�

ROSARIO, 24 NOVIEMBRE 2023

�JORGE KOHANOFF

INSTITUTO DE FUSION NUCLEAR “GUILLERMO VELARDE”

UNIVERSIDAD POLITECNICA DE MADRID

SPAIN

ELECTRONIC STOPPING IN

LIQUID WATER, VAPOR, ICE,

AND SOLVATED DNA�

CONTENTS

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• Motivation and background
• Stopping power calculations via real-time TDDFT
• Trajectory sampling: liquid water
• Additivity: water vapor
• Electronic-mediated nuclear stopping: ice
• Solvation effects: DNA
• Stopping of electrons in water
• Summary

IRRADIATION: �A MULTI-SCALE PHENOMENON

• Physical stage (~fs to ns)
• Ionization: secondary electrons and holes. Excitations.
• Electron, hole and excitation transport: elastic and inelastic.
• Electron capture, impact ionization, recombination, …
• Chemical stage (~ps to ms)
• Bond breaking and formation
• Heating, diffusion
• Damage
• Biological/engineering scale (~ns to hours/days-years)
• Functional and structural modifications, phase transformations.
• Failure: death, fracture, explosion.

THE PHYSICAL STAGE

Electronic excitation and ionisation

ρ(r,t’’)

V(r,t’’)

ρ(r,t=0)

ρ(r,t)

V*(r,t)

Inelastic electron transport

Non-adiabatic dynamics: TDDFT, Ehrenfest and beyond

MODELLING THE PHYSICAL STAGE

• In principle this requires non-adiabatic quantum dynamics:
• Excited electrons
• Electron-phonon interactions
• Ehrenfest dynamics: or

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• TDDFT-Ehrenfest (Artacho et al): limited to ~500 atoms and 1 ps
• TB-Ehrenfest (Race et al): limited to ~20,000 atoms and weak excitations (band width)
• Ehrenfest is a mean-field theory. No collisions; No spontaneous phonon emission; No electron-electron or electron-phonon equilibration.

ELECTRONIC FRICTION: �STOPPING POWER

• Stopping power: energy deposited per unit length travelled by a projectile (S=dE/dx)

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• Nuclear: dominates at low energies
• Electronic:
• Metals: for v→0, S ~ v (e-h pairs). Decreases like v^-2 for large v (Bethe)
• Insulators: threshold due to band gap v_th≈ 0.1−0.2 a.u.

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SRIM/TRIM�WWW.SRIM.ORG

• Is a Monte Carlo code that calculates the transport of ions in matter, and gives a spatial distribution of ions

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• Uses nuclear and electronic stopping power tables
• Binary collision approximation
• Many other Monte Carlo codes exist, for specific applications

ELECTRONIC STOPPING TABLES USED BY SRIM/TRIM �H THROUGH AL (WWW.SRIM.ORG)

• Can we predict/challenge SRIM Stopping curves from first-principles?

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• Stopping in compounds are based on additivity rules. Can we validate?

DYNAMICS WITH EXCITED ELECTRONS

• First-principles molecular dynamics (Born-Oppenheimer or Car-Parrinello) treats the electronic system as a “slave” of the nuclear configuration. There is no “true” electronic dynamics, and hence no electronic excitations
• Time-dependent DFT (TDDFT) introduces electronic excitations via the time-dependent Kohn-sham equations:

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• The timescale of the electronic motion is much shorter than that of the nuclear motion (by a factor > 1000). Therefore, real-time TDDFT simulations are limited to less than 1 ps and small systems.

COMPUTATIONAL METHODS�FOR REAL-TIME ELECTRONIC DYNAMICS

• REAL-TIME ELECTRONIC DYNAMICS VIA TDDFT
• ADIABATIC GGA (AGGA) APPROXIMATION TO TIME-DEPENDENT XC
• ONLY THE PROJECTILE IS MOVED “BY HAND”
• CHANNELING

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• TIME-DEPENDENT KOHN-SHAM EQUATIONS IMPLEMENTED IN SIESTA

A. TSOLAKIDIS, D. SANCHEZ-PORTAL AND R. M. MARTIN, PHYS. REV. B 66, 235416 (2002)

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KOHN-SHAM ORBITALS EXPANDED IN ATOMIC ORBITAL BASIS (LOCAL BASIS)

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[NEW INTEGRATOR: J. F. K. HALLIDAY AND E. ARTACHO, PHYS. REV. RESEARCH 3, 043134 (2021)

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COMPUTATIONAL METHODS�FOR REAL-TIME ELECTRONIC DYNAMICS

• TIME-DEPENDENT KOHN-SHAM EQUATIONS IMPLEMENTED IN CP2K

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KOHN-SHAM ORBITALS EXPANDED IN GAUSSIAN AND AUGMENTED PW (GAPW)

S. ANDERMATT ET AL, JCTC 12, 3214 (2016)

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REAL-TIME PROPAGATOR WITH CORRRECTION FOR MOVING BASIS

T. TODOROV, JPCM 13, 10125 (2001); T. KUNERT AND R. SCHMIDT, EPJD 25,15 (2003)

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SIMILAR APPROACH BY I. MALIYOV, J.-P CROCCOMBETTE, F. BRUNEVAL; EPJD 91,172 (2018)

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Ab initio MD equilibration, initial electronic state for real-time propagation;

All electron. Basis sets: 6-311G**, pob-TZVP with MOLOPT;

ETRS integration, ds=0.01 Å

ELECTRONIC STOPPING VIA TDDFT

• Electronic stopping in LiF (E_g= ~ 13 ev)
• Threshold: v=0.1 a.U. (Exp) or v=0.2 a.U. (Theory)

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M. Draxler et al., Phys. Rev. Lett. 95, 113201 (2005)

M. Pruneda et al., Phys. Rev. Lett. 99, 235501 (2007)

ELECTRONIC STOPPING OF PROTONS IN AL

A. Correa, J. Kohanoff, E. Artacho, D. Sanchez-Portal and A. Caro, Phys. Rev. Lett. 108, 213201 (2012)

ELECTRONIC STOPPING OF �PROTONS AND 𝛼-PARTICLES IN AU

M. A. Zeb, J. Kohanoff, D. Sanchez-Portal, A. Arnau, I. Juaristi, and E. Artacho, PRL 108, 225504 (2012).

SELF-IRRADIATION OF NI (PKA)� �Ullah, Artacho and Correa, Phys. Rev. Lett. 121, 116401 (2018)

• Core electrons of the projectile are important at higher energies
• Ab initio vs SRIM: SRIM for heavy projectiles is based on scaling

SELF-IRRADIATION OF FE (PKA)

• Remnant charge in Fe atoms closest to the track
• Initial stage of Coulomb explosion
• A. A. Correa (unpublished)

STOPPING OF 50 KEV PROTONS �IN LIQUID WATER

Bin Gu, B. Cunningham, D. Muñoz-Santiburcio, F. Da Pieve, E. Artacho, and J. Kohanoff,

J. Chem. Phys. 153, 034113 (2020)

50keV-bulk.avi

ELECTRONIC STOPPING OF �PROTONS IN LIQUID WATER

Bin Gu, B. Cunningham, D. Muñoz-Santiburcio, F. Da Pieve, E. Artacho, and J. Kohanoff,

J. Chem. Phys. 153, 034113 (2020)

Statistical averaging over trajectories is an important issue

• Randomly or uniformly distributed trajectories converge slowly
• Pick sequence of trajectories that optimize p-O and p-H distance distribution
• Semi-empirical tables (SRIM-PSTAR) need careful consideration

BRAGG ADDITIVITY RULE (BAR) FOR COMPOUNDS: �PROTONS IN WATER VAPOR

Bin Gu, D. Muñoz-Santiburcio, F. Da Pieve, F. Cleri, E. Artacho, and J. Kohanoff

Radiat. Phys. Chem. 193, 109961 (2022).

• 2 H₂ + O₂ → 2 H₂O
• BAR applicable to rt-TDDFT without scaling (SRIM 0.94)

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• Core and Bond:

ELECTRONIC-MEDIATED NUCLEAR STOPPING�IN WATER ICE

D. Muñoz-Santiburcio, J. Kohanoff, and E. Artacho, https://arxiv.org/abs/2303.12975

• Estudio de la irradiación de hielo con protones de alta energía (1 keV – 1.6 MeV)

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• Visión tradicional: el proceso de transferencia de energía del ión proyectil al target tiene 2 componentes independientes:

S_total = S_electrónico + S_nuclear

ELECTRONIC-MEDIATED NUCLEAR STOPPING�IN WATER ICE

• Modelo: 144 H₂O, hielo Ih, slab exponiendo cara (001)
• Ehrenfest MD (Real-Time Time-Dependent DFT), timestep = 1.14 – 0.0285 as
• Código CP2K, funcional PBE, all-electron. 19 trayectorias, 11 velocidades

ELECTRONIC-MEDIATED NUCLEAR STOPPING�IN WATER ICE

Evolución de la densidad electrónica en espacio real, t_sim= 0.8 fs

ELECTRONIC-MEDIATED NUCLEAR STOPPING�IN WATER ICE

• Se da una enorme distorsión de la densidad electrónica en la inmediata vecindad de los núcleos.
• Aparecen fuerzas muy elevadas

ELECTRONIC-MEDIATED NUCLEAR STOPPING�IN WATER ICE

• Fuerzas hacen trabajo suficiente para que los H adquieran gran energía cinética
• Transferencia de energía cinética del proyectil al target sin colisiones binarias
• Efecto desconocido en la literatura (99% de simulaciones no calculan fuerzas)

ELECTRONIC-MEDIATED NUCLEAR STOPPING�IN WATER ICE

• Esto explica un efecto isotópico descrito experimentalmente, pero ignorado hasta la fecha (se asumía que el H₂O y D₂O eran equivalentes a este respecto)

ELECTRONIC-MEDIATED NUCLEAR STOPPING�IN WATER ICE

• Por otra parte, la transferencia de energía a los grados de libertad electrónicos es idéntica a la experimental

ELECTRONIC-MEDIATED NUCLEAR STOPPING�IN SOLVATED DNA

Implicaciones y siguientes pasos:

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Es esperable que este efecto ocurra en cualquier sistema acuoso o rico en H (tejidos biológicos), con implicaciones en:

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• Hadronterapia en tratamiento del cáncer

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• Protección de la radiación en entornos

espaciales

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• Tecnología para la industria nuclear

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ELECTRONIC STOPPING OF �PROTONS IN SOLVATED AND DRY DNA

• DNA decamer: CCATTAATGG

(structure 1WQZ in PDB)

• 40% GC content (human genome: 35-60%, average 41%)
• Periodic box of 34 × 26 × 26 Å
• 544 H₂O and 20 Na⁺
• Total 2288 atoms
• Density 1.186 g/cm³

D. Muñoz-Santiburcio, Bin Gu, F. Da Pieve, E. Artacho, and J. Kohanoff (unpublished)

METHODS

• CP2K code
• TZV2P basis sets, GTH pseudopotentials
• System equilibrated during 4 ps via Born-Oppenheimer MD in the NVT ensemble at 300 K with revPBE+D3 plus 0.75 ps with revPBE
• Projectile H⁺ starts with a given velocity: 0.5, 1, 1.72, 3, 8 au
• Ehrenfest MD (timesteps from 0.91 to 0.057 as, Δs = 0.005 Å)
• Forces are explicitly considered in the calculation (as opposed in the constant-velocity approximation usually taken in the field).
• For v = 0.5 au we also carried out constant velocity calculations.

TRAJECTORY SAMPLING

• 20 trajectories of the projectile sampled.

Geometry pre-sampling scheme:

Bin Gu et al, JCP 153, 034113, 2020

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• Same 20 trajectories sampled for dry DNA (both anionic and saturated)

ELECTRONIC STOPPING

• Electronic stopping power at v = 1.72 au

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• Averages:

0.181 au (DNA/water)

0.066 au (DNA, anionic)

0.060 au (DNA, neutral)

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0.181 is close to (but below) the stopping of pure water scaled by the density,

0.164 × 1.186 = 0.194

HOLE/EXCITATION DISTRIBUTION

• Obtained by projecting the non-adiabatic wavefunctions on the ground state wavefunction
• Maximum of the excitation distribution around 10 eV

(IE of H₂O = 11 eV)

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Theory: Zeb et al, PRL 108, 225504 (2012)

HOLE/EXCITATION DISTRIBUTION

• Projectile velocity dependence

Maximum of the excitation distribution around 10 eV, but they extend to much higher energy

HOLE/EXCITATION ENERGY DISTRIBUTION

• Total hole and excitation populations are the same (conservation)
• Total hole energy is smaller than total excitation energy
• Sum of the two is roughly equal to Kohn-Sham energy
• Hole contribution to stopping peaks at lower velocity than excitations and Kohn-Sham (total)

RESULTS

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• Wet vs dry DNA: slow projectile

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v = 0.5 au

20 / 2 / 2023

Proton irradiation of DNA in physiological conditions by ab initio simulations – D. Muñoz-Santiburcio –

RESULTS

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• Wet vs dry DNA: Bragg peak velocity

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v = 1.72 au

20 / 2 / 2023

Proton irradiation of DNA in physiological conditions by ab initio simulations – D. Muñoz-Santiburcio –

RESULTS

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• Wet vs dry DNA: fast projectile

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v = 8 au

20 / 2 / 2023

Proton irradiation of DNA in physiological conditions by ab initio simulations – D. Muñoz-Santiburcio –

DEPOPULATION OF ELECTRONIC STATES �BY PROTON IRRADIATION OF DNA

D. Muñoz-Santiburcio, Bin Gu, F. Da Pieve, E. Artacho, and J. Kohanoff (unpublished)

• Maximally localized Wannier functions
• Depopulation dynamics
• Water molecules more depopulated than DNA states
• No preference between bonds and lone pairs in water, or bonds in DNA
• Solvation water changes populations in DNA bonds.

SUMMARY AND CHALLENGES

• TDDFT reproduces SRIM and experimental stopping curves with very good accuracy for a wide range of materials, and gives additional insight. In some cases it challenges SRIM.
• Bragg additivity rule (BAR) and Core and Bond (CAB) model, used in SRIM for compounds, can also be used with rt-TDDFT, and without ad hoc scaling.
• New effect discovered: electron-mediated nuclear stopping. Important close to Bragg peak.
• Irradiation of solvated DNA shows that water affects the population dynamics. The most depopulated states can be different.
• Stopping of electrons: important in biological context
• Can we obtain cross-sections from these simulations?

ELECTRONIC STOPPING IN WATER

N. Koval, F. Da Pieve, Bin Gu, D. Muñoz-Santiburcio, J. Kohanoff, and E. Artacho,

Phys. Rev. Research 5, 033063 (2023)

Protons

Electrons

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– interacting response function

K – interaction kernel, RPA

m – occupied

n – unoccupied

E – energy loss

q – momentum transfer

E, q

– dielectric function

• 1^st step:

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• Ground-state eigenvalues with DFT

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• 2^nd step:

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• ELF with LR-TDDFT

http://mbpt-domiprod.wikidot.com

ENERGY LOSS FUNCTION (ELF) FROM LR-TDDFT

ENERGY LOSS FUNCTION (ELF) FOR WATER

N. Koval, P. Koval, F. Da Pieve, J. Kohanoff, E. Artacho, and D. Emfietzoglu,

R. Soc. Open Sci. 9, 212011 (2022)

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R. Garcia-Molina et al., Surface and Interface Analysis 49, 11-17 (2016) D. Emfietzoglou et al., Radiation Research 164, 202 (2005) S. Incerti et al., Medical Physics 45, e722-e739 (2018)

ELECTRONIC STOPPING (ELECTRONS) FROM ELF

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• External electron is modeled as a classical negative point charge (Gaussian sphere)

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• Bragg peak is shifted to lower energy

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• Classical electron → quantum nature of the external electron is not accounted for while being extremely important at low energies

ELECTRONIC STOPPING (ELECTRONS) FROM RT-TDDFT

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• External electron modelled as a classical negative point-charge. No quantum nature.

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• In constant velocity simulations, mass is infinite

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• In LR stopping formulae, mass enters in the integration limits

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• Changing the integration limits in LR to infinite mass reproduces height, but not location of peak.

ELECTRONIC STOPPING (ELECTRONS) FROM RT-TDDFT

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• In RT simulations at constant velocity, mass is irrelevant. The difference is due to opposite charges: Barkas effect

ELECTRONIC STOPPING: PROTONS VS ELECTRONS

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• Can we calculate doubly differential cross sections, as a function of E and q (or angle)? These can be compared to CDW-EIS scattering calculations by Mariel in project MAMBA

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• Can we extract cross sections from Real-time TDDFT simulations?

SINGLE DIFFERENTIAL CROSS SECTIONS:

TOWARDS MONTE CARLO CODES

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• LR-TDDFT provides sufficiently accurate ELF without the need of any experimental input

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• LR-TDDFT gives access to finite momentum transfer, low energies, and targets other than water

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• Nonlinear effects in the electronic stopping of electrons at low energies are important: Mass vs nonlinear effects

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CHALLENGES

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• Quantum effects in the electron scattering: What is the lowest energy for which electrons can be treated classically? Dependence on impact parameter. Ehrenfest dynamics (CP2K)

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• Can we simulate an incoming quantum electron?

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We hope to be able to address these in project MAMBA

SUMMARY: STOPPING OF ELECTRONS

COLLABORATORS

• Emilio Artacho (San Sebastian – Cambridge)

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• Natalia Koval (San Sebastian)

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• Bin Gu (Nanjing, China - QUB)

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• Daniel Muñoz Santiburcio (Nanogune-UPM)

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Thank you!