mapping chemical interactions and dynamics with x-ray
TRANSCRIPT
Mapping chemical interactions and dynamics with x-ray spectroscopy Philippe Wernet Institute for Methods and Instrumentation for Synchrotron Radiation Research Helmholtz-Zentrum Berlin für Materialien und Energie Ultrafast X-ray Summer School (UXSS) 2016, SLAC National Accelerator Laboratory (Stanford, USA), June 2016
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LCLS on Sunday, 12 June 2016
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Outline Part I What are we talking about? Some fundamentals…
Part II
Part III
Only x-ray spectroscopy!
See the other talks and extra slides for other x-ay methods and methods without x-rays.
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See the atoms move
http://www.molecularmovies.com/
http://www.molecularmovies.com/
Michael Odelius, Stockholm University , Sweden
Molecular motors
Photochemistry Organic chemistry
Water
Lars Ojamäe, Linköping University , Sweden
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A. Zewail, Nobel price in Chemistry (1999) „… for his studies of the transition states of chemical reactions using femtosecond spectroscopy.“
Nobel Lecture http://nobelprize.org/nobel_prizes/chemistry/laureates/1999/zewail-lecture.html# „Femtochemistry: Atomic-Scale Dynamics of the Chemical Bond Using Ultrafast Lasers“
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Wait a second: What do we actually talk about? Chemical dynamics deals with the atomic-scale view of the elementary steps of a chemical reaction (pico- to femtoseconds and Ångstrom).
This could be a triggered reaction (pump-probe) or a non triggered reaction (e.g. thermally activated). Most often, photoreactions (triggered) are studied!
We talk about the atomic-scale dynamics of chemical interactions See extra slides on the delineation of kinetics and dynamics!
Ener
gy
Generalized reaction coordinate
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃
∆𝐺‡
𝑇𝑃𝑅𝑅𝑃𝑇𝑃𝑇𝑃𝑅 𝑃𝑃𝑅𝑃𝑅 (𝑅𝑃𝑃𝑇𝑎𝑅𝑃𝑅𝑃 𝑃𝑃𝑐𝑐𝑐𝑅𝑐)
∆𝐺
𝐾 =[𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃][𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃] = 𝑅−
∆𝐺𝑅𝑅 𝑘 = υ ∙ 𝑅−
∆𝐺‡
𝑅𝑅
Thermodynamics Transition state theory
• Thermodynamic properties of the transition state (∆𝐺‡) and the crossing frequency (𝜐) determine the reaction rate (𝑘) (the kinetics) of thermal reactions
• The energy potential landscapes determine the reaction mechanisms (the dynamics) of photochemical reactions
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A textbook example for molecular dynamcis A. Zewail
Real-time tracking of nuclear dynamics (1000 m/s = 1 Å/100 fs)
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But wait, why bother?
Fundamental questions in solar energy conversion
Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
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But wait, why bother?
Fundamental questions in solar energy conversion
Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
• Electronic excitation • (Ultrafast) relaxation
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Electronic excitation
Demtröder, Molekülphysik
Energy
Electronic excitation
Svanberg Atomic and Molecular Spectroscopy
Electronic ground state
Electronic excited state
Vibrational levels
Vibrational levels
Rotational levels
Rotational levels
Distance
Schematic representation of the energy levels of a diatomic molecule
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The Born-Oppenheimer Approximation
• When we describe chemical interactions
• By forming molecular from atomic orbitals
• We often assume the nuclei to be at rest
• We need to drop this approximation!
• Nuclear movements are now part of the problem!
• This problem is not analytically solvable…
• We need the Born-Oppenheimer approximation!
Max Born and Robert Oppenheimer: Zur Quantentheorie der Molekeln, Annalen der Physik, 389 (20), p. 457–484 (1927).
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The Born-Oppenheimer Approximation
Solve the Schrödinger equation for the electrons in the static potential of the fixed nuclei (“Produktansatz”). The electronic part of the wavefunction depends on the nuclear coordinates BUT as a parameter, NOT as a variable!
Ψmolecule = Ψe · Ψn
Ψe = Ψe(re, Rn)
Ψn = Ψn(Rn)
By neglecting the coupling of nuclear and electron motions we can treat the motion of nuclei and electrons independently. Masses of electrons and nuclei are so different (104) that the nuclei appear to be fixed while electrons are moving!
Within the adiabatic approximation (“electrons follow nuclear motions instantaneously”) we can solve the Schrödinger equation for Ψe at fixed Rn repeatedly for many Rn:
[Te + Ve] Ψe(re, Rn=const) = Ee Ψe(re , Rn=const) Te/Ve = kinetic/potential energy of electrons
By plotting the resulting set of solutions Ee versus Rn we build potential energy curves!
(surfaces, landscapes, depending on the number of parameters/reaction coordinates)
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The Born-Oppenheimer Approximation
Demtröder, Molekülphysik
The potential energy curve Ee versus Rn
corresponds to the “electronic part” of the
total energy of the molecules plus the energy
arising from repulsion of the nuclei (sum of
kinetic and potential energy of electrons plus
potential energy of nuclei)
Vibrational and rotational energies of the
molecule are missing!
Electronic excitation
Electronic ground state
Electronic excited state
Vibrational levels
Vibrational levels
Rotational levels
Rotational levels
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The Franck-Condon Principle For the transition between state X and A with vibrational levels υ the transition probability
(electronic dipole transition) is proportional to:
The electronic dipole moment times the Franck-Condon factors
│< ΨeA │ de │ Ψe
X > │2 · │< Ψnυ │ Ψn
0 > │2
│∫ ΨeA de Ψe
X dre│2 · │∫ Ψnυ Ψn
0 dRn│2
Haken/Wolf Molecular Physics and Elements of Quantum Chemistry Atkins/Friedman Molecular Quantum Mechanics
The QM harmonic oszillator
X
A
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The Franck-Condon Principle For the transition between state X and A with vibrational levels υ the transition probability
(electronic dipole transition) is proportional to:
The electronic dipole moment times the Franck-Condon factors
│< ΨeA │ de │ Ψe
X > │2 · │< Ψnυ │ Ψn
0 > │2
│∫ ΨeA de Ψe
X dre│2 · │∫ Ψnυ Ψn
0 dRn│2
Haken/Wolf Molecular Physics and Elements of Quantum Chemistry Atkins/Friedman Molecular Quantum Mechanics
X
A
Those overlap integrals S are the famous Franck-Condon factors
X
A
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Electronic excitation and (ultrafast) relaxation
Haken/Wolf Molecular Physics and Elements of Quantum Chemistry
Red-shift of fluorescence!
Sequence of radiationless transitions
• Internal conversion
• Intramolecular vibrational redistribution
• Intersystem crossing
• Thermalization
(see extra slides for details)
Fluorescence Absorption
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Electronic excitation and (ultrafast) relaxation
Haken/Wolf Molecular Physics and Elements of Quantum Chemistry
Typical energy level diagram of a molecule
Singlet states
Triplet states
Intersystem crossing
Abso
rptio
n
Fluo
resc
ence
Abso
rptio
n
Phos
phor
esce
nce
Internal conversion
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Femtosecond Absorption Spectroscopy of Transition Metal Charge-Transfer Complexes, James K. McCusker, Acc. Chem. Res. 36, 876-887 (2003).
Absorption
Fluorescence (Emission)
Electronic excitation and (ultrafast) relaxation A famous example
Wai-Yeung Wong (Ed.) Organometallics and Related Molecules for Energy Conversion
19 Femtosecond Absorption Spectroscopy of Transition Metal Charge-Transfer Complexes, James K. McCusker, Acc. Chem. Res. 36, 876-887 (2003).
If we can make this state live long, we can efficiently make use of the electron-hole pair!
Electronic excitation and (ultrafast) relaxation A famous example
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(Nuclear) Wavepackets
A. Zewail
Superposing plane waves
For dissociative states
• Coherent superposition of vibrational states • Formation of a nuclear wavepacket • The wavepacket is evolving in time (nuclei are
moving)! • Wavepackets to describe particles confined in space
t1
t2 t3
For free particles
Atkins, Friedman, Molecular quantum mechanics
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Reaction coordinate
Pote
ntia
l ene
rgy
Pump
Ground state X
Excited state A
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Reaction coordinate
Pote
ntia
l ene
rgy
kT
Intermediate state
Transition state („activated complex“)
Transient configurations
Transient configurations
Pump
Probe
…and phenomena that happen on their way (IVR, IC, ISC, …)
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Quiz part I
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How many drawings in one?
1. Shape of the harmonic potential V(x)
2. Energy levels E of the harmonic oscillator
3. Nuclear wavefunctions plotted vs. x
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Reaction coordinate
Pote
ntia
l ene
rgy
Pump
Ground state X
Excited state A
What is “wrong” here (within the BOA)?
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Spot the different pictures/concepts/approximations!
Haken/Wolf Molecular Physics and Elements of Quantum Chemistry
Singlet states
Triplet states
Intersystem crossing
Abso
rptio
n
Fluo
resc
ence
Abso
rptio
n
Phos
phor
esce
nce
Internal conversion
• One-electron orbital picture • Many-electron total energy picture
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What time-resolution do I need to resolve molecular motion?
Take the speed of sound • Resolve corresponding displacements • “Speed of atoms” several 100-1000 m/s • This corresponds to resolving 100-1000·1010 Å/1015 fs = 0.1-1 Å/100 fs Take the oscillation period of a molecule • Resolve the oscillatory motion • E.g. 3500 cm-1 (wavelength of ~3 µm) for the O-H stretch vibration in H2O • T = 1/f, c=λ·f T = λ/c T = 3·10-6m/(3·108m/s) = 10-14 s • This corresponds to a duration of the vibrational period of ~10 fs Take the Brownian motion • Resolve the mean square displacement 𝑐2 𝑃 with time 𝑃: 𝑐2 𝑃 = 𝑘𝐵𝑅
3π𝑅η𝑃
• With 𝑇 temperature, 𝑅 particle radius, η viscosity • Albert Einstein (1905), Marian Smoluchowski (1906) and Paul Langevin (1908) • Displacement of 1 Å takes 100 fs (𝑅 = 1 Å, η = 20·10-6 kg/(m·s) for O2, ideal gas)
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How fast do electrons move?
• Ask yourself: Why do I care? • Do I want to observe electron motion?
– Classically, the electron takes 150 as to circulate the proton (and there are other ways to
estimate those time scales…)
– I need as time resolution!
• Do I want to follow the rearrangements of electrons as nuclei are moving? – I need fs time resolution!
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How big is the “mistake” we make in calculating the energy of a state within the Born-
Oppenheimer approximation? • That’s not an easy question! • Think about it (slide 11): “Masses of electrons and nuclei are so different (104)…”
Within the adiabatic approximation (“electrons follow nuclear motions instantaneously”) we can solve the Schrödinger equation for Ψe at fixed Rn repeatedly for many Rn:
[Te + Ve] Ψe(re, Rn=const) = Ee Ψe(re , Rn=const) Te/Ve = kinetic/potential energy of electrons
Haken/Wolf Molecular Physics and Elements of Quantum Chemistry
Electrons
Nuclei
Neglect within BOA
Schrödinger equation with „Produktansatz“:
Masses of nuclei mn
Mass of elctron me
The neglected part of the energy is smaller by m0/m1, 2 = me/mn = 10-4
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What do you see?
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Molecular dynamics simulation Michael Odelius (Stockholm University)
Fe(CO)5 dissociation in ethanol
Where are the electrons?
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Outline Part I What are we talking about? Some fundamentals…
Part II
Part III
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Gilles Bachelet My Cat, The Silliest Cat in the World
How did that happen?
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Resolving transient and intermediate states Why short pulses and what is „short“?
Palo Alto, CA 1872
Leland Stanford Eadweard Muybridge
The birthday of ultrafast technology (?)
Bet: Does a galloping horse have all 4 hooves in the air at any instant?
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Why short pulses and what is „short“?
Théodore Géricault - Derby in Epsom (Wikipedia)
Resolving transient and intermediate states
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Why short pulses and what is „short“?
Palo Alto, CA 1872
Leland Stanford Eadweard Muybridge
Temporal resolution: 1/60 s (17 ms)
Resolving transient and intermediate states
The birthday of ultrafast technology (?)
Bet: Does a galloping horse have all 4 hooves in the air at any instant?
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Why short pulses and what is „short“?
Animations based on photos from E. Muybridge, Wikipedia
Resolving transient and intermediate states
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Gilles Bachelet My Cat, The Silliest Cat in the World
How did that happen?
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So let’s first have a look at how we can resolve transient states
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41 Ph. Wernet, Electronic structure in real time: Mapping valence electron rearrangements during chemical reactions, Phys. Chem. Chem. Phys. 13, 16941 (2011).
Mapping valence electron rearrangements
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Mapping valence electron rearrangements
Experiment
• Valence photoelectron spectroscopy
• Femtosecond VUV/XUV (23 eV) pulses from HHG
Wernet, Odelius, Godehusen, Gaudin,Schwarzkopf, Eberhardt, PRL 103, 013001 (2009).
43 Wernet, Odelius, Godehusen, Gaudin,Schwarzkopf, Eberhardt, PRL 103, 013001 (2009).
Experiment Theory
A‘ and B‘ of excited molecule
3P of free atom
Mapping valence electron rearrangements
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Experiment
Theory
Wernet et al. PRL 103, 013001 (2009).
Mapping valence electron rearrangements
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Quiz part II
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Which one applies? Detect the differences…
Ph. Wernet, Electronic structure in real time: Mapping valence electron rearrangements during chemical reactions, Phys. Chem. Chem. Phys. 13, 16941 (2011).
Binding energy O
rbita
l ene
rgy
Koopmans' theorem: The first ionization energy of
a molecular system is equal to the negative of the
orbital energy of the highest occupied molecular
orbital (HOMO).
Nuclear distance
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What time-resolution do we need?
Fourier-transform limited pulses
ΔtFWHM · ΔEFWHM = 1.85 eV fs
e.g. 20 fs, 0.1 eV
Svanberg Atomic and Molecular Spectroscopy
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Now let’s see how we resolve intermediate states and why that is important
49
Outline Part I What are we talking about? Some fundamentals…
Part II
Part III
50
Richard Avedon – Photographs 1946-2004
51 Shibuya Hachiko crossing, Tokio – Photographed from Starbucks (2009)
52 Shibuya Hachiko crossing, Tokio – Photographed from Starbucks (2009)
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How can we pick out a specific signal from a soup of molecules?
How can we be selective?
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Why x-rays? Spectroscopy
Element-, site- and orbital-specific
• Coordination • Charge density • Spin density
Tuneable, short and intense soft x-ray pulses
55 Reaction coordinate
Ener
gy
Ground state
Valence excited states
Core-excited state 2
Core-excited state 1
Chemical species 1
Chemical species 2
This is what we are interested in! This where the
chemistry happens!
Femtosecond x-ray spectroscopy with FELs
56 Reaction coordinate
Ener
gy
Ground state
Valence excited states
Core-excited state 2
Core-excited state 1
Chemical species 1
Chemical species 2
This „just“ the probing!
Femtosecond x-ray spectroscopy with FELs
57 Reaction coordinate
Ener
gy
Ground state
Valence excited states
Core-excited state 2
Core-excited state 1
Chemical species 1
Chemical species 2
Femtosecond x-ray spectroscopy with FELs
5-10 eV
5-10 eV
Our probe needs to be sensitive to the ev-sub eV scale!
58
Fe C O
In
Out
1 mol/l in EtOH
K. Kunnus et al. Rev. Sci. Instrum. 83, 123109 (2012). Ph. Wernet et al. Nature 520, 78-81 (2015).
<100 fs
„16-electron catalyst“
Time-resolved RIXS at LCLS
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Resonant inelastic (x-ray) scattering
Demtröder, Laser Spectroscopy (Fig. 8.1)
X-rays (RIXS) UV/Vis
Selecting elements and sites by tuning to their core-electron resonances!
Molecule
Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
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Selecting elements and sites by tuning to their core-electron
resonances!
Core electron binding energy Chemical shift (ESCA, K. Siegbahn)
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Resonant inelastic x-ray scattering
Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
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Resonant inelastic x-ray scattering
Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
63
Selecting orbitals by tuning to the respective resonances!
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Bonding in Fe(CO)5
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RIXS of Fe(CO)5
Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
66
RIXS of Fe(CO)5
Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
2π*
5σ
dσ*
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5σ*
2π*
dσ*
dπ dπ
ΔE
Fe(CO)5 CO Fe(CO)4
2π*
5σ
dσ*
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5σ*
2π*
dσ*
dπ dπ
ΔE
Fe(CO)5 CO Fe(CO)4
2π*
5σ
dσ*
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5σ*
2π*
dσ*
dπ dπ
ΔE
Fe(CO)5 CO Fe(CO)4
2π*
5σ
dσ*
70
5σ*
2π*
dσ*
dπ dπ
ΔE
Fe(CO)5 CO Fe(CO)4
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Fe(CO)5 Fe(CO)4 fragments
Ph. Wernet, K. Kunnus, I. Josefsson, I. Rajkovic, W. Quevedo, M. Beye, S. Schreck, S. Grübel, M. Scholz, D. Nordlund, W. Zhang, R. W. Hartsock, W. F. Schlotter, J. J. Turner, B. Kennedy, F. Hennies, F. M. F. de Groot, K. J. Gaffney, S. Techert, M. Odelius, A. Föhlisch, Nature 520, 78-81 (2015).
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Fe(CO)5 Fe(CO)4 fragments Early Late
Ph. Wernet, K. Kunnus, I. Josefsson, I. Rajkovic, W. Quevedo, M. Beye, S. Schreck, S. Grübel, M. Scholz, D. Nordlund, W. Zhang, R. W. Hartsock, W. F. Schlotter, J. J. Turner, B. Kennedy, F. Hennies, F. M. F. de Groot, K. J. Gaffney, S. Techert, M. Odelius, A. Föhlisch, Nature 520, 78-81 (2015).
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RIXS from an excited state (Anti-Stokes)
RIXS from a triplet species (intersystem crossing)
RIXS from ligated species (solvent ligation)
75
Solvent ligation
76
Temporal evolution
77
Simple, conceptual conclusions
Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
78 Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
79 Book chapter in „X-Ray Free Electron Lasers“ by J. Yano, V. Yachandra, U. Bergmann (Eds.), Royal Society of Chemistry Energy and Environment Series, Ph. Wernet (2016).
Radial probability distributions in H
a0 = Bohr radius (~0.5 Å)
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Quiz part III
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What is shown here?
Ene
rgy
Inversion
3 2
1
4
Stimulated emission
1 2 3 4
82
What is the difference?
83 Reaction coordinate
Ener
gy
Ground state
Valence excited states
Core-excited state 2
Core-excited state 1
Where are the arrows for XAS and RIXS?
Delay Δt XAS
RIXS
Core-hole lifetime
84
See the atoms move
http://www.molecularmovies.com/
http://www.molecularmovies.com/
Michael Odelius, Stockholm University , Sweden
Molecular motors
Photochemistry Organic chemistry
Water
Lars Ojamäe, Linköping University , Sweden
But don‘t forget the electrons!
85
Further reading
„X-Ray Free Electron Lasers“, Royal Society of Chemistry Energy and Environment Series, J. Yano, V. Yachandra, U. Bergmann (Eds.) (2016).
Haken/Wolf Molecular Physics and Elements of Quantum Chemistry
Demtröder Laser Spectroscopy
Atkins/Friedman Molecular Quantum Mechanics
86
Some extra slides
87
Chemical and thermodynamic equilibrium 1
Chemical equilibrium:
𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃 ⇄ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 e.g.: 𝐴 + 𝐵 ⇄ 𝐶 + 𝐷
𝑅𝑅𝑃𝑅 =𝑃[𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃]
𝑃𝑃 α [𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃]
α: Proportional to
[…]: Concentration of …
e.g. 𝑅𝑅𝑃𝑅 = 𝑑[𝐶]𝑑𝑑
α 𝐴 ∙ [𝐵] (neglecting stochiometry and reaction order)
Proportionality constant 𝒌 (rate konstant):
𝑅𝑅𝑃𝑅 = 𝑘 ∙ [𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃]
The rate (the rate konkstant) quantifies the speed/the efficiency of the reaction.
88
Chemical and thermodynamic equilibrium 2
𝑅𝑅𝑃𝑅 𝑓𝑃𝑃𝑓𝑅𝑃𝑃 = 𝑅𝑅𝑃𝑅 𝑏𝑅𝑃𝑘𝑓𝑅𝑃𝑃
𝑘𝑓𝑓𝑓𝑓𝑓𝑓𝑑 ∙ 𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃 = 𝑘𝑏𝑓𝑏𝑘𝑓𝑓𝑓𝑑 ∙ [𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃]
⇒ 𝑘𝑓𝑓𝑓𝑓𝑓𝑓𝑑𝑘𝑏𝑓𝑏𝑘𝑓𝑓𝑓𝑑
= [𝑃𝑓𝑓𝑑𝑃𝑏𝑑𝑃][𝑅𝑅𝑓𝑏𝑑𝑓𝑅𝑑𝑃]
≡ 𝐾 Equilibrium constant
Thermodynamic equilibrium:
∆𝐺 = −𝑅𝑇 ∙ 𝑐𝑅𝐾
⇒ 𝐾 =[𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃][𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃]
= 𝑅−∆𝐺𝑅𝑅
with ∆𝐺 = 𝐺 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝐺(𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃)
Ener
gy
Generalized reaction coordinate
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃 ∆𝐺
89
Chemical and thermodynamic equilibrium 3
Ener
gy
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃
𝐾 = 𝑅−∆𝐺𝑅𝑅 with ∆𝐺 = 𝐺 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝐺(𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃)
∆𝐺 < 0 𝐾 ≫ 1
∆𝐺 > 0 𝐾 ≫ 1
∆𝐺 = 0 𝐾 = 1
Energy is released Energy is consumed
90
Transition state theory 1 En
ergy
Generalized reaction coordinate
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃
∆𝐺‡
𝑇𝑃𝑅𝑅𝑃𝑇𝑃𝑇𝑃𝑅 𝑃𝑃𝑅𝑃𝑅 (𝑅𝑃𝑃𝑇𝑎𝑅𝑃𝑅𝑃 𝑃𝑃𝑐𝑐𝑐𝑅𝑐)
∆𝐺‡ = 𝐺 𝑃𝑃𝑅𝑅𝑃𝑇𝑃𝑇𝑃𝑅 𝑃𝑃𝑅𝑃𝑅 − 𝐺(𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃)
𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃 ⇄ 𝑇𝑃𝑅𝑅𝑃𝑇𝑃𝑇𝑃𝑅 𝑃𝑃𝑅𝑃𝑅 → 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
𝑅𝑅𝑃𝑅 = υ ∙ [𝑇𝑃𝑅𝑅𝑃𝑇𝑃𝑇𝑃𝑅 𝑃𝑃𝑅𝑃𝑅]
υ: Crossing frequency (frequency of crossing the barrier)
Collision frequency (for treatment within collision theory)
Neglecting molecular orientation (steric effects)
91
Transition state theory 2
With 𝑅𝑅𝑃𝑅 = 𝑘 ∙ 𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃 it follows:
υ ∙ 𝑇𝑃𝑅𝑅𝑃𝑇𝑃𝑇𝑃𝑅 𝑃𝑃𝑅𝑃𝑅 = 𝑘 ∙ 𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃
Or
𝑘 = υ ∙ 𝐾 = υ ∙ [𝑅𝑓𝑓𝑅𝑃𝑇𝑑𝑇𝑓𝑅 𝑃𝑑𝑓𝑑𝑅][𝑅𝑅𝑓𝑏𝑑𝑓𝑅𝑑𝑃]
Sometime called the Eyring equation.
With 𝐾 = 𝑅−∆𝐺‡
𝑅𝑅 we arrive at:
𝑘 = υ ∙ 𝑅−∆𝐺‡
𝑅𝑅 Sometime called the Eyring equation.
92
Transition state theory 3 En
ergy
Generalized reaction coordinate
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
𝑅𝑅𝑅𝑃𝑃𝑅𝑅𝑃𝑃
Connecting with the Arrhenius equation
𝑘 = 𝐴 ∙ 𝑅−𝐸𝑎𝑅𝑅
𝐸𝑓 is the activation energy.
What is 𝐴 (besides an experimental parameter)?
Does 𝐴 have a physical meaning?
Can 𝐸𝑓 be calculated?
𝐸𝑓
Transition state theory gives answers:
• 𝐴 could be the collision frequency: 𝐴 = υ
• 𝐴 could be the collosion frequency including a factor 𝜌 accounting for steric
effects (such as the relative orientation of molecules): 𝐴 = υ ∙ 𝜌
• 𝐸𝑓 = ∆𝐺‡ can be calculated
93
Transition state theory 4 Limitations
• Its original goal was to calculate absolute rate konstants („absolute-rate theory“).
• TST turned out to be more successful in calculating the thermodynamic properties of
the transition state from measured rate constants (calculating the Gibbs energy ∆𝐺‡ as
well as the enthalpy and entropy).
• TST neglects the possibility of tunneling through the barrier (it assumes that the
reaction does not occur unless particles collide with enough energy to form the
transition structure).
• TST can fail for high temperatures when high vibrational modes are populated and
transition states far from the lowest energy saddle point are formed.
• TST assumes that intermediates (reactants and products, see above, of elementary
steps in a multi-step reaction) are long-lived (reaching a Boltzmann distribution of
energies) and thus TST fails for short-lives intermediates.
• TST generally fails for photochemical reactions (that are determined by the energy
potential landscape rather than the thermodynamics properties of TSs).
94
Femtochemistry (A. Zewail)
95
96
Important technical terms (see EXTRA SLIDES for more detailed explanations)
• Reaction intermediate (next slide) • Transition state (next slide) • Transient species (next slide) • Intramolecular vibrational redistribution (IVR)
– Redistribution of vibrational energy within one electronic state of the molecule
• Internal conversion (IC) – Conversion from electronic into vibrational energy within one molecule (vibronic coupling)
• Intersystem crossing (ISC) – Change of multiplicity of the molecule (e.g. change from singlet to triplet)
See the IUPAC (International Union of Pure and Applied Chemistry) Gold Book: http://goldbook.iupac.org/index.html See the review paper by I. Hertel and W. Radloff: Rep. Prog. Phys. 69, 1897 (2006).
97 Reaction coordinate
Pote
ntia
l ene
rgy
kT
Intermediate state
Transition state („activated complex“)
Transient configurations
Transient configurations
Pump
Probe
…and phenomena that happen on their way (IVR, IC, ISC, …)
Reaction intermediates, transition states and transient configurations
98
Reaction intermediates
Reaction intermediate / intermediate / reactive intermediate:
• Most chemical reactions occur in a step wise fashion with more than one elementary step
• Reaction intermediate = – Molecular entity formed from the reactants or preceding intermediates in each step (except for
the last one)
– With a lifetime appreciably longer than a molecular vibration corresponding to a local potential
energy minimum of depth greater than kT
– Reacts further to give the directly observed products of the reaction (last step)
• Are usually short lived and seldom isolated • Need to be distinguished from transition states (through their lifetime)
99
Transition states
Transition states: • The state through which the molecule must pass as it changes from reactants or
intermediates to products or another set of intermediates • The state corresponding to the highest energy along the corresponding reaction
coordinate (nuclear distance e.g.) • The molecules goes through transient states / transient species / transient
configurations as they pass the transition state • This often involves bond breaking and making • See also transition state theory, transition state method, activated complex (slightly
different from the transition state)
See also:
100
Intramolecular vibrational redistribution (IVR)
• Redistribution of vibrational energy within one electronic state of the molecule
• IVR leads from the initially populated vibrational states (e.g. a particular stretch vibration) to a broad distribution of vibrational energy equilibrated over all vibrational degrees of freedom (e.g. all stretch and bend vibrations)
• Usually occurs on a time scale of ps to ns
• Can be described by coupled anharmonic oscillators
which exchange energy
• IVR is a sequence of radiationless transitions
• IVR leads to the fact that the fluorescence is
red-shifted compared to the absorption (see figure)
• If the molecule is embedded in a medium (e.g. a liquid),
IVR is faster than in the isolated molecule and leads to
a thermalization with respect to the temperature of the
medium
radiationless
Photon energy
Fluorescence
101
Internal conversion (IC)
• Conversion from electronic into vibrational energy within one molecule • Radiationless transition within one molecule • Usually occurs on a time scale of fs to ps • Mediated by vibronic coupling (see figure) between different electronic states of
the same molecule
Model for vibronic coupling
102
Intersystem crossing (ISC)
• Change of multiplicity of the molecule (e.g. change from singlet to triplet) • Radiationless transition within one molecule • Usually occurs on a time scale of ps to µs • Mediated by spin-orbit coupling (often/fast when heavy atoms are present) • Often results in phosphorescence (see figures comparing fluorescence and
phosphorescence)
From: Atkins, Friedman, Molecular quantum mechanics
103
Environments: Where chemical dynamics occurs • Gas phase
– Low concentrations (1010-1013 molecules/cm3) limits the set of methods
– Elementary reactions, “simple” systems, isolated systems
– Hard to get certain molecules into the gas phase (biomolecules e.g.)
• Liquids – Solutes (note the higher concentrations than in the gas phase: 1 mol/l corresponds to
approximately 1020 molecules/cm3)
– Solvents (hydrogen-bond dynamics, collective phenomena)
– Their interactions
• Bio-molecules – Special case (depending on sample preparation in the gas phase, in solution or in solids)
• Surfaces/interfaces – Often treated in a specialized community due to very different sample preparation in
vacuum
– Very few x-ray methods applied to chemical dynamics on surfaces (W. Wurth/Hamburg, A.
Nilsson/Stanford)
• Rarely: In solids (e.g. crystals, rare gas matrices)
104
Other methods (not using x-rays)
• “Laser femtochemistry” Sets the stage! – Pump-probe methods with laser/optical/IR/THz wavelengths
– Detecting ions, fluorescence etc.
• Laser photoelectron spectroscopy Strong overlap with x-ray PES – Special case of “laser femtochemistry”
• Electron scattering Hard to get fs pulses but high spatial resolution! – Gas phase
– Surfaces
• fs-IR spectroscopy Strong overlap with XAS/EXAFS (in terms of insight) – Liquids
– Surfaces
Why x-rays?
105
Why x-rays? Why x-rays for studying chemical dynamics?
• X-ray absorption spectroscopy (XAS/XANES/NEXAFS) and Extended X-ray Absorption Fine Structure (EXAFS)
– Element and chemical state selective (“chemical shift”)
– Local (starts from a core level which is well localized in space)
– Symmetry sensitive (dipole selection rule implies sensitivity to orbital symmetry)
– Combines sensitivity to electronic and geometric structures
• X-ray scattering – Structural probe with atomic scale resolution (both in time and space)
– Global view (e.g. solute and solvent)
• Photoelectron spectroscopy (PES) – See XAS/EXAFS
– In contrast to laser PES: Reach all possible ionic final states (complete picture)
• Resonant Inelastic X-ray Scattering (RIXS) – See XAS/EXAFS
Study fundamental aspects of elementary reactions
106
X-ray methods: How to probe chemical dynamics
• X-ray absorption spectroscopy (XAS/XANES/NEXAFS) and Extended X-ray Absorption Fine Structure (EXAFS) Mostly hard x-rays
– Liquids
– Bio-molecules
– [Surfaces/interfaces]
• X-ray scattering Exclusively hard x-rays – Liquids
– [Gas phase, future at FELs (holography from within)?]
• Photoelectron spectroscopy (PES) Mostly laser and VUV wavelengths – Gas phase
– Surfaces/interfaces
– [Liquids]
• Resonant Inelastic X-ray Scattering (RIXS) Well suited for liquids – Liquids
– Solids/surfaces/interfaces
– Not yet for gas phase and bio-molecules (target density)