Introduction to Nonadiabatic Dynamics

The following sections are included:

• Contents
• Preface

Nonadiabatic transition, generally implying a transition between adiabatic states, represents a very basic mechanism of state change. “Adiabaticity” or “adiabatic state” is a 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 fixed values of one set of variables which change slowly compared to the fast changing other set. This slowly varying set of variables are called “adiabatic parameter” 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. 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 way, then no state change happens at all and nothing exciting occurs. Namely, no state change happens. 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, when the adiabatic parameter changes. 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 other 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 occurs in many places, 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 transition represents one of the very basic mechanisms of the mutability of the universe…

As was mentioned in the previous chapter, the concept of nonadiabatic transition is very much multi-disciplinary and plays essential roles in state/phase changes in various fields of physics, chemistry, and biology. Not only in natural sciences, but also even in social sciences the concept must be useful to analyze various phenomena. In this chapter some practical examples in physics, chemistry, and biology are presented to emphasize the significance of the concept and to help the reader’s deep understanding. Possible applications in economics will also be touched upon briefly.

The following sections are included:

• One-Dimensional Curve Crossing Problem
• Three-Dimensional Problem
• Semiclassical Theory
  • Scattering Matrix in Two-Channel Problem
    • Inelastic Scattering
    • Elastic Scattering with Resonances
    • Perturbed Bound State
  • Multi-Channel Problem
    • Semiclaasical Theory of Multi-Channel Problem
    • Exactly Solvable Model
  • Basic Theories of Time-Independent Two-State Problems
    • Landau Method and Landau–Stückelberg Theory of Curve Crossing
    • Non-Curve-Crossing Case
    • Exponential Potential Models
• References

The following sections are included:

• Time-Dependent Linear Potential Model
• Landau–Zener Formula in the Time-Independent Case
• References

Nonadiabatic tunneling is a phenomenon of transmission through the potential barrier in the nonadiabatic tunneling type of potential energy curve crossing shown in Fig. 2.2. The peculiarity in this case is the fact that the potential barrier is created by the diabatic coupling between the two diabatic potential energy curves with opposite signs of slopes. This means that the lower adiabatic potential barrier, thus the transmission through that potential barrier, is affected by the upper adiabatic potential. This indicates that the transmission must be different from the ordinary single potential barrier transmission. This feature is schematically shown in Fig. 5.1. The red line shows the transmission probability in the case of two adiabatic potential energy curves like the one depicted in Fig. 2.2 and the blue line is the transmission probability in the case of single potential barrier with the existence of the upper adiabatic potential well neglected. As is well known, the ordinary tunneling probability monotonically increases up to unity. On the other hand, the nonadiabatic transmission probability interestingly oscillates at high energies E > E_b, where E_b is the bottom of the upper adiabatic potential well. Besides, the probability becomes exactly zero at certain discrete energies, E₁, E₂, E₃,…. This is a very interesting phenomenon called complete reflection and will be discussed in more detail in Chapter 7. It should also be noted that the red line is always lower than the blue line. The magnitude of the nonadiabatic tunneling probability actually depends on the diabatic coupling strength. As is easily understood, the probability is zero, namely, no transmission, when the diabatic coupling is zero…

As was mentioned in the last part of Chapter 5, it is quite common to use classical trajectories to treat multi-dimensional dynamics. That is to say, the nuclear coordinates R are considered to be time-dependent variables R(t) and the time-dependent Schrödinger equation is solved. This is formulated as follows. The Schrödinger equation is given by

The following sections are included:

• Explanation of the Phenomenon in the Case of Nonadiabatic Tunneling
• Complete Reflection in Other Types of Potentials
  • Diabatically Avoided Crossing
  • Double Crossing Nonadiabatic Tunneling Type
    • Conditions of Complete Reflection
    • Numerical Examples
• Intriguing Possibilities
  • Bound States in the Continuum
  • Molecular Switching
    • Basic Idea
    • One-Dimensional Model
    • Two-Dimensional Model
• References

The following sections are included:

• Analysis of Basic Coupled Equations of Linear Curve Crossing Problem
  • Landau–Zener Type
  • Nonadiabatic Tunneling Type
• Generalization to General Curved Potential Case
  • Landau–Zener Type
  • Nonadiabatic Tunneling Type
• Classically Allowed Transition
  • Landau–Zener Type (E ≥ E₂(R₀)) (see Fig. 2.1)
  • Nonadiabatic Tunneling Type (E ≥ E_b) (see Fig. 2.2)
• Classically Forbidden Transition
  • Landau–Zener Type (E ≤ E₂(R₀)) (see Fig. 2.1)
  • Nonadiabatic Tunneling Type (see Fig. 2.2)
• Numerical Examples
• References

Time-dependent dynamics is described naturally by the time-dependent Schrödinger equation,

Since most of physical, chemical, and biological dynamic processes in reality proceed in multi-dimensional configuration space, it is naturally very important to develop useful methodologies to deal with multi-dimensional processes [1]. Unfortunately, however, it is almost impossible to develop any intrinsically multi-dimensional analytical theory and thus it is necessary to figure out some useful approximate methods with the help of accurate one-dimensional theories. There are two ways to do that: one is to reduce any multi-dimensional problem to a one-dimensional multi-channel problem by expanding the total wave function in terms of appropriate internal states, as usually done in the quantum mechanical numerical solutions of Schrödinger equations. The second, which is more approximate but may be practically more useful, is to use classical trajectories which define curvilinear one-dimensional space, and try to incorporate quantum mechanical effects as much as possible. The ordinary multi-dimensional semiclassical mechanics belong to this category [2–5]. The one-dimensional semiclassical theories of nonadiabatic transitions discussed so far can be incorporated in both methods, in principle…

The theoretical methods presented in the previous chapter have been employed to clarify the mechanisms of various chemical and biological nonadiabatic dynamics. These are reported here, covering a wide range of areas from chemical reactions and absorption spectrum of tri- and tetra-atomic systems, reaction rate constant of electron transfer, photochemical dynamics of polyatomic molecules, photo-isomerization of retinal in vacuo and in protein environment, and chemical reactions in solutions. These are the works done by the author’s research group and collaborators [1]. There are many other investigations carried out by many other authors with use of the Zhu–Nakamura theory. Examples are the ZN-TSH study of reactions in ${\text{DH}}_2^{+}$ and ${\text{HD}}_2^{+}$ systems [2] and ab initio molecular dynamics studies of azobenzene [3–6], anion and neutral chromophore in green fluorescent protein [7, 8], arylazoimidazole [9], sacicylidene methylamine [10], trans-urocanic acid [11], 2-1(1-(methylimino)methyl-6-chlorophenol [12], hemithioindigohemistilbene [13], trans-1,4-diphenyl-1,3-butadiene [14, 15], and 5-diazo Meldrum’s acid [16].

Control of molecular processes or chemical dynamics, in general, is one of the most active fields in chemical physics recently, and attracts much attention theoretically as well as experimentally. This is basically because the recent progress of laser technology is remarkable and has opened new possibilities of controlling various molecular processes. Many theoretical approaches have been proposed so far. One of these is the coherent control method originated by Brumer and Shapiro [1]. The essence of this method is to utilize the quantum mechanical interference effect to control the branching of a prepared state into a possible final channel with use of the phase and intensity of two laser pulses. The second example is the pump-dump scheme proposed by Tannor and Rice [2]. Their original idea is to use a laser pulse to create a localized wave packet on a bound excited state. When the wave packet comes to an appropriate position of the excited state, the second laser pulse is used to make a transition to a final desirable state. The use of linearly chirped (or frequency swept) laser pulse originally proposed by Chelkowski and Bandrauk [3] is also one of the effective methods and has been studied by many authors [4–8]. Most of them utilize the so called adiabatic rapid passage (ARP) [4, 9]. That is to say, by sweeping the field slowly, the system is kept to stay on the same adiabatic state. Methods to find the best pulse shape of lasers to control wave packet dynamics (optimal control theory) have been formulated and discussed by Kosloff et al. [10], Rabitz and coworkers [11], Kohler et al. [12], and Fujimura and coworkers [13]. The pulse shapes are optimized under various conditions. Unfortunately, however, the optimized pulses have sometimes rather complicated structures in frequency as well as time domains. The so-called π-pulse method has also been proposed to induce resonant transitions efficiently [14, 15]. In the above mentioned approaches, nonadiabatic transitions, in general, play important roles, because molecular processes induced by an external field, not only transitions among electronic states but also rovibrational transitions, may be considered as nonadiabatic transitions with the external field regarded as the adiabatic parameter. For instance, a rovibrational transition induced by a chirped laser field can be regarded as a nonadiabatic transition among adiabatic Floquet states [16–18] by taking the laser frequency as the adiabatic parameter…

Numerical applications are presented here based on the methods described in Sections 12.2–12.4. These are not yet experimentally realized, but demonstrate the possibilities theoretically…

The importance of nonadiabatic transitions which represent basic mechanisms of various dynamic processes in a variety of natural sciences was emphasized and their basic theories were presented. There are basically two types of transitions, namely, time-dependent and time-independent ones. In the former case, the most fundamental Landau–Zener–Stückelberg linear potential model has revived and plays significant roles in various physical processes such as quantum dots, quantum computation and interferometry. A generalization beyond the linear potential model was shown to be made by the Zhu–Nakamura theory in a semiclassical way…

The following section is included:

• Index