. 6.1(a))
  • Diabatically avoided crossing case: β ≤ 0 (see Fig. 6.1(b))

• Other Exactly Solvable Models

The following sections are included:

• Diagrammatic Technique

• Inelastic Scattering

• Elastic Scattering with Resonances and Predissociation

• Perturbed Bound States

• Time-Dependent Periodic Crossing Problems

As was mentioned in Introduction, nonadiabatic transitions due to curvecrossings play important roles in condensed matter also. In this case, effects of dissipation and fluctuation should be taken into account. Theoretical studies have been carried out only for the time-dependent Landau–Zener model. Here analytical as well as numerical studies done by Kayanuma and his collaborators are summarized [89–93]…

The following sections are included:

• Exactly Solvable Models
  • Time-independent case
  • Time-dependent case

• Semiclassical Theory of Time-Independent Multi-Channel Problems
  • General framework
    • Case of no closed channel (m = 0)
    • Case of m ≠ 0 at energies higher than the bottom of the highest adiabatic potential
    • Case of m ≠ 0 at energies lower than the bottom of the highest adiabatic potential
  • Numerical example

• Time-Dependent Problems

The following sections are included:

• Classification of Surface Crossing
  • Crossing seam
  • Conical intersection
  • Renner–Teller effect

• Reduction to One-Dimensional Multi-Channel Problem
  • Linear Jahn–Teller problem
  • Collinear chemical reaction
  • Three-dimensional chemical reaction

• Semiclassical Propagation Method

The following sections are included:

• One NT-Type Crossing Case

• Diabatically Avoided Crossing (DAC) Case

• Two NT-Type Crossings Case
  • At energies above the top of the barrier: (E_u, ∞)
  • At energies between the barrier top and the higher crossing: (E₊, E_u)
  • At energies in between the two crossing regions: (E₋, E₊)
  • At energies below the crossing points: (−∞, E₋)
  • Numerical examples

The following sections are included:

• Basic Idea

• One-Dimensional Model
  • Transmission in a pure system
  • Transmission in a system with impurities
  • Molecular switching

• Two-Dimensional Model
  • Two-dimensional constriction model
  • Wave functions, matching, and transmission coefficient

• Numerical Examples

The following sections are included:

• Control of Nonadiabatic Transitions by Periodically Sweeping External Field

• Basic Theory
  • Usage of the Landau–Zener–Stueckelberg type transition
  • Usage of the Rosen–Zener–Demkov type transition
  • General case

• Numerical Examples
  • Spin tunneling by a magnetic field
  • Vibrational and tunneling transitions by laser
    • Landau–Zener–Stueckelberg type transition
    • Rosen–Zener–Demkov type transition
    • General case

• Laser Control of Photodissociation with Use of the Complete Reflection Phenomenon

As was pointed out frequently, nonadiabatic transitions play very important roles in various fields of natural sciences and, in a sense, they are controlling this world. Thus, it is very significant that the LZS (Landau–Zener–Stueckelberg) type problems, which constitute the most fundamental parts of nonadiabatic transitions, have been solved completely…

The following sections are included:

• Appendix A Final Recommended Formulas for General Time-Independent Two-Channel Problem
  • Landau–Zener Type
    • E ≥ E_X (crossing energy) (b² ≥ 0)
    • E ≤ E_X (b² ≤ 0)
    • Total scattering matrix
  • Nonadiabatic Tunneling Type (see Fig. A.2)
    • E ≥ E_b
    • E_b ≥ E ≥ E_t
    • E ≤ E_t

• Appendix B Time-Dependent Version of the Zhu—Nakamura Theory

• Bibliography

• Index