Assessment of Real-Time Time-Dependent Density Functional Theory

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A: Spectroscopy, Photochemistry, and Excited States

Assessment of Real-Time TDDFT in Radiation Chemistry: Ionized Water Dimer Jan Chalabala, Frank Uhlig, and Petr Slavicek J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b01259 • Publication Date (Web): 07 Mar 2018 Downloaded from http://pubs.acs.org on March 8, 2018

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The Journal of Physical Chemistry

Assessment of Real-Time TDDFT in Radiation Chemistry: Ionized Water Dimer Jan Chalabala1, Frank Uhlig1,3 and Petr Slavíček1,2,* 1

Department of Physical Chemistry, University of Chemistry and Technology, Technická 5, 16628 Prague, Czech Republic

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Jaroslav Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolejškova 3, 18200 Prague, Czech Republic 3

Institute for Computational Physics, University of Stuttgart, Allmandring 3, 70569 Stuttgart, Germany *Corresponding author: [email protected]

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Abstract Ionization in the condensed phase and molecular clusters leads to a complicated chain of processes with coupled electron-nuclear dynamics. It is difficult to describe such dynamics with conventional nonadiabatic molecular dynamics schemes since the number of states swiftly increases as the molecular system grows. It is therefore attractive to use a direct electron and nuclear propagation such as the real-time time-dependent density functional theory (RT-TDDFT). Here we report a RT-TDDFT benchmark study on simulations of singly and doubly-ionized states of a water monomer and dimer as a prototype for more complex processes in a condensed phase. We employed the RT-TDDFT based Ehrenfest molecular dynamics with a generalized gradient approximate (GGA) functional and compared it with wavefunction-based surface hopping (SH) simulations. We found that the initial dynamics of a singly HOMO ionized water dimer is similar for both the RT-TDDFT/GGA and the SH simulations, but leads to completely different reaction channels on a longer time scale. This failure is attributed to the self-interaction error in the GGA functionals and it can be avoided by using hybrid functionals with large fraction of exact exchange (represented here by the BHandHLYP functional). The simulations of doubly ionized states are reasonably described already at the GGA level. This suggests that the RT-TDDFT/GGA method could describe processes following the autoionization processes such as Auger emission, while its applicability to more complex processes such as intermolecular coulombic decay remains limited.

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Introduction The progress in XUV/X-ray induced chemistry, facilitated by newly available light sources, such as high-harmonic generation, X-ray plasma or free electron lasers,1-3 paves the way for observing dynamical phenomena below the femtosecond resolution. New experimental data require theoretical interpretation; however, existing theoretical tools are still rather limited. We need to describe scattering, bound and highly excited states and also coupled electron nuclear dynamics. More problems arise for systems with a large number of particles, e.g. solvated molecules. Among the available theoretical tools, time dependent density functional theory in real-time (RTTDDFT)4 emerged as a viable tool offering an option to directly model high energy processes for systems with many particles.5-7 This is the situation frequently encountered in radiation chemistry and X-ray photochemistry.8,9 For example, the Auger process lead to a formation of a doubly charged molecule embedded in liquid environment, a situation hardly tractable by conventional ab initio techniques. Therefore, are the RT-TDDFT simulations in radiation chemistry trustworthy with the currently available technologies? In the present work, we investigate the applicability of the RT-TDDFT to simple examples of ionized water monomer and dimer. At this moment, we do not aim to describe the interaction of the molecule with the ionizing field. The ionization initiates nuclear dynamics which in turns stimulate electronic motion. The most straightforward approach from a chemist’s perspective is to start with Born−Oppenheimer approximation followed by non-adiabatic corrections. For low-energy ionization, we can model the system on the ground electronic state of the ion.10-12 Such an approach is justified for the systems with an ultrafast internal conversion compared with a fragmentation or autoionization.13 The dynamics on higher-lying excited states require including non-adiabatic corrections14,15 using either quantum dynamics calculations on precomputed potential energy surface16-19, on-the-fly schemes with moving Gaussian20-23, trajectories based Ehrenfest dynamics24-26, or surface hopping27-30 scheme. Unfortunately, nonadiabatic simulations become impractical as the number of states rapidly increases for extended systems. An illustrative example is an ionized molecule surrounded by a solvent where a positive charge or excitation can hop from the molecule to a solvent molecule. Further complications arise from emerging autoionizing states and although it is possible to evaluate the rates of the autoionization31-33, the molecule is still in a highly excited electronic arrangement which is 3



difficult to obtain by a systematic diagonalization of the electronic Hamiltonian. Alternatively, we can simulate directly the coupled electronic/nuclear dynamics. Electron dynamics itself can be treated at various levels, such as time-dependent configuration interaction method34, coupled clusters35 or complete active space self-consistent field (CASSCF) method.36 The studies on electron dynamics typically ignore ionic motion. Electron propagation within the RT-TDDFT framework appears to be an appealing option for extended systems.5-7,37-39 There are several RT-TDDFT implementations for treating the electron propagation (see Foglia et. al.40 and Lopata et. al.7 and references therein) using real space grids (as implemented e.g. in the Octopus code41), plane wave basis sets (as implemented e.g. in the CPMD code42,43), atom centered basis sets44,45 (as implemented e.g. in the NWChem code46), or mixed basis set47,48 with Gaussian orbitals and auxiliary plane waves (GPW) as implemented e.g. in the CP2K.49,50 While atom centered orbitals seem to be a natural choice for a chemist, plane waves or real space grids are better suited for description of highly delocalized electrons. We can start with a well-defined initial electronic structure, following the subsequent electronic and nuclear dynamics. While the propagation of the electronic wavefunction requires very short propagation time-steps, the DFT framework offers fast and efficient evaluations of each step. The RT-TDDFT approach is mostly used in the context of linear and non-linear spectroscopies7,44,51-54, yet its scope was extended toward computational photodynamics.14 It was used in order to investigate dynamics following Auger decay in water or nucleic acid bases55,56, to model low energy collisions between atoms and molecules57, radiation damage in materials58, or stopping power in the context of hadron therapies59-61 and for observing charge migration in molecules and dye-sensitized materials.62,63 RT-TDDFT is on the other hand no panacea and there are several issues to be addressed. First, it is not guaranteed that the adiabatic approximation in the TDDFT, ignoring the functional dependence on the history of the density, is appropriate. Secondly, the calculations of exact exchange in hybrid functionals are still expensive unless we use atom centered basis40,64,65. Therefore, the generalized gradient approximation family (GGA) are typically used. These functionals are known to suffer from various deficiencies, particularly the self-interaction error and wrong asymptotic form of the exchange potential.66,67 The problems are more severe for open-shell electronic structure.68,69 Finally, the coupling between the electronic and nuclear degrees of freedom in RT-TDDFT is mediated via 4


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the Ehrenfest approach70 in which the atomic nuclei move (classically) on an averaged potential energy surface.71,72 All these features might naturally lead to spurious results. We performed testing of the RT-TDDFT approach for singly and doubly ionized water monomer and dimer which can be compared to high level ab initio calculations27,73-77 and experimental data73,78-81. An ionized water dimer exhibits non-adiabatic effects27 and it is therefore interesting to see whether the RT-TDDFT method is capable to capture them. We use the surface hopping method with the CASSCF wavefunction as a reference calculation. The main issue here is whether the RT-TDDFT in its present state can describe radiation induced processes, i.e. whether the ultrafast dynamics triggered by ionization of one of the water units is faithfully described for situations in which wavefunction-based non-adiabatic simulations can be performed.

Methods We have studied the dynamics of a singly and doubly ionized water monomer and dimer in several electronic states. This testing was performed on a set of geometries corresponding to the thermal equilibrium in the ground state. For each geometry, we initiated the coupled electron/nuclear dynamics (RT-TDDFT and SH simulations) by preparing a non-equilibrium state with a well-defined physical meaning (e. g. with a hole in a certain orbital). For ground state ionizations, we also performed the Born−Oppenheimer molecular dynamics (BOMD) simulations in which the electronic Hamiltonian was diagonalized at each step and nuclei were classically propagated on a single DFT potential energy surface (PES) using the velocity Verlet algorithm. Initial conditions The initial geometries of the water dimer were sampled using the path integral molecular dynamics (PIMD) simulations accelerated with the so-called quantum thermostat based on a generalized Langevin equation (the scheme known as PI+GLE82,83) employing 4 replicas. This scheme was shown to be suitable for the description of hydrogen bonded systems.84 The temperature was set to 180 K which guaranteed the stability of a water dimer with respect to its dissociation. The total simulation time was 200 ps with 20 a.u. timestep. From this trajectory, 47 geometries were extracted. Velocities of these initial geometries were neglected. The water 5



monomer geometries were taken from the dimer sampling removing one of the water units. RT-TDDFT simulation protocol The RT-TDDFT simulations were performed with the Perdew–Burke−Ernzerhof85 (PBE) and Becke−Half−and−Half (BHandHLYP)86 functionals which belongs to the GGA and hybrid classes of functionals, respectively. Most of the calculations were performed in the real space grid OCTOPUS 6 code37,38,41,87 with the PBE functional. Note that GGA functionals have also been typically used in the dynamical studies performed so far62,88,89, but their application is problematic for open shell systems. It is well known that the ionized water dimer described with GGA functionals spuriously favour the so-called hemi-bonded structures (HB, see Table 1) due to the self-interaction error and artificial charge delocalization related to it.74,90 We therefore tested the hybrid BHandHLYP functional, but these simulations were performed in CP2K 5.1 code using the GPW approach.48-50,91 Comparison of both codes in the simulations of singly ionized water dimer using the same functional can be seen in Figure S5 (see the supporting information). In all RT-TDDFT simulations, we started with the optimized wavefunctions of the neutral system. In the water dimer, this assures that orbitals at the PBE level are mainly localized either on the donor or acceptor water unit (see Figure 1). Starting with the PBE orbitals optimized for the ionic systems would be problematic due to the charge delocalization as discussed above. Next, we removed (annihilated) an electron(s) from some orbital(s) neglecting the electronic relaxation of the initial state, i.e. no further SCF optimization was performed. The starting electronic configuration therefore does not represent a stationary electronic state (unlike the states obtained by the CASSCF method for the SH simulations). The Kohn-Sham orbitals were always propagated in a field-free mode from the non-stationary initial state. In OCTOPUS simulations, the optimized effective potential regime (OEP)92 with the Troullier-Martins93 norm-conserving pseudopotential for oxygen 1s electrons was used. We used a spherical box with a radius of 15 Bohr. The energy cutoff for the PBE functional was 158 Ry. These cutoffs were obtained by checking the energy convergence as a function of different grid spacing. We did not find a significant difference between the Approximated Enforced TimeReversal Symmetry (A-ETRS) and the full ETRS evolution propagators38 in the terms of energy conservation and calculation cost. The maximal timestep which guaranteed energy conservation 6


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was found to be 0.03 a.u. Nuclei were propagated classically on the Ehrenfest mean potential. In CP2K simulations, we used cubic box with 10 Ångstrom size and 0.05 a.u. timestep. The mixed plane-wave and basis set was used with DZVP-GTH basis set and GTH pseudopotentials.94 The plane wave grid cut-off was 400 Ry and the ETRS propagator was used. Surface-hopping simulations We used a trajectory surface hopping scheme as implemented in our in-house code ABIN95 using Tully’s fewest switches29,96 approach with empirical decoherence according to Granucci and Persico.97 Nuclei were propagated using the velocity Verlet algorithm and the electronic structure (non-adiabatic couplings, energies and gradients) was obtained from the state-averaged complete active space self-consistent field method (SA-CASSCF) with the 6-31+G** basis set. In the simulations of the singly ionized water dimer, the CASSCF active space comprised 7 electrons in the 4 highest occupied orbitals and averaging over 4 states (SA4-CASSCF(7,4)). The CASSCF method lacks the dynamical correlation yet the CASSCF based PES follows key features of more advanced electronic structure methods.27 For the doubly ionized water dimer in ground triplet state, the SA6-CASSCF(10,6) active space was found to be appropriate and stable during simulations. This active space comprised 6 highest occupied orbitals averaged over 6 states, containing thus all outer valence orbitals occupied in the neutral water dimer. The time-dependent Schrödinger equation was integrated via the 5th order Butcher integrator98 with 100 substeps for each nuclei propagation. The ground state of the singly and doubly ionized water dimer was also simulated via Born‒Oppenheimer molecular dynamics (BOMD) with a diagonalization at each step using the PBE/6-31++G** and PBE0/6-31++G** method. For the SH and BOMD simulations and 2D PES scans, on the fly electronic structure quantities (e.g. energies, gradients and nonadiabatic couplings) were obtained from the Gaussian 09 code99 (BOMD simulations and PES scans with the PBE, PBE0 and BHandHLYP functionals) and MOLPRO 2012100 (CASSCF for SH simulations and PES scans). Computational codes and data analysis In the figures 2-7, the populations represent the fraction of the trajectories with certain structural characteristics. We used following criteria: first, we tested whether any of hydrogen atoms were dissociated (O−H distances > 4.5 Å). If more than 2 hydrogen atoms were dissociated, we 7



checked whether molecular hydrogen is formed (H−H distance < 1.5 Å). Then we checked how many hydrogen atoms were connected to each oxygen atom (atom H is connected to first oxygen atom if O1−H distance < O2−H distance and the shorter O−H < 2 Å). Finally, the structures with the O−O distance larger than 4.5 Å were considered as dissociated. These thresholds were varied in order to obtain the smallest fluctuation in populations during the time evolution. In figures 2-7, the RT-TDDFT calculations are simply denoted as ED (Ehrenfest dynamics), Born‒Oppenheimer molecular dynamics as BOMD. In further text, RT-TDDFT means RT-TDDFT/PBE and BOMD means BOMD/PBE if not stated otherwise. In the case of HOMO ionized dimer, RTTDDFT/PBE and RT-TDDFT/BHandHLYP are always distinguished.

Results & Discussion Before presenting the actual results, we briefly summarize the dynamical events following ionization in water dimer. Depending on the photon energy, the electrons are ionized either from a valence shell (1b1, 3a1 or 1b2 electrons with the binding energies 12.6, 14.8 and 18.6 eV respectively), inner-valence (2a1, 32.6 eV) or from the 1a1 core orbital (i.e. 1s orbital on oxygen with energy 539.8 eV).101,102 Upon the 1b1 ionization from the HOMO orbital (highest occupied molecular orbital), an ultrafast proton transfer (PT) is observed as the main reaction channel in liquid water as well as in finite size molecular clusters: H O ∙∙∙ H O → HO∙ ∙∙∙ H O Proton transfer was suggested to take place within ~40 fs after the ionization in liquid water103, however, polarization anisotropy measurements104 found H2O+ lifetime is about ~200 fs. The longer reaction time is consistent simulations for water clusters.27 The ionization of deeper-lying valence electrons is followed by an ultrafast population transfer into lower electronic states and the products might be different at that point.27 At higher energies (inner valence 2a1 orbital or core 1a1 orbital ionization), autoionization channels start to compete with internal conversion and fragmentation relaxations and the autoionization channel dominates eventually. With the autoionization time estimated in the order of tens of femtoseconds, there is still enough time for nuclear dynamics taking place on the 2a1 and satellite states. Note at this point that the autoionization process (Auger decay)

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H O → H O + e

is energetically closed upon inner-valence electron (2a1 orbital) ionization in isolated water, i.e. the energy of the inner valence ionized state is lower than the energy of the doubly ionized state in an isolated water molecule.105 The doubly charged water cation can thus be formed only upon the 1a1 (i.e. 1s electron) ionization. In molecular clusters, several non-local autoionization processes, such as Intermolecular Coulomb Decay (ICD)106-108 or Electron Transfer Mediated Decay (ETMD), are possible.109 During the ICD, the inner-valence or core hole is filled with a valence electron and this relaxation energy is used to ionize a neighbouring water unit: H O ∙∙∙ H O → H O + H O + e

It has been found that the non-local autoionization processes (ICD and ETMD) are significantly catalysed by the proton transfer during the so-called Proton Transfer Mediated Charge Separation process.110 This leads to various final states, including OH+···H3O+. Similar processes were observed for other hydrogen bonded systems.110,111 All the ICD or ETMD final products are highly reactive particles which may play role in the radiation damage of biomolecules.108,112,113 Dynamics of singly ionized water First, we address how the choice of an initial electronic wavefunction affects the overall dynamics. We compared the BOMD with two RT-TDDFT simulations of the ionized water monomer, first starting from the initial neutral wavefunction (with annihilated electron) and the second starting from the optimized wavefunction of water monomer ion. The OH distances in the BOMD simulation are identical to the RT-TDDFT simulation starting with the wavefunction optimized for the ionized state (see Fig. S2). On the other hand, the RT-TDDFT simulation starting with the wavefunction of the neutral molecule leads to larger vibrational amplitude of the OH bonds. This is not surprising as we start from an unrelaxed wavefunction and the system thus experiences an additional electronic energy. Nevertheless, the general dynamical features for the unrelaxed RT-TDDFT simulations remain similar to the BOMD (e.g. causing fluctuations along the O−H bond with a similar frequency). Therefore, it appears meaningful to start from the unrelaxed electronic wavefunction. This is an important conjecture as (i) we are forced to use the wavefunction of the neutral system due to problematic 9



SCF convergence for higher ionized states and (ii) the ground state PBE electronic wavefunction describes more faithfully the ground electronic structure of the ionized state as we show below. Dynamics of singly ionized water dimer Next, we focus on the water dimer in its lowest ionized state. In the water dimer, we distinguish the hydrogen bond donor and hydrogen bond acceptor units. As the water HOMO orbital corresponds to a lone electron pair on oxygen, the electron on the water acceptor unit should be energetically favoured due to the electrostatic interaction between the hydrogen bonding atom and the lone pair on the acceptor molecule. Indeed, the HOMO orbitals calculated both at the Hartree‒Fock (Figure 1d) and PBE (Figure 1b) levels are localized on the hydrogen bond donor. This is the same for the position of the electron hole in the ionized water dimer at the HF level. On the contrary, the hole in the ionized water dimer delocalizes at the PBE level (Figure 1a) due to the DFT self-interaction error.69 Therefore, it would not be appropriate to start the electron dynamics with an optimized ionized-state wavefunction using GGA functional. a) ELECTRON HOLE (PBE ion 1+)

b) HOMO (PBE neutral)

c) ELECTRON HOLE (HF ion 1+)

d) HOMO (HF neutral)

Figure 1: Electron hole (a, c) of the ionized water dimer and HOMO orbital (b, d) of the neutral water dimer calculated at the HF and PBE levels with 6-31++g** basis set. The geometries correspond to the energy minima of the neutral molecule at the PBE/6-31++g** and HF/6-31++g** levels of theory.

The self-interaction error in the GGA functionals has further consequences for the dynamical processes following the ionization. The global minimum of the HOMO ionized water dimer is known78 to be proton transferred with positive charge localized at H3O+ moiety (Table 1, PT). 10


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There is also a secondary minimum, the hemi-bonded structure (Table 1, HB) where the two water units are bound via oxygen atoms and the charge is delocalized over both units.75 While this structure is energetically disfavoured, the PBE functional predicts the hemi-bonded structure to be the global minimum. Table 1: Energy Differences, Mulliken Charges and Geometries of the Singly Ionized Water Dimer in the Proton Transferred (PT) and Hemi-Bonded (HB) Structures and the Optimized Neutral Water Dimer (N). summed Mulliken charges on oxygen atoms

energy

/ multiples of elementary charge ePT

method /

∆E(HB‒PT) /

6-31++g**

eV

HB

N

H 3O +

OH⋅

H 2O

H 2O

H 2O

H 2O

(acceptor)

(donor)

PBE

−0.35

0.82

0.18

0.50

0.50

0.40

0.60

PBE0

0.04

0.85

0.15

0.50

0.50

0.29

0.71

−0.05

0.85

0.15

0.50

0.50

0.33

0.67

BHandHLYP

0.40

0.88

0.12

0.50

0.50

0.06

0.94

MP2

0.19

0.90

0.10

0.50

0.50

0.04

0.95

B3LYP

Structures:

It is therefore clear that starting the RT-TDDFT simulations of the water dimer with the PBE ionic state wavefunction would lead to an erroneous behaviour from the very beginning. We therefore decided to start with the wavefunction of the neutral water dimer where the charge is correctly localized on the water donor unit upon the HOMO electron ionization. The importance of the proper choice of the initial wavefunction was recently demonstrated for the attosecond charge transfer dynamics.114 The price we pay is the additional excess energy deposited into the system. The coupled electron/nuclear dynamics is still controlled by the PBE functional. The 11



hypothesis is that the coupled electron/nuclear dynamics may still follow the “true” dynamics even with the approximate GGA functional. First, we compare the RT-TDDFT/PBE and the SH simulations on the CASSCF potential energy surface. In the SH simulations (Figure 2c), we observe a very fast proton transfer which is essentially complete within the first 100 fs. Subsequently, a gradual dissociation of the complex leads to a formation of the H3O+ ion and OH· radical. Such a conclusion is similar to results from both adiabatic10,11 and non-adiabatic simulations.27 In RT-TDDFT/PBE simulations (Figure 2a), we observe the fast proton transfer dynamics as well, but the quantitative features are different. The proton transfer here is faster than in the SH simulations; yet, the hydrogen atoms immediately bounce back to the original oxygen atom. The complex starts to dissociate into the two water units. It is evident that the PBE electron/nuclear dynamics provide qualitatively incorrect results on a longer timescale. While the short time dynamics is controlled by the initial electronic distribution (supporting the proton transfer), the electronic relaxation forces the system to localize the proton in-between the oxygen atoms. There is still a substantial difference between the RT-TDDFT/PBE and BOMD/PBE dynamics. While the BOMD/PBE (Figure S3 Figure 2in supporting information) leads to a minor proton transfer population, this channel is dominant for the RT-TDDFT/PBE dynamics (Figure 2a). Also in the BOMD dynamics with the PBE0 hybrid functional (Figure S3 Figure 2in supporting information), we observe a significant proton transfer. However, the dissociation into the H3O+ and OH fragments is in all three cases (BOMD/PBE, BOMD/PBE0, RT-TDDFT/PBE) negligible. The self-interaction error of the pure GGA functionals seems to be the most probable cause of the spurious RT-TTDFT dynamics. While there are additional differences between the SH and RT-TDDFT simulations (single determinant description in RT-TDDFT vs. multideterminant within the CASSCF), these differences should not play an important role for the HOMO ionization. Using hybrid functionals with high fraction of exact exchange (e.g. BHandHLYP functional) improves significantly the agreement with the SH simulations (Figure 2b). In fact, the RT-TDDFT/BHandHLYP dynamics follows the SH results almost quantitatively. Using hybrid functionals seems therefore almost mandatory for water radiation chemistry applications where open shell ions are frequently formed. Note, that the present BHandHLYP 12


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data were obtained with a mixed Gaussian and plane waves basis set. The calculations would be too much computationally demanding in plane wave or real space grid formalism. a) RT-TDDFT/PBE

d) PBE/6-31++G**

b) RT-TDDFT/BHandHLYP

e) BHandHLYP/6-31++G**

c) SH CASSCF(7,4)/6-31++G**

f) DW-CASSCF(7,4)/6-31++G**

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Figure 2: Dynamics and PES scans of the HOMO ionized water dimer: a) RT-TDDFT/PBE, b) RT-TDDFT BHandHLYP and c) SH/SA4-CASSCF(7,4)/6-31+g** dynamics. In the panel a,b and c panels: the blue curve corresponds to the water dimer dissociation yielding two separate water units, the black curves shows the total proton transfer including both dissociated and bound H3O+ and OH· fragments. The red curve shows only dissociated H3O+ and OH· fragments. The PES scans of the ionized water dimer along the O−O and proton transfer coordinates were calculated at the a) PBE/6-31++G**, b) BHandHLYP/6-31++g** and c) DWCASSCF(7,4)/6-31++G** levels of theory. Dynamical weighting (DW) of states with damping parameter equal to 3 was applied in order to obtain smoother PES.115 Remaining degrees of freedom were kept fixed with the values corresponding to the PBE/6-31++G** optimized proton transfer geometry. The proton moves along O˗O line (Y-axis) highlighted by dashed line.

The difference between the dynamics on the three potential energy surfaces shown above can be understood from a 2D PES scans along the O···O and proton transfer coordinate. It is clear that the proton transfer is energetically convenient at the CASSCF (Figure 2f) and BHandHLYP levels (Figure 2e) while the PBE functional (Figure 2d) exhibits a rather flat region between the hydrogen bonded and proton transferred structures. 14


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Dynamics upon ionization of deeper-lying electrons Analogously to the HOMO ionization, we asked whether the GGA type PBE functional would not perform better for deeper ionizations, bringing more energy into the system. We made a comparison with the SH calculations. In the case of the SH dynamics, proton transfer is the dominant channel upon the ionization of the HOMO−1 (Figure 3a) and HOMO−2 (Figure 3Figure 3b) electrons (corresponding to ionization of the 1b1 and 3a1 electrons on the water acceptor and donor unit respectively). Although the overall proton transfer process is slower compared with the HOMO ionization (Figure 2), it is still the most important reaction route, consistent with a previous study.27 On the contrary, the proton transfer is observed only negligibly for the RT-TDDFT dynamics. Instead, we observe a dissociation of the ionized water dimer into the H2O+ and H2O fragments as the major channel. This conclusion is in agreement with the failure of the RT-TDDFT/GGA approach for the HOMO ionization, the more excess energy therefore does not seem to improve the situation.

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a) HOMO−1

b) HOMO−2

Figure 3: Dynamics of the water dimer upon the a) HOMO−1 and b) HOMO−2 ionizations from RTTDDFT/PBE and SH/SA4-CASSCF(7,4)/6-31+g** simulations. In both panels: the blue (RT-TDDFT) curves correspond to the water dimer dissociation yielding two separate water units, red (RT-TDDFT) and black (SH) curves show the overall proton transfer including both the dissociated and bound H3O+ and OH· fragments. The purple (SH) curve shows a dissociated state of the H3O+ and OH· fragments. The corresponding channel in the RT-TDDFT simulations is not shown due to the low probability of the channel.

For even higher energy states, more issues are to be considered. A previous study27 has shown 16


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that the dynamics upon the ionization of the 1b2 electron is already rather complex non-adiabatic process, leading either to a direct dissociation into H2O+ and H2O fragments or into the H2O+ + OH + H fragments. We cannot expect the single-determinant description within the RT-TDDFT to describe reliably such a process. Some fraction of the trajectories undergoes a proton transfer.27 The RT-TDDFT simulations resulted almost exclusively yielded dissociation into the H2O+ + H2O fragments (Figure 4). The RT-TDDFT dynamics thus predict a different dynamical behaviour compared to lower ionization, i.e. the immediate proton transfer disappears. It is unable to describe, however, the reaction details and splitting between different reaction channels. The potential source of errors can stem from several issues: (i) the self-interaction error in the GGA functional (ii) the single determinant framework and (iii) the Ehrenfest dynamics used for solving the coupled electron/nuclear equations of motion.

Figure 4: Dynamics of the water dimer upon the HOMO−4 (1b2) from the RT-TDDFT/PBE simulations. The blue curve corresponds to the water dimer dissociation and the black curve shows the overall proton transfer including bounded H3O+ and OH· complex.

We also performed dynamical simulations for the ionization of the 2a1 electron in the hydrogen bond donor unit (HOMO‒7 orbital). This system is already difficult to simulate due to the large energy generated in the system upon the ionization, requiring the smaller 0.005 a.u. timestep to ensure energy conservation (see Figure 1a in the supporting information). The simulation here cannot be easily compared with the SH simulations as the 2a1 state is embedded in a large 17



number of satellite states. Furthermore, the 2a1 state is already susceptible to non-local electron autoionization79,108 which cannot be reliably described within RT-TDDFT.116 Its lifetime is thus limited to several tens of femtoseconds and only the ultrafast dynamics is of interest in this context. We can expect, however, that the dynamics following the 2a1 ionization (i.e. ionization from the oxygen 2s orbital) should proceed analogically to the dynamics following the 1s electron ionization.117 In the latter case, an ultrafast proton transfer immediately after the ionization is observed. The initial dynamics emerging from the RT-TDDFT simulations (Figure 5) seems to be consistent with this picture. In the RT-TDDFT, we also observed dissociation of the molecule yielding atomic hydrogen atoms, originating apparently from the coupling of the 2a1 state with the satellite states.

Figure 5: Dynamics of the water dimer upon the HOMO−7 (2a1) ionization from the RT-TDDFT/PBE simulations. The results were averaged over 15 trajectories. The black curve corresponds to proton transfer leading to the H3O+ and OH· complex. The orange curve represents the dissociation of the water molecule to 2 protons/hydrogen atoms and the oxygen atom which is bonded to the second water molecule, e.g. (O+H2O).

Dynamics of doubly charged water monomer Next, we study if the RT-TDDFT method can describe the doubly ionized water molecules. In fact, the simulations of the processes following the Auger decay in water have been performed previously.88 We start with a simple case of doubly ionized water monomer in triplet electronic state as the singlet state is problematic to simulate due to problems with a near-degeneracy of 18


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orbitals. We compare the BOMD trajectories with the RT-TDDFT trajectories for the same state (Figure 6Figure 6).

Figure 6: Dynamics of the doubly ionized water monomer in the triplet electronic state showing BOMD/PBE/6-31++g** (solid curves) and RT-TDDFT/PBE (dashed curves) simulation. The black and red curves represent one and two dissociating O-H bonds, respectively.

For majority of the population within the BOMD dynamics, we observed water dissociated to a single proton and the OH+ cation H O → HO T + H and some dissociation of both protons leaving oxygen radical in a triplet state behind H O → O T + 2H The second reaction is, however, the major reaction channel in the RT-TDDFT simulations even though this reaction is energetically less favourable by 5.13 eV at the PBE/6-31++g** level compared to the first reaction. Dissociation in the RT-TDDFT was sequential; the second proton starts to dissociate after the first one. The fact that we observed this channel points to much higher initial energy in the RT-TDDFT simulation if we start from the unrelaxed initial wavefunction. Indeed, the energy relaxation is in this case higher when compared with the single electron ionization (~3 eV vs ~5 eV). The observed discrepancy does not pose a major problem 19



for the liquid phase simulations as the OH+ cation is a very strong acid and the proton is transferred very fast upon the OH+ formation.117 In the case of doubly ionized water dimer in the triplet electronic state, the two holes correspond to the 1b1 ionization on the two neighbouring water units. Such a state is the lowest electronic state of double cation and we can directly compare the simulations either with the BOMD/PBE simulations or with the CASSCF simulations. The electronic structure is essentially identical for all of these cases and the dominant reaction channel is a dissociation of the water dimer into two singly charged water units, H2O+ + H2O+, i.e. Coulomb explosion of the two ionized water units (Figure 7). This process is very fast and is completed in less than 100 fs. The RT-TDDFT simulations predict also small fractions of hydrogen transfer between the water units and dissociation of one water molecule.

Figure 7: Dynamics of the lowest electronic state of the doubly ionized water dimer in triplet electronic state showing the RT-TDDFT/PBE, BOMD/PBE/6-31++g** and SH/SA6-CASSCF(10,6)/6-31+g** simulations. Dynamics leads mainly to the dimer dissociation showed as the red curve for BOMD, black for SH and blue for RT-TDDFT simulations. The green (RT-TDDFT) curve corresponds to the H3O+ and OH· bonded complex and the orange (RT-TDDFT) curve represents the dissociation of one OH bond.

Finally, we present the results of the RT-TDDFT calculations for a doubly ionized water dimer with the positive holes localized in a single molecular orbital (see Figure S4 in the supporting information); such a state is typically formed within the Auger process. Here, the 20


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direct comparison with the SH simulations is not possible as the state of interest represents a highly excited state embedded in a large number of states with a delocalized charge. In this case, we observed an immediate proton transfer to a neighbouring molecule followed by a swift back transfer. Eventually, the proton dissociates from molecule and significant fraction of the population shows even two dissociating protons.

Conclusions We studied the applicability of the RT-TDDFT approach to simulate X-ray/VUV induced processes in water, using water monomer and dimer models. The present calculations were confined to valence and inner valence ionized states. We conclude that the general features of short-term dynamics are captured for an ionized water dimer with the GGA-based RT-TDDFT simulations. Immediately upon ionization of the donor molecule, the RT-TDDFT/GGA simulations seem to correctly predict the proton transfer. However, the proton transfer is too fast and it is not accompanied by oxygen-oxygen contraction. The final reaction outcome is the dissociation into H2O+ and H2O. This contrasts with the CASSCF based SH dynamics which points to a dominant formation of H3O+ and OH fragments. Furthermore, the RT-TDDFT/GGA simulations cannot capture minor dissociation channels as previously observed upon the ionization of the 1b2 electron. The unphysical behaviour is related to the approximate nature of the GGA functional, suffering from the well-known self-interaction error.66 Even if we start with a correct electronic structure with a charge localized on a single unit, the electrons will swiftly relax towards the (incorrect) equilibrium arrangement of the electrons. This situation limits the applicability of the RT-TDDFT/GGA in the radiation chemistry, because single ionization is an important step in many radiation processes. We showed that using hybrid functionals should improve the situation, however, the price we pay is higher computational demands forcing us to use atom centered basis sets. This might be a severe limitation for the current applicability of the RT-TDDFT in radiation chemistry. Other approaches such as self-interaction correction scheme can represent an alternative solution more suitable for the plane wave formalism.118 Ehrenfest based RT-TDDFT dynamics can also suffer from additional problems for more energetic states (e.g. limited ability to describe multiple bond breaking), but correct treatment of the selfinteraction error is essential for simulations in radiation chemistry field. 21



On the other hand, the RT-TDDFT/GGA calculations were found to be able to simulate doubly ionized states as it has already been demonstrated.56,89 As we can select an arbitrary nonequilibrium arrangement as a starting point of the simulation, we can start with an electron ionized from any pair of (localized) orbitals.119 In this way, the RT-TDDFT/GGA simulations allow us to simulate processes following the Auger process. Simulating similar processes such as ICD or ETMD120 is more challenging as the states of interest represent two singly charge molecular units. Caution should be paid to the spurious orbital near-degeneracy with approximate density functionals. For this reason, the case of a doubly ionized water dimer can only be reliably studied in the triplet state. Our final objective is simulating radiation processes in liquids. This brings us to a difficult conundrum. Ideally, the simulations start with a liquid in the ground state and the ionization process as well as the follow up processes would be modelled within a single simulation. While processes involving formation of free electron would require using plane wave basis or real space grid (and therefore GGA functionals), the subsequent electron-nuclear dynamics needs to be described with hybrid functionals (and therefore preferably with atom centered basis set). Some other processes such as Auger decays are hard to describe with RT-TDFT at all.116 The different stages of radiation chemistry simulations therefore have to be performed separately with the current stage of technologies.

Supporting Information Manuscript is accompanied by the supporting information file consisting: the code compilation details, timestep and initial wavefunction testing, comparison of the Octopus and CP2K codes’ dynamics, comparison of BOMD and RT-TDDFT dynamics of several singly and doubly ionized states of water monomer and dimer.

Acknowledgments The financial support of Czech Science Foundation (GA 13-34168S and 18-23756S) and MŠMT project COST CZ No. LD15025 is gratefully acknowledged. The computational resources were provided through a project of the Ministry of Education, Youth and Sports from the Large 22


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Infrastructures

for

Research,

Experimental

Development

and

Innovations

project

“IT4Innovations National Supercomputing Center – LM2015070“. Jan Chalabala is a student of International Max Planck Research School for "Many-Particle Systems in Structured Environments" and acknowledges a financial support from specific university research (MSMT No 20-SVV/2018).

References

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