Chemical Theory beyond the Born-Oppenheimer Paradigm Nonadiabatic Electronic and Nuclear Dynamics in Chemical Reactions

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1 Chemical Theory beyond the Born-Oppenheimer Paradigm Nonadiabatic Electronic and Nuclear Dynamics in Chemical Reactions

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3 Chemical Theory beyond the Born-Oppenheimer Paradigm Nonadiabatic Electronic and Nuclear Dynamics in Chemical Reactions Kazuo Takatsuka Takehiro Yonehara Kota Hanasaki Yasuki Arasaki The University of Tokyo, Japan World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Chemical theory beyond the Born-Oppenheimer paradigm : nonadiabatic electronic and nuclear dynamics in chemical reactions / Kazuo Takatsuka, Takehiro Yonehara, Kota Hanasaki, Yasuki Arasaki, The University of Tokyo, Japan. pages cm Includes bibliographical references and index. ISBN (hardcover : alk. paper) 1. Born-Oppenheimer approximation. 2. Chemical reactions. 3. Charge exchange. I. Takatsuka, Kazuo. II. Yonehara, Takehiro. III. Hanasaki, Kota. IV. Arasaki, Yasuki. QC A66C ' dc British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore

5 Preface Acknowledgments Contents 1. The Aim of This Book: Where Are We? Potential energy surfaces and nonadiabatic transitions Electronic state theory Nonadiabatic transitions A brief overview Necessity of nonadiabatic dynamical electron theory Progress in laser chemistry Chemistry without potential energy surfaces: Highly quasi-degenerate electronic states General theory of mixed quantum and classical dynamics Structure of this book Basic Framework of Theoretical Chemistry Born Huang expansion Born Oppenheimer approximation Bound states and notion of potential energy surface Stationary-state scattering theory for electrons by molecules Validity of the BO approximation Generalization of the adiabatic electronic states xiii xvii v

6 vi Chemical Theory Beyond the Born-Oppenheimer Paradigm 3. Nuclear Dynamics on Adiabatic Electronic Potential Energy Surfaces Classical nuclear dynamics: Ab initio molecular dynamics Coupling of electronic and nuclear motion in tautomerization dynamics Molecular machine? Nuclear quantum dynamics on an adiabatic potential surface Time-propagation with split operator formalism An example of three dimensional dynamics Polynomial expansion method Multiconfigurational time-dependent Hartree (MCTDH) approach Eigenfunctions extracted from wavepacket dynamics: Energy screening Probing the dynamics with time-resolved photoelectron spectroscopy Hamiltonian describing interaction with an external field Coupled dynamics in electronic excitation Nuclear wavepacket description of femtosecond time-resolved photoelectron spectroscopy Geometry-dependent photoionization matrix elements Calculation of the photoelectron orbitals Numerical time-propagation describing photoionization Efficient diagonalization of the interaction matrix The external fields introduced Photoelectron spectra from the dynamics and their transient counterparts History of dynamics in the transient photoelectron spectrum Velocity map imaging and its time derivative An example: The Na 2 double minimum state... 50

7 Contents vii 4. Breakdown of the Born Oppenheimer Approximation: Classic Theories of Nonadiabatic Transitions and Ideas behind Theories for one-dimensional curve crossing problem The Landau Zener theory of curve crossing model Quantum phase arising from nonadiabatic transitions Zhu Nakamura theory Mixed quantum classical formulation of electron-nucleus coupled nonadiabatic dynamics Pechukas path integrals Mean-field path representation: Semiclassical Ehrenfest theory Quantum variables mapped to classical ones: Meyer Miller method Initial value representation of semiclassical estimate of nonadiabatic transition amplitudes Surface hopping scheme and beyond Surface hopping model Surface hopping driven by several types of state couplings Tully s fewest switch surface hopping method anditsvariants Spawning method of Martínez Remixing of electronic states to incorporate the quantum nature of nuclear dynamics and interference among the paths Coherence and decoherence before and after nonadiabatic interaction Decay of mixing with coherence switching Notion of decoherence in quantum subsystems by contact with classical subsystems and decoherence time Some specific methods recently proposed for nonadiabatic dynamics Hybrid methods for nonadiabatic dynamics in large molecular systems

8 viii Chemical Theory Beyond the Born-Oppenheimer Paradigm 5. Direct Observation of the Wavepacket Bifurcation due to Nonadiabatic Transitions How does the Born Oppenheimer approximation break down? Nuclear wavepacket bifurcation as observed with time-resolved photoelectron spectroscopy Coupled nuclear dynamics on diabatic potential energy surfaces Wavepacket bifurcation in the NaI system Photoelectron signals arising from the NaI dynamics Control of nonadiabatic chemical dynamics Various time domains of external field control An example: Fluctuating potential curves Shift of conical intersection and replacement byavoidedcrossing Conical intersection and wavepacket dynamics there The NO 2 system Time-resolved photoelectron spectroscopy of the conical intersection dynamicsintheno 2 system Monitoring the effect of a control pulse on a conical intersection by time-resolved photoelectron spectroscopy High-harmonic spectroscopy to monitor nonadiabatic transition High-harmonic generation and associated phases Transient grating and interferometry Transient grating interferometry of the conical intersection dynamics in NO Electron and nucleus dynamics tracked with pulse train in time-resolved photoelectron spectroscopy Generation of pulse train A case study on LiH molecule Pulse train induced dynamics Transient photoelectronspectrum Roles of individual components Photoemission arising from electron transfer within a molecule

9 Contents ix 6. Nonadiabatic Electron Wavepacket Dynamics in Path-branching Representation Path-branching representation for electron wavepacket propagation Theoretical background: A representation of the total wavefunctions Nonadiabatic electron wavepackets along branching paths Dynamics in the electron-nuclear quantum-classical mixed representation The semiclassical Ehrenfest theory as a special case Methods of averaging and branching Electronic state mixing along branching paths The electronic wavepackets on the branching paths Branching conditions Energy-conserving path-branching with the force averaging What is the decoherence in nonadiabatic transitionsafterall? Numerical examples of branching paths and transition probability Systems of two and three electronic states nonadiabatically coupled Practices in the semiclassical Ehrenfest theory and the full quantum dynamics of nuclear wavepacket dynamics Nonadiabatic transition probability Force diabatization Geometry of branching paths An example in which the semiclassical Ehrenfest fails Highly degenerate coupled electronic states System functions and computational details Dynamics in five state model Nonadiabatic dynamics in fifteen state model

10 x Chemical Theory Beyond the Born-Oppenheimer Paradigm 6.5 Electronic phase interference between different branching paths: Dynamics around conical intersections Quantum effects manifesting in the nuclear branching paths Interactions and initial conditions Practices of the dynamic calculations Surmounting a potential barrier by lower energy paths; a behavior looking like quantum tunneling Trapping above the potential barrier: Time-delay in reaction dynamics Full-quantum dynamics to verify the branching phenomena Quantization of non-born Oppenheimer paths Action Decomposed Function (ADF) Normalized Variable Gaussians (NVG) as a simple approximation Illustrative application to a two-state model Appendix A: Reduction of muti-dimensional PSANB to one-dimensional (1D) approximation Appendix B: Quantum chemical calculations of the matrix elements of nonadiabatic interactions On diabatic representation Evaluation of the nuclear derivative coupling matrix elements with canonical molecular orbitals Nuclear derivative coupling elements in CSF representation Practical calculation of XIJ k Nonadiabatic coupling without use of a nuclear derivative Appendix C: Tracking the continuity of molecular orbitals along a nuclear path Concept of unique-continuity of molecular orbitals Practical implementation of unique-continuity of molecular orbitals

11 Contents xi 7. Dynamical Electron Theory for Chemical Reactions Electron flux in chemical reactions Definitions An example: Collision of Na and Cl Real-time dynamics of electron migration in a model water cluster anion system Nonadiabatic dynamics of hydrated electron Mechanisms of migration of the hydrated electron Complex-valued natural orbitals in electron wavepacket dynamics Nuclear motion inducing nonadiabatic transitions Isotopeeffects Single and relayed proton transfer in peptide Quantities characterizing the electron dynamics Computational details Proton transfer in the ground state Reversal electron current against the proton motion Rearrangement of π-bonds: Dynamical manifestation of the Pauling resonance structures Stabilization of the zwitter ionic resonance structure by the mediating water Double proton transfer in formic acid dimer Electronic configurations and initial conditions Electron dynamics in dimerization process Net electron flow across a geometric cross-section Electron dynamics in double proton transfer Summary of the mechanism Excited-state proton-electron simultaneous transfer Electronic configurations and initial conditions Mechanism of transitions Quantities related to electron density Electron dynamics The successive nonadiabatic transitions Analysis of electron dynamics Summary

12 xii Chemical Theory Beyond the Born-Oppenheimer Paradigm 7.6 Chemical dynamics for systems where notion of potential energy surfaces loses sense Molecular Electron Dynamics in Laser Fields Experimental progress and theoretical issues Attosecond laser tracking of molecular electronic states Laser manipulation to create new states Secondary effects of an induced electromagnetic field by external laser fields Dressed electronic states and nonadiabatic nuclear dynamics on them driven by laser fields Introduction to the standard Floquet analysis Generalization of the Floquet theory to treat dynamics in pulse lasers Generalization of path-branching representation for arbitrary optical and nonadiabatic transitions Mixed quantum-classical Hamiltonian in an optical field Coupled dynamics of electrons and nuclei: Generalized semiclassical Ehrenfest dynamics Applications: Electron jump in laser fields Control of nonadiabatic transition in NaCl Collision of LiH + H Diborane in laser field Dynamics of photoionization Electron flux and photoelectron signals Electron dynamics of ionizing states Summary Epilogue 401 Bibliography 403 Index 423

13 Preface Since the publication of the seminal paper by Born and Oppenheimer in 1927, the theoretical foundation of molecular science in the 20th century, and even to date, had been dominated by the so-called Born Oppenheimer approximation, which dynamically separates electronic and nuclear motions under an assumption that electrons can follow the nuclear dynamics almost instantaneously. This idea led to the fixed nuclei approximation and the notion of electronic stationary-states that adjust themselves to any nuclear configurations in space. However, nonadiabatic interactions and associated quantum transitions arising from the breakdown of the Born Oppenheimer separation are critically important in chemical reaction dynamics. This is because almost all the interesting chemical and even biological processes involve nonadiabatic transition events in them. In particular, dynamics of molecules optically pumped to electronically excited states shows very characteristic quantum phenomena, which are vital to understanding the nature of chemistry and chemical reactions. Many theoretical methods for treating such nonadiabatic transitions have been proposed, most of which are about quantum transition of nuclear dynamics between nonadiabatically coupled electronicenergy surfaces (the so-called potential energy surfaces). With no doubt their contributions to the progress of elementary dynamical processes of chemical reactions have been and continue to be critical for developments in chemical science. There have been already published so many excellent books on the standard theories of nonadiabatic transitions, and we are not trying to add a new volume into the list of them. Instead, the present book has been written based on our thought about chemical dynamics, which is briefly summarized as follows. (i) Chemical xiii

14 xiv Chemical Theory Beyond the Born-Oppenheimer Paradigm properties and dynamical processes of molecules are primarily determined by the electronic states in them. Hence the primary aim of chemical reaction theories is to track the qualitative change of electronic states in courses of chemical reactions. This is why quantum chemistry (molecular electronic structure theory) is so important. (ii) However, it should not be disregarded that electronic states themselves couple with nuclear motions not only dynamically (through potential energy) but also kinematically (through the quantum momentum operators). Besides, electrons move with finite speed in molecules. This is an obvious fact, not to mention the theory of relativity, and even energetics is under control by the energy-time uncertainty principle through such an elapsed time taken by electronic-state change. To place a chemical theory in a correct context, therefore, the study on dynamics of electrons is indeed critical. (iii) Thus, chemistry can be better understood in terms of the direct notion of nonadiabatic dynamical electrons, that is, a theory of time-dependent electron wavepacket dynamics that nonadiabatically couples with nuclear motions. A little more concrete reasoning behind the necessity of nonadiabatic dynamical electron theory is as follows. In the current status of the experimental progress, we are often faced with complicated and difficult situations that have not been studied before. For instance, due to advances in laser technology, an intense electromagnetic vector potential has become available, which now makes it possible to modify the molecular electronic states easily, which in turn induces novel nonadiabatic couplings. Also, chemistry of molecules in highly fluctuating electronic states due to high quasi-degeneracy, being coupled with nuclear motions, will become one of the most exciting fields in the next era of chemistry, which can be termed as non-born Oppenheimer chemistry or beyond-born Oppenheimer chemistry. Thus this book describes the recent theories of chemical dynamics beyond the Born Oppenheimer framework from a fundamental perspective of quantum wavepacket dynamics. To formulate these issues on a clear theoretical basis and to develop the novel theories beyond the Born Oppenheimer approximation, however, we should first learn a basic classical and quantum nuclear dynamics on an adiabatic (the Born Oppenheimer) potential energy surface. So we learn much from the classic theories of nonadiabatic transition such as the Landau-Zener theory and its variants. Subsequently our characteristic way of discussion on nonadiabatic transition enters, paying a particular attention to the direct (experimental) observation of the nuclear wavepacket bifurcation in the passage of nonadiabatic regions. We show that the instant of nonadiabatic transition

15 Preface xv can be indeed observed experimentally and the wavepacket bifurcation caused by quantum entanglement between electronic and nuclear motions. We claim that the wavepacket bifurcation represents the very essential feature of nonadiabatic transitions. Then the recent notion of nonadiabaticity in electron dynamics is introduced. To be consistent with the wavepacket bifurcation, we introduce the method of electron wavepacket dynamics that undergoes bifurcation while being carried along the so-called non-born Oppenheimer paths, which also branch due to nonadiabatic interactions. We will further proceed to the discussion about the interaction of molecular nonadiabatic states with intense laser fields. In this way, we penetrate on one hand into unknown domains of molecular properties such as (1) electron-nuclear quantum entanglement due to nonadiabatic transitions and its experimental observation, (2) coherence and decoherence of electron and nuclear wavepackets, which qualitatively dominate the quantum mechanical probabilities of quantum transition dynamics, (3) characteristic phenomena arising from the timedependent fluctuation of molecular electronic states, (4) the physics of interference between the nonadiabatic dynamics and external fields, and so on. On the other hand, the nonadiabatic electron wavepacket dynamics is extremely vital not only in chemical dynamics in laser fields but also for the theory of chemical reactivity in the absence of laser field. In particular, for analysis of chemical reactions in electronically excited states the nonadiabatic electron dynamics is indispensable. Moreover, there exist rather ubiquitously systems that have manifolds composed of highly quasi-degenerate electronic state, for which it is almost meaningless to single out each potential energy surface as usually done in the Born Oppenheimer approximation. This is what we call chemistry without potential energy surfaces. Even in these extreme cases, the nonadiabatic electron dynamics evolves in time along non-born Oppenheimer branching paths. Finally we note that one of the vital factors that are necessary to actually perform deeper research beyond the textbook knowledge about formal theories is practical capability to cope with materials in hand. Theoretical understanding alone is quite often not sufficient to knock the doors of and penetrate into the secrets of nature. We therefore describe in this book technical issues to some extent, which we think are useful in actual applications. The authors believe that only through such quests for the new and fundamental approaches along with technical expertise such as the art of computing, one of the doors to the next stage of chemistry will open.

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17 Acknowledgments This book has been made up with many scientific materials and computational results that have been studied and carried out by one of the authors (KT) and his colleages and former graduate students. He deeply appreciates their sincere and valiant attitudes towards a new field of chemical dynamics. Consequently some of the materials chosen for this book are reorganized and instructive excerpts from those original works. The authors are grateful to Professors Hiroshi Ushiyama and Satoshi Takahashi not only for their scientific contributions but to their efforts in management of the research enviroments. Very special thanks are due to Professor Vincent McKoy at California Institute of Technology. Most of the theoretical results from the studies of time-resolved photoelectron spectroscopy in Section 5 had been attained only by very close collaboration between the McKoy and Takatsuka groups. We are deeply grateful to Professor Jörn Manz for his long standing collaboration in scientific activities, in particular in the studies of electron dynamics and electron flux in chemical reactions. Last but not least, writing this was supported in part by a Grant-in- Aid for Scientific Research from the Ministry of Education and Science in Japan (MEXT). We thank the MEXT for the long-standing supports of our scientific activities. xvii