**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* **

**S**_{2 }

**S**_{0 }

**hydrogen transfer ** **HONO rotation **

**5.45 eV **

**0.0 eV **

**1.17 eV ** **3.20 eV **

**0.90 eV ** **1.15 eV **

**n → π* **

**S**_{1 }^{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. [Fe^{III}(C_{2}O_{4})_{3}]^{3− hυ}[Fe^{III} C_{2}O_{4})_{3}^{3− ∗}→ [(• C_{2}O_{4})Fe^{II} (C_{2}O_{4})_{2}]^{3−}

2. [Fe^{III}(C_{2}O_{4})_{3}]^{3− hυ}[Fe^{III} C_{2}O_{4})_{3}^{3− ∗}→ [Fe^{III}(C_{2}O_{4})_{3}]^{2−} + e_{solv}
3. [Fe^{III}(C_{2}O_{4})_{3}]^{3− hυ}[Fe^{III} C_{2}O_{4})_{3}^{3− ∗}→ [Fe^{III}(C_{2}O_{4})_{2}]^{−} + 2CO_{2}^{•−}

4. [Fe^{III}(C_{2}O_{4})_{3}]^{3−} S = ^{5}_{2} ^{hυ}[Fe^{III} C_{2}O_{4})_{3}^{3− ∗} S = ^{5}_{2} → [Fe^{III}(C_{2}O_{4})_{3}]^{3−} S = ^{3}_{2}
5. [Fe^{III}(C_{2}O_{4})_{3}]^{3−} S = ^{5}_{2} ^{hυ}[Fe^{III} C_{2}O_{4})_{3}^{3− ∗} S = ^{5}_{2} → [Fe^{III}(C_{2}O_{4})_{3}]^{3−} S = ^{1}_{2}

2Fe^{III}(C_{2}O_{4})_{3}^{3−} 2Fe^{II}(C_{2}O_{4})_{2}^{2−}

+2CO_{2} + C_{2}O_{4}^{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**

**E**_{0}

**2 **

**E**_{0}

**1 **

**S**_{0 }**D**_{0 }**D**_{1 }

* h*ν

**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 **

**≈ **

**Ene****rgy** **≈ **

**hν **

**decay: e**^{− }**(or hν)**

**M**^{++}

**t**_{0 }

**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 **

### (

^{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) **

**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*

*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 **

**Quantum and Classical (Adiabatic) Dynamics **

**Classical dynamics approximates quantum dynamics **

d

*dt** R* =

*m*

**P**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!

**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 **

**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 **

**H**_{2}**O **

**D**_{2}**O **

**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* *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* ∇_{α}Ψ

*J*

*e*
**r**

*k*^{IJ}**(R)** = Ψ

*I*

*e* ∇^{2}Ψ

*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**

**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

, , ,

*N**S*

*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 **

Quantum

nucleus Classical nuclei

Swarm of trajectories

*Ab initio energies*

**Statistical evaluation!**

Classical approximation

**Semiclassical Approach: Surface Hopping **

**Semiclassical Approach: Ehrenfest method **

### ( ) ( ) ( ( ) )

1

, , ,

*N**S*

*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 **

**Example: Water Ionization **

**. ** **. ** **. ** **. **

1b_{1 }
3a_{1 }

2a_{1 }
1b_{2 }

1a_{1 }

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

H_{3}O^{+ }+
OH^{• }

H^{+}

transfer

H_{3}O^{+ }+
OH^{• }
H_{2}O^{•+ }+
H_{2}O

**Inspection of PES ** **MD simulation **

Initial state: D_{2 } Initial state: D_{3 }

**DONOR** **ACCEPTOR**

**H**_{3}**O**^{+ }**+ OH**^{●}
**(H**_{2}**O)**_{2 }**3a**_{1}

**H**_{2}**O**^{+ }**+ H**_{2}**O **
**(H**_{2}**O)**_{2 }**3a**_{1}

**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 **

H_{2}CO 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 H**_{2}**O**^{+}**…H**_{2}**O**^{+}** 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: H**_{2}**O**^{2+}** **

**PTM-Auger: H**_{3}**O**^{+ }**… OH**^{+}** **
**ICD, PTM-ICD: H**_{2}**O**^{•+ }**… H**_{2}**O**^{•+}** **

**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 **