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Research Collection
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
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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
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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
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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
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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
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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
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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
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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
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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
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Chapter 1. Introduction
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
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-
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
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Chapter 1. Introduction
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
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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
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Chapter 1. Introduction
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
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-
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,
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Chapter 1. Introduction
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
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-
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]
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Chapter 1. Introduction
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)
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-
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
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Chapter 1. Introduction
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)
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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)
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Chapter 1. Introduction
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
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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.
Diss. ETH 21307 — draft compiled 20.03.2014 19:43 — 17
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Chapter 1. Introduction
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.
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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.
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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-
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-
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
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Chapter 2. Intramolecular vibrational energy redistribution
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〉
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-
2.1. Effective hamiltonian and IVR
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
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-
Chapter 2. Intramolecular vibrational energy redistribution
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
26 — draft compiled 20.03.2014 19:43 — A. Kushnarenko
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2.1. Effective hamiltonian and IVR
|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-
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-
Chapter 2. Intramolecular vibrational energy redistribution
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)
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2.1. Effective hamiltonian and IVR
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.
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-
Chapter 2. Intramolecular vibrational energy redistribution
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]
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2.1. Effective hamiltonian and IVR
|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
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Chapter 2. Intramolecular vibrational energy redistribution
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.
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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.
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Chapter 2. Intramolecular vibrational energy redistribution
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
34 — draft compiled 20.03.2014 19:43 — A. Kushnarenko
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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
à =h
8π2cIA, B̃ =
h
8π2cIB, C̃ =
h
8π2cIC. (2.24)
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Chapter 2. Intramolecular vibrational energy redistribution
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) =