^{1,a)}, Satoshi Takahashi

^{1,b)}and Kazuo Takatsuka

^{1,c)}

### Abstract

A practical quantum theory for unifying electronic and nuclear dynamics, which were separated by the Born–Oppenheimer approximation, is proposed. The theory consists of two processes. Nonadiabatic (quantum) electron wavepacket dynamics on branching (non-Born–Oppenheimer) nuclear paths are first constructed. Since these paths are not the classical trajectories, most of the existing semiclassical theories to generate quantum wavepacket do not work. Therefore, we apply our own developed semiclassical wavepacket theory to these generated non-Born–Oppenheimer paths. This wavepacket is generated based on what we call the action decomposed function, which does not require the information of the so-called stability matrix. Thus, the motion of nuclei is also quantized, and consequently the total wave function is represented as a series of entanglement between the electronic and nuclear wavepackets. In the last half of the article, we show the practice to demonstrate how these independent theories can be unified to give electron-nuclear wavepackets in a two-state model. The wavepackets up to the phases and resultant transition probabilities are compared to the full quantum-mechanical counterparts. It turns out that the lowest level approximation to the wavepacket approach already shows a good agreement with the full quantum quantities. Thus, the present theoretical framework gives a basic method with which to study non-Born–Oppenheimer electronic and nuclear wavepacket states relevant to ultrafast chemical events.

This work was supported by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

I. INTRODUCTION

II. THEORETICAL BACKGROUND: REPRESENTATION OF THE TOTAL WAVE FUNCTIONS

A. The BO representation of molecular total wave function

B. Path branching in nonadiabatic transition

III. NONADIABATIC ELECTRON WAVEPACKETS ALONG BRANCHING PATHS

A. The total Hamiltonian represented in nuclear-configuration and electronic-Hilbert spaces

B. Dynamics in the electron-nuclear quantum-classical mixed representation

1. Dynamics of quantum electron wavepackets

2. Classical nuclear motions in nonadiabatic transitions: A generalization of classical mechanics

C. Electronic state mixing along branching paths

1. Eigenforce and branching paths

2. Averaging over the paths to extract a (few) representative path(s) in the coupling region

3. Branching of the averaged path

D. The electronic wavepackets on the branching paths

IV. ASSOCIATING NUCLEAR WAVEPACKETS WITH THE BRANCHING

A. Semiclassics in terms of ADF

1. Equation of motion

2. Solving the ADF equation up to the velocity gradient term

B. Normalized variable Gaussians as a simple approximation

1. Rescaling the exponents according to the Gaussian height

2. WKB velocity gradient term revisited

3. Quantum diffusion

V. ILLUSTRATIVE APPLICATION TO A TWO-STATE MODEL: PRACTICE AND PERFORMANCE

A. System examined

1. Potential functions

2. Initial preparation of nuclear wavepackets

3. Integrators

B. Branching paths

C. Propagating ADF-NVG along the branching paths

1. Evaluation of the action integral

2. Treatment at the turning points

3. The single NVG approximation

D. Numerical tests

1. Nuclear wavepacket

2. Electronic state population

3. Miscellaneous quantities estimated in terms of the wavepackets

VI. CONCLUDING REMARKS

### Key Topics

- Semiclassical theories
- 43.0
- Non adiabatic reactions
- 27.0
- Wave functions
- 25.0
- Electroluminescence
- 14.0
- Non adiabatic couplings
- 14.0

## Figures

The upper panel: Two adiabatic potential curves that couple with each other through the coupling element (the Gaussian-like curve). The lower panel magnifies the coupling function, which illustrates the starting point of the phase-space averaging and the exit from which for two branching paths to emerge.

The upper panel: Two adiabatic potential curves that couple with each other through the coupling element (the Gaussian-like curve). The lower panel magnifies the coupling function, which illustrates the starting point of the phase-space averaging and the exit from which for two branching paths to emerge.

PSANB paths (the leftmost column), the space-time plot of (the middle column), and that of (the rightmost column). These paths are tracked along (a) the vibrational decay event with and (b) the collision event with . The green small circles on the horizontal straight lines denote the branching points for the PSANB paths.

PSANB paths (the leftmost column), the space-time plot of (the middle column), and that of (the rightmost column). These paths are tracked along (a) the vibrational decay event with and (b) the collision event with . The green small circles on the horizontal straight lines denote the branching points for the PSANB paths.

Comparison of the real parts of (blue and broken curves) and (red and solid curves) for the vibrational decay case with the initial condition . The wavepackets on the lower (upper) straight line is running on the lower (upper) potential surface. are slightly shifted below for clearer presentation. Time at which each snapshot is taken is indicated in each box.

Comparison of the real parts of (blue and broken curves) and (red and solid curves) for the vibrational decay case with the initial condition . The wavepackets on the lower (upper) straight line is running on the lower (upper) potential surface. are slightly shifted below for clearer presentation. Time at which each snapshot is taken is indicated in each box.

Comparison of the real parts of (blue and broken curves) and (red and solid curves) for the collision case with the initial condition . The wavepackets on the lower (upper) straight line is running on the lower (upper) potential surface. are slightly shifted below for clearer presentation. Time at which each snapshot is taken is indicated in each box.

Comparison of the real parts of (blue and broken curves) and (red and solid curves) for the collision case with the initial condition . The wavepackets on the lower (upper) straight line is running on the lower (upper) potential surface. are slightly shifted below for clearer presentation. Time at which each snapshot is taken is indicated in each box.

The variation of the population on the upper electronic state, of Eq. (85) estimated with the PSANB without nuclear wavepacket (red and solid curve), PAN (PSANB-ADF-NVG, green dashed curve), and the full quantum method (blue circle). (a) Vibrational decay case with , (b) Collision case with

The variation of the population on the upper electronic state, of Eq. (85) estimated with the PSANB without nuclear wavepacket (red and solid curve), PAN (PSANB-ADF-NVG, green dashed curve), and the full quantum method (blue circle). (a) Vibrational decay case with , (b) Collision case with

Four miscellaneous quantities that are calculated in terms of the quantum nuclear wavepackets given in the present nonadiabatic dynamics. [Panels (a1) and (b1)] A time for a wavepacket to stay in the upper electronic surface. [(a2) and (b2)] Flux to the dissociation channel on the ground state (the values are multiplied by 10). [(a3) and (b3)] Accumulated population to the dissociation channel, which is given by the time integration of the above flux. [(a4) and (b4)] The dispersion of the wavepacket around its center, which is monitored on the upper surface. The left and right panels are for the vibrational decay and collision , respectively.

Four miscellaneous quantities that are calculated in terms of the quantum nuclear wavepackets given in the present nonadiabatic dynamics. [Panels (a1) and (b1)] A time for a wavepacket to stay in the upper electronic surface. [(a2) and (b2)] Flux to the dissociation channel on the ground state (the values are multiplied by 10). [(a3) and (b3)] Accumulated population to the dissociation channel, which is given by the time integration of the above flux. [(a4) and (b4)] The dispersion of the wavepacket around its center, which is monitored on the upper surface. The left and right panels are for the vibrational decay and collision , respectively.

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