Review pubs.acs.org/CR Simulating Coherent Multidimensional Spectroscopy of Nonadiabatic Molecular Processes: From the Infrared to the X‑ray Regime † ‡ § ‡ ⊥ † Markus Kowalewski,*, , Benjamin P. Fingerhut,*, , Konstantin E. Dorfman,*, Kochise Bennett, † and Shaul Mukamel*, † Department of Chemistry and Department of Physics and Astronomy, University of California, Irvine, California 92697-2025, United States § Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, D-12489 Berlin, Germany ⊥ State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China ABSTRACT: Crossings of electronic potential energy surfaces in nuclear configuration space, known as conical intersections, determine the rates and outcomes of a large class of photochemical molecular processes. Much theoretical progress has been made in computing strongly coupled electronic and nuclear motions at different levels, but how to incorporate them in different spectroscopic signals and the approximations involved are less established. This will be the focus of the present review. We survey a wide range of time-resolved spectroscopic techniques which span from the infrared to the X-ray regimes and can be used for probing the nonadiabatic dynamics in the vicinity of conical intersections. Transient electronic and vibrational probes and their theoretical signal calculations are classified by their information content. This includes transient vibrational spectroscopic methods (transient infrared and femtosecond off-resonant stimulated Raman), resonant electronic probes (transient absorption and photoelectron spectroscopy), and novel stimulated X-ray Raman techniques. Along with the precise definition of what to calculate for predicting the various signals, we outline a toolbox of protocols for their simulation. CONTENTS 2.3.2. Continuous Frequency Switch 12175 2.4. Frequency-Dispersed Infrared Detection of 1. Introduction 12166 Nonadiabatic Relaxation 12177 1.1. Conical Intersections and Their Theoretical 2.4.1. UV−vis Pump IR Probe (UV/IR) 12177 Description 12167 2.4.2. Transient Two-Dimensional Infrared 1.2. Nuclear Dynamics Simulation Protocols 12168 − Spectroscopy (T-2DIR) 12178 2. Simulation Protocols for UV vis Pump and Broad- 2.4.3. Two-Dimensional Electronic−Vibrational Band IR Probe (UV/IR) Signals 12169 Spectroscopy (2DEV) 12178 2.1. Loop Diagram Representation of Frequency- 3. Electronically off-Resonant Stimulated Raman Dispersed Transmission Signals 12169 Spectroscopy (SRS) 12179 2.2. Simulation Toolbox for Nonlinear Optical 3.1. Simulation Protocols 12179 Signals with Broad-Band Probe Pulses 12171 3.2. Detection Schemes for off-Resonant Raman 2.2.1. First Simulation Protocol: Numerical Signals 12180 Propagation of the Wave Function 12171 3.2.1. Signals Linear in the Probe 12180 2.2.2. Second Simulation Protocol: Snapshot 3.2.2. Signals Quadratic in the Probe 12180 Limit; Sum Over States (SOS) Approx- 3.3. Loop Diagram Representation of Raman imation 12172 Signals 12181 2.2.3. Third Simulation Protocol: Frequency 3.3.1. Interplay of Temporal and Spectral Modulation Induced by Coupling to a Resolution 12183 Classical Bath 12172 3.3.2. Additional Variants of the SRS Technique 12184 2.2.4. Fourth Simulation Protocol: Stochastic 3.3.3. Comparison of Various SRS Signals 12189 Liouville Equation (SLE) for Vibronic Line Shapes 12173 2.3. Interplay of Temporal and Spectral Resolu- tion in UV/IR Signals 12174 Received: February 6, 2017 2.3.1. Linear Matter Chirp 12175 Published: September 26, 2017 © 2017 American Chemical Society 12165 DOI: 10.1021/acs.chemrev.7b00081 Chem. Rev. 2017, 117, 12165−12226 Chemical Reviews Review 4. Resonant Electronic State Detection in the Visible special cases. With the advent of modern quantum chemistry and the Ultraviolet 12190 methods and the computer age, it is now established that CoIns 4.1. Transient Absorption (TA) 12190 can be found in almost any molecule.4 They are the rule rather 4.1.1. Transient Absorption of a Shaped Pulse; than the exception and play an essential role in photochemistry, Linear Variant of FSRS 12191 photophysics, and nature. CoIns enable ultrafast, sub-100 fs 4.2. Two-Dimensional Electronic Spectroscopy nonradiative relaxation pathways, which control product yields (2DES) 12193 and rates of a large class of photochemical processes (for a 4.2.1. Two-Dimensional Electronic Spectrosco- general review of CoIns see ref 5). At a CoIn, electronic and py in the UV (2DUV) 12193 nuclear frequencies are comparable and become strongly 4.2.2. Phase-Modulated 2D Fluorescence (PM- coupled since the Born−Oppenheimer approximation,6 which Fl) 12195 normally allows their separation, breaks down. 5. X-ray Probes for Valence Electronic States 12195 The celebrated Woodward−Hofman rules7 in organic 5.1. X-ray Hybrid Stimulated Raman Detection: chemistry relate the stereochemistry of pericyclic reactions to Attosecond Stimulated X-ray Raman Spec- orbital symmetry. The photoinduced reaction pathway involves − troscopy (ASRS) 12196 CoIns, as has been shown on, e.g., cyclohexadiene.8 15 Artificially 5.2. Linear off-Resonant X-ray Raman 12197 designed molecular systems like optical switches10,16 rely on 5.3. Time-Resolved Photoelectron Spectroscopy 12198 CoIns as a basic functional principle. Many important light- 5.4. Other Methods 12199 induced biological processes, like the photosynthesis of Vitamin 5.4.1. Auger Electron Spectroscopy 12199 D,17 the main event of vision in retinal,18 photodamage of − 5.4.2. Time-Resolved High Harmonic Spectros- deoxyribonucleic acid (DNA),19 and DNA repair,20 22 are copy 12200 decided by CoIns. This review surveys the spectroscopic 6. Summary and Conclusions 12200 techniques that can be employed for the monitoring of CoIns 7. Outlook 12202 and outlines protocols for their simulation. Appendix 12205 The strong mixing of the nuclear and the electronic degrees of Time-Gated Signals 12205 freedoms in the vicinity of a CoIn affects vibrational and Frequency-Domain UV/IR Signal Expression 12205 electronic properties. Two strategies are possible for their Spontaneous Signals 12206 spectroscopic detection: One can either use transient vibrational Frequency-Resolved Spontaneous Raman Sig- spectroscopy or probe properties which are attributed to nal (FR-SPRS) 12206 electronic states. Vibrational techniques rely mainly on the fact Mapping of off-Resonant Interactions to Effective that the shape of the potential energy is heavily affected by the Polarizabilities 12207 CoIn, which results in a temporal variation of vibrational mode Stimulated Signals 12208 frequencies of spectator modes. This can be probed by, e.g., FSRS 12209 femtosecond stimulated Raman spectroscopy (FSRS), coherent TG-ISRS and TR-ISRS 12209 anti-Stokes Raman spectroscopy (CARS), or transient infrared Stochastic Liouville Equation (SLE) Based Signals 12210 (IR). More subtle vibrational properties are the temporal change Two-State Jump Model 12210 of intermode couplings, which require multidimensional vibra- FSRS for General Multistage Jump and Arbi- tional probe schemes, such as transient 2D-IR, to observe. trary Temperature 12210 Changes in the vibrational frequency, intensity of a vibrational Closed Expressions for the Raman Signals in transition, or intermode couplings during the time evolution of the SML 12211 an excited state wavepacket can serve as signatures for a CoIn. Associated Content 12211 The second major class of spectroscopic techniques detect Special Issue Paper 12211 properties attributed to electronic states. As the system passes Author Information 12211 through a CoIn, population is transferred between the electronic Corresponding Authors 12211 states involved in the surface crossing. The character of the ORCID 12211 adiabatic electronic wave function changes rapidly, as do Author Contributions 12211 properties such as transition dipole moments. The energy gap Notes 12211 between the electronic states varies strongly down to degeneracy Biographies 12211 of both curves at the CoIn. These changes can be probed by Acknowledgments 12212 addressing the electronic degrees of freedom either by resonant Acronyms 12212 transient absorption with visual/ultraviolet (UV) light, in a References 12213 Raman process with extreme ultraviolet (XUV) of X-ray light, or tracked by time-resolved photoelectron spectroscopy. Moreover, the branching of the nuclear wavepacket creates a short-lived 1. INTRODUCTION electronic coherence that can be probed by sufficiently short It has been widely believed that electronic states of the same pulses. In a landmark femtosecond experiment, the population symmetry cannot cross. Hund, von Neumann, and Wigner first dynamics of electronic states was spectroscopically observed by − realized1,2 that the strict noncrossing rule for potential energy Zewail23 25 in sodium iodide molecules and theoretically surfaces (PESs) of the same symmetry only holds for systems described by means of wave packet dynamics.26,27 Since this with one vibrational degree of freedom. Teller described these system only has a single vibrational degree of freedom, it degenerate points as double cones3 and coined the term conical constitutes an avoided crossing28,29 rather than a CoIn. intersection (CoIn). CoIns in crossings are possible if there are Commercial ultrafast laser sources with sub-100 fs pulse two or more vibrational modes. It was thought that such lengths cover the IR to the near UV regime. More recent electronic degenerate points between two PESs are very rare and developments30 pushed