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Doctoral Thesis

Investigation of intramolecular vibrational energy flow inpolyatomic molecules by the femtosecond pump-probetechnique

Author(s): Kushnarenko, Alexander

Publication Date: 2013

Permanent Link: https://doi.org/10.3929/ethz-a-010108536

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Diss. ETH No. 21307

Investigation of intramolecular vibrationalenergy flow in polyatomic molecules by the

femtosecond pump-probe technique

A thesis submitted to attain the degree of

Doctor of Sciences of ETH Zurich

(Dr. sc. ETH Zurich)

presented by

Alexander Kushnarenko

Master of Physics,

Saint-Petersburg State University

born on 27.09.1981

citizen of the Russian Federation

accepted on the recommendation of

Prof. Dr. Dr. h.c. Martin Quack, examiner

Prof. Dr. Hans Jakob Wörner, co-examiner

2013

Светлой памяти Крылова Виталия Николаевича (1947–2009)

In memory of Vitaly Krylov (1947–2009)

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Contents

Abstract ix

Zusammenfassung xiii

1 Introduction 1

1.1 Historical perspective and the role of IVR in reaction kinetics . . . 1

1.2 Theoretical approaches to unimolecular reaction dynamics . . . . 3

1.2.1 General classical mechanical model . . . . . . . . . . . . . . 3

1.2.2 Quantum dynamical treatment of intramolecular motion . . 5

1.2.3 The Rice-Ramsperger-Kassel-Marcus model . . . . . . . . . 5

1.2.4 Statistical adiabatic channel model . . . . . . . . . . . . . . 6

1.3 Evidence of IVR in multiphoton excitation and dissociation . . . . 7

1.4 Timescales of intramolecular processes . . . . . . . . . . . . . . . . 8

1.5 Intramolecular quantum dynamics and Schrödinger equation . . . 12

1.6 Investigation of IVR . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7 Motivation and survey of the present thesis . . . . . . . . . . . . . 18

2 Intramolecular vibrational energy redistribution 21

2.1 Effective hamiltonian and IVR . . . . . . . . . . . . . . . . . . . . . 222.1.1 Effective hamiltonian . . . . . . . . . . . . . . . . . . . . . . 222.1.2 Hierarchy of states . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.3 Structure of the effective hamiltonian . . . . . . . . . . . . . 262.1.4 Zero-order states . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.5 First-order states . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Dynamics of perturbed zero-order states . . . . . . . . . . . . . . . 32

2.3 Rovibrational dynamics . . . . . . . . . . . . . . . . . . . . . . . . 34

Diss. ETH 21307 v

Contents

2.4 Statistical modeling of the dynamics . . . . . . . . . . . . . . . . . 38

2.5 Reaction rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5.1 Three state kinetic model . . . . . . . . . . . . . . . . . . . . 44

2.5.2 Parallel coupling . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5.3 Collisional deactivation . . . . . . . . . . . . . . . . . . . . . 53

2.6 Vibrational temperature of a selected mode . . . . . . . . . . . . . 56

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Femtosecond pump-probe experiments 59

3.1 Pulsed femtosecond radiation . . . . . . . . . . . . . . . . . . . . . 60

3.1.1 Gaussian beam . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1.2 Bandwidth-limited pulses . . . . . . . . . . . . . . . . . . . . 62

3.2 Temporally and spectrally resolved detection . . . . . . . . . . . . 65

3.2.1 Pump-probe efficiency . . . . . . . . . . . . . . . . . . . . . . 653.2.2 Temporal resolution in a pump-probe experiment . . . . . . 67

3.2.3 Spectral resolution in pump-probe experiment . . . . . . . . 71

3.2.4 Pulse distortion due to absorbing medium . . . . . . . . . . 72

3.3 Probing IVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.1 Probing with IR-pulses . . . . . . . . . . . . . . . . . . . . . 75

3.3.2 Probing with UV-pulses . . . . . . . . . . . . . . . . . . . . . 77

3.4 Nonadiabatic molecular alignment . . . . . . . . . . . . . . . . . . 81

3.4.1 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4.2 Recurrence time and rotational constants . . . . . . . . . . . 83

3.4.3 Improving the temporal resolution . . . . . . . . . . . . . . . 89

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Hollow waveguides 93

4.1 Monochromatic radiation in hollow waveguides . . . . . . . . . . 95

4.1.1 Cylindrical hollow waveguide . . . . . . . . . . . . . . . . . 95

4.1.2 Bending of a hollow waveguide . . . . . . . . . . . . . . . . . 99

4.1.3 Hollow waveguides with finite walls . . . . . . . . . . . . . . 102

4.2 Transmission of femtosecond laser pulses . . . . . . . . . . . . . . 108

4.2.1 Dispersion of femtosecond laser pulses in a hollow waveguide109

vi — draft compiled 20.03.2014 19:43 — A. Kushnarenko

Contents

4.2.2 Use of hollow waveguide in femtosecond pump-probe

experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5 Experimental setup 119

5.1 Femtosecond laser system . . . . . . . . . . . . . . . . . . . . . . . 119

5.1.1 Generation and amplification of femtosecond pulses . . . . 120

5.1.2 Parametric generation of frequency tunable femtosecond

pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Pump-probe signal detection . . . . . . . . . . . . . . . . . . . . . 122

5.2.1 Optical delay line . . . . . . . . . . . . . . . . . . . . . . . . 124

5.2.2 Sample cell with a hollow waveguide . . . . . . . . . . . . . 124

5.2.3 Reference beam . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2.4 Polychromator . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2.5 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2.6 Data transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3 Characterisation of femtosecond laser radiation . . . . . . . . . . . 127

5.3.1 Spatial beam profile . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.2 Pulse duration . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.3.3 Radiation spectrum . . . . . . . . . . . . . . . . . . . . . . . 129

5.4 Software to operate the experiment . . . . . . . . . . . . . . . . . . 130

5.4.1 Software architecture . . . . . . . . . . . . . . . . . . . . . . 130

5.4.2 Software usage . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.5 Samples and materials . . . . . . . . . . . . . . . . . . . . . . . . . 133

6 Experimental results 135

6.1 Trifluoromethane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2 Iodomethanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.2.1 CH3I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.2.2 CHD2I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.2.3 CH2DI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.3 Other molecules with a single CH-group . . . . . . . . . . . . . . . 152

6.3.1 CHD3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Diss. ETH 21307 — draft compiled 20.03.2014 19:43 — vii

Contents

6.3.2 CHD2F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.3.3 CF3CHFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.3.4 CHBrClF and CHBrFI . . . . . . . . . . . . . . . . . . . . . . 156

6.4 Acetylenes with a single CH chromophore . . . . . . . . . . . . . . 157

6.4.1 General aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.4.2 Cyanoacetylene . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.4.3 Propyne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.4.4 Trifluoropropyne . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.5 Bichromophoric propargyl halides . . . . . . . . . . . . . . . . . . 164

6.6 Towards statistical IVR in terminal acetylenes . . . . . . . . . . . . 170

6.7 Benzene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7 Conclusions and outlook 179

A Source codes 183

A.1 Rotational coherence effects . . . . . . . . . . . . . . . . . . . . . . 183

B Pump-probe signals 195

C Normal modes 207

Nomenclature 219

List of figures 231

List of tables 235

Bibliography 237

Scientific publications 273

Other scientific contributions 275

Acknowledgments 281

viii — draft compiled 20.03.2014 19:43 — A. Kushnarenko

Abstract

The present studies are devoted to the investigation of the intramolecular vibra-

tional energy redistribution (IVR), the processes of transfer of energy from some

initially excited vibrational state to other nearly isoenergetic states. IVR with a

great variety of redistribution scenarios and the corresponding timescales is one

of the most important primary processes for chemical kinetics. In the common

quasi-equilibrium theories of reaction dynamics, such as transition state theory,

RRKM theory (Rice-Ramsperger-Kassel-Marcus) and SACM (statistical adiabatic

channel model), it is assumed that IVR is much faster than chemical reaction and

thus IVR leads to quasi-equilibrium prior to reaction. However, understanding

the principles of vibrational energy flow may open a path for new developments

in reaction kinetics such as mode-selective chemistry. Indeed, IVR might be used

as a powerful instrument for controlling chemical reactions.

Different approaches for the description of the IVR processes have been used.One theoretical approach would be classical molecular dynamics on ab initio

or empirical potential energy hypersurfaces. A more exact numerical treatment

consists in the numerical integration of the Schrödinger equation with the mo-

lecular hamiltonian drawn from ab initio calculations. Such an approach is lim-

ited to small molecules of perhaps four to six atoms at most and the accuracy is

severely limited by the limited accuracy of ab initio potentials. A more realistic

time evolution in IVR can be obtained experimentally with the molecular hamil-

tonian derived from the analysis of high-resolution infrared spectra. Finally,

statistical models are sometimes used for cases of relatively large systems when

the molecular hamiltonian can not be obtained in explicit form. The relaxation

dynamics can then be obtained experimentally from spectroscopic line shapes

or kinetic measurements. Theoretically it can be calculated in this case from the

Diss. ETH 21307 ix

Abstract

integration of kinetic equations.

In the introduction to our experiments, we discuss the problems of temporal

and spectral resolution in femtosecond pump-probe experiments. The limiting

factors for time-resolution such as the duration of the pump and probe pulses as

well as the effects of nonadiabatic molecular alignment are considered. The con-sequences of the simultaneous detection of multiple transitions are also taken

into account.

The use of a hollow waveguide in a femtosecond pump-probe experiment to

increase the interaction volume is developed in the present thesis for the first

time. The propagation of femtosecond radiation in dielectric and metallic hollow

waveguides of circular cross-section is studied theoretically and experimentally.

The optimal conditions for the use of hollow waveguides in femtosecond pump-

probe experiments are determined. Enhancements of the signal-to-noise ratio

by factors of up to twenty are obtained with the implementation of the hollow

waveguides.

The setup for a femtosecond pump-probe experiment with near infrared exci-

tation and infrared/ultraviolet probing for investigations of IVR processes is

presented. Different pump-probe schemes for selective detection of the depop-ulation of the initially excited states and the population of coupled states are

introduced using a hollow waveguide. A detector array is placed behind a poly-

chromator for spectral resolution of the probe radiation. The operation software

exclusively developed for the setup is also able to treat in real time the collected

data.

IVR processes after excitation in the region of the first overtone of CH-stretching

vibrations are investigated in a number of methane and benzene derivatives

as well as in terminal acetylenes. A great variety of IVR timescales from sub-

picosecond up to several nanoseconds is observed for different functional groups.Triexponential decays with the characteristic times in the range from 300 fs up

to 2 ns are observed after initial excitation of the methyl-CH chromophore in

trifluoromethane, iodomethane and its isotopomers 13CH3I, CHD2I and triply

x — draft compiled 20.03.2014 19:43 — A. Kushnarenko

Abstract

deuterated methane, and with excitation around 6000 cm−1 in benzene. Biex-ponential decays with characteristic times in the range from 5 ps up to 400 ps

are observed after initial excitation of the methyl-CH chromophore in CH2DI

and CHD2F, the acetylenic CH-chromophore in cyanoacetylene and trifluoro-

propyne, and with excitation around 6000 cm−1 in C6H5D and C6HD5. Mo-noexponential relaxation is observed for initial excitation of the methyl-CH

chromophore in CHFI CF3, CHBrClF and CHBrFI, and the acetylenic CH-

chromophore in propyne. The bichromophoric halopropynes CH2Cl C CH,

CH2Br C CH and CH2I C CH show chromophore-selective energy redistri-

bution: fast redistributions on a single sub-picosecond timescale are observed

for depopulation of the initially excited alkylic-CH chromophore and much

slower triexponential (biexponential for propargyl iodide) relaxations are ob-

served on timescales from 6 ps up to 600 ps for initial excitation of the acetylenic

CH-chromophore. The fast IVR is related to the presence of relatively strong re-

sonances, while relatively long timescales are explained by a model with sequen-

tially coupled sets of nearly isoenergetic states and approximate vibrationally

adiabatic separation of the acetylenic CH-stretching vibration, initially. An os-

cillatory IVR behaviour with periods of oscillations from 400 fs up to 130 ps is

observed for the initial excitation of superposition states in CH3I, CHD2I and

N C C CH. The damping of the oscillations is explained by decoherence of

the broad rotational ensemble.

The possibilities for further investigations of IVR processes are discussed in the

outlook.

Diss. ETH 21307 — draft compiled 20.03.2014 19:43 — xi

Zusammenfassung

Die hier vorgelegte Untersuchung beschäftigt sich mit der intramolekularen Um-

verteilung von Schwingungsenerige (intramolecular vibrational energy redistri-

bution, IVR), also dem Prozess der Übertragung der Energie von einem anfangs

angeregten Schwingungszustands auf andere, näherungsweise isoenergetische

Zustände. Aufgrund der grossen Zahl an möglichen Übertragungswegen und

den betreffenden Zeitskalen, ist IVR einer der wichtigsten Primärprozesse derchemischen Kinetik. Üblicherweise, etwa in Quasi-Gleichgewichtstheorien wie

der Theorie des Übergangszustands, RRKM-Theorie (Rice-Ramsperger-Kassel-

Marcus) oder dem statistischen adiabatischen Kanalmodell (SACM), wird an-

genommen, dass IVR viel schneller als die chemische Reaktion stattfindet und

daher vernachlässigt werden kann. Ein Verständnis der Prinzipien nach denen

Schwingungsenergie umverteilt wird, kann jedoch neue Entwicklungen in der

Reaktionskinetik eröffnen, wie etwa die modenselektive Chemie. IVR könnte da-bei ein leistungsfähiges Instrument zur Kontrolle chemischer Reaktionen sein.

Verschiedene Beschreibungen von IVR-Prozessen sind denkbar: Auf theoreti-

scher Ebene könnte die klassische Moleküldynamik auf ab initio berechneten

oder empirisch bestimmten Potentialhyperflächen benutzt werden. Eine exak-

tere, numerische Behandlung besteht in der Integration der entsprechenden

Schrödingergleichung mit molekularen Hamiltonoperatoren aus ab initio Rech-

nungen. Ein solcher Ansatz ist auf die Behandlung kleiner Moleküle mit ma-

ximal vier bis sechs Atomen begrenzt, wobei die Genauigkeit stark durch die

Qualität des benutzten ab initio Potentials limitiert ist. Eine realistischere Dy-

namik der IVR kann experimentell bestimmt werden, indem der molekulare

Hamiltonoperator aus einer Analyse hochaufgelöster Infrarotspektren bestimmt

wird. Schliesslich finden statistische Modelle im Falle grosser Systeme, in denen

Diss. ETH 21307 xiii

Zusammenfassung

der molekulare Hamiltonoperator nicht explizit behandelt werden kann, An-

wendung. Die Relaxationsdynamik kann experimentell aus spektroskopischen

Linienformen oder kinetischen Messungen gewonnen werden.

In der Einleitung zu unseren Experimenten diskutieren wir Probleme der zeitli-

chen und spektroskopischen Auflösung von “pump-probe”-Experimenten mit

Anregungs- und Nachweislaser Pulsen im Femtosekundenbereich. Limitierende

Faktoren für die Zeitauflösung werden berücksichtigt, wie etwa die Pulsdauer

sowie nicht-adiabatische Effekte der molekularen Ausrichtung. Auch die Effektebei gleichzeitiger Detektion mehrerer Übergänge werden berücksichtigt.

Die Verwendung von Hohlwellenleitern in einem pump-probe-Experiment im

Femtosekundenbereich zur Maximierung des Interaktionsvolumens wird zum

ersten Mal in dieser Arbeit behandelt. Die Ausbreitung von Strahlung im Fem-

tosekundenbereich in dielektrischen und metallischen Hohlwellenleitern mit

kreisförmiger Querschnittfläche wird theoretisch und experimentell untersucht.

Optimale Bedingungen für den Gebrauch solcher Wellenleiter für pump-probe

Experimente werden bestimmt. Das Verhältnis von Signal zu Rauschen kann

um bis zu einem Faktor 20 durch die Verwendung solcher Wellenleiter verbes-

sert werden.

Der experimentelle Aufbau eines pump-probe-Experimentes mit Nahinfrarot-

Anregung und Infrarot- oder UV-Nachweis von IVR-Prozessen im Femtosekun-

denbereich wird vorgestellt. Unterschiedliche pump-probe-Schemata zum selek-

tiven Nachweis der Depopulierung der ursprünglich angeregten Zustände und

die Population der gekoppeltem Zustände unter Benutzung von Hohlwellenlei-

tern werden eingeführt. Eine Detektorzeile wird hinter einem Polychromator

platziert, zur spektralen Auflösung der Nachweis-Strahlung. Eine geeignete Soft-

ware wurde speziell für diesen Aufbau entwickelt und kann die gesammelten

Daten in Echtzeit verarbeiten.

IVR-Prozesse nach Anregung in der Region des ersten Obertones der CH-

Streck-Schwingung werden in einer Zahl von Methan- und Benzolderivaten

sowie in terminalen Acetylenen untersucht. Eine grosse Zahl an Zeitskalen

xiv — draft compiled 20.03.2014 19:43 — A. Kushnarenko

Zusammenfassung

für IVR-Prozesse, von sub-Pikosekunden bis zu mehreren Nanosekunden, wer-

den für unterschiedliche funktionelle Gruppen beobachtet. Ein triexponenti-

eller Zerfall mit charakteristischen Zeiten von 300 fs bis 2 ns wird nach ur-

sprünglicher Anregung des Methyl-CH Chromophors in Trifluoromethan, Io-

domethan und Isotopomeren 13CH3I, CHD2I und dreifachdeuteriertem Methan

sowie nach Anregung bei ca. 6000 cm−1 in Benzol beobachtet. Ein biexpoen-tieller Zerfall mit charakteristischen Zeiten von 5 ps bis 400 ps wird nach ur-

sprünglicher Anregung des Methyl-CH Chromophors in CH2DI und CHD2F,

im CH-Chromophor des Acetylens in Cyanoacetylen und Trifluoropropin, so-

wie bei Anregung um 6000 cm−1 in C6H5D und C6HD5 beobachtet. Ein mo-noexponentieller Zerfall wird nach ursprünglicher Anregung des Metyhl-CH-

chromophors in CHFI CF3, CHBrClF, und des CH-Chromophors des Acetylens

in Propin beobachtet. Die bichromophoren Halopropine oder Propargylhaloge-

nide CH2Cl C CH, CH2Br C CH und CH2I C CH zeigen eine chromophor-

selektive Energieumverteilung: ein schnelle Umverteilung auf einfacher sub-ps-

Zeitskala wird beobachtet bei der Entvölkerung des ursprünglich angeregten

Alkyl-CH-Chromophors, und eine viel langsamere triexponetielle (biexponen-

tiel für Propargyl-Iodid) Relaxation auf Zeitskalen von 6 ps bis 600 ps bei ur-

sprünglicher Anregung des CH-Chromophors in Acetylen. Ein schnelles IVR ist

an das Vorhandensein starker Resonanzen gekoppelt, während lange Zeitskalen

durch ein Modell mit sequentiell gekoppelten Zuständen bei näherungsweiser

gleicher Energie und näherungsweiser adiabatischer Separation der CH-Streck-

Schwingung in Acetylen erklärt wird. Ein oszillierendes Verhalten des IVR mit

Perioden von 400 fs bis 130 ps wird für ursprüngliche Anregung eines Superpo-

sitionszustands in CH3I, CHD2I und N C C CH beobachtet. Das Abklingen

der Oszillationen wird durch Dekohärenz des Ensembles an Rotationszustän-

den erklärt.

Die Möglichkeit weiterer Untersuchungen von IVR-Prozessen wird als Ausblick

behandelt.

Diss. ETH 21307 — draft compiled 20.03.2014 19:43 — xv

Chapter 1

Introduction

πάντα χωρεῖ καὶ οὐδὲν μένει.1

Ηράκλειτος ὁ Εφέσιος

1.1 Historical perspective and the role of

intramolecular vibrational redistribution in

reaction kinetics

The idea that molecules, which undergo chemical reaction, must have a certain

excitation energy goes back to Arrhenius [1889] leading to the Arrhenius equa-

tion for the thermal rate constant

k(T ) = A(T )exp(−EART

). (1.1)

The Arrhenius activation energy EA is roughly equivalent to the excitation en-

ergy required to achieve reaction (for a more precise modern definition of Ar-

rhenius parameter see [Quack and Jans-Bürli 1986; Quack 2012; Cohen et al.2007]). More specifically, for unimolecular reactions, F. A. Lindemann (see his

1Everything changes and nothing remains still. Heraclitus (c. 535 – c. 475 BCE), cited after[Sherbakova 2012]

Diss. ETH 21307 1

Chapter 1. Introduction

remark in [Lindemann et al. 1922]) introduced the mechanism today called afterhim

M + A A∗ + M, (1.2)A∗ + M A + M, (1.3)

A∗ B. (1.4)

This mechanism contains the bimolecular processes of activation (1.2) and de-

activation (1.3) and a pure unimolecular step (1.4) (for details on the history

see [Quack 2011a, 2012]). From the current point of view, molecules can react

from many excited quantum states above a certain minimum critical energy E0.

One thus can define for a given excited state A∗(E,J, . . .) at some total energy E,angular momentum J and possibly other quantum numbers (. . .) a specific rate

constant for unimolecular decay

kuni(E,J, . . .) =−1

[A∗(E,J, . . .)]d[A∗(E,J, . . .)]

dt. (1.5)

If one has a simple first order rate law, kuni(E,J, . . .) would be independent of

time, but more generally kuni(E,J, . . .) might also depend upon time.

In most common treatments of unimolecular reaction dynamics, it is assumed

that kuni(E,J, . . .) can be derived as a time independent rate constant arising from

an ensemble with a preestablished microcanonical equilibrium. This assump-

tion is the basis for the so called statistical theories of unimolecular reactions,

such as transition state theory (or activated complex theory), quasi-equilibrium

theory, RRKM theory or the statistical adiabatic channel model (SACM), some

of which we shall discuss in more detail below. These theories constitute the

most widely used “generally accepted” approaches. The common assumption

in these approaches is that intramolecular microcanonical equilibrium is es-

tablished in polyatomic molecules on a sub-picosecond timescale by processes

of intramolecular energy flow or intramolecular vibrational energy redistribu-

tion (IVR). However in the past few decades, it has become clear, that in some,

perhaps many cases, intramolecular energy flow may become rate determining

leading to “nonstatistical” unimolecular reaction rate, see [Quack 1981c; Mar-

2 — draft compiled 20.03.2014 19:43 — A. Kushnarenko

1.2. Theoretical approaches to unimolecular reaction dynamics

quardt and Quack 2001]. It thus has become important to study the processes

of intramolecular energy flow in more detail, which is the subject of the present

thesis. In the following subsections of the introduction we shall discuss some of

the current theoretical approaches to unimolecular reaction dynamics and then

outline the main contents of the present thesis.

1.2 Theoretical approaches to unimolecular

reaction dynamics

1.2.1 General classical mechanical model

A common approach in studying the dynamics of molecules consists in first

treating electronic motion quantum mechanically with fixed positions of the

nuclei (or “atoms”) in the molecule. The resulting electronic energy generates

a potential energy hypersurface for the motion of atoms. This is the essence of

the Born-Oppenheimer approximation. The time-dependent microscopic rate

coefficient can then be calculated by classical mechanical theory in the followingway. Consider a large ensemble of N -atomic molecules prepared with a defined

energy E. Every molecule is represented by a single point in a 2(3N − 6) dimen-sional phase space (coordinate and momentum for each of the 3N − 6 internaldegrees of freedom, excluding that of the center of mass and rotation of the

molecule). Knowing the initial coordinates and momenta for every molecule,

one can calculate the evolution of these 2(3N − 6) parameters for any time inthe future by solving the Hamilton’s equations of motion with a predefined mo-

lecular potential surface. The solution of the Hamilton’s equations can be rep-

resented by a motion of a point along some trajectory in the phase space, fully

defined by the initial conditions, as schematically shown in figure 1.1 [Gilbert

and Smith 1990]. As soon as the trajectory crosses the surface dividing reactant

and product, the so called dividing or transition state surface, it is assumed that

it appears irrevocably on the product side and the reaction has happened with

the reaction time treac. The rate coefficient basically represents the ratio of thenumber of trajectories irreversibly crossing the transition state surface per unit

Diss. ETH 21307 — draft compiled 20.03.2014 19:43 — 3


pi

qi

(pi ,qi , t0)reactant product

t0 + treac,i

Figure 1.1: Schematic representation of 2(3N − 6)-coordinate phasespace. The curve shows the temporal evolution of the molecule pre-pared in state (pi ,qi) at time t0. Dashed line denotes the transitionstate surface.

time to the number of remaining trajectories

kuni(E,t) =

"reactantsurface

∣∣∣Eδ (t0 + treac(p,q)− t) g(p,q)dpdq

"reactantsurface

∣∣∣Eh(t0 + treac(p,q)− t) g(p,q)dpdq

, (1.6)

where the integration is done along the whole phase space for the reactant with

the energy E, g(p,q) is the distribution function of the initial molecular ensemble

at time t0, δ(t) is the delta function and h(t) is the Heaviside step function

h(t) def= limξ→0

11 + e−t/ξ

, (1.7)

δ(t) def=ddt

h(t). (1.8)

A special chemical model due to N. B. Slater in addition assumes a normal mode

treatment, i.e. absence of IVR. This is not considered realistic today. Classical

molecular dynamics is widely used, but the use of classical mechanics for atomic


1.2. Theoretical approaches to unimolecular reaction dynamics

and nuclear motion in molecules is questionable [Quack and Troe 1981a; Manca

et al. 2008].

1.2.2 Quantum dynamical treatment of intramolecular

motion

Classical molecular dynamics in terms of the common current classical trajec-

tory calculations for the motion of atoms (or nuclei) on Born-Oppenheimer po-

tential hypersurfaces is certainly not quantitatively valid for intramolecular dy-

namics and unimolecular decay in general [Manca et al. 2008]. A more accurateapproach is clearly to treat nuclear motion quantum mechanically on the given

potential energy hypersurfaces. Theories of this kind can be classified as fully

quantum dynamical theories or quantum statistical theories, which we will dis-

cuss in more detail below in subsections 1.2.3 and 1.2.4.

1.2.3 The Rice-Ramsperger-Kassel-Marcus model

One of the widely used simplified calculations of the reaction rate coefficientfor the reaction step (1.4) is the Rice-Ramsperger-Kassel-Marcus (RRKM) the-

ory [Rice and Ramsperger 1927; Kassel 1928a,b; Marcus and Rice 1951; Marcus

1965], which is based on two assumptions: intramolecular microcanonical equi-

librium of the reacting molecules and fixed transition state assumption. The

microcanonical equilibrium assumption implies that all parts of the phase space

are equally populated on the average over the timescale of reaction with iden-

tical probability, i.e. the vibrational energy is statistically redistributed within

the whole molecule and a statistical distribution is reached much faster than the

reaction takes place. The transition state assumption presumes that all reaction

trajectories from the reactant area are crossing the transition state surface only

once. Considering a microcanonical ensemble of various possible reactant states

with a defined energy E, the RRKM theory leads to the reaction rate coefficient



calculated as

kuni(E) =

E−E0U0

ρ‡(E′)dE′

hρ(E), (1.9)

where ρ is the vibrational density of states of the molecular reactant, ρ‡ is thedensity of states on the transition state surface. Equation (1.9) basically repre-

sents a ratio of the total number of states of the excited molecule on the transition

surface with the energy above the critical energy E0 and up to a given energy E

to the density of states at this energy. Thus the RRKM approach is applicable

for molecules which can be described by a quasi-continuous density of states

above the critical energy E0. The validity of RRKM theory breaks down for sys-

tems where the microcanonical equilibrium assumption can not be applied, for

example for those molecules where a mode-selective energy redistribution on

the reaction timescale is expected [Quack 1990a]. Limitations of RRKM theory

arise from both the microcanonical equilibrium assumption and from assum-

ing a fixed transition state. On the positive side, RRKM theory is inherently a

quantum mechanical theory.

1.2.4 Statistical adiabatic channel model

An alternative statistical model which is free from the fixed transition state as-

sumption is the statistical adiabatic channel model (SACM) [Quack and Troe

1974, 1975a,b, 1977a]. It is also based on the statistical assumption of micro-

canonical equilibrium of the reacting molecules, and in addition the presence

of rovibrational adiabaticity for reaction dynamics leading from reactants to

products, i.e. the rovibrational adiabatic channel potentials connecting specific

reactant and product states are equally populated if they are adiabatically open.

The microscopic reaction rate coefficient is obtained somewhat similar to that ofRRKM theory, but now counting individual, adiabatically open reaction chan-

nels taking constants of the motion such as angular momentum J into account


1.3. Evidence of IVR in multiphoton excitation and dissociation

explicitly.

kuni(E,J) =

∑i

h(E −Emaxi

)hρ(E,J)

=WRS(E,J)hρ(E,J)

, (1.10)

where the summation goes over all reaction channels accessible with reactant

energy E and rotational quantum number J , and Emaxi is the maximum of the

adiabatic potential along the reaction coordinate (the numerator WRS(E,J) in

equation (1.10) is nothing but the total number of adiabatically open reaction

channels for given E and J). The advantage of the SACM is that it does not refer to

a fixed transition state and allows for state selectivity [Quack and Troe 1981b]. It

is a generalized transition state theory, however, it also assumes microcanonical

equilibrium at energy E and angular momentum J (including also possibly other

constants of the motion) prior to reaching a reaction channel. The SACM theory

is inherently a quantum statistical model.

1.3 Evidence of intramolecular vibrational energy

redistribution in infrared multiphoton

excitation and dissociation of polyatomic

molecules

While originally unimolecular reaction rate theory applied to reaction after colli-

sional excitation under thermal conditions, more recently excitation with strong

infrared lasers has been studied. There were already early indications that the

statistical theory may be applicable to unimolecular dissociation after multi-

photon excitation, assuming that the IVR process is faster than the excitation

itself [Quack 1978; Schulz et al. 1979]. It is already a quarter of a century sincethe experiment on bichromophoric 1,4-difluorobutane-1-d [Quack and Thöne

1987] with the selective activation of one of the terminal chromophores and

the consequent detachment of HF showed the following results: after multiple

photon excitation of vibrations in the R CHDF chromophore by a 70 ns laser



pulses at 950 cm−1

CH2F CH2 CH2 CHDFnhν CH2F CH2 CH2 CHDF

∗(vCD=n), (1.11)

the two possible reaction channels

CH2F CH2 CH2 CHDF∗(vCD=n)

CH2F CH2 CH CHD + HF,

CH2 CH CH2 CHDF + HF

(1.12)

(1.13)

did not show any notable product selectivity up to the timescale of some 10 ps.

The latter timescale was obtained by a study with collisional quenching as a

competing process. At the same time the experiment proved the validity of the

chromophore principle for multiphoton excitation [Lupo and Quack 1987]. The

possible explanation for such results is that the rate kIVR of intramolecular vibra-

tional energy redistribution (IVR) for vibrationally excited molecules is higher

than 1010 s−1 to 1011 s−1, giving an upper limit for the characteristic timescaleτIVR = 1/kIVR. Since the energy transfer can not be faster than the speed of light,

the lower limit for the IVR timescale is about 3 as, if energy has to migrate over

0.9 nm, or, assuming that the vibrational energy transfer can not be faster than

the motion of nuclei, gives the lower limit of about 1 fs. At this time it was only

possible to obtain the estimation 1fs < τIVR < 10ps, but it was not clear how fast

IVR actually happens. Moreover it was not clear whether the energy flow is mode

specific and whether it leads to a truly statistical relaxation and microcanonical

equilibrium.

1.4 Timescales of intramolecular processes

Intramolecular processes cover a very broad range of timescales [Quack 1995a]

from sub-attosecond for electronic motion [Krausz and Ivanov 2009; Wörner

and Corkum 2011; Gallmann and Keller 2011] up to an estimated second or

even kilosecond timescale for parity violation [Quack 2002, 2011b] as shown in

figure 1.2. The timescales are determined by the hierarchy of the corresponding


1.4. Timescales of intramolecular processes

Electronic motion

Vibrational motion

Vibrational energy flow and redistribution

Nuclear spin symmetry change

Parity violation

τ/s

as 10−18

fs 10−15

ps 10−12

ns 10−9

µs 10−6

ms 10−3

s 100

ks 103

Figure 1.2: Typical timescales for intramolecular processes.

contributions to the molecular hamiltonian operator [Quack 1983]

Ĥ = T̂e + V̂nn + V̂ne + V̂ee + T̂n +ĤSO +ĤSS +Ĥrel +ĤHFS +Ĥweak + . . . , (1.14)

where T̂e and T̂n are the electronic and nuclear kinetic energy operators, V̂nn,

V̂ne, and V̂ee are operators of internuclear, nucleus-electron, and interelectronic

Coulomb potentials responsible for electronic and vibronic motion; ĤSO and

ĤSS are operators of spin-orbital and spin-spin interaction, Ĥrel and ĤHFS are

relativistic and hyperfine structure hamiltonian contributions responsible for

nuclear spin symmetry change; and at last the electro-weak term Ĥweak respon-

sible for parity violation. The contributions may be roughly ordered by size as in

equation (1.14). The larger the contributing term is in the total hamiltonian, the

faster occurs the corresponding process. Since the intramolecular vibrational

energy redistribution involves resonances between different vibrational modes,



it can not occur faster than the timescale of the corresponding vibrational mo-

tion of the nuclei [Zewail 2000; Albert et al. 2011], i.e. not faster than severalfemtoseconds. In the absence of strong resonances the vibrational energy can

stay in the selected mode for a relatively long period of time, say hundreds of

picoseconds, but nevertheless not infinitely long because there is always some

weak coupling between the modes due to the break down of the normal mode

model [Iung et al. 2004]. Finally the IVR behaviour is perturbed by nuclear spinsymmetry change [Chapovsky and Hermans 1999; Puzzarini et al. 2005] whichtakes place on the timescale from nanoseconds to some seconds [Quack 1977,

1995a, 2011b].

The direct experimental observation of IVR on the timescale longer than several

nanoseconds meets a number of experimental difficulties. To avoid intermolec-ular collisions, the experiment has to be carried out at relatively low pressure.

At atmospheric pressure with typical gas kinetic cross-sections the mean time

between collisions is estimated to be about 1 ns [Quack 2012]. However, cross-

sections can be larger, so that to have the mean collision time at least τcoll = 10 ns

for CHF3 at room temperature, the sample pressure has to be lower than 1.5 kPa,

or 15 mbar (calculated from [Birnbaum et al. 1968] and [Tretyakov et al. 2006]),what requires several orders higher sensitivity for pump-probe measurements

as compared to the liquid phase. Collision-free measurements can be carried

out in molecular beams, but they require an even higher detection sensitivity.

For femtosecond and picosecond ranges the delay between the pump and probe

pulses is changed by a mechanically driven elongation of the pathway of one of

the beams, a delay of ∆t = 10 ns requires the construction of a 1.5 m long reliable

translation stage. In some cases the problem of long delays may be overcome

by use of consecutive pulses from the same laser system, then the delay can be

obtained with increments equal to the round trip time of the laser resonator.

Of course, it is also possible to generate delays by electronic means as in the

original flash photolysis kinetic spectroscopy of Norrish and Porter [1954].

Having a measured spectrum covering the range ∆ν measured as full width at

half maximum (FWHM), it is not forbidden in accordance with the uncertainty


1.4. Timescales of intramolecular processes

Table 1.1: Typical bandwidths of different spectroscopic techniquesand the corresponding temporal resolution.

Spectroscopic technique ∆ν ∆E ∆ν̃/cm−1 δt

NMR 1 GHz 4 µeV 0.033 0.5 ns

ESR 100 GHz 0.4 meV 3 5 ps

femtosecond pump-probe 4.5 THz 0.02 eV 150 100 fs

FTIR ... FT-VIS 750 THz 3 eV 25000 500 as

UV ... VUV 7.5 PHz 30 eV 2.5× 105 50 asx-rays ... soft γ-rays 7.5 EHz 30 keV 2.5× 108 50 zshigh energy physics 25 YHz 100 GeV 8× 1014 0.02 ys

principle to obtain a temporal resolution down to δt

δt >2ln2π

1∆ν

=2ln2π

h∆E

=2ln2πc

1∆ν̃

, (1.15)

where the relation (1.15) is determined for a gaussian spectrum (see section 3.1.2

for more details). The values theoretically reachable for the temporal resolution

of different spectroscopic techniques are reviewed in table 1.1. The analysis ofIR-spectra within the range ∆ν̃ = 25000 cm−1 can in principle reveal moleculardynamics with δt = 500 as, or even better resolution [Quack 2003]. But even hav-

ing rather moderate spectral resolution compared to other spectroscopic meth-

ods, the femtosecond pump-probe technique is the only kind of measurements

where the spectral and temporal resolution are simultaneously and naturally

obtained. The shorter the pulse duration the broader is the bandwidth and the

more superposition states are reachable for a coherent excitation. Nowadays the

generation of 100 fs pulses even in the IR range has become a standard technique

with commercially available lasers. The corresponding temporal resolution is

enough for the investigation of many IVR processes. However the fastest known

IVR process driven by a Fermi-type resonance [Fermi 1931; Herzberg 1945] in

alkylic CH-chromophores [Dübal and Quack 1984a; von Puttkamer et al. 1983]



is expected on a sub-100 fs timescale and one needs much shorter pulse duration,

say 10 fs, to investigate this process. This becomes already problematic for the

IR region around ν̃ = 2850 cm−1 (region of vCH = 2→ vCH = 3 transition) due tothe required bandwidth of ∆ν̃ = 1500 cm−1. It should cover the relatively broadrange ν̃ = 2100 . . .3600 cm−1.

1.5 Intramolecular quantum dynamics and

time-dependent Schrödinger equation

It is now almost hundred years since the development of quantum mechanics

to study the dynamics of atoms and molecules. We use here the Schrödinger

picture together with the time evolution operator approach following [Merkt

and Quack 2011].

The molecule is considered as consisting of positively charged nuclei and neg-

atively charged electrons which are moving in the common electromagnetic

field of all particles together with an applied external field. The molecular sys-

tem is described by the wavefunctionΨ (x1, y1, z1,x2, y2, z2, . . . ,xN , yN , zN , t), which

depends on the coordinates of all particles and time. Further we will use the

general notation for the coordinate vector r of N particles

Ψ (r, t) def= Ψ (x1, y1, z1,x2, y2, z2, . . . ,xN , yN , zN , t). (1.16)

The absolute square of the wavefunction is a probability density function

P (r, t) =Ψ (r, t) ·Ψ ∗(r, t) = |Ψ (r, t)|2 , (1.17)

where Ψ ∗(r, t) is the complex conjugate. P (r, t) corresponds to the probability tofind the system in the stateΨ (r, t) at time t and the position r. The wavefunction

is the solution of the time dependent Schrödinger equation [Schrödinger 1926]

ih

2π∂∂tΨ (r, t) = Ĥ(t)Ψ (r, t), (1.18)


1.5. Intramolecular quantum dynamics and Schrödinger equation

where the hamiltonian operator Ĥ of the isolated molecule is the sum of the

kinetic and potential energy operators T̂ and V̂ correspondingly

Ĥ = T̂ + V̂ . (1.19)

For N particles the kinetic operator is

T̂ =12

N∑i=1

p̂2imi, (1.20)

where p̂i is the momentum operator for i-th particle. Here and below we omit in

our notation the implied dependence of the wavefunction on r unless specially

needed. The solution of (1.18) one can advantageously find in the form

Ψ (t) = Û (t, t0)Ψ (t0), (1.21)

where Ψ (t0) is the partial solution of (1.18) for the specific time t0 and the

unitary operator Û (t, t0) is the time evolution operator, which obeys the equation

ih

2π∂∂tÛ (t, t0) = Ĥ(t)Û (t, t0) . (1.22)

For a time independent hamiltonian

Û (t, t0) = exp[−i2πh

(t − t0)Ĥ]

(1.23)

and the solution of (1.18) is simplified

Ψ (t) = Û (t, t0)Ψ (t0) = exp[−i2πh

(t − t0) · Ĥ]Ψ (t0). (1.24)

But in the case of a time-dependent hamiltonian the search for a solution of the

Schrödinger equation depends on specific properties of Ĥ(t).

The concept for the understanding of IVR processes includes different ap-proaches based on time-independent and time-dependent experimental and

theoretical studies. The approach is justified by the fact that the molecular hamil-

tonian may be decomposed into a zero-order hamiltonian Ĥ0 and a perturbation



operator V̂ (generally time dependent), following [Merkt and Quack 2011]

Ĥ = Ĥ0 + V̂ , (1.25)

ih

2π∂Ψ (t)∂t

= Ĥ Ψ (t) =(Ĥ0 + V̂

)Ψ (t). (1.26)

The solution of the Schrödinger equation for the zero-order hamiltonian is as-

sumed to be known

Ĥ0ψi = Eiψi . (1.27)

Then the solution of equation 1.26 in a complete basis {ψi} is

Ψ (t) =∑i

bi(t)ψi exp[−2πih

Eit]. (1.28)

Substituting the solution (1.28) into the time-dependent Schrödinger equation

we obtain a set of coupled differential equations

ih

2π

dbj(t)

dt=

∑i

Vj ibi(t)exp[iωj it

], (1.29)

where the following notations are done

Vj i =〈ψj

∣∣∣V̂ ∣∣∣ψi〉 , (1.30)ωj i =

2πh

(Ej −Ei

). (1.31)

Defining a matrix H ′ with elements

H ′j i = Vj i exp(iωj it

), (1.32)

the set of coupled differential equations (1.29) may be written in matrix form

ih

2πdb(t)

dt=H ′b(t), (1.33)

with the coefficient vector b(t) = (b1, b2, . . . , bm, . . .)T. Using substitution

ai = bi exp[−2πih

Eit]

(1.34)


1.5. Intramolecular quantum dynamics and Schrödinger equation

and defining the diagonal matrix Ediag with elements Ei on its diagonal, one

obtains

ih

2πda(t)

dt=

(Ediag +V

)a(t). (1.35)

For time-independent hamiltonian the solution of (1.35) is

a(t) = exp[−2πih

(t − t0)(Ediag +V

)]a(t0). (1.36)

The problem is considered in more detail in section 2.1.

The hamiltonian of a molecule in an external laser field is of interest in the

present work. Such a hamiltonian consists of a time independent term of the

molecular hamiltonian Ĥm and a perturbation term arising from the interaction

with an oscillating electric field of a laser in electric dipole approximation (here

we neglect the interaction with the magnetic component of the laser field, which

is much smaller under the conditions of our experiments)

Ĥ(t) = Ĥm− µ̂ Ê(t), (1.37)

where µ̂ is the molecular electric dipole moment operator, Ê(t) is the operator

of the external electric field. To separate the space coordinates and the time-

dependent part of the problem one can use the basis of eigenstates {φi} of thefield-free molecular hamiltonian [Quack 1998].

Ĥmφi = E(m)i φi . (1.38)

The wavefunction Ψ (r, t) can be decomposed in the basis of {φi} with time-dependent coefficients ci(t) as

Ψ (r, t) =∑i

ci(t)φi(r). (1.39)

In analogy with the derivation of equations (1.29)–(1.35) we obtain a hamil-

tonian matrix with eigenstates E(m)i on the diagonal and the time-dependent

off-diagonal elements, whose evolution is determined by the electric field as

V ′j i = −〈φj

∣∣∣µ∣∣∣φi〉 ·E(t) (1.40)



with vector representations of the dipole moment µ and the electric field E(t).

1.6 Investigation of IVR

There are several methods used for investigation of IVR dynamics. The first

method involves the thorough line-by-line analysis of the highly resolved time-

independent infrared molecular spectra [Albert et al. 2011] with subsequentconstruction of an effective hamiltonian operator Ĥeff based on the effectivesymmetries and least-square fitting of spectroscopic parameters for the best

reproduction of the observed spectra [Albert et al. 2011; Niederer 2011]. Thesimulated spectral lines can be assigned to specific molecular states. At this

point one can use the information obtained from ab initio studies. Solving the

electronic Schrödinger equation with the ab initio hamiltonian operator Ĥab initiothe ab initio potential hypersurface can be obtained and then used for a numeri-

cal simulation of the wave-packet dynamics [Marquardt and Quack 2001]. Also

the derived ab initio potential hypersurface can be used for understanding and

construction of the real molecular hamiltonian operator Ĥm based on observed

spectra and symmetries from the effective hamiltonian [Dübal and Quack 1984a;Quack 1990b, 1995a]. In this way the effective and ab initio hamiltonian ope-rators are approximations to the real one, but Ĥeff is not able to reproduce the

potential hypersurface and Ĥab initio is not able to reproduce the observed spec-

tral lines. The real hamiltonian Ĥm contains the parameters of the molecular

potential hypersurface and is able to reproduce the observed spectra. The solu-

tion of the time-dependent equation of motion with the time evolution operator

Û (t, t0) based on the real molecular hamiltonian Ĥm gives the most reliable

wave-packet dynamics. The complete concept of comprehensive IVR investiga-

tion is shown in figure 1.3.

Another experimental approach uses time resolved spectroscopy for the direct

observation of the IVR dynamics of initially populated zero- or first-order states,

which are not molecular eigenstates, but may be represented as a superposition

of the molecular eigenstates. There are many kinds of time-dependent measure-

ments and many of them use pulsed lasers to initially prepare the system in


1.6. Investigation of IVR

Investigation of IVR

high-resolutionmolecular

spectroscopy

effectivehamiltonian

Ĥeff

molecularhamiltonian

Ĥm

time evolu-tion operator

Û (t, t0)

time dependentwavepackets and

all observables

ab initiocalculations

ab initiopotential energy

hypersurfaceand hamilto-nian Ĥab initio

femtosecondpump-probeexperiment

time dependentabsorption spectra

final co

mp

aris

on

Figure 1.3: Schematic diagram for the comprehensive investigationof intramolecular energy redistribution. The concept shows the co-operation between different methods based on time-dependent andtime-independent measurements.



some specific state and then to probe subsequently the dynamics by time re-

solved spectroscopy. Depending on the detection principle the interpretation

of obtained temporally resolved spectra gives versatile information about the

molecular dynamics. The population dynamics of the selected states can be ob-

served by IR-absorption [Yoo et al. 2004a] and multiphoton ionization [Ebataet al. 2001], for the observation of the population dynamics of lower modesthe UV-absorption technique [Bingemann et al. 1997; Charvat et al. 2001; Elleset al. 2004; Krylov et al. 2004] can be used. The molecular dynamics simulatedwith the molecular hamiltonian Ĥm can be verified in time-dependent measure-

ments, and vice versa the experimentally observed time-dependent spectra may

be interpreted with the knowledge of resonances obtained from the analysis of

time-independent spectra. Thus the described methods are rather complemen-

tary to each other.

1.7 Motivation and survey of the present thesis

The examples given above shows how important the knowledge of timescales

and general mechanisms of IVR is for molecular reaction dynamics [Quack and

Troe 1977b, 1981a]. There is still a number of open questions: is energy flow of

IVR processes mode specific, what are the timescales, and does it lead ultimately

to statistical relaxation and microcanonical equilibrium? The answers to these

questions may change the contemporary concept of unimolecular reaction kinet-

ics. The application of the knowledge of IVR may create possibilities for control

of chemical reactions and unprecedented access of mode-selective chemistry

[Quack 1990b; Crim 1996; Assion et al. 1998]. The present work is inspired anddriven by this ambitious and challenging motivation.

In chapter 2 we model IVR dynamics for zero-, first- and higher-order states

using an effective molecular hamiltonian. Also we consider IVR processes witha quantum mechanical, statistical model. Section 2.5 is devoted to a considera-

tion of IVR in the framework of a kinetic model. The derivation of the effectivetemperature of a small subsystem in a microcanonical ensemble is described in

section 2.6.


1.7. Motivation and survey of the present thesis

Chapter 3 is devoted to description of femtosecond pump-probe experiments

for the investigation of the IVR processes. Different probing schemes are con-sidered. Special attention is paid to the problem of temporal resolution in these

experiments.

In chapter 4 a new technique for the investigation of IVR is proposed with the

use of a hollow waveguide in femtosecond pump-probe experiments. Advan-

tages and limitations of this technique are considered in detail.

In chapter 5 an experimental setup used in the present studies is described, and

the results for IVR in a variety of chemically interesting molecules with one or

two relevant infrared chromophores as obtained with this setup are presented in

chapter 6. Programming source codes, some additional graphs and the normal

modes of the molecules investigated are presented in appendices.


Chapter 2

Intramolecular vibrational energy

redistribution

It can scarcely be denied that the supremegoal of all theory is to make the irreducible

basic elements as simple and as few aspossible without having to surrender the

adequate representation of a single datum ofexperience.1

Albert Einstein (1879–1955)

The phenomenon of intramolecular vibrational energy redistribution is of fun-

damental importance in physical chemistry. It is a central aspect and a condition

for most statistical reaction theories, where it is assumed that redistribution pro-

ceeds faster than reaction so that microcanonical intramolecular equilibrium

is reached prior to reaction [Quack and Troe 1981b; Pilling 1987; Quack and

Troe 1977b, 1981a]. The question concerning the time scale of IVR became also

important when the first IR-multiphoton experiments had been made, where

a mode selective chemistry can be expected, if the unimolecular reaction step

at a certain excitation energy is faster than the IVR process [Quack and Sut-

cliffe 1984; Quack 1990b, 1995a, 2001; Marquardt and Quack 2001; Shapiroand Brumer 2003]. In the present chapter, we consider different quantum, as

1The Herbert Spencer Lecture, delivered at Oxford, June 10, 1933, see [Einstein 1933]

Diss. ETH 21307 21

Chapter 2. Intramolecular vibrational energy redistribution

well as classical, models for the description of IVR processes and discuss the

applicability of these models.

2.1 Effective hamiltonian and IVR

A real molecular system in principle has an infinite basis of eigenstates

(rank(Hm)→∞), and in spite of the fact, that the number of really populatedstates up to some energy is finite, its dimension grows drastically with the in-

crease of the number of particles. Moreover the exact derivation of the corre-

sponding eigenfunctions is complicated by lack of information about the exact

molecular hamiltonian taking into account relativistic effects and perhaps theelectric weak interaction. So to obtain the exact solution is a quite challenging

(if not to say impossible) task, but one can make a number of approximations to

estimate it in a much easier way.

2.1.1 Effective hamiltonian

First we apply the Born-Oppenheimer approximation [Born and Oppenheimer

1927] to separate the nuclear and electronic wavefunctions and we follow here

the notation of [Merkt and Quack 2011]

Φn(rnuc,rel) = ψ(nuc)m(n) (rnuc)φ

(el)n (rnuc,rel), (2.1)

where ψ(nuc)m(n) is the nuclear wavefunction, which depends on the coordinates

of the nuclei rnuc and associated with a given electronic state n (index m is

for distinction of different states of nuclear motion), and φ(el)n is the electronicwavefunction, which depends on the electronic coordinates rel at fixed (para-

metrically given) nuclear coordinates. The nuclear wavefunction in turn can be

represented as a combination of vibrational and rotational wavefunctions

ψ(nuc)m(n) (rnuc) = ψ

(vib)n (rnuc)ψ

(rot)n (rnuc). (2.2)

Since we consider the nuclear wavefunction in the electronic ground state ψ(nuc)m(n)is implied below without mentioning the superscript (nuc) and index (n). Vibra-


2.1. Effective hamiltonian and IVR

tional wavefunctions ψ(vib)n = |vn〉 are eigenfunctions of the anharmonic oscil-lator which are characterized by the vibrational quantum number v which is

provided by an index to distinguish the type of vibrations (vs for a stretching vi-

bration, vb for a bending vibration, and so on). The vibrational quantum number

v can take integer values from 0 up to some maximum value which is defined

in reality by the corresponding dissociation energy. Rotational wavefunctions

are eigenfunctions of the rigid rotor which are characterized by several quan-

tum numbers. The wavefunction ψ(rot)JM of a linear molecule is indexed by the

total angular momentum quantum number J and the quantum number M (the

projection component of J to the space axis). A symmetric top molecule has

the wavefunction ψ(rot)JKM , where K is the projection component of J to the mo-

lecular axis. An asymmetric top molecule is characterized by J and τ and M

and the corresponding wavefunction ψ(rot)JτM can be found as a superposition of

wavefunctions of a symmetric top molecule

ψ(rot)JτM =

∑K

aJKMψ(rot)JKM . (2.3)

The J number can take integer values from 0 up to some number limited by the

ionization energy, K and τ can take 0,±1,±2, . . . ,±J . In the absence of an externalmagnetic field all rotational wavefunctions are 2J + 1 times degenerate since M

can take values 0,±1,±2, . . . ,±J .

It makes sense to restrict the basis of molecular wavefunctions ψi so that

1 6 i 6 imax, restricting the quantum numbers referred only to substantially pop-

ulated states and states coupled to them. Now the matrix representation of the

hamiltonian has finite rank.

2.1.2 Hierarchy of states

The transitions from the ground state to a specific vibrational state are quite easy

for the identification in molecular IR-spectra. As a rule the more quanta are in

the vibrational mode the smaller is the transition moment from the ground state

and every additional quantum reduces the absorption intensity (proportional

to the square of the transition moment) by one or two orders of magnitude. As



the harmonic frequencies for the XH-stretching vibration (X = C,N,O) are rela-

tively large, high excitation energies are reached through the absorption of two

or more quanta. To obtain similar excitation energies for the other vibrational

modes, a much higher number of quanta is necessary. For this reason the over-

tones of the XH-stretching vibration are easily identified in the near-IR spectra

of polyatomic molecules, due to their comparably high band strength. How-

ever, a detailed analysis of the near-IR spectra shows for many molecules that

the number of identified vibrational transitions is much higher than expected

from a simple separable oscillator model. An example is shown in figure 2.1

for the region of the first overtone of the CH-stretching vibration in CHF3. In

5200 5400 5600 5800 60000.0

0.5

1.0

1.5

~

~

~

~

~

~ν |22〉 = 5710 cm-1

ν |21〉 = 5959 cm-1

ν |0,4〉 = 5407 cm-1

ν |2,0〉 = 5913 cm-1

abso

rban

ce

ν /cm-1

ν |1,2〉 = 5691 cm-1

~ν |23〉 = 5337 cm-1

Figure 2.1: Spectrum of CHF3 in the region of the first overtone ofCH-stretching vibration. The wavenumbers of the observed transi-tions (shown by arrows) of the spectroscopic states ν̃2i are shiftedwith respect to the calculated wavenumbers of the uncoupled nor-mal modes ν̃|2−n,2n〉 (dashed lines), see also [Dübal and Quack1984a; Albert et al. 2011].

contrast to one expected line, three transitions are observed. In a normal mode

description, the three states can be represented by the basis function |vs,vb〉,where vs and vb are the quanta in the CH-stretching and CH-bending vibration

respectively. For our example, we can identify the three states by |2,0〉, |1,2〉



and |0,4〉. In a zero order picture, the band strength for the first state is muchhigher, as only two vibrational quanta have to be absorbed, whereas the other

involve three or four quantum steps. Anharmonic vibrational couplings are re-

sponsible for the experimentally observed band strengths in these bands. Due

to these vibrational couplings, the observed states are eigenstates of an effectivehamiltonian including these couplings, where the band strength of the individ-

ual states is determined by the expansion coefficient related to the |2,0〉 zeroorder state in a first order description [Dübal and Quack 1984a]. Taking into

account the coupling between vibrational modes, one can calculate corrected

energies, which are much closer to the observed ones but still differ from them.Further investigation might show that there are other, much weaker, couplings

to other states, which also have to be taken into account. Figure 2.2 shows how

the consequent inclusion of higher-order coupling terms makes the perturbed

states shift against each other. Every next perturbation term is much smaller

Ene

rgy

0th-order states 1st-order states 2nd-order states

}

∆E(1)

∆E(2)

Figure 2.2: Hierarchy of states: the consequent inclusion of higher-order coupling terms makes perturbed states repel from each other.Dotted lines – uncoupled states, solid lines – coupled states, cou-pling is shown by gray arrows, see [Quack 1981a].

than the preceding one and the related energy shift of perturbed states is corre-

spondingly much smaller. Excitation of lower-order states requires a minimal

bandwidth to cover the whole superposition of coupled higher-order states. In



order to excite zero-order states, one needs a bandwidth not less than ∆E(1),

while for the excitation of the selected first-order state an energy width of about

∆E(2) would be enough (see figure 2.2). In practice one can neglect coupling

elements much smaller than the vibrational energy separation in the basis of the

normal modes (zero-order states). Consequent for the first-order states, one can

neglect coupling elements with smaller separation of the corresponding rovibra-

tional states. Ultimately, at sufficiently high densities of states such sequentialcouplings might generate global vibrational states and a spectrum where all

vibrational levels appear, perhaps weakly, see [Quack 1981a].

2.1.3 Structure of the effective hamiltonian

In the following we restrict the molecular hamiltonian to couplings of vibra-

tional levels belonging to the same polyad. The polyad classification groups the

vibrational levels of selected modes with roughly the same energy. For example

the frequency of the CH-stretching normal mode in CHX3 molecule is quite

close to the doubled frequency of the CH-bending vibrations [Dübal and Quack

1984a,b; Dübal et al. 1989] and we assume that all the levels with vs quantain the stretching mode and vb quanta in the bending mode belong to the same

polyad if they have the same number N = vs +12vb (for more examples see [Beil

et al. 1996; Pochert and Quack 1998]). The reason for neglecting inter-polyadinteractions is that usually few-quanta exchange couplings are much stronger

than the ones for many-quanta exchange. More generally, the couplings between

blocks of the polyad hamiltonian are removed by appropriate transformation

[Beil et al. 1996]. As a result of such an arbitrary rearrangement, the hamiltonianmatrix representation for the vibrational problem has a block-diagonal shape,

where every block corresponds to some polyad. Effectively off-diagonal elementsof the rearranged field-free hamiltonian matrix arise from an imperfection of

the simplified potential hypersurface, where higher terms in the expansion have

been neglected. Such a polyad-restricted molecular hamiltonian Ĥeffm is called

“effective”. An example of an effective hamiltonian matrix with polyad structurefor a symmetric top molecule is presented in figure 2.3 after [Dübal and Quack

1984a]. The vibrational levels are characterized by the number of quanta in



|vs,vb〉

−1√2ksbb

−1√2ksbb

−ksbb−ksbb

−√2ksbb

−√2ksbb

−√62 ksbb

−√62 ksbb

−2ksbb−2ksbb

−3√2ksbb

−3√2ksbb

H011

|0,0〉|0,0〉0

0 H111

|1,0〉

|1,0〉0

0 H122

|0,2〉

|0,2〉

0

0

0

0

0

0 H211

|2,0〉

|2,0〉

0

0

0

0

0

0 H222

|1,2〉

|1,2〉

0

0

0

0

0

0

0

0 H233

|0,4〉

|0,4〉

0

0

0

0

0

0

0

0

0

0

0

0 H311

|3,0〉

|3,0〉

0

0

0

0

0

0

0

0

0

0

0

0 H322

|2,2〉

|2,2〉

0

0

0

0

0

0

0

0

0

0

0

0

0

0 H333

|1,4〉

|1,4〉

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 H344

|0,6〉

|0,6〉

Figure 2.3: The effective hamiltonian matrix with polyad structurefor the CH-vibrational levels of a symmetric top molecule. All lev-els have not mentioned quantum number lb = 0 because of shownpolyads are integer (2N = even). Diagonal elementsHNj j are orderedby energy within each polyad. Off-diagonal coupling elements withthe Fermi resonance parameter ksbb are for the terms with ∆vs = ∓1and ∆vb = ±2.

the stretching and bending modes |vs,vb〉 and the vibrational angular momen-tum quantum number lb for the degenerate two-dimensional, approximately

isotropic bending oscillator. The diagonal elements HNjj are indexed with re-

spect to the polyad quantum number N and the sequential number j within the

polyad, ordering the levels within each polyad according to their energy. Off-diagonal factor ksbb is the effective potential cubic constant for the terms with∆vs = ∓1, ∆vb = ±2 and ∆lb = 0 (for more detail see [Dübal and Quack 1984a]).One sees that the hamiltonian has tridiagonal structure, i.e. only the close levels

with minimum number of exchanged quanta are coupled directly.

Up to now we have considered only the vibrational structure of the effectivehamiltonian matrix. To include also the rotational structure, we restrict our-



selves to one selected polyad. If we consider only couplings to the states which

have the same symmetry, i.e. to states with the same set of rotational quantum

numbers, then every polyad-referred block of the hamiltonian matrix appear as

in the upper part of figure 2.4. The diagonal of the matrix consists of energies of

unperturbed rovibrational states Ev(J), where v and J are the generalized vibra-

tional and rotational quantum numbers characterizing the corresponding levels,

off-diagonal elements Vvi vj (J), defined as

Vvi vj (J) =〈vi , J

∣∣∣Ĥ ∣∣∣vj , J〉 , (2.4)correspond to couplings between vibrational levels vi and vj having the same

set of rotational quantum numbers (numbered as J). The rearrangement of this

hamiltonian with respect to rotational states lead to the block-diagonal shape

as presented in the lower part of figure 2.4. Below we consider separately the

coupling of vibrational states for the same rotational level and take into account

the whole rotational ensemble.

2.1.4 Zero-order states

A choice of the basis set functions is in general quite arbitrary, and this choice

can affect the efficiency and convenience of the following calculations and theinterpretation of results. Often, a useful, simple and natural choice of the ba-

sis set is related to the set of vibrational normal modes of the corresponding

molecule,and we shall use this here for the first approach.

As a basis set we choose the product of wavefunctions φvi (ri) which are eigen-

functions of the corresponding harmonic oscillators along i-th normal mode

depending on the coordinate ri (one-dimensional ri = ri , or n-dimensional

ri = (ri1 , . . . , rin)T in the case of n-degeneracy) where vi is a set of quanta in the

i-th mode (i.e. vi = (vi1 , . . . , vin) for n-dimensional case) [Marquardt 1989]

Ψ (r1, . . . ,rm) =∑v1

· · ·∑vm

bv1,...,vm

m∏i=1

φvi (ri), (2.5)



E1(0)

E1(1). . .

E1(n)

E2(0)

E2(1). . .

E2(n). . .

Em(0)

Em(1). . .

Em(n)

V21(0)

V12(0)

Vm1(0)

V1m(0)

V21(1)

V12(1)

Vm1(1)

V1m(1)

. . .

. . .

. . .

. . .

V21(n)

V12(n)

Vm1(n)

V1m(n)

Vm2(0)

V2m(0)

Vm2(1)

V2m(1)

. . .

. . .

Vm2(n)

V2m(n)

E1(0) V21(0) · · · Vm1(0)V12(0) E2(0) · · · Vm2(0)...

.... . .

...

V1m(0)V2m(0) · · · Em(0)E1(1) V21(1) · · · Vm1(1)V12(1) E2(1) · · · Vm2(1)...

.... . .

...

V1m(1)V2m(1) · · · Em(1)

E1(n) V21(n) · · · Vm1(n)V12(n) E2(n) · · · Vm2(n)...

.... . .

...

V1m(n)V2m(n) · · · Em(n)

. . .

Figure 2.4: A polyad-block of the effective hamiltonian matrix: up-per part – ordering with respect to vibrational states, lower part– ordering by states having the same symmetry. Notation Hvivj (J)with vibrational quantum numbers in indices, and rotational inparentheses.



where the summation goes over all possible sets of numbers for each vi . Here

we introduce another notation for the product of wavefunctions φvi (ri)

m∏i=1

φvi (ri) = φv1(r1)φv2(r2) · . . . ·φvm(rm)def= |v1,v2, . . . ,vm〉

=∣∣∣∣v11 ,v12 , . . . , v1n1 ,v21 ,v22 , . . . , v2n2 , . . . , vm1 ,vm2 , . . . , vmnm 〉 . (2.6)

The states |v1,v2, . . . ,vm〉 are obtained simply straightforward and this choice ofthe basis wavefunctions are conventionally called “zero-order states”.

As an example of such zero-order states, we consider the normal modes of CH-

chromophore in CHF3. There are two modes which have to be taken into account:

the one-dimensional CH-stretching and the two-dimensional CH-bending mode,

the latter is doubly degenerate. Zero-order basis wavefunctions can be con-

structed as a product of the corresponding stretching and bending wavefunc-

tions or |vs,vb, lb〉, where vs and vb are the numbers of stretching and bendingquanta, and lb is the vibrational angular momentum of the degenerate bend-

ing oscillator [Dübal and Quack 1984a]. Sometimes polyads are conventionally

labeled by the effective number of quanta in a selected mode, so with respectto CH-stretching vibrations in the considered CHF3 molecule the polyad quan-

tum number is N = vs +12vb. The polyad with N = 2 corresponding to the first

overtone range of CH-stretching vibrations has just three coupled states |2,0,0〉,|1,2,0〉, and |0,4,0〉 for every rotational level. Due to the fact that these statesbelong to the same species and have nearly the same energy they perturb each

other [Bernstein and Herzberg 1948], in other words they are in Fermi-type re-

sonance [Fermi 1931; Herzberg 1945]. Due to anharmonicity of CH-stretching

vibrations, the resonance is more pronounced for higher overtones [Dübal and

Quack 1984a; Carrington et al. 1987]. Figure 2.5 shows the numerical valuesfor the CH-polyad-structured hamiltonian of CHF3 (see also figure 2.3 for more

details). In actual practice one uses adiabatically separable anharmonic oscilla-

tor states as basis functions, which leads to the term value formula [Dübal and

Quack 1984a]



|vs,vb〉0

|0,0〉|0,0〉0

0 3018

|1,0〉

|1,0〉0

0

75

75 2730

|0,2〉

|0,2〉

0

0

0

0

0

0 5913

|2,0〉

|2,0〉

0

0

0

0

0

0

106

106 5691

|1,2〉

|1,2〉

0

0

0

0

0

0

0

0

150

150 5407

|0,4〉

|0,4〉

0

0

0

0

0

0

0

0

0

0

0

0 8684

|3,0〉

|3,0〉

0

0

0

0

0

0

0

0

0

0

0

0

130

130 8528

|2,2〉

|2,2〉

0

0

0

0

0

0

0

0

0

0

0

0

0

0

212

212 8311

|1,4〉

|1,4〉

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

225

225 8033

|0,6〉

|0,6〉

Figure 2.5: The effective hamiltonian matrix for CH-vibrational lev-els of CHF3 given in cm

−1 (after [Dübal and Quack 1984b], see alsofigure 2.3 for more details).

2.1.5 First-order states

The presence of higher terms in the representation of the potential energy sur-

face in the zero-order states basis leads to couplings between these states, for

example Fermi resonance coupling. Also vibrational levels can be coupled due

to rotationally induced interactions, so called Coriolis coupling [Herzberg 1945;

Albert et al. 2011]. These couplings lead to the appearance of off-diagonal ele-ments in the effective hamiltonian matrixH effm written in a zero-order-state-basis.The matrix can be diagonalized by a unitary transformation

Z−1H effm Z = diag(E1,E2, . . . ,Esbas

)def= ΛH, (2.7)

where the Z -matrix consist of the eigenvectors of the H effm matrix, ΛH is a diago-

nal matrix of eigenvalues of the H effm matrix, and sbas is the size of the zero-order



state basis, the number of basis elements

sbas =m∑n=1

in, (2.8)

where nj are from equation (2.6). The unitary matrix Z transforms the basis

wavefunctions |v1,v2, . . . ,vm〉 to the new basis set as{Nj

}= Z {|v1,v2, . . . ,vm〉} , (2.9)

where N is the polyad notation number and j is the sequential number of the

state within the polyad [Albert et al. 2011]. These states are conventionally called“first-order states” and they have corresponding first-order energy eigenvalues Ei .

The previously mentioned zero-order states of theN = 2 polyad are transformed

now into the first-order states 21, 22, and 23 (an example is shown in figure 2.1,

see also [Dübal and Quack 1984a]).

2.2 Dynamics of perturbed zero-order states

Consider now the external field-free temporal evolution of the zero- and first-

order state population after the selective excitation of specific states. Equa-

tion (1.24) shows that the U (t, t0) matrix can be calculated for the time-

independent hamiltonian as

U (t, t0) = exp[−i2πh

(t − t0)H]. (2.10)

Consider the matrix representation in the basis of zero-order states and the

zero-order effective hamiltonian as a hamiltonian for the complete set of states.Equation (2.10) can be written as

U (t, t0) = exp[−i2πh

(t − t0)H effm]

= Z exp[−i2πh

(t − t0)ΛH]Z−1 (2.11)

= Z diag(exp

[−i2πh

(t − t0)E1], . . . ,exp

[−i2πh

(t − t0)Esbas])Z−1.


2.2. Dynamics of perturbed zero-order states

The fractional population vector p(t) can be represented as a vector with com-

ponents

pi(t) = |bi(t)|2 = bi(t) · b∗i (t). (2.12)In accordance with equation (1.36) the evolution of the decomposition coeffi-cient vector b(0)(t) for the first-order states is

b(0)(t) = U (t, t0)b(0)(t0) = Z exp

[−i2πh

(t − t0)ΛH]Z−1b(0)(t0). (2.13)

Since the U (t, t0)-matrix is non-diagonal, the vector b(0)(t) may periodically

change with the period of oscillations Tosc

Tosc = LCM(hEi

), (2.14)

where LCM({ξi}) is a function which finds the least common multiple for theset of numbers {ξ1,ξ2, . . . ,ξN }. Since any complex function can be representedas a product of a real amplitude and a complex phase function, the compo-

nents of b(0)(t) can be written as b(0)i (t) = Ai(t)eiφi(t), where Ai(t) and φi(t) are

real periodic functions with a period of Tosc. The corresponding component of

the fractional population vector p(0)(t) depends only on the squared amplitude

function p(0)i (t) = A2i (t), but the function A

2i (t) may oscillate with even shorter

period of Tosc/n, n ∈Z+, i.e. a submultiple of Tosc. If all A2i (t) have a period ofTosc/n0 then p(0)(t) oscillates with the same period, which is shorter than for

corresponding b(0)(t). If a majority of substantially populated Ei is located at

around some Eex value, one can estimate Tosc roughly as [Quack 1981c]

Tosc > h〈ρ(Eex)〉, (2.15)

where ρ(E) is the density of spectroscopic states, calculated by direct state

counting algorithm [Beyer and Swinehart 1973], 〈·〉 denotes the expectationvalue within some interval. For example after the excitation of 2ν1 in CHF3,

we have average density of states 〈ρ̃(5960cm−1)〉 ≈ 3cm and the correspond-ing Tosc & 100ps, whereas for the analogous excitation in CF3 C CH we get〈ρ̃(6550cm−1)

〉≈ 5900 cm and the corresponding Tosc & 200ns.



At the same time one can see that the components of the decomposition coeffi-cient vector in the first-order state basis b(1)(t) have just the oscillating complex

phases

b(1)(t) = Z−1U (t, t0)Zb(1)(t0) = exp[−i2πh

(t − t0)ΛH]b(1)(t0) (2.16)

={

exp[−i2πh

(t − t0)E1]b

(1)1 (t0), . . . ,exp

[−i2πh

(t − t0)Esbas]b

(1)sbas(t0)

}T,

where sbas is number of elements in the basis. The corresponding fractional

population vector p(1)(t) does not depend on time

p(1)i (t) = b

(1)i (t0) · b

(1)i

∗(t0) =

∣∣∣∣b(1)i (t0)∣∣∣∣2 (2.17)in the basis of the first-order states, which act as eigenstates in this basis.

2.3 Rovibrational dynamics

For the model considered above, the effective molecular hamiltonian matrix hasa block-diagonal form, the different blocks are related to the different polyads.The polyad block described above is also block-diagonal with blocks related

to different values of the generalized angular momentum quantum number Jsince we assume a restriction on the change of the angular momentum quantum

number ∆J = 0 for the couplings. In principle the probe laser radiation could be

dispersed behind the experiment to obtain rotational resolution. But to obtain

a satisfactory signal-to-noise ratio the practical resolution is limited to some

cm−1. From this point of view it is useful to consider the population dynamicsof the initially excited vibrational state taking into account contributions from

all rotational levels. For simplification we consider a symmetric top molecule

and assume that the selected vibrational level is characterized by the set of

vibrational quantum numbers {vi} and rotational quantum numbers J , K andM (here we neglect the nuclear spin quantum numbers and the corresponding

statistics). Then the total fractional population of the vibrational state p{vi }(t) issimply the sum of fractional populations p{vi }(J,K,M,t) for all corresponding


2.3. Rovibrational dynamics

rotational states

p{vi }(t) =∑J

∑K

∑M

p{vi }(J,K,M,t). (2.18)

In the general case the dynamics of p{vi }(J,K,M,t) can be quite complicated de-pending on the number of coupled states. To get some insight into the influence

of the rotational energy separation on the population dynamics we investigate

a simplified model. We consider a symmetric top molecule with initially pop-

ulated vibrational state |i〉 which is coupled to another unpopulated single vi-brational state |f〉 with the restriction ∆J = 0, ∆K = 0, ∆M = 0. In this case thehamiltonian matrix has the block-diagonal form (as the blocks of rearranged

hamiltonian in figure 2.4) with the blocks Ei(J,K) Vi f(J,K)Vf i(J,K) Ef(J,K) = E(J,K)− δ(J,K) V (J,K)V (J,K) E(J,K) + δ(J,K)

(2.19)related to different J- and K-quantum numbers, without dependence on the M-quantum number, where V (J,K) is the coupling element for the corresponding

states, and the following notations are introduced

E(J,K) def=Ef(J,K) +Ei(J,K)

2, (2.20)

δ(J,K) def=Ef(J,K)−Ei(J,K)

2. (2.21)

Neglecting higher order terms in the expansion of the rotational energy we have

Evi (J,K) = Evi (0,0) + hc[B̃vi J(J + 1) + (Ãvi − B̃vi )K2

]for prolate top, (2.22)

Evi (J,K) = Evi (0,0) + hc[B̃vi J(J + 1) + (C̃vi − B̃vi )K2

]for oblate top, (2.23)

where Ãvi , B̃vi and C̃vi are rotational constants for the vibrational ground state,

which are related to the principal moments of inertia IA, IB and IC of the corre-

sponding state [Bauder 2011] as

Ã =h

8π2cIA, B̃ =

h

8π2cIB, C̃ =

h

8π2cIC. (2.24)



The two-level problem for Fermi-type coupling with the above mentioned initial

conditions has the following solution in accordance with equation (2.13)

b(0)i (J,K, t) =

investigation of intramolecular vibrational energy flow in … · 2020-05-09 · investigation of...

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