# Quantum Chaos in the Dynamics of Molecules

## Abstract

**:**

## 1. Introductory Remarks: What Makes Molecules Special in Quantum Science and Chaos

## 2. Brief Introduction to the Theoretical Framework of Molecules: Born-Oppenheimer Approximation to Separate Electronic and Nuclear Motions

#### 2.1. The Born-Oppenheimer (BO) Approximation

#### 2.2. The Born-Huang Expansion

## 3. Implication of Chaos in Chemical Dynamics

#### 3.1. Determinicity versus Stochasticity in Molecules

#### 3.2. Intramolecular Vibrational Energy Redistribution (IVR)

#### 3.3. Onset of Statistical Properties: Liquid-Like Clusters

## 4. Quantum Dynamics in the Quasi-Separatrix of Two Dimensional Molecular Vibration

#### 4.1. Quantum Wavepackets Embedded in the Quasi-Separatrix

#### 4.2. Energy Spectra and Eigenfunctions in the Quasi-Sparatrix

#### 4.2.1. Energy Screening of Quantum Wavepackets

#### 4.2.2. Chaotic Eigenfunctions and Spectra

#### 4.2.3. Dependence on the Magnitude of the Planck Constant

#### 4.3. Time-Dependent Spectrum of Very Long-Time Dynamics

## 5. Quantization of Chaos with Semiclassical Wavepackets

#### 5.1. The Gutzwiller Periodic Orbit Theory

#### 5.2. Wavepacket Semiclassics with Action Decomposed Function

#### 5.2.1. Short Time Dynamics of the Maslov Type Wavefunction

#### 5.2.2. Semiclassical Approximation

#### 5.2.3. Gaussian Wavepacket Approximation

#### 5.3. On the Quantization Condition: Creation and Elimination of the Spectral Peaks by the Phases

#### 5.3.1. Prior Quantization Conditions Based on the Periodic Orbits

#### 5.3.2. Peaks Arising from Individual Orbits and an Extended Quantization Condition

#### 5.3.3. Destructive Interference Suppressing the Spurious Energy Peaks: Another Essential Role of the Phases for Quantization

#### Case in which $\mathbf{E}$ Is Slightly Shifted from a True Eigenvalue (Near-Resonance Peaks)

#### Case of Generation of Harmonics in $\mathbf{E}$

#### 5.4. Amplitude-Free Energy Quantization of Classical Chaos

#### 5.5. Chaos Mediated by Dynamical Tunneling

#### 5.5.1. Semiclassical Tunneling Theory

#### Connection Problem

#### 5.5.2. Statistical Redistribution of Tunneling Paths

## 6. Chaos Arising from Repeated Bifurcation and Merge of the Quantum Wavepackets on Nonadiabatically Coupled Potential Basins

#### 6.1. Experimental Observation of Bifurcation and Merge of the Quantum Wavepackets

#### 6.2. Chaotic Eigenfunctions in Nonadiabatically Coupled Potential Functions

#### 6.3. Need for Measures of Chaoticity in Quantum Wavepackets

## 7. Chaos in Nonadiabatic Electron Dynamics of Molecules

#### 7.1. The Hamiltonian Studied for Electron Dynamics

#### 7.1.1. Electron Dynamics

#### 7.1.2. Nuclear Dynamics: Path Branching

#### 7.1.3. Nuclear Dynamics: Mean-Field Approximation (Semiclassical Ehrenfest Theory)

#### 7.2. Longuet–Higgins (Berry) Phase and Lorentz-like Force

#### 7.3. Chaotic Electron Dynamics in Densely Quasi-Degenerate Electronic-State Manifold; B${}_{12}$ Cluster as an Example

#### 7.3.1. Nearest Neighbor Level Spacing Distribution (NNLSD)

#### 7.3.2. Diffusive Dynamics in the State Space: Fractional Brown Motion

#### 7.3.3. Shannon Entropy

#### 7.3.4. Lyapunov Exponents for the Loss of Electronic State Memory

#### 7.3.5. Turbulent Electron Flow in the Cluster

#### 7.4. The Long Life-Time of Dynamical Chemical Bonding: Hyper Resonance

#### 7.5. Intra-Molecular Nonadiabatic Electronic Energy Redistribution

#### 7.5.1. Huge Inflation of Phase-Space Volume

#### 7.5.2. Intra-Molecular Nonadiabatic Electronic Energy Redistribution (INEER): Nonadiabatic Interaction to Close Dissociation Channels

## 8. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Stable structure of 5-methyltropolone (5MTR). Hydrogen atoms attached on C${}_{3}$, C${}_{4}$, C${}_{6}$, and C${}_{7}$ are not shown. (

**b**) The transition state on the way. (

**c**) Proton has ben shifted from O${}_{2}$ to O${}_{1}$ inducing the internal rotation of the methyl group at the bottom. [Taken from Ushiyama, H.; Takatsuka, K. Angew. Chem. Int. Ed.

**2005,**44, 2 with permission].

**Figure 2.**Four isomers on the single Lennard-Jones potential energy surface. The names and the bottom energy of each potential basins are presented. [Drawn by Dr. C. Seko].

**Figure 3.**Schematic presentation of a unit of reaction tube in a potential basin and its bifuraction structure. The circle schematically models a potential basin, each of which supports one isomer. Only one of tubes is depicted and truncated at some points. The reaction tubes lie behind classical chaos in structural isomerization dynamics, which is one of the so-called many-valey problems. The similar tube units are extended from every neighboring potential basins, and one basin is filled with many tubes. Time-revesal symmetry gives an idea of how the reaction tubes are reconnected with one another to form thicker tubes and get out of the basin to next ones. [Drawn by Dr. C. Seko].

**Figure 4.**Panels in the top row for the potential functions used (the left one for a perspective view and the right for the contour plot). The bottom row, left panel indicates the classical Poincaré surface of section at $x=0$ and ${p}_{x}>0.$ The dark area with many dotts displays the quasi separatorix separating the major tori. The symbols “lib” and “rot” characterize the vibrational modes of the individual tori. The bottom right panel exemplifies one of the torus modes, a libratation. [Taken from Hashimoto, N.; Takatsuka, K. J. Chem. Phys.

**1998,**108, 1893 with permission].

**Figure 5.**Time propagation of a quantum wavepacket in the quasi-separatrix. Time passes from panel (

**a**–

**d**). The upperrow indicates ${\left|\psi \left(x,y,t\right)\right|}^{2}$ at selected times, while the lower row shows the corresponding quantum Poincaré section, in which the wavepacket is seen to be running on the classical quasi-separatrix. [Taken from Hashimoto, N.; Takatsuka, K. J. Chem. Phys.

**1998,**108, 1893 with permission].

**Figure 6.**A pair of almost degenerate eigenfunctions lying in the quasi-separatrix. In the left panel is an odd function with respect to the symmetry axis $x=0$ [energy $E=0.152094$], while the right being even [$E=0.152500$]. The slight energy difference and the separation of the symmetry are due to what we call dynamical tunneling of the second kind. See the text. [Taken from Hashimoto, N.; Takatsuka, K. J. Chem. Phys.

**1998,**108, 1893 with permission].

**Figure 7.**Quantum Poincaré surface of section. Panels at the top row reprsent an quasi-separatorix eigenfunction given by $\hslash =0.005$ ($[{E}_{1},{E}_{2}]=[0.150.0.153]$). The bottom panels are the eigenfunction at $\hslash =0.010$ ($[{E}_{1},{E}_{2}]=[0.150.0.156]$). Left column (

**a**) for the perspective views and the right (

**b**) for contour plots. [Taken from Hashimoto, N.; Takatsuka, K. J. Chem. Phys.

**1995,**103, 6914 with permission].

**Figure 8.**An illustration for geometrical implication of $\sigma \left(t\right)$ defined in Equation (36). A deviation determinant $\sigma \left(t\right)$, which is an orientable volume, withrespect to the main path is formed with the help of nearby classical paths. Time passes from panel (

**a**–

**d**). [Taken from Takahashi, S.; Takatsuka, K. Phys. Rev. A

**2004,**69, 022110 with permission].

**Figure 9.**(

**a**) Classical Poincaré surface of section at $x=0$, ${p}_{x}=0$ for an ensemble of trajectories with the total energy ${E}_{cl}$$=0.157$. $\hslash =0.005$. (

**b**) The nearest neighbor level spacing of the full quantum spectrum in this energy region, which is subject to the Wigner distribution, an indication of quantum chaos. The right panel shows the spectrum taken from the amplitude free correlation function (green solid) and full quantum counterpart (red). Only the spectral positions should be compared. [Taken from Takahashi, S.; Takatsuka, K. J. Chem. Phys.

**2007**, 127, 084112 with permission].

**Figure 10.**The right panel of the upper row shows the Poincaré section ($x=0$ and ${p}_{x}>0$) along with the vibrational modes (left) belonging to the tori. $E=0.09.$ The botton row shows a long time plot at the Poincaré section, which wanders from one torus to another through tunneling. Statistical distribution through tunneling is recognized. [Taken from Ushiyama, H.; Takatsuka, K. Phys. Rev. E

**1996,**53, 115 with permission].

**Figure 11.**Photoelectron kinetic energy spectrum as the wavepackets bifurcate and merge at the avoided crossing. (The potential curves in this figure are graphcally represented in the diabatic representation.) The polarization vector of the probe laser light is perpendicular to that of the pump, while the latter is set parallel to the line between Na and I. In the upper row are shown the absolute square of the wavepackets at selected times (red (blue) one on the red (blue) potential curve). The bottom row represents the corresponding energy-resolved photelectron signals. Figure drawn by Dr. Y. Arasaki. [Taken from Takatsuka, K. Bull. Chem. Soc. Jpn.

**2021**, 94, 1421 with permission].

**Figure 12.**A schematic representation of Heller’s model system. The distance between the centers of the potentials is 2a, and the angle between the relevant crossing seam (dotted line) and the primary axis of each potential (dashed line) is $\theta .$ Inset: The perspective view of the present system. The energy difference of the potential bottoms is $\Delta \u03f5={\u03f5}_{B}-{\u03f5}_{A}$. [Taken from Fujisaki, H.; Takatsuka, K. J. Chem. Phys.

**2001,**114, 3497 with permission].

**Figure 13.**First row from the top: Eigenfunctions squared (from left to right, 800th, 810th, and 820th) of Heller’s model system on surface A (see Figure 12) with $J=1.5$ (intermediate coupling) and $\theta =\pi /6$. Second row: The corresponding eigenfunctions (squared) on surface B. Third row: The corresponding amplitude distributions of the eigenfunctions onsurface A. The dashed envelope curves represent Gaussian functions. [Taken from Fujisaki, H.; Takatsuka, K. Phys. Rev. E

**2001,**63, 066221 with permission].

**Figure 14.**A symmetry breaking by pyramidalization of a coplanar molecular system, leading to molecular chirality. (

**a**) Atom A is approaching the planar molecule on the same plane. (

**b**) The Lorentz-like nonadiabatic force triggers to guide the incident atom to the direction out of the plane. (

**c**) The Hellmann-Feynman force follows to make the pyramidalization complete. [Taken from K. Takatsuka, J. Chem. Phys.

**146**, 084312 (2017) with permission].

**Figure 15.**A particle coming in with a velocity vector $\mathbf{v}$ into the ${\mathbf{Z}}_{IJ}$ field feels a force ${\mathbf{f}}_{IJ}$, which is orthogonal to both $\mathbf{v}$ and ${\mathbf{Z}}_{IJ}$, according to ${\overrightarrow{f}}_{IJ}^{a}\left(\mathbf{R}\right)={\overrightarrow{v}}^{a}\times {\overrightarrow{Z}}_{IJ}^{{}_{a}}\left(\mathbf{R}\right).$ See the text for the definition of ${\mathbf{Z}}_{IJ}$.

**Figure 16.**Diffusion–like redistribution of the electronic states in the excited electronic state manifold of B${}_{12}$ cluster. It is actually a fractional Brownian motion [164] in the state space. The dynamics of the log${\left|{C}_{I}\left(t\right)\right|}^{2}$ of Equation (108) is tracked with the SET and is charted in color code. An initial state has been prepared at the 300th adiabatic state. The extensive diffusive state is realized well before 10 fs. [Figure drawn by Dr. Y. Arasaki. Taken from Takatsuka, K. Bull. Chem. Soc. Jpn.

**2021**, 94, 1421 with permission].

**Figure 17.**Time evolution of the Shannon entropy with respect to the adiabatic state population ${\left|{C}_{I}\left(t\right)\right|}^{2}$ involved in the wavepackets, each starting from the adiabatic state #100, #200, and #300, respectively. [Provided by Dr. Yasuki Arasaki].

**Figure 18.**Loss of memory of nonadiabatic states in the Hilbert space. (

**a**) Quick decay of ${\left|{C}_{I}\left(t\right)\right|}^{2}$ from 1.0 to ${C}_{I}(t+\Delta t)$ with $\Delta t=0.5$ fs at selcted points of t for B${}_{12}$ cluster SET dynamics starting from the 300th adiabatic state. Dynamics starting at $t=0.0$, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 fs. (

**b**) Exponent of memory loss, defined in Equation (128), for the initial adiabatic component weight, $\xi (t,\Delta t)$, starting from either the 300th (red) or the 100th (black) adiabatic state. $\Delta t=0.5$ fs. Lines connect calculated points only for visualization. [Provided by Dr. Yasuki Arasaki].

**Figure 19.**Spontaneous electron flux arising from the wavepackets in the SET dynamics starting from the 74th adiabatic state at (

**a**) 0.6 fs and (

**b**) 1.0 fs, and from the 75th adiabatic states at (

**c**) 0.6 fs and (

**d**) 1.0 fs. dynamics. The flux vectors are systematically multiplied by $2.0\times {10}^{4}$ for visualization. These fluxes are driven by nonadiabatic interactions. Soon after $1.0$ fs, the regular-looking fluxes become turbulent. [Taken from Yonehara, T; Takatsuka, K. J. Chem. Phys.

**2016**, 144, 164304 with permission].

**Figure 20.**B${}_{12}$ bonding structures along a trajectory under (

**a**) adiabatic dynamics, and (

**b**) nonadiabatic dynamics, starting from the same initial conditions (initial adiabatic state 47). Time t in fs is indicated beside each structure. At $t=110$ fs, atom 1 is in the dissociation channel in the adiabatic dynamics, while this atom is pulled back to the cluster site in the nonadiabatic dynamics. [Taken from Arasaki, Y.; Takatsuka, K. J. Chem. Phys.

**2019,**150, 114101 with permission].

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Takatsuka, K.
Quantum Chaos in the Dynamics of Molecules. *Entropy* **2023**, *25*, 63.
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Takatsuka K.
Quantum Chaos in the Dynamics of Molecules. *Entropy*. 2023; 25(1):63.
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Takatsuka, Kazuo.
2023. "Quantum Chaos in the Dynamics of Molecules" *Entropy* 25, no. 1: 63.
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