ADVANCES IN CHEMICAL PHYSICS

VOLUME 150

EDITORIAL BOARD

Moungi G. Bawendi, Department of Chemistry, Massachusetts Institute of Technology,Cambridge, Massachusetts, USA

Kurt Binder, Condensed Matter Theory Group, Institut für Physik, Johannes Gutenberg-Universität Mainz, Mainz, Germany

William T. Coffey, Department of Electronics and Electrical Engineering, Trinity College,University of Dublin, Dublin, Ireland

Karl F. Freed, Department of Chemistry, James Franck Institute, University of Chicago,Chicago, Illinois, USA

Daan Frenkel, Department of Chemistry, Trinity College, University of Cambridge,Cambridge, United Kingdom

Pierre Gaspard, Center for Nonlinear Phenomena and Complex Systems, Université Libre deBruxelles, Brussels, Belgium

Martin Gruebele, School of Chemical Sciences and Beckman Institute, Director of Center forBiophysics and Computational Biology, University of Illinois at Urbana-Champaign, Urbana,Illinois, USA

Jean-Pierre Hansen, Department of Chemistry, University of Cambridge, Cambridge, UnitedKingdom

Gerhard Hummer, Chief, Theoretical Biophysics Section, NIDDK-National Institutes ofHealth, Bethesda, Maryland, USA

Ronnie Kosloff, Department of Physical Chemistry, Institute of Chemistry and Fritz HaberCenter for Molecular Dynamics, The Hebrew University of Jerusalem, Israel

Ka Yee Lee, Department of Chemistry and The James Franck Institute, The University ofChicago, Chicago, Illinois, USA

Todd J. Martinez, Department of Chemistry, Stanford University, Stanford, California, USAShaul Mukamel, Department of Chemistry, University of California at Irvine, Irvine,

California, USAJose Onuchic, Department of Physics, Co-Director Center for Theoretical Biological Physics,

University of California at San Diego, La Jolla, California, USASteven Quake, Department of Physics, Stanford University, Stanford, California, USAMark Ratner, Department of Chemistry, Northwestern University, Evanston, Illinois, USADavid Reichmann, Department of Chemistry, Columbia University, New York, New York,

USAGeorge Schatz, Department of Chemistry, Northwestern University, Evanston, Illinois, USANorbert Scherer, Department of Chemistry, James Franck Institute, University of Chicago,

Chicago, Illinois, USASteven J. Sibener, Department of Chemistry, James Franck Institute, University of Chicago,

Chicago, Illinois, USAAndrei Tokmakoff, Department of Chemistry, Massachusetts Institute of Technology,

Cambridge, Massachusetts, USADonald G. Truhlar, Department of Chemistry, University of Minnesota, Minneapolis,

Minnesota, USAJohn C. Tully, Department of Chemistry, Yale University, New Haven, Connecticut, USA

ADVANCES INCHEMICAL PHYSICS

VOLUME 150

Edited bySTUART A. RICE

Department of Chemistryand

The James Franck InstituteThe University of Chicago

Chicago, Illinois

AARON R. DINNER

Department of Chemistryand

The James Franck InstituteThe University of Chicago

Chicago, Illinois

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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10 9 8 7 6 5 4 3 2 1

CONTRIBUTORS TO VOLUME 150

Mark A. Berg, Department of Chemistry and Biochemistry, University of SouthCarolina, Columbia, SC 29208, USA

Bernard R. Brooks, Laboratory of Computational Biology, National Heart,Lung, and Blood Institute (NHLBI), National Institutes of Health (NIH),5635 Fishers Lane, Bethesda, MD 20892-9314, USA

Ana Damjanovic, Laboratory of Computational Biology, National Heart, Lung,and Blood Institute (NHLBI), National Institutes of Health (NIH), 5635 Fish-ers Lane, Bethesda, MD 20892-9314, USA; Department of Biophysics, JohnsHopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA

Igor Goychuk, Institute of Physics, University of Augsburg, Universitätsstr. 1,D-86135 Augsburg, Germany

Taekjip Ha, Department of Physics and the Center for the Physics of Living Cells,University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; HowardHughes Medical Institute, Urbana, IL 61801, USA

Kyung Suk Lee, Department of Physics and the Center for the Physics of LivingCells, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Szilvia Pothoczki, Grup de Caracterització de Materials, Departament de Física iEnginyeria Nuclear, ETSEIB, Universitat Politècnica de Catalunya, Diagonal647, 08028 Barcelona, Catalonia, Spain

László Pusztai, Research Institute for Solid State Physics and Optics, HungarianAcademy of Sciences (RISSPO HAS), H-1121 Budapest, Konkoly Thege út29-33, Hungary

Kousik Samanta, Department of Chemistry, Texas A&M University, College Sta-tion, TX 77843, USA; Department of Chemistry, Rice University, Houston,TX 77005, USA

László Temleitner, Japan Synchrotron Radiation Research Institute (SPring-8/JASRI), 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5198, Japan

Reza Vafabakhsh, Department of Physics and the Center for the Physics ofLiving Cells, University of Illinois at Urbana-Champaign, Urbana, IL 61801,USA

v

vi contributors to volume 150

Xiongwu Wu, Laboratory of Computational Biology, National Heart, Lung, andBlood Institute (NHLBI), National Institutes of Health (NIH), 5635 FishersLane, Bethesda, MD 20892-9314, USA

Danny L. Yeager, Department of Chemistry, Texas A&M University, CollegeStation, TX 77843, USA

PREFACE TO THE SERIES

Advances in science often involve initial development of individual specializedfields of study within traditional disciplines, followed by broadening and over-lapping, or even merging, of those specialized fields, leading to a blurring of thelines between traditional disciplines. The pace of that blurring has accelerated inthe last few decades, and much of the important and exciting research carried outtoday seeks to synthesize elements from different fields of knowledge. Examplesof such research areas include biophysics and studies of nanostructured materials.As the study of the forces that govern the structure and dynamics of molecularsystems, chemical physics encompasses these and many other emerging researchdirections. Unfortunately, the flood of scientific literature has been accompaniedby losses in the shared vocabulary and approaches of the traditional disciplines,and there is much pressure from scientific journals to be ever more concise in thedescriptions of studies, to the point that much valuable experience, if recorded atall, is hidden in supplements and dissipated with time. These trends in scienceand publishing make this series, Advances in Chemical Physics, a much neededresource.

The Advances in Chemical Physics is devoted to helping the reader obtain gen-eral information about a wide variety of topics in chemical physics, a field that weinterpret very broadly. Our intent is to have experts present comprehensive anal-yses of subjects of interest and to encourage the expression of individual pointsof view. We hope that this approach to the presentation of an overview of a sub-ject will both stimulate new research and serve as a personalized learning text forbeginners in a field.

Stuart A. Rice

Aaron R. Dinner

vii

CONTENTS

1Multidimensional Incoherent Time-Resolved Spectroscopy andComplex Kinetics

By Mark A. Berg

103Complex Multiconfigurational Self-Consistent Field-BasedMethods to Investigate Electron-Atom/Molecule ScatteringResonances

By Kousik Samanta and Danny L. Yeager

143Determination of Molecular Orientational Correlationsin Disordered Systems from Diffraction Data

By Szilvia Pothoczki, László Temleitner, and László Pusztai

169Recent Advances in Studying Mechanical Properties of DNA

By Reza Vafabakhsh, Kyung Suk Lee, and Taekjip Ha

187Viscoelastic Subdiffusion: Generalized LangevinEquation Approach

By Igor Goychuk

255Efficient and Unbiased Sampling of BiomolecularSystems in the Canonical Ensemble: A Review ofSelf-Guided Langevin Dynamics

By Xiongwu Wu, Ana Damjanovic, and Bernard R. Brooks

327Author Index

345Subject Index

ix

MULTIDIMENSIONAL INCOHERENTTIME-RESOLVED SPECTROSCOPY AND

COMPLEX KINETICS

MARK A. BERG

Department of Chemistry and Biochemistry, University of South Carolina,Columbia, SC 29208, USA

CONTENTS

I. IntroductionA. Multidimensional Kinetics Versus Multidimensional Coherent SpectroscopyB. MUPPETS Approach to Multidimensional KineticsC. Overview

II. Multidimensional Correlation FunctionsA. Frequency Correlation Functions in Coherent Spectroscopy

1. Spectral Line Broadening and Motional Narrowing2. 2D Coherent Echoes: Separating Homogeneous and Inhomogeneous Line

Broadening3. Stimulated Echo Spectroscopy: Measuring Spectral Diffusion

B. Rate Correlation Functions in Incoherent Spectroscopy1. Rate Dispersion and Motional Narrowing of Rates2. 2D Rate “Echoes”: Separating Homogeneous and Heterogeneous Rate Dispersion3. Rate “Stimulated Echoes”: Measuring Exchange Times

C. Equilibrium Versus Nonequilibrium ProcessesD. Rate Cross-Correlation Functions

III. Spectral Representations of Kinetic DataA. One-Dimensional Rate Spectra

1. Time Decays Versus Rate Spectra2. Uniqueness of Rate Spectra3. Homogeneous–Heterogeneous and Similarity Approximations4. Differences Between Rate and Frequency Spectra

B. Representations of Multidimensional Kinetic Data1. Time–Time Representations: Rate Filtering2. Time–Rate Representations: Rate Hole-Burning Spectra

Advances in Chemical Physics, Volume 150, First Edition. Edited by Stuart A. Rice andAaron R. Dinner.© 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

1

2 mark a. berg

3. Rate–Rate Representations: Rate Correlation SpectraIV. Theory of MUPPETS Measurements

A. General Incoherent Experiments1. Hilbert Spaces: Incoherent, Coherent, and Liouville2. Defining the Incoherent Hilbert Space3. Perturbation Pathway Description of MUPPETS

B. Optical MUPPETS1. Basis Sets for Optical States2. Optical Transition and Detection Operators3. Two-Level Systems and Effective Transition Operators4. Visualizing Complex States and Signals5. Evaluating Pathways for Optical MUPPETS

C. Examples of New Theoretical Results1. Magic Angles and Orientational Gratings in 2D2. Thermal Gratings: Pathway Cancellation3. Enantiometric Pathways and Incoherent Entangled States4. MUPPETS “Stimulated Echoes”

V. Experimental Implementation of MUPPETSA. Optical Design Criteria

1. Current MUPPETS Design2. Why Transient Gratings?3. Why Heterodyne Detection?4. Why Diffractive Optics?5. Why Lenses?

B. Heterodyne Detection with Two Detectors1. Symmetric Detection with Automatic Phase Calibration2. Differential Detection to Eliminate Third-Order Bleaching Signals

VI. Experimental Examples of Analyzing MUPPETS DataA. Electron Trapping in CdSe Nanoparticles: Graphical AnalysisB. Reaction Dynamics in an Ionic Liquid: Model FittingC. Dye Mixtures: Multicomponent Inversion

VII. Future of MUPPETSAcknowledgmentsReferences

I. INTRODUCTION

A. Multidimensional Kinetics Versus MultidimensionalCoherent Spectroscopy

Chemical kinetics dates back to the start of quantitative measurement in chem-istry. From our current perspective, those measurements were both incoherent andone dimensional (1D). They are one dimensional in the sense that a single pertur-bation is applied to the system, followed by a single period of evolution, beforethe final state is measured. Coherent time-resolved spectroscopy started with thediscovery of the spin echo in 1950 [1]. This field has since evolved into a powerful

muppets and complex kinetics 3

array of techniques. They are applicable in the fields of NMR [2], electronic [3–5],Raman [6], and infrared [7–10] spectroscopies and have been used to extract manydifferent structural and dynamic properties from many different systems. The sur-prising abilities of these methods are often attributed to the properties of quantummechanical coherence. However, in addition to exploiting coherence, these tech-niques are also multidimensional: more than one excitation is used to prepare thesystem, and there are multiple periods of evolution before detection. Yet whilethe field of multidimensional coherent spectroscopy has expanded vigorously,interest in multidimensional measurements of incoherent states has been onlysporadic.

This chapter will review a research program to define and develop the potentialof multidimensional incoherent spectroscopy [11–20]. In principle, one mightretrace the development of 1D kinetics by first developing multidimensionalmeasurements of slow reactions, for example, by working on multidimensionalstopped-flow methods. In fact, the theoretical and experimental methods devel-oped for coherent spectroscopy are very powerful, and we have relied heavily onborrowing methods and ideas from that field. As a result, it has been easier tostart by developing the multidimensional version of ultrafast kinetics. However,there is no fundamental barrier to extending these methods to longer timescalesfor slower process. Our approach to multidimensional kinetics has been namedmultiple population period transient spectroscopy (MUPPETS) to recognize itsconnections to multidimensional coherent spectroscopy.

The analogy between coherent and incoherent spectroscopies can be seen in asimple way. Coherent time evolution is described by factors of e±iωt , where ω isa transition frequency, whereas incoherent time evolution is describe by factors ofe−kt , where k is a decay rate. Wherever a property of a spectral transition can bemeasured by coherent spectroscopy, there is an analogous property of rates that canbe measured by an incoherent experiment. The first coherent spectroscopy, the spinecho [1], measured a homogeneous line shape within an inhomogeneously broad-ened line. In kinetics, the analogue of a broadened spectral line is a nonexponentialdecay.

Exponential kinetics and a single rate constant are easily justified for elemen-tary unimolecular processes. However, nonexponential kinetics are increasinglycommon as the material examined becomes more complex [21–27]. Polymers,supercooled liquids and glasses [28–32], and biomolecules [33, 34] are classicexamples of systems with nonexponential relaxation; nanoparticles [35, 36] andionic liquids [37] are more recent ones. A nonexponential decay appears to havemultiple rate constants and so is also called rate dispersion. One possible explana-tion is rate heterogeneity: each molecule in the sample has an exponential decay,but different molecules have different rate constants. Often, one can propose analternative mechanism in which every molecule has a nonexponential decay—inother words, homogeneous rate dispersion.

4 mark a. berg

One-dimensional kinetics cannot distinguish between these mechanisms, justas 1D coherent spectroscopy cannot distinguish between homogeneous and in-homogeneous line broadening mechanisms. However, a 2D kinetics experimentcan measure a homogeneous decay within a system with rate heterogeneity, justas a 2D coherent experiments can measure a homogeneous line shape within aninhomogeneous band. Thus, the difference between coherent and incoherent spec-troscopies is whether frequencies or rates are measured. The ability to detect het-erogeneity is a property of a multidimensional experiment, whether coherent orincoherent.

The comparison to coherent spectroscopy suggests that multidimensional ki-netics can be of both intellectual interest and practical utility. A further comparisonindicates some of the challenges of turning these concepts into robust experiments.Two-dimensional (2D) coherent optical spectroscopy is generally a χ(3) process.Although experiments can be done with only two input beams, to reach its fullpotential, a four-beam experiment is needed. In comparison, a 2D kinetics experi-ment is a χ(5) process. It can be done with as few as three beams, but to reach its fullpotential, six beams are needed. Three-dimensional (3D) kinetics experiments arealso attractive, and they would be χ(7) processes requiring up to eight beams. Thedifficulty with such high-order experiments is partly the small size of the signals,but just as important is the complexity of building and maintaining the necessaryoptical apparatus.

Experiments using six or more optical beams have been performed previously[38–43] but have a reputation for being heroic experiments aimed at specific, high-value questions. In contrast, kinetics with rate dispersion is a broad issue coveringdiverse systems, processes, and timescales. Thus, it is important to develop ex-perimental methods that do not work on just one system but that are robust andadaptable to many problems.

Developing MUPPETS has required simultaneous progress along severalfronts: the concepts of homogeneous and heterogeneous rates needed to be re-fined, a general theory of incoherent spectroscopy in multiple dimensions had tobe devised, experimental methods that are practical on a broad array of systemshad to be developed, and methods to quantitatively analyze the results had to beimplemented. Much remains to be done in each of these areas. The completedstudies focus on distinguishing homogeneous and heterogeneous contributions tothe electronic relaxation of two-state systems—the incoherent analogue of thespin-echo experiment, the simplest multidimensional coherent experiment.

However, the theory developed for MUPPETS makes it clear that a muchbroader array of possibilities exists, given a reasonable amount of ingenuity inadvancing the experimental methods. Heterogeneity in molecular reorientation,energy transport, and molecular diffusion can be measured by straightforward ex-tensions of existing MUPPETS methods. These fundamental molecular processes

muppets and complex kinetics 5

can be related to other processes using suitable probe molecules. When coherentexperiments are extended to multilevel systems, they generate cross-correlationspectra between different transitions. Analogous rate cross-correlation spectra arepossible in kinetic schemes with multiple transitions. A 3D MUPPETS analogue ofthe coherent stimulated echo experiment has been predicted to measure exchangebetween different rate subensembles. With more experimental creativity, almostany process measurable by 1D kinetics can be the object of a multidimensionalkinetics experiment. Coherent experiments are confined to ultrafast timescalesby the speed of coherent decay. However, with a suitable change in experimentaltechnology, MUPPETS should be applicable on any timescale on which kineticsoccur. Thus, the purpose of this chapter is not to document an established field,but to outline the potential for future expansion of an emerging technique.

B. MUPPETS Approach to Multidimensional Kinetics

The concept of spectroscopy using multiple excitation pulses is so broad, it isnot surprising that it has been introduced many times in different contexts. Theconcept of multiple excitations to deal with rate heterogeneity was introducedby Frauenfelder in the context of CO recombination on nanosecond and longertimes [44, 45]. This work was followed up by Post [46, 47] on long timescalesand by Dlott [48] and Champion [49, 50] on the picosecond timescale. At theother extreme of timescales, Ediger used multiple photobleaching experiments tolook for rate heterogeneity near the glass transition on the kilosecond time range[29]. Multiple pulse anisotropy experiments have been demonstrated for lookingat rate heterogeneity in energy transport [51, 52] and to isolate ground-state rota-tional dynamics [53]. Outside the area of optical spectroscopy, nonlinear dielectric[54, 55], mechanical [56, 57], and spin relaxation [58] experiments have lookedfor rate heterogeneity in those processes.

A variety of studies have used multiple excitation pulses in multilevel sys-tems with the goal of accessing states that are difficult to reach in a single transi-tion, rather than looking for rate heterogeneity [59–65]. A substantial literature on“pump–dump–probe” spectroscopy and its variants [66–70] fits in this category.These experiments are based on the same essential concepts as MUPPETS, justexpressed by different experimental methods and applied to different problems.Subject to a few restrictions, any pump–dump–probe experiment should have ananalogous MUPPETS experiment and visa versa.

What distinguishes MUPPETS from these other multidimensional experi-ments? First, many of these previous studies were focused on a particular systemand a particular problem. The connections between the different studies were notobvious, and the ultimate potential for expanding any of them was not clear. Wehave attempted to establish a general theoretical framework for MUPPETS that

6 mark a. berg

is developed from first principles. As a result, the broadest possible range of con-ceivable multidimensional incoherent experiments is defined. Connections can beseen between experiments that appear to be quite different at first.

The second defining property of MUPPETS is that it uses weak excitations.Many other multidimensional experiments rely on strong excitations. A strong-field experiment attempts to come as close as possible to saturating a transition.Multidimensional NMR often uses strong fields, for example, π/2 and π pulses.The obvious advantage is to maximize the size of the nonlinear interaction betweenexcitations. The disadvantage is that the theory to describe these experiments isbased on strongly coupled nonlinear differential equations.

In contrast, MUPPETS is a weak-field approach in which populations arechanged by �10% in each excitation. As a result, perturbation theory can beused, just as it is in optical coherent spectroscopy. The nonlinear problem reducesto a multilinear one: the results can be calculated as a sum of pathways of alter-nating excitation and free evolution, each step of which is a linear problem. Fromthe perspective of a perturbation theory, strong-field methods mix together resultsup to very high orders, complicating the interpretation. The relative simplicity ofweak-field theory allows MUPPETS theory to be very general and also leads torelatively simple methods for designing and interpreting the experiments. The dis-advantage of the weak-field approach is that the desired nonlinear signal is small,both in an absolute sense and in comparison to all lower order processes.

The MUPPETS answer to this problem is to use spatially nonuniform excita-tions, that is, high-order transient gratings. In a grating experiment, each excitationand the detection are performed by a pair of simultaneous pulses of the same fre-quency. One-dimensional transient grating experiments using three or four pulsesare well known [71–75], but their extension to multiple dimensions and morepulses is not. In a grating experiment, phase-matching conditions must be metto create a signal. Again, borrowing an idea from coherent optical spectroscopy,the phase-matching pattern can be designed to eliminate signal from competinglower order processes and to limit the signal to a specific term in the perturbationtheory. Traditional transient grating experiments relied on homodyne detectionof a diffracted probe beam [71–73]. MUPPETS again follows modern opticalcoherent spectroscopy in using heterodyne detection [74, 75] to avoid these prob-lems. In general, the resulting N-dimensional MUPPETS experiment requires 2Ndifferent pulses.

The use of transient gratings leads to other advantages as well. Transient gratingsallow the full range of polarization conditions. High-order orientational gratingsare examples of experiments that cannot be performed with single-beam exci-tations (Section IV.C.1). In addition, rate heterogeneity in spatial diffusion, thatis, anomalous diffusion, can be investigated. Incoherent entanglement (SectionIV.C.3) is another interesting phenomenon that can only be observed in a gratingconfiguration.

muppets and complex kinetics 7

The context of MUPPETS would not be complete without considering its re-lationship to single-molecule spectroscopy (SMS). Single-molecule spectroscopyis widely used to deal with heterogeneous samples, including those with heteroge-neous rates [34, 76–80]. Ideally, single-molecule spectroscopy can characterize thestatistical properties of the instantaneous rate by performing a time average overits equilibrium fluctuations, including measurements of multiple-time correlationfunctions of the rate. MUPPETS is a nonequilibrium measurement that also yieldsmultiple-time correlation functions of the rate. Analogies to linear response theorysuggest a fundamental connection between MUPPETS and SMS, but a rigoroustheory is still lacking. On the practical level, there are certainly differences. As anonequilibrium method, MUPPETS cannot make measurements after the longestliving state of the system. On the other hand, the inherently low rate of photoncollection limits SMS to slowly evolving rates: fundamentally, to times longerthan the longest lifetime of the system; practically, to times longer than 100 �sin most experiments. The low signal levels of SMS also mean that observationsare often limited to high quantum yield states and to photostationary, not trueequilibrium, states.

The MUPPETS measurements on CdSe nanoparticles (Section VI.A) provideseveral clear examples of these differences. Single-particle measurements havefocused on similar core–shell nanoparticles that have a high quantum yield. TheMUPPETS measurements are on bare particles, with quantum yields from 5% to0.005%. The most prominent dynamics seen in SMS are the slow blinking in andout of an unobservable dark state from microseconds to seconds [35]. MUPPETSmeasurements extend from 0.3 ps to 2 ns. It is generally believed that the dark statesseen in SMS are a photoionized state [81]. They are generated in very low yieldbut have a very long lifetime. As a result, they are prominent in the photostationarystate measured in SMS but play no obvious role in the MUPPETS measurements.In many ways, MUPPETS and SMS provide contrasting and complementary in-formation on heterogeneous systems.

C. Overview

MUPPETS is still an emerging field. Broad concepts reach further than detailed the-oretical work, and theoretical predictions reach further than experimental demon-strations. This chapter is organized to reflect these facts, starting with the broaderand more abstract ideas and working toward the narrower and more concrete real-izations of those ideas.

Sections II and III look at general aspects of multidimensional kinetics froma theoretical and an experimental perspective, respectively. Section II developscorrelation functions to describe the multidimensional kinetics of systems withrate dispersion. It uses this perspective to emphasize the analogy between coher-ent and incoherent experiments. Section III looks at the empirical description of

8 mark a. berg

nonexponential data. It focuses on the transformation of time domain kinetic datainto spectral representations, a parallel of the transformation of time domain coher-ent data into frequency spectra. Neither Section II nor III discusses any specificson how the correlation functions or time domain data are to be measured.

Section IV begins to address this question by developing the theory of MUP-PETS measurement. The methods that previously have been used to calculate 1Dexperiments could be extended to multiple dimensions. However, the complex-ity of such calculations is daunting and impedes insight into the design of newexperiments. Section IV develops a new, “Hilbert space” approach to describingmultidimensional incoherent experiments. It results in perturbation pathways thatare essentially similar to the Liouville pathways used to describe coherent exper-iments. The extent to which phenomena in quantum mechanics are mimicked inincoherent processes is surprising: complex states and operators, pathway inter-ference and cancellation, and even entangled states have incoherent analogues.

Section V looks at the problem of designing a practical apparatus to makeMUPPETS measurements. The specifications of our instrument have been doc-umented previously, so this section focuses on the reasoning behind those spec-ifications: what the critical design choices are and why we made the decisionswe did.

Section VI looks at three systems that have been studied in detail with MUP-PETS. These results validate the results of the earlier sections. However, the focusis on approaches to data analysis: How can real MUPPETS data be inverted tothe distribution and decay kinetics of subsets of molecules within a heterogeneoussample?

Throughout this chapter, areas for further improvement and extension of MUP-PETS are pointed out. The chapter concludes in Section VII with several evenlonger range and more speculative ideas on how MUPPETS can be extended andhow it can interact with other approaches to multidimensional kinetics and withother methods for analyzing complex materials.

II. MULTIDIMENSIONAL CORRELATION FUNCTIONS

This section looks at nonexponential kinetics from a molecular perspective by de-riving the time correlation functions involved in multidimensional kinetics [18].These rate correlation functions are closely connected to the frequency correlationfunctions that govern multidimensional coherent spectroscopy, and this connec-tion provides useful interpretations of the incoherent functions. The essentials offrequency correlation functions are reviewed in Section II.A before defining analo-gous rate correlation functions in Section II.B. A significant difference between theprocesses that typically drive frequency and rate correlation functions is discussedin Section II.C. Section II.D briefly considers rate cross-correlation functions,

muppets and complex kinetics 9

which are not as well developed as the autocorrelation functions discussed in theremainder of this section.

A. Frequency Correlation Functions in Coherent Spectroscopy

1. Spectral Line Broadening and Motional Narrowing

The semiclassical theory of coherent spectroscopy is based on the idea of a time-dependent frequency ω(t) [82–84]. If the off-diagonal density-matrix element be-tween states a and b is ρab(t), the frequency of the corresponding transition is

ω(t) = −i

ρab(t)

d

dtρab(t) (1)

In a classical system, ρab(t) is replaced with the appropriate oscillating variable.The states a and b are not eigenstates, and the frequency of the transition is notconstant, due to perturbations of the transition by its bath. In a 1D experiment, thecoherent observable is related to the ensemble-averaged value of ρab(t), which isproportional to a correlation function C(1)

ω (τ) that involves the frequency:

C(1)ω (τ1) = 〈ρab(t1)ρab(t0)〉/ ⟨ρ2

ab(t0)⟩

=⟨

exp(i∫ t1t0

ω(t) dt)⟩ (2)

The initial excitation of the coherence is at time t0, and the final measurement ofthe coherent observable is at time t1. Throughout this chapter, absolute times willbe denoted by ti and time intervals by

τi = ti − ti−1 (3)

An inverse Fourier transform yields the standard frequency spectrum C(1)ω (�).

This spectrum can be defined implicitly by the forward Fourier transform

C(1)ω (τ1) =

∫ ∞

−∞C(1)

ω (�) e−i�τ1 d� (4)

Note that the transformed variable � is not the same as the molecular frequencyω. If C(1)

ω (τ1) is an undamped oscillation, then the frequency spectrum is a deltafunction; decay of C(1)

ω (τ1) represents line broadening. This decay is due to afrequency that is not constant. Thus, the linewidth is determined by the propertiesof ω(t).

The time-dependent frequency ω(t) can be characterized by a range �ω and acharacteristic time for variation Tω, whereas the decay of C(1)

ω (τ1) is characterizedby its half-life τ1/2. If Tω is much longer than τ1/2, the decay of C(1)

ω (τ1) is onlydue to the range of frequencies in the ensemble at t0. The line is inhomogeneously(heterogeneously) broadened.

10 mark a. berg

On the other hand, if ω(t) varies during τ1/2, the integral in Eq. (2) averagesthe frequency and reduces its line broadening effect. The linewidth is less than �ω

and becomes narrower, the faster Tω is. The line is motionally narrowed. Becausethe frequency varies before an observable drop in the coherent can build up, it isimpossible to observe the instantaneous value of ω(t). The line is homogeneous butstill has a width. In the limit Tω → 0, that is, when the frequency retains no memoryof its past, the linewidth approaches zero. The frequency spectrum is broadenedbecause the bath perturbing the frequency has memory for a significant time.

From a measurement of τ1/2, it is impossible to uniquely determine Tω and �ω,or more generally, it is impossible to invert Eq. (2) to obtain ω(t) from C(1)

ω (τ1).Moreover, more than one process may be perturbing the frequency, each with adifferent Tω and �ω. Some processes may be in the homogeneous limit and othersin the heterogeneous limit, but both contribute indistinguishably to the decay ofC(1)

ω (τ1) and to the linewidth. Thus, a 1D coherent experiment cannot separatehomogeneous and inhomogeneous line broadening.

The standard theory of frequency domain line broadening assumes that ω(t)is a stationary stochastic process reflecting only equilibrium fluctuations of thesystem [82–84]. However, it is also possible to consider that the time dependenceof the frequency is wholly or in part due to a nonequilibrium, deterministic processinitiated by the initial excitation of the coherent. A common example is the Stokesshift of a solvating excited state [85]. The analysis above holds even under thisbroader set of circumstances.

2. 2D Coherent Echoes: Separating Homogeneous andInhomogeneous Line Broadening

Distinguishing between homogeneous and inhomogeneous line broadening re-quires a multidimensional measurement. Using two excitations separated by atime τ1 before detection of the coherent observable at a time τ2 after the secondexcitation creates a two-pulse echo experiment. Its 2D correlation function is

C(2)ω (τ2, τ1) =

⟨exp

(−i

∫ t2

t1

ω(t) dt + i

∫ t1

t0

ω(t) dt

)⟩(5)

The important feature of C(2)ω (τ2, τ1) is that the frequency of a single molecule is

sampled over two periods, τ1 and τ2, before ensemble averaging.If all the molecules in the sample behave identically, or more precisely, if the

memory of any difference in their initial frequency ω(t0) is lost between the twosamplings, the two integrals can be performed and averaged separately. In thatcase,

C(2)ω (τ2, τ1) =

⟨exp(−i∫ t2t1

ω(t) dt)⟩⟨

exp(+i∫ t1t0

ω(t) dt)⟩

= C(1)∗ω (τ2)C(1)

ω (τ1)(6)

muppets and complex kinetics 11

and the 2D correlation function becomes the product of two 1D correlation func-tions. This result is characteristic of homogeneous line broadening.

On the other hand, if the frequency of each molecule is constant over the totalduration of the experiment (Tω � τ1 + τ2), then

C(2)ω (τ2, τ1) = 〈exp [iω(t0) (τ1 − τ2)]〉

= C(1)ω (τ1 − τ2)

(7)

The 2D correlation function reduces to a single 1D correlation function. In thiscase, each molecule has a delta function spectral line. Any linewidth is due toaveraging over the ensemble.

Equations (6) and (7) show that characteristically different results occur in thelimits of homogeneous and inhomogeneous line broadening. Thus, the effects offrequency dynamics and the initial distribution of the frequency can be separatedin a 2D experiment. A variety of experimental methods exist for measuring thiscorrelation function [86]. These experiments can determine whether a spectral lineis homogeneously or inhomogeneously broadened, and if both types of broadeningare present, they can separate the two contributions to the linewidth.

3. Stimulated Echo Spectroscopy: Measuring Spectral Diffusion

The 2D echo experiment is generalized by placing the system in a population stateduring a time τ2 in between the two coherent measurements during times τ1 andτ3. The resulting 3D correlation function is

C(3)ω (τ3, τ2, τ1) =

⟨exp

(−i

∫ t3

t2

ω(t) dt + i

∫ t1

t0

ω(t) dt

)⟩(8)

This correlation function describes a stimulated echo experiment [86]. During τ2,molecules can change their frequency, that is, lose memory of their frequencyduring τ1. This process is called spectral diffusion. If the second period is longerthan the spectral diffusion time, τ2 > Tω, the line will appear to be homogeneous

C(3)ω (τ3, τ2, τ1) =

⟨exp(−i∫ t3t2

ω(t) dt)⟩⟨

exp(+i∫ t1t0

ω(t) dt)⟩

= C(1)∗ω (τ3)C(1)

ω (τ1)(9)

[cf. Eq. (6)], even if there is inhomogeneity. By varying τ2, the spectral diffusiontime can be measured.

It now appears that the definition of homogeneity varies with the value of τ2.Other experiments to measure homogeneity have a similar timescale that playsthe role of τ2, even when it is not explicit [87]. The 2D echo [Eq. (5)] is thelimiting case of all experiments in which τ2 → 0. Instantaneous frequencies thatcannot be distinguished in a 2D echo are motionally averaged; they cannot be

12 mark a. berg

distinguished by any experiment. Thus, the 2D echo has the most rigorous criterionfor homogeneity and should be used as the definition of homogeneity.

B. Rate Correlation Functions in Incoherent Spectroscopy

1. Rate Dispersion and Motional Narrowing of Rates

The time-dependent rate of a single molecule k(t) is defined to be similar to thetime-dependent frequency [Eq. (1)], but using a diagonal element of the densitymatrix ρbb(t),

k(t) = −1

ρbb(t)

d

dtρbb(t) (10)

The rate is not constant due to interactions with the environment that retain memoryfor some time. Standard kinetics are 1D measurements of the ensemble-averagedvalue of ρbb(t) and can be related to a correlation function C

(1)k (τ1) that involves

an ensemble average of the single-molecule rate:

C(1)k (τ1) = 〈ρbb(t1)ρbb(t0)〉/ ⟨ρ2

bb(t0)⟩

=⟨

exp(− ∫ t1

t0k(t) dt

)⟩ (11)

Comparing Eqs. (10) and (11) to Eqs. (1) and (2), one can see that the issue ofnonexponential incoherent decay (rate dispersion) is formally identical to the issueof line broadening in coherent spectroscopy.

Defining a Laplace spectrum C(1)k (κ) implicitly through a Laplace transform,

C(1)k (τ1) =

∫ ∞

0C

(1)k (κ) e−κτ1 dτ1 (12)

makes the point clearer [cf. Eq. (4)]. If C(1)k (τ1) decays exponentially, then the

Laplace spectrum is a delta function; nonexponential decays represent line broad-ening of the Laplace spectrum. This decay is due to a rate that is not constant.Thus, the Laplace linewidth is determined by the properties of k(t).

The time-dependant rate k(t) can be characterized by a range �k and a charac-teristic time for variation Tk, whereas the decay of C

(1)k (τ1) is characterized by its

half-life τ1/2. If Tk is much longer than τ1/2, the decay of C(1)k (τ1) is only due to

the range of rates existing in the ensemble at t0. The line is heterogeneously (inho-mogeneously) broadened. The system is assumed to be ergodic; given sufficienttime, every molecule will visit every rate within the ensemble. The case of speciesthat never exchange is taken as the limit of Tk → ∞.

On the other hand, if k(t) varies during τ1/2, the integral in Eq. (11) averages

the rate. The decay of C(1)k (τ1) becomes closer to exponential, and the Laplace

muppets and complex kinetics 13

linewidth is less than �k, the faster Tk is. The Laplace spectrum is motionallynarrowed. Because the rate varies before an observable drop in the population canbuild up, it is impossible to observe the instantaneous value of k(t). The decay ishomogeneous. In the limit Tk → 0, that is, when the rate retains no memory of itspast, the Laplace linewidth approaches zero. The Laplace spectrum is broadenedbecause the bath perturbing the rate has memory for a significant period of time.

This conclusion is familiar from the theory of stochastic processes. An ex-ponentially decaying correlation function (Markov process) is special because itrepresents the limit of an underlying process with no memory. Nonexponential de-cays imply the existence of memory. Similarly, line broadening of Laplace and fre-quency spectra are linked to persistent memory in the underlying time-dependentrate or frequency, respectively.

From a measurement of τ1/2, it is impossible to uniquely determine Tk and �k,

or more generally, it is impossible to invert Eq. (11) to obtain k(t) from C(1)k (τ1).

Moreover, more than one process may be perturbing the rate, each with a differentTk and �k. Some may be in the homogeneous limit and others in the heterogeneouslimit, but both contribute indistinguishably to the rate dispersion of the decay ofC

(1)k (τ1) and to the Laplace linewidth. Thus, a 1D kinetics experiment cannot

distinguish between homogeneous and heterogeneous causes of rate dispersion.

2. 2D Rate “Echoes”: Separating Homogeneous and HeterogeneousRate Dispersion

To develop a 2D approach to kinetics analogous to 2D coherent spectroscopy(Section II.A), we need to ask for a joint probability: What is the probability of anexcited molecule surviving a time period τ2 if it has already survived a period τ1?Averaging over the ensemble of different molecules gives a 2D rate correlationfunction defined as

C(2)k (τ2, τ1) = 〈ρbb(t2)ρbb(t1)ρbb(t1)ρbb(t0)〉/ ⟨ρ2

bb(t1)ρ2bb(t0)

⟩=⟨

exp(− ∫ t2

t1k(t) dt − ∫ t1

t0k(t) dt

)⟩ (13)

This correlation function requires that a molecule be in the perturbed state b at timet0 and survive until time t1, a measurement of the decay rate. The measurementis immediately repeated; the existence of the molecule in state b at time t1 isreestablished, and the survival to time t2 is measured. The key feature is that bothmeasurements occur on the same molecule. The joint probabilities from individualmolecules are then averaged over the ensemble. This correlation function is ananalogue of the coherent echo correlation function [Eq. (5)] under the substitution±iω → −k. Section IV.A.3 will show that it is measurable in experiments.

If all the molecules behave identically, or, more precisely, if any memory ofany difference in their initial rate k(t0) is rapidly lost, then the ensemble average

14 mark a. berg

over each period can be done separately:

C(2)k (τ1, τ2) =

⟨exp(− ∫ t2

t1k(t) dt

)⟩⟨exp(−i∫ t1t0

k(t) dt)⟩

= C(1)k (τ2)C(1)

k (τ1)(14)

The 2D correlation function is a product of 1D correlation functions. This is thecase of no heterogeneity. Any rate dispersion is due to the time dependence of thesingle-molecule rate k(t). The result is analogous to the result for homogeneousline broadening [Eq. (6)].

On the other hand, if the rate of each molecule is constant over the total durationof the experiment (Tk � τ1 + τ2), then

C(2)k (τ1, τ2) = 〈exp [−k(0) (τ1 + τ2)]〉

= C(1)k (τ1 + τ2)

(15)

The 2D correlation function reduces to a single 1D correlation function. This isthe case of an exponential single-molecule decay. Any rate dispersion is only dueto averaging over a heterogeneous ensemble.

The 2D rate correlation function is characteristically different for decays that arenonexponential for homogeneous [Eq. (14)] or heterogeneous [Eq. (15)] reasons.A 2D measurement can classify the mechanism when there is a single mechanism,or separate the contributions of homogeneous and heterogeneous mechanismswhen there is more than one mechanism. Equation (15) is analogous to Eq. (7) forinhomogeneous line broadening, except for the sign used in combining the times.Although the signal never peaks at a nonzero time, and there is no true echo, inother regards, a 2D kinetics measurement is analogous to a 2D coherent echo.

If there is no discrimination between homogeneous and heterogeneous mech-anisms, that is, if

C(1)k (τ1 + τ2) = C

(1)k (τ2)C(1)

k (τ1) (16)

then there is no rate dispersion, that is, the 1D decay C(1)k (τ1) is an exponential.

This case corresponds to a Laplace line that has no width, and thus no broadeningmechanism to measure.

3. Rate “Stimulated Echoes”: Measuring Exchange Times

The 2D rate correlation function in Eq. (13) can be viewed as a limiting case of afull 3D correlation function:

C(3)k (τ3, τ2, τ1) = 〈ρbb(t3)ρbb(t2)ρbb(t1)ρbb(t0)〉/ ⟨ρ2

bb(t2)ρ2bb(t0)

⟩=⟨

exp(− ∫ t3

t2k(t) dt − ∫ t1

t0k(t) dt

)⟩ (17)

muppets and complex kinetics 15

[cf. Eq. (8)]. Here, a variable time τ2 is allowed between the end of the first ratemeasurement, which is made during τ1, and the second rate measurement, whichis made during τ3. During τ2, it is possible for molecules to exchange betweendifferent rate subensembles. If this period is longer than the characteristic time forexchange, τ2 > Tk, the results will appear to be homogeneous,

C(3)k (τ3, τ2, τ1) =

⟨exp(− ∫ t3

t2k(t) dt

)⟩⟨exp(−i∫ t1t0

k(t) dt)⟩

= C(1)k (τ3)C(1)

k (τ1)(18)

[cf. Eq. (14)], even if there is heterogeneity. By varying τ2, the exchange timecan be measured. These properties are analogous to the coherent stimulated echo[Eq. (9)], with exchange between rate subensembles playing the role of spectraldiffusion.

The 2D rate correlation function [Eq. (13)] is the τ2 → 0 limit of the 3D corre-lation function [Eq. (17)]. Just as the 2D coherent echo provides the most rigorousand correct definition of frequency homogeneity, the 2D rate correlation functiongives the most rigorous and correct definition of rate heterogeneity.

Confusion can result if the roles of the exchange time Tk and the primaryrate k are not clearly distinguished. The case of permanent heterogeneity, such aswith different chemical species, represents the limit of an infinite Tk. When Tk isfinite, the important timescale of an experiment is how long it takes to measurek(t) compared to Tk. For example, a single-molecule experiment can measure ananosecond fluorescence decay time, but may require 1 ms to collect the photonsneeded to make that measurement. If Tk is faster than this time resolution, theheterogeneity will not be resolved, and the apparently “homogeneous” decay willbe nonexponential due to the unresolved heterogeneity. Defining homogeneousand heterogeneous rate dispersion on this basis is dependent on the experiment,not on the fundamental properties of the system.

The important comparison is with a time inherent to the sample: Is the exchangetime Tk faster than the primary decay time τ1/2? If so, the differences in the instan-taneous rate k(t) average before there is a significant change in population, andk(t) cannot be observed by any experiment. The effect is analogous to motionalnarrowing in coherent spectroscopy. In a 2D MUPPETS experiment, the measure-ment is made as rapidly as allowed by the primary rate. If a decay is not seen asheterogeneous in a MUPPETS experiment, it will not appear to be heterogeneousin any experiment.

The coherent stimulated echo relies on storing the information on the coherentdecay gained during τ1 in a population grating during τ2 [88]. There is no generalanalogue for this information exchange that applies to incoherent processes. How-ever, under certain specific circumstances, it is possible to exchange informationbetween two incoherent coordinates, resulting in the same effect (Section IV.C.4).

16 mark a. berg

Thus, it is possible to measure a rate “stimulated echo” and thereby measure therate exchange time Tk.

C. Equilibrium Versus Nonequilibrium Processes

The rate correlation functions developed in Section II.B provide a definition ofhomogeneous and heterogeneous rate dispersion in terms of a time-dependent ratek(t). More consideration must be given to why the rate is time dependent. It wasstated that the bath has to have a memory. In the language of stochastic processes,the bath must contain a “hidden variable” whose time dependence constitutes thememory of this system. The hidden variable can be one of two types: a variabledescribing equilibrium fluctuations in the ground state of the system θ(t) or oneundergoing nonequilibrium motion initiated by the initial excitation of the systemϕ(t). Thus, the time-dependent rate can be written more explicitly as k(θ(t), ϕ(t)).

Examples of equilibrium fluctuations are illustrated in Fig. 1a and b. Figure1a shows a system whose excited-state relaxation rate depends on a continuousbath variable θ(t). This variable might represent the local solvent configuration,the conformation of a protein containing the reacting system or an internal twistof the reacting molecule itself. The important feature is that there is an equilib-rium distribution of this variable before the excited state is created. Because this

Figure 1. Several ki-netic schemes illustrate howhomogeneous rate dispersioncan arise. An electronic tran-sition (dashed arrow) excitesthe system, and the recoveryof the ground state (0) is mea-sured. Solid arrows of differ-ent thickness indicate relax-ation at different rates. Grayboxes indicate kinetic states.(a and b) Stochastic fluctua-tions in the ground state causeheterogeneous rate dispersion.(c–h) Deterministic dynamicsin the excited state cause ho-mogeneous rate dispersion.

muppets and complex kinetics 17

distribution is in equilibrium, movement within it is stochastic with a characteristicexchange time Tk. If this time is long relative to the excited-state relaxation rate,Tkk(θ) � 1, there will be a heterogeneous distribution of decay rates. Figure 1bshows two ground-state conformers with different relaxation rates, ka and kb. Inthis example, the bath variable is discrete, θ(t) ∈ {a, b}. Still, if the exchange timeTk is long, heterogeneous rate dispersion results.

If the exchange time in either of these cases is fast, the observed decay ishomogeneous, but it is also exponential. This type of homogeneity is not a concernfor MUPPETS. There is an intermediate regime, Tkk ∼ 1, where the 1D decayis initially nonexponential and then switches to exponential at long times. Thisregime is intermediate between homogeneous and heterogeneous. Thus, ground-state fluctuations can cause heterogeneous or intermediate rate dispersion but nothomogeneous rate dispersion.

Homogeneous rate dispersion comes from nonequilibrium dynamics of the bath.Several examples are shown in Fig. 1c–g. In Fig. 1c, a continuous bath variable ϕ(t)controls the rate of excited-state relaxation k(ϕ). The temperature is assumed to below, so that the molecule is confined to a narrow range of ϕ(t) in the ground state.In the excited state, the equilibrium position of ϕ(t) changes, so both ϕ(t) and k(ϕ)evolve deterministically after excitation with a characteristic time Tk = k1. If thistime is either very slow or very fast, Tkk(ϕ) � 1 or Tkk(ϕ) 1, relaxation occursfrom only one configuration and with only one rate, either k(ϕ(0)) or k(ϕ(∞)),respectively. Rate dispersion only occurs in the intermediate range Tkk(ϕ) ∼ 1.

This situation can be analyzed by considering a converse of the 2D correlationfunction defined in Eq. (13):

C(2)k (τ2, τ1) =

⟨exp

(−∫ t2

t1

k(t) dt

)[1 − exp

(−∫ t1

t0

k(t) dt

)]⟩(19)

whereas C(2)k (τ2, τ1) asks what is the probability of a molecule surviving τ2 if it

survives τ1, C(2)k (τ2, τ1) asks what is the probability of a molecule surviving τ2 if it

does not survive τ1. In either case, the molecule is reestablished in the excited stateat the start of τ2, implying a second excitation in the case of C

(2)k (τ2, τ1). Whether

the condition for homogeneity [Eq. (14)] is satisfied or not depends on whether therelaxation in a second excitation–relaxation cycle is influenced by the relaxationtime during an initial excitation–relaxation cycle. If the ground state reequilibratesrapidly after relaxation from the excited state (Fig. 1c), the answer is no, andthe process is homogeneous. Thus, nonequilibrium dynamics can cause homoge-neous rate dispersion. An experimental example similar to Fig. 1c will be seen inSection VI.B.

A similar analysis shows that a variety of multilevel kinetic schemes can causehomogeneous dispersion. Figure 1d shows a case where an excited-state confor-mational change must occur before relaxation to the ground state. In this case, the

18 mark a. berg

states are discrete, and the hidden variable can be taken as the ratio of the twopopulations: ϕ(t) = Pa(t)/Pb(t).

In Fig. 1c, the ground-state recovery rate k(t) is initially low and increases withtime. The signal will show an induction period. It is concave downward on a log–linear plot, is steeper than an exponential on a linear–log plot, and has negativecomponents in a rate spectrum (Fig. 2d). As the rate of internal conversion normallyincreases as the energy gap decreases, Fig. 1d would have the same behavior.Heterogeneity cannot produce any of these features, although it can mask them.An experimental example will be seen in Section VI.B.

In Fig. 1, parts (f) and (g) are similar to (c) and (d) but show opposite behaviorsof k(t). In Fig. 1f, stabilization of state 1 with time increases the barrier height tothe relaxing state 2, and thus the recovery rate decreases with time. In Fig. 1g, if k1,the equilibration rate of the bright state 1 with the dark state 2, is faster than k2, therelaxation rate from state 1 to state 0, then the net ground-state recovery rate willslow down with time. In Fig. 1e, an example is shown with a nonmonotonic k(t).There is an induction period while the initially excited state 1 branches into states2 and 3. The instantaneous rate rises as states 2 and 3 are populated and then dropsas the faster relaxing state 2 is depopulated (see Section VI.B for an example).

Each of these cases involves several eigenstates of the system and more thanone relaxation process. Each relaxation may be exponential individually. However,a complete characterization of the population of each eigenstate is often not exper-imentally possible. In these examples, the different excited states or ground statesare not spectroscopically distinguishable. The experimental observable combinesthe eigenstates into effective kinetic states, as shown by the gray boxes in Fig. 1.This grouping creates the hidden variable, and dynamics within the kinetic statecreates the memory of this variable. The formal reduction from eigenstates tokinetic states is described in Section IV.B.3.

D. Rate Cross-Correlation Functions

The examples in Fig. 1a–g all reduce to two kinetic states connected by a singleoptical transition. Single-wavelength experiments are sufficient for these systems.Figure 1h is an example with multiple optical transitions—in this case, betweensingly and doubly excited states. Such systems must be described by multiple ki-netic states, and experiments with multiple wavelengths may be able to address thetwo transitions separately. Pump–dump–probe experiments have begun to explorethese possibilities [66–70]. Even in the absence of rate heterogeneity, these ex-periments have been used to access kinetic transitions that would be unobservablewith a single excitation, much as pump–dump experiments in the gas phase canaccess spectroscopic transitions that are otherwise inaccessible.

For multilevel systems with rate heterogeneity, the rate correlation functionneeds to be considered. The 2D and 3D correlation functions given above need to be