X-ray Photodynamics

(1)

Petr Slavíček

Department of Physical Chemistry, University of Chemistry and Technology, Prague and

Jaroslav Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic

X-ray Photodynamics

(2)

E [eV]

π → π*

S₂

S₀

hydrogen transfer HONO rotation

5.45 eV

0.0 eV

1.17 eV 3.20 eV

0.90 eV 1.15 eV

n → π*

S₁ ^{4.15 eV}

2.81 eV

Computational Photodynamics

(SA3-6/5 MRCI / 6-31g*, dynamics at CASSCF level)

HONO dissociation

(3)

X-ray Photons Probing UV Photodynamics

Possible mechanisms:

1. [FeÎII(C₂O₄)₃]^{3− hυ}[FeÎII C₂O₄)₃^{3− ∗}→ [(• C₂O₄)FeÎI (C₂O₄)₂]³⁻

2. [FeÎII(C₂O₄)₃]^{3− hυ}[FeÎII C₂O₄)₃^{3− ∗}→ [FeÎII(C₂O₄)₃]²⁻ + e_solv 3. [FeÎII(C₂O₄)₃]^{3− hυ}[FeÎII C₂O₄)₃^{3− ∗}→ [FeÎII(C₂O₄)₂]⁻ + 2CO₂^•−

4. [FeÎII(C₂O₄)₃]³⁻ S = ⁵₂ ^hυ[FeÎII C₂O₄)₃^{3− ∗} S = ⁵₂ → [FeÎII(C₂O₄)₃]³⁻ S = ³₂ 5. [FeÎII(C₂O₄)₃]³⁻ S = ⁵₂ ^hυ[FeÎII C₂O₄)₃^{3− ∗} S = ⁵₂ → [FeÎII(C₂O₄)₃]³⁻ S = ¹₂

2Fe^III(C₂O₄)₃³⁻ 2Fe^II(C₂O₄)₂²⁻

+2CO₂ + C₂O₄²⁻

?

not very likely small quantum yield

Struct. Dyn. 2, (2015), 034901.

(4)

Radiation Chemistry

Under the action of ionizing radiation, without specification of mechanism

𝐺𝐺 = Number of molecules 100 𝑒𝑒𝑒𝑒

The science of chemical effects brought about by the absorption of ionizing radiation in matter, mainly due to electronic processes (different from radiochemistry).

(5)

X-rays: Radiation Damage

(6)

X-ray Photons as Reactants

Specific bond cleavage

(7)

X-ray Photons as Reactants

Radiosenzitization and activation of nanoparticles

Formation of new species in astrochemical environments X-ray photochromism

X-ray photoreduction

(8)

X-ray Photons as Reactants

X-ray uncaging

(9)

Understanding Spectroscopy

Isotope effects in XES Isotope effects in Auger

(10)

Molecules and Radiation

σ E

E₀

2

E₀

1

S₀ D₀ D₁

hν

(11)

Molecules and High Energy Radiation

Photoexcitation Photoionization Auger decay X-ray fluorescence

Large number of electronic states New processes

(12)

Nuclear motion

Non-adiabatic transitions

Autoionization Fluorescence

≈

Energy ≈

hν

decay: e⁻(or hν)

M⁺⁺

t₀

M^*+

M

Molecules and High Energy Radiation

Absorption

(13)

Secondary Processes: The Main Channel

Primary versus secondary processes in water

Auger Cascade

Stupmf, Gokhberg, Cederbaum Nature Chemistry 8, 237–241 (2016)

Slavicek, Kryzhevoi, Aziz, WinterJ. Phys. Chem. Lett., 7 , 234–243 (2016)

(14)

Exact Solution of Schrodinger Equation

(

^{r R t}^{, ,}

) (

^ˆ

( )

^, ^ˆ^Coulomb

( )

^,

) ⁽

^{, ,}

⁾

i T r R V r R r R t

t

∂Ψ = + Ψ

∂

 

  

Limitations to several particles (including electrons)

(15)

Solving Problem Step by Step

Promoting molecule into excited state Time evolution on single PES

Population transfer between electronic states Coupling to continuum

Follow up dynamic

(16)

Quantum (Adiabatic) Dynamics

( , , ) ( , ) ( ; )

T e

I I

t χ t

Ψ r R = R Ψ r R

e e e( ) e

I I I

H Ψ = E R Ψ

( ( )) ( , )

( , )

N e

I I

I

T E R R t

i R t

t

χ χ

+ =

∂

t=0

t

Born-Oppenheimer approximation

Potential Energy Surface

Calculating vibrational states

(T

^N

+ E

_I^e

) χ

_I

= E

^T

χ

_I

Wavepacket evolution

(17)

Quantum and Classical (Adiabatic) Dynamics

Classical dynamics approximates quantum dynamics

d

dtR = mP

e

d I

d d

E t = − d

P (R)

R

Quantum Dynamics

Solving by e.g. expansion into a basis

We need (initial) positions and momenta!

(18)

Wigner distribution function, definition

y e

t y x t

y x t

p x

ipy

w 1 ( , ) ( , ) d

) , , (

2

* 

 + −

=

∫

^∞

∞

−

ψ π ψ

ρ

q e

t q p t

q p t

p x

iqx

w 1 ( , ) ( , ) d

) , , (

2

* 



∞ −

∞

−

− Φ +

Φ

= _π

∫

ρ

Properties of Wigner distribution function 1. It is a real function

2.

3.

)2

, ( d

) , ,

(x p t p x t

w ψ

ρ =

∫

∞

−

)2

, ( d

) , ,

(x p t x p t

w = Φ

∞

∫

∞

−

ρ

ψ χ ρ

ρ^χ( , , ) ^ψ ( , , )d = |

∞∫

∞

−

xdp t

p x t

p

x _w

w

It almost behaves as a

probability distribution function

…..however, WD can be negative Positive distributions exist (Husimi)

Quasiclassical Approach: Wigner distribution

(19)

Equation of motion for the Wigner distribution

From which we can derive underlying “Hamilton” equations

Hamilton function is no more a constant along the trajectory

Quasiclassical Approach: Wigner distribution

(20)

Semiclassical Description: Wavepacket Dispersion

The process is

essentially 1D in this case

1D quantum description vs. fully dimensional

semiclassical description

H₂O

D₂O

(21)

How to Get Potential Energy Surface?

Variational collapse for highly excited states Low-lying excited states

Whole plethora of available methods

Selecting proper states e.g. by Maximum Overlap Method

Preselected excitations

RAS-SCF method, TDDFT approach…

Thrifty solution: Z+1 method

Simulating (slightly) different system

(22)

Beyond Born-Oppenheimer

( 1 )

1 ( 2 )

2

N II e

I I

N

IJ IJ

J J I

J I

T K E

K i

t

µ χ

χ χ χ

≠

µ

+ + +

+ − ⋅ ∇ + = ∂

∑

^f ∂ f_α^IJ (R) = Ψ

I

e ∇_αΨ

J

e r

k^IJ (R) = Ψ

I

e ∇²Ψ

J

e r

Derivative couplings connects the different electronic states

If the expansion is not truncated the wavefunction is exact since the set Ψ_I^e is complete.

N

I

e I I

T

a

Ψ

=

Ψ

∑

=1

)

; ( )

( )

,

(r R χ R r R

(23)

For real wavefunctions

The derivative coupling is inversely proportional to the energy difference of the two electronic states. Thus the smaller the difference, the larger the coupling. If ∆E=0 f is infinity.

Beyond Born-Oppenheimer

(24)

Semiclassical Approach: Surface Hopping

( ) ( ) ( ( ) )

1

, , ,

NS

k k

k

t c t φ t

=

Ψ ^{r R} =

∑

^{r R}

( )

¹

( ) ( ) ( ( ) ) ^{( )}

k c c

k j kj

d c t

i E t c t R t t

d t

= − ^− −

∑

^F ⋅^v

( )

^*

*

max 0, 2 Re ^c ^c

l k l k kl

l l

P t c c

→ c c

 ∆ 

=  − ⋅ 

 F v 

With know trajectory R(t), at each point, we solve electron SE

Solving TDSE

Random hops reflecting the electronic population 𝐻𝐻�_𝑒𝑒∅_𝑘𝑘(𝒓𝒓, 𝑹𝑹 𝑡𝑡 ) = 𝐸𝐸_𝑘𝑘(𝑅𝑅 𝑡𝑡 )∅_𝑘𝑘(𝒓𝒓, 𝑹𝑹 𝑡𝑡 ) The total wavefunction can then be expanded

(25)

 Quantum

nucleus Classical nuclei

Swarm of trajectories

Ab initio energies

Statistical evaluation!

Classical approximation

Semiclassical Approach: Surface Hopping

(26)

Semiclassical Approach: Ehrenfest method

( ) ( ) ( ( ) )

1

, , ,

NS

k k

k

t c t φ t

=

Ψ ^{r R} =

∑

^{r R}

( )

¹

( ) ( ) ( ( ) ) ^{( )}

k c c

k j kj

d c t

i E t c t R t t

d t

= − ^− −

∑

^F ⋅^v

With know trajectory R(t), at each point, we solve electron SE

Solving TDSE

Moving on average potential

𝐻𝐻�_𝑒𝑒∅_𝑘𝑘(𝒓𝒓, 𝑹𝑹 𝑡𝑡 ) = 𝐸𝐸_𝑘𝑘(𝑅𝑅 𝑡𝑡 )∅_𝑘𝑘(𝒓𝒓, 𝑹𝑹 𝑡𝑡 ) The total wavefunction can then be expanded

(27)

Example: Water Ionization

. . . .

1b₁ 3a₁

2a₁ 1b₂

1a₁

12.6 eV 14.8 eV 18.6 eV

32.6 eV

541 eV Inner valence electrons

Core electrons Valence electrons

(28)

Example: Water Ionization

Svoboda, Hollas, Ončák, Slavíček, PhysChemChemPhys. 15 (2013) 11531.

H⁺

transfer

H₃O⁺+ OH^•

H⁺

transfer

H₃O⁺+ OH^• H₂O^•++ H₂O

Inspection of PES MD simulation

Initial state: D₂ Initial state: D₃

DONOR ACCEPTOR

H₃O⁺+ OH^● (H₂O)₂ 3a₁

H₂O⁺+ H₂O (H₂O)₂ 3a₁

(29)

Different reaction channels

Svoboda, Hollas, Ončák, Slavíček, PhysChemChemPhys. 15 (2013) 11531.

Example: Water Ionization

(30)

Can We Use Dynamics on the Ground State?

Is internal conversion fast enough? Ammonia dimer

Electronic populations

Proton transfer population

(31)

Treating Highly Excited States

Highly excited states are formed within ICD

Real time electronic propagation

(32)

Electronic Dynamics with TDDFT

Real-time TDDFT: propagating electronic densities

Involving laser pulse or non-equilibrium initial state

Time-dependent functional Adiabatic approximation

Coupling with nuclear motion: Ehrenfest dynamics

Example: Charge migration in glycine

Lara-Astiaso et al., Farad. Disc. 2016, DOI: 10.1039/C6FD00074F

(33)

Complex Hamiltonian

Random hops into the final state

For example: Instantaneous Auger spectrum

Monte Carlo Wavepackets method

Including Decaying States

(34)

Including Decaying States

Uncertainity principle

Missing Interferences

Auger spectra (Non-resonant) XES

Carrol and Thomas, J. Chem. Phys. 86, 5221 (1987)

(35)

Including Decaying States

Classical trajectories with a phase

(36)

Coupling to Continuum: Fano Theory

Decaying state

Continuum of final states

(37)

Fano profile

(38)

Simplest Case: Hartree-Fock Wavefunction

Initial state represented by a single Slater determinant Final state

Decay width

represented by N−1 electron function and outgoing electron wave

(39)

Decay Rates With Quantum Chemistry

Technology of quantum chemistry is developed, including excited states

Yet the spectra are (i) discrete (ii) they have wrong normalization

(40)

HF neutral state Initial state Final state

Decay Rates With Quantum Chemistry

(41)

Solution: Stieltjes imaging

Technology of quantum chemistry is developed, including excited states

Yet the spectra are (i) discrete (ii) they have wrong normalization

Analogic procedure for decay widths

Decay Rates With Quantum Chemistry

(42)

Non-Hermitian Quantum Chemistry

Quantum chemistry with complex absorbing potential

Imaginary potential is placed on the boundaries of the system

Extrapolation to zero CAP

The energies of metastable state are complex

(43)

Autoionization within RT-TDDFT

Overcoming exponential bottleneck with DFT Can TDDFT describe properly auto-ionization?

Problem: Adiabatic approximation

(44)

Thrifty Estimate of the Lifetime: Population Analysis

Auger decay is dominated by on-site processes

Cooking the Auger intensities based on the atomic contributions

H₂CO Relative importance of

different channels

“Core hole clock”

(45)

Electronic Force Field Approach

Su and Goddard: “Classical” electron

Modelling Auger processes

(46)

Spectroscopy Signatures of Nuclear Motion

X-ray photoemission: Water and ice

(47)

Auger Spectra of Liquid Water

Bernd Winter, BESSY Berlin

Normal Auger

Spectator Auger

Delocalized states of H₂O⁺…H₂O⁺ type?

(48)

Auger Spectra of Liquid Water: Solvent Effects

(49)

Photoelectrons Auger electrons

Auger Spectra of Liquid Water: Isotope Effects

(50)

Auger Spectra of Liquid Water

Bernd Winter, BESSY Berlin CDFT calculations

Strong isotope dependence of the fast electron peak

(51)

Electron and Nuclear Dynamics in Water

Normal Auger

Intermolecular Coulomb Decay (ICD)

Proton Transfer Mediated Charge Transfer (PTM-CS)

Thürmer, Ončák, Ottosson, Seidel, Hergenhahn, Bradforth, Slavíček, Winter, Nat. Chemistry 5 (2013) 590.

(52)

+

Ultrafast Proton Transfer on Core Ionized State

Flat PES of core ionized state Dispersion of the wavepacket

Morrone, Car PRL 101 (2008) 017801

(53)

Calculated Auger Spectrum

Experiment Calculations

Slavíček, Winter, Cederbaum, Kryzhevoi, JACS, 136 (2014) 18170.

(54)

Various final states Auger: H₂O²⁺

PTM-Auger: H₃O⁺… OH⁺ ICD, PTM-ICD: H₂O^•+… H₂O^•+

Time evolution Relaxation processes

at different snapshots

Slavíček, Winter, Cederbaum, Kryzhevoi, JACS, 136 (2014) 18170.

Entangled Electron and Nuclear Dynamics

(55)

Entangled Electron and Nuclear Dynamics

Ammonium cation… …with double proton transfer

(56)

Double proton transfer observed

Entangled Electron and Nuclear Dynamics

(57)

Probing strength of hydrogen bond

Liquid Structure via PTM-CS

(58)

Closing and Opening ICD by Nuclear Motion

(59)

Summary

Promoting molecule into excited state Time evolution on single PES

Population transfer between electronic states Coupling to continuum

Follow up dynamic

(60)

Experiment

U. Hergenhahn, MPI Garching B. Winter, BESSY Berlín

Acknowledgement

Theoretical Photodynamics Group

Theory

Nikolai Kryzhevoi, Heidelberg Lenz Cederbaum, Heidelberg

Nicolas Sisourat, Paris Daniel Hollas Jan Chalabala Eva Muchová

X-ray Photodynamics

Full text

Petr Slavíček

X-ray Photodynamics

Computational Photodynamics

X-ray Photons Probing UV Photodynamics

?

Radiation Chemistry

X-rays: Radiation Damage

X-ray Photons as Reactants

X-ray Photons as Reactants

X-ray Photons as Reactants

Understanding Spectroscopy

Molecules and Radiation

Molecules and High Energy Radiation

Molecules and High Energy Radiation

Secondary Processes: The Main Channel

Exact Solution of Schrodinger Equation

(

) (

( )

( )

) (

)

Solving Problem Step by Step

Promoting molecule into excited state Time evolution on single PES

Population transfer between electronic states Coupling to continuum

Follow up dynamic

Quantum (Adiabatic) Dynamics

(T

+ E

) χ

= E

χ

Quantum and Classical (Adiabatic) Dynamics

∫

∫

∫

∫

Quasiclassical Approach: Wigner distribution

Quasiclassical Approach: Wigner distribution

Semiclassical Description: Wavepacket Dispersion

How to Get Potential Energy Surface?

Beyond Born-Oppenheimer

µ χ

χ χ χ

µ

∑

∑

Beyond Born-Oppenheimer

Semiclassical Approach: Surface Hopping

( ) ( ) ( ( ) )

∑

( )

( ) ( ) ( ( ) ) ( )

∑

( )

Semiclassical Approach: Surface Hopping

Semiclassical Approach: Ehrenfest method

( ) ( ) ( ( ) )

∑

( )

( ) ( ) ( ( ) ) ( )

∑

Example: Water Ionization

. . . .

Example: Water Ionization

Example: Water Ionization

Can We Use Dynamics on the Ground State?

Treating Highly Excited States

Highly excited states are formed within ICD

Real time electronic propagation

Electronic Dynamics with TDDFT

Including Decaying States

Including Decaying States

Including Decaying States

Coupling to Continuum: Fano Theory

Fano profile

Simplest Case: Hartree-Fock Wavefunction

Decay Rates With Quantum Chemistry

) ⁽

⁾

( ) ( ) ( ( ) ) ^{( )}

( ) ( ) ( ( ) ) ^{( )}