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

E [e

V]

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

X-ray Photons Probing UV Photodynamics

Possible mechanisms: 1. [FeIII(C2O4)3]3−

hυ[FeIII C2O4)3

3− ∗→ [(• C2O4)FeII (C2O4)2]3−

2. [FeIII(C2O4)3]3−hυ

[FeIII C2O4)33− ∗

→ [FeIII(C2O4)3]2− + esolv

3. [FeIII(C2O4)3]3−hυ

[FeIII C2O4)33− ∗

→ [FeIII(C2O4)2]− + 2CO2•−

4. [FeIII(C2O4)3]3− S = 52

hυ[FeIII C2O4)3

3− ∗S = 5

2→ [FeIII(C2O4)3]3− S = 3

2

5. [FeIII(C2O4)3]3− S = 52

hυ[FeIII C2O4)3

3− ∗S = 5

2→ [FeIII(C2O4)3]3− S = 1

2

2FeIII(C2O4)33− 2FeII(C2O4)2

2− +2CO2 + C2O4

2− ?

not very likely

small quantum yield

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

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).

X-rays: Radiation Damage

X-ray Photons as Reactants

Specific bond cleavage

X-ray Photons as Reactants

Radiosenzitization and activation of nanoparticles

Formation of new species in astrochemical environments

X-ray photochromism

X-ray photoreduction

X-ray Photons as Reactants

X-ray uncaging

Understanding Spectroscopy

Isotope effects in XES Isotope effects in Auger

Molecules and Radiation

σ

E

E02

E01

S0

D0

D1

hν

Molecules and High Energy Radiation

Photoexcitation Photoionization Auger decay X-ray fluorescence

Large number of electronic states

New processes

Nuclear motion

Non-adiabatic transitions

Autoionization

Fluorescence ≈

≈

Ener

gy

hν

decay: e− (or hν)

M++

t0

M*+

M

Molecules and High Energy Radiation

Absorption

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)

Exact Solution of Schrodinger Equation

( ) ( ) ( )( ) ( ), , ˆ ˆ, , , ,Coulomb

r R ti T r R V r R r R t

t∂Ψ

= + Ψ∂

Limitations to several particles (including electrons)

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 eI It tχΨ = Ψr R R r R

( )e e e eI I IH E RΨ = Ψ

( ( )) ( , )

( , )

N eI I

I

T E R R t

i R tt

χ

χ

+ =∂∂

t=0

t

Born-Oppenheimer approximation

Potential Energy Surface

Calculating vibrational states

(T N + EIe )χ I = ET χ I

Wavepacket evolution

Quantum and Classical (Adiabatic) Dynamics

Classical dynamics approximates quantum dynamics ddt m

=R P

eIdd

dE

t d= −

P (R)R

Quantum Dynamics

Solving by e.g. expansion into a basis

We need (initial) positions and momenta!

Wigner distribution function, definition

yetyxtyxtpxipy

w d),(),(1),,(2

*

−+= ∫

∞

∞−

ψψπ

ρ

qetqptqptpxiqx

w d),(),(1),,(2

*

−∞

∞−

−Φ+Φ= ∫πρ

Properties of Wigner distribution function 1. It is a real function 2. 3.

2),(d),,( txptpxw ψρ =∫∞

∞−

2),(d),,( tpxtpxw Φ=∫∞

∞−

ρ

ψχρρ ψχ |d),,(),,( =∫∞

∞−

xdptpxtpx ww

It almost behaves as a probability distribution function …..however, WD can be negative Positive distributions exist (Husimi)

Quasiclassical Approach: Wigner distribution

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

Semiclassical Description: Wavepacket Dispersion

The process is essentially 1D in this case

1D quantum description vs. fully dimensional semiclassical description

H2O

D2O

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

Beyond Born-Oppenheimer

1( )

1 ( 2 )2

N II eI I

NIJ IJ

J J IJ I

T K E

K it

χµ

χ χ χµ≠

+ + +

∂+ − ⋅ ∇ + =

∂∑ f

fαIJ (R) = Ψ

I

e ∇αΨJ

e

r

k IJ (R) = ΨI

e ∇2ΨJ

e

r

Derivative couplings connects the different electronic states

If the expansion is not truncated the wavefunction is exact since the set ΨI

e is complete.

N

I

eII

Ta

Ψ=Ψ ∑=1

);()(),( RrRRr χ

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

Semiclassical Approach: Surface Hopping

( ) ( ) ( )( )1

, , ,SN

k kk

t c t tφ=

Ψ = ∑r R r R

( ) ( ) ( ) ( )( ) ( )1k c ck j kj

dc ti E t c t R t t

dt−= − − ⋅∑ F v

( )**

2max 0, Re c cl k l k kl

l l

tP c cc 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 ) = 𝐸𝐸𝑘𝑘(𝑅𝑅 𝑡𝑡 )∅𝑘𝑘(𝒓𝒓,𝑹𝑹 𝑡𝑡 )

The total wavefunction can then be expanded

Quantum nucleus

Classical nuclei

Swarm of trajectories

Ab initio energies

Statistical evaluation!

Classical approximation

Semiclassical Approach: Surface Hopping

Semiclassical Approach: Ehrenfest method

( ) ( ) ( )( )1

, , ,SN

k kk

t c t tφ=

Ψ = ∑r R r R

( ) ( ) ( ) ( )( ) ( )1k c ck j kj

dc ti E t c t R t t

dt−= − − ⋅∑ 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 ) = 𝐸𝐸𝑘𝑘(𝑅𝑅 𝑡𝑡 )∅𝑘𝑘(𝒓𝒓,𝑹𝑹 𝑡𝑡 )

The total wavefunction can then be expanded

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

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

Different reaction channels

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

Example: Water Ionization

Can We Use Dynamics on the Ground State?

Is internal conversion fast enough?

Ammonia dimer Electronic populations

Proton transfer population

Treating Highly Excited States

Highly excited states are formed within ICD

Real time electronic propagation

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

Complex Hamiltonian

Random hops into the final state

For example: Instantaneous Auger spectrum

Monte Carlo Wavepackets method

Including Decaying States

Including Decaying States

Uncertainity principle

Missing Interferences

Auger spectra (Non-resonant) XES

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

Including Decaying States

Classical trajectories with a phase

Coupling to Continuum: Fano Theory

Decaying state

Continuum of final states

Fano profile

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

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

HF neutral state Initial state Final state

Decay Rates With Quantum Chemistry

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

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

Autoionization within RT-TDDFT

Overcoming exponential bottleneck with DFT

Can TDDFT describe properly auto-ionization?

Problem: Adiabatic approximation

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”

Electronic Force Field Approach

Su and Goddard: “Classical” electron

Modelling Auger processes

Spectroscopy Signatures of Nuclear Motion

X-ray photoemission: Water and ice

Auger Spectra of Liquid Water

Bernd Winter, BESSY Berlin

Normal Auger

Spectator Auger

Delocalized states of H2O+…H2O+ type?

Auger Spectra of Liquid Water: Solvent Effects

Photoelectrons Auger electrons

Auger Spectra of Liquid Water: Isotope Effects

Auger Spectra of Liquid Water

Bernd Winter, BESSY Berlin CDFT calculations

Strong isotope dependence of the fast electron peak

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.

+

Ultrafast Proton Transfer on Core Ionized State

Flat PES of core ionized state Dispersion of the wavepacket

Morrone, Car PRL 101 (2008) 017801

Calculated Auger Spectrum

Experiment Calculations

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

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

Entangled Electron and Nuclear Dynamics

Ammonium cation… …with double proton transfer

Double proton transfer observed

Entangled Electron and Nuclear Dynamics

Probing strength of hydrogen bond

Liquid Structure via PTM-CS

Closing and Opening ICD by Nuclear Motion

Summary

Promoting molecule into excited state Time evolution on single PES Population transfer between electronic states Coupling to continuum Follow up dynamic

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