Rydberg-type electronic excited state dynamics in small sodium–water clusters

Equilibrium geometries and thermodynamic potentials of the neutral and ionised species of the $ \hbox {Na}\cdots (\hbox {H}_{2}\hbox {O})_{{n}} $ Na⋯(H2O)n ( $ {n}=1,..,8 $ n=1,..,8) mixed clusters were computed at MN15/def2-TZVPD level of density functional theory (DFT). The vertical and adiabatic ionisation energies and enthalpies were computed and their cluster size dependence was discussed. Laser-induced ionisation involves electronic excitation through Rydberg-type excited states, which have been characterised using the TDDFT method, including the ωB2PLYP double-hybrid exchange-correlation functional. Ab initio molecular dynamics calculations were performed on a time scale of 20 picoseconds. Fluctuations of the charge and the sodium–oxygen atomic distances predict that, the $ 3\hbox {s}^{1} $ 3s1 electron of the sodium atom are transferred from the delocalised Rydberg orbitals to the Rydberg orbitals around the water molecules and the sodium atom becomes positively charged with around 0.6e after the first $ 10\,\hbox {ps} $ 10ps. On the other hand, some of the water molecules can move away up to 5 Å  from the sodium with a significant negative charge on them. It has been shown that non-radiative relaxation cannot be excluded, they can mostly occur for cases $ n \geqslant 4 $ n⩾4. The results confirm that the adiabatic photo-ionisation can occur on the basis of cluster disintegration. GRAPHICAL ABSTRACT


Introduction
The specific conditions for the formation of a solvated electron and its subsequent dynamical behaviour have been the focus of scientific attention for many years [1][2][3][4].Solvated electrons are typically created by radiolysis or photoionisation of molecules in clusters or in solutions [5,6].The escaping of this solvated electron from its microenvironment can lead to phenomena such as nucleation and aerosol growth in the high Studies on structure and dynamics of clusters carried out in molecular beam experiments are particularly suitable for understanding the structures and properties of the smallest molecular aggregates formed in the initial stage of the nucleation phenomena [26,27].Mass spectrometry combined with Na-doping and UV photoionisation has been proposed as a method for evaluating aerosol dimensions [28], being based on soft-ionisation that allows the detection of mass distributions with little changes due to photofragmentation [29].Using conventional UV lasers, the sodium-doped small water clusters are easily detectable in the mass spectrometry and it was extensively studied, due to their photoionisation mechanism and the key role of the 3s 1 electron of the sodium atom in the formation of the ionised cluster forms [18,[30][31][32].
In photoionisation experiments the shape of the photoionisation efficiency (PIE) curve contains information about the nature of the process [33,34].Sharp PIE profile is mostly characteristic for the vertical ionisation, while the broader one can be assigned to those ionisation processes which shows large structural rearrangement upon ionisation, known as adiabatic ionisation.In the latter case, Gao and Liu [35] showed that, by the removal of the sodium's 3s 1 electron, extensive structural reorganisation takes place.This structural rearrangement often means that there is an energetically significant difference (larger than 1 eV) between vertical and adiabatic ionisation potentials [23][24][25].Moreover, the excitation energy does not need to reach the ionisation potential barrier, it is enough to excite only the higher Rydberg orbitals to obtain an ionised cluster structure after relaxation.However, the formation of an ionised cluster structure also implies that the detached electron leaves the system together with a smaller cluster fragment [35].On the other hand, the process by which the vertically excited electron leaves the sodium atom, passes through the water molecules and, together with one of the molecules, moves away from the nucleation centre has not yet been fully understood.Theoretical analysis based on the quantum mechanical treatment can help to better understand the phenomenon in detail, but the simple 'static' ab initio calculations cannot fully answer this question, as they do not take into account the dynamical behaviour of the electrons and nuclei inside the cluster system.
Tha aim of the present work is to characterise in detail the nature of Rydberg-type electron transitions involved in the photo-ionisation process of the small sodium-water clusters, as well as to follow through the initial phase of adiabatic relaxation in time by means of ab initio molecular dynamics study.

Ionisation potential
The equilibrium geometries and thermodynamic potentials of the neutral and ionised species of the Na • • • (H 2 O) n (n = 1, .., 8) mixed clusters were obtained in the framework of the Density Functional Theory (DFT) implemented in the PSI4 code [36].Accordingly, the MN15 [37][38][39] exchange-correlation functional together with the def2-TZVPD [40,41] basis set were employed for the electronic structure calculations.The vertical-(VIE) and adiabatic ionisation (AIE) energies were computed considering the total energy differences between the neutral (E(N)) and its cationic (E(N-1)) forms.In order to include the thermal effects in the adiabatic ionisation energy calculation, the enthalpy values for the neutral and ionic species were also computed via the thermochemistry module of the vibrational normal mode analysis.

Ab initio molecular dynamics (AIMD)
Ab initio molecular dynamics of the electronic excited states of the Na • • • (H 2 O) n (n = 1, .., 5) mixed clusters were performed considering the molecular dynamics (MD) module implemented in the ORCA programme package [42,43].The MD module was developed in the framework of the Born-Oppenheimer molecular dynamics (BOMD) simulations, by solving the timeindependent Schrödinger equation to compute the gradients and the new coordinates of the atoms [44,45].Since, the ionisation energy of the Na-water mixed clusters were measured in the supersonic expansion in the vacuum, periodic boundary conditions or buffer gas effects were not taken into account.However, a harmonic repulsive wall around the system was considered in order to to keep the molecules inside of a well-defined volume.The wall was considered to be spherical in shape and its radius was chosen to be 2-2.5 Å larger from the furthest atom taken from the sodium.For the spring constant the value of 20 kJ mol −1 Å −2 was considered.A sphere with an extra width of 2-2.5 Å and the soft wall was chosen in order to allows the detaching fragment to return to the system without any velocity increase.The force defined by this spring always acts on the atoms in the system which are outside of the cell.The molecular dynamics simulations were done at room temperature (298 K), considering the Nosé-Hoover chains thermostat [46,47].The time step of the dynamics was set to 0.5 fs, while the strength of the cluster system-thermostat coupling was set to 10 fs.The whole duration of the dynamics was considered to be 20 ps (40000 time step).The first 50 fs dynamics were considered for the system thermalisation by starting with the randomly initialised velocities, then the 20 ps MD run was started considering these thermalised velocities.On the other hand, the 0.5 fs is considered a reasonable time step in order to maintain the linear regime of the time propagation between two consecutive time steps [48].
The electronic excited states were computed using the time-dependent DFT (TD-DFT) method considering the ωB2PLYP [49] range-separated double-hybrid exchangecorrelation (XC) functional implemented in the same ORCA programme package [42,43].The ωB2PLYP XC functional has shown improved performance not only for long-range excitations (Rydberg and charge-transfer excitations) but also for local-valence excitations and the difficult to treat first two excitations in polycyclic aromatic hydrocarbons [49].In the case of the excited state calculations the RIJCOSX approximation [50] as well as the Def2/J [51] and def2-TZVPP/C [52] auxiliary basis sets were also taken into account.In order to analyse the role of the diffuse functions on the electronic excited state energies, TD-DFT calculation with the and aug-cc-pVQZ [53] basis set was also considered.For the analysis of the charge population (Löwdin atomic charges) of a given electronically excited state, a posterior TDDFT calculation using the corresponding excited state density was performed based on the trajectory coordinates obtained from the original MD run.To have a clearer view of the true underlying behaviour of the dynamics for excited state energies, charges or bond distances, instead of the original value an averaged value was taken, considering the adjacent averaging smoothing algorithm implemented in the OriginLab software [54].Each smoothed x i data point is computed from data points within a moving window of {x i−n/2 , . . ., x i+n/2 }.Accordingly, each x i was computed using the following expression: x j In the present case, n is taken to be 100, which means that the averaging is considered for a time interval of 5 fs.

Experimental conditions
Before starting the theoretical study of the cluster systems, it is important that the experimental conditions are being well defined.The scheme of the experimental apparatus has been described in detail in Ref.
-s [22,24] (See Figure 1(a) in these two previous research papers).The experimental phenomenon is briefly described as follows: first, the water vapours are expanded in ultrahigh vacuum forming a supersonic molecular jet.Then the molecular beam is sent to a small stainless-steel oven, kept at 200 • C, which vaporised sodium flakes.In the oven the Na picked-up the water clusters and the jet is introduced in the mass spectrometer where is radiated by an UV laser source.The ionised clusters, after travelling the free flight region, are detected and their mass spectra are acquired at different ionisation energies in the range 4.66 -3.01 eV.Thus, a model can be understood as a molecular cluster in vacuum that does not interact with its environment, only the coupling with the thermostat was taken into account in order to consider temperature effects during the dynamics.For neutral Na mixed clusters starting when the laser-induced vertical excitation is produced, the time of flight in the mass spectrometer, after that their ionised forms are detected, is about 10-50 microseconds, depending on the mass of the clusters.

Cluster structures, ionisation potentials and electronic excited states
As first step, the equilibrium geometry structures, vibrational normal modes and thermodynamic potentials of the neutral and ionised forms of the Na ., 8) mixed clusters were computed at DFT level of theory considering the MN15 exchange-correlation functional and def2-TZVPD basis set.The molecular structures of the optimised cluster structures both in neutral and ionised electronic configurations are shown in figures of Table S1 of the supplementary material (SM).Furthermore, the vertical-(VIE) and adiabatic energies (AIE) as well as the adiabatic ionisation enthalpies (AIH) of the Na-water mixed clusters have been also performed.
Different ionisation potential values obtained using theoretical calculations as well as the experimental values taken from Ref. [24] are presented in Table 1.The socalled 'appearance energy' seen in the mass spectrum can be associated with the adiabatic ionisation potential and therefore can be compared with the experimental IP values.
Apart from the n = 1 case, it can be observed that the experimental results obtained for the ionisation potential are between the vertical (VIE) and the adiabatic (AIE) theoretical IP energy values, slightly closer to the AIE values than to the VIE.If, however, thermodynamic effects are taken into account (IPs based on enthalpies called AIH), one obtains values that are more in line with the experimental values.It is also important to note that in addition to the cluster configuration with a global energy minimum, there are several local minimum configurations that are energetically very close to the global minimum, but differ in structure.This is particularly true when the size of the cluster increases.However, the combined effect of these is very difficult to take into account and it is therefore hardly possible to expect an exact  Note: a Experimental values were taken from Ref. [24].
agreement between experimental and theoretical values, especially in the case of n 6.For VIE, it is observed that for n 4 the values decrease and then start to increase for n > 4, while for AIE there is a monotonic decrease as the cluster dimension increases.This in turn means that as the cluster dimension n increases, the difference between the vertical and adiabatic IP values will also increase and exceed 1 eV for n 8.
Since the main goal of this study is to provide a comprehensive description of the ionisation mechanism in the Rydberg-type electronically excited state, it is also worthwhile to investigate the ability of the ωB2PLYP XC functional to reproduce the experimental IP values.Accordingly, the equilibrium geometry structures, vibrational normal modes and thermodynamic potentials of the neutral and ionised forms of the Na In the experimental results presented in Ref [24], it can be also observed that it is not necessary to excite the cluster system above the first ionisation potential, but it is sufficient to get to an intermediate energy level after which the system reaches the ionised state during the geometry relaxation over the electronic excited states.It is important to note that 3.50 eV laser energy was sufficient to ionise the Na • • • (H 2 O) n mixed cluster, while the excitation energy of the Na • • • (H 2 O) n sodium-formic acid-water mixed cluster was 3.97 eV [24].Accordingly, the first six low lying electronic excited states of each Na • • • (H 2 O) n (n = 1, .., 8) cluster configuration were computed considering the TD-DFT framework by using the ωB2PLYP range-separated double-hybrid XC functional and def2-TZVPD basis set.Energy values up to the sixth excited state level are collected in Table 2, while the UV-Vis absorption spectra are shown in Table S2 in the SM file.Based on the values presented in Table 2 and the spectral figures shown in Table S2 in the SM file, it can be concluded that for each cluster there is a stronger and a weaker absorption spectral region.The stronger part is formed by the excited electron states S 1 -S 3 , while the weaker part is formed by the higher excited states above S 4 .A gap slightly wider than 1 eV is observed between the two spectral regions.Next, the nature of the electron on the last shell was investigated.In order to do this, localised molecular orbits obtained from the Pipek-Mezey algorithm [55] were determined rather than canonical molecular orbits.The localisation method does not result in a narrow range of spaces confined around an atom (e.g.sodium), but gives a very spread out orbital profile (See Table S3 in the SM file).Furthermore, analysing the electron transitions (See their natural difference orbitals in Table S4 in the SM file), it can be observed that the first six electron states whose energy values are shown in Table 2 are all Rydberg-type excitations, with no valence-type excitations appearing in the low-lying spectral region.In addition, it can be also observed that the Rydberg electron, which is proportionally small around the sodium atom, tends to be concentrated around the water molecules during excitation.It is known that the size of the diffuse basis function included in the basis set can considerable change the Rydberg-type excited electronic levels [56].Accordingly, the first six low lying electronic excited states of each Na • • • (H 2 O) n (n = 1, .., 8) cluster configuration were also computed considering the TD-DFT framework by using the same ωB2PLYP range-separated double-hybrid XC functional, but at this time setting the aug-cc-pVQZ basis set.The results are collected in the second part of Table 2. Using larger diffusion functions through the aug-cc-pVQZ basis set results in energies of the S 3 -S 6 excited states that are on average 1-1.5 eV lower than those obtained with the def2-TZVPD basis set.Furthermore, at the end of Table 2, in the last row, the order of the smallest excited state with an energy above 3.5 eV is shown.It can be observed that the order of these states for the aug-cc-pVQZ basis is much higher (for n = 3, .., 8 is the 16th order or higher) than that obtained for def2-TZVPD.Unfortunately, this information is not enough to explain how ionisation can occur during relaxation, so the time evolution of the system dynamics must also be investigated.
Table 2.The first eight electronic excited states of the Na • • • (H 2 O) n mixed clusters computed at TD-DFT/ωB2PLYP/(def2-TZVPD and aug-cc-pVQZ) levels of theories (energies are given in eV, while the corresponding wavelengths in nm in parenthesis).

Rydberg-type electronic excited state relaxation
Given that one wants to follow electron (ground or excited) states and their energy and charge characteristics, we have to consider the nuclear and electron dynamics defined in the framework of the quantum mechanics.This, in turn, implies a time step below femtoseconds for electron dynamics and significantly limits the whole time interval for the total nuclear dynamics (few tens of picoseconds).In the light of these considerations, combined electron and nuclear dynamics have been performed only for the n = 1, .., 5 cases of Na • • • (H 2 O) n cluster configuration, considering the optimised geometries as the starting configurations for the dynamics.Furthermore, since for smaller clusters (n = 1,2) the electron states S 4 and for larger clusters (n = 3,. . .,5) the electron states S 5 are closest to the excitation energy for which adiabatic ionisation was experimentally observed, the fifth excitation electron state is considered as the initial state for the dynamics of each cluster.Monitoring the energy of the one lower excited state (that of S 4 compared with S 5 ) is necessary in order to follow the possible transitions between states, which thus provide the possibility of radiationless relaxation [57] through the Rydberg states.On the other hand, from the computational cost point of view, several trajectories or large basis sets cannot be afforded, thus only one trajectory for each cluster case was considered.This, unfortunately, means that we do not have a complete picture of the dynamics, but we expect to see trends in a trajectory that is typical of such clusters.

Na • • • (H 2 O) 1 cluster
Ab initio molecular dynamics for the 1:1 sodium-water cluster configuration was performed for 20 ps time scale.The excitation energy changes of the fourth and fifth electronic excited states are given in Figure 1(a), while those for the Mulliken effective charges of the sodium atom and water molecule as well as the Na • • • O bond distance is shown in Table S5 in the SM file.During the 20 picosecond dynamics, it can be observed that the charges on the sodium atom and the water molecule are distributed in an average ratio of 0.5e:-0.5e.Exceptions are the time intervals of around 10 and 12-14 ps when the sodium atom moves away from the water molecule (up to 5 Å) and the charge polarisation becomes larger (0.7e:-0.7e).During these time intervals, the values of the S 4 and S 5 excited energies also spike a bit.Overall, it can be said that during the 20 picosecond dynamics, the charge after excitation is evenly distributed between the sodium atom and the water molecule, and except for a small time interval, the Na-water complex remains together.

Na • • • (H 2 O) 2 cluster
In the case of 1:2 mixed cluster stoichiometry, the starting geometry shows a hydrogen-bonded dimer configuration, where one of the lone-pair electron orbitals of each O atoms are oriented towards the sodium (See the second row form Table S1 in the SM file).For this case, as for the previous one, a 20 ps dynamic was performed.As for the temporal behaviour of the excited states, it is observed that both S4 and S5 states fluctuate around a well-defined mean value (see Figure 1(b)).They are: < S 4 >= 3.15 eV and < S 5 >= 4.08 eV.By analysing the temporal behaviour of the effective charges on the sodium atom and on the two water molecules separately, it can be seen that during the first 5 ps the charge of the sodium atom varies between 0.5e and 0.6e, then decreases to 0.4e for a few picoseconds and jumps back to 0.6e for a few moments until the dynamic end of 20 ps (See the first row form Table S6 in the SM file).If we look at water molecules, we can see that the charge fluctuations of one of the two water molecules behave in a similar way to the sodium atom.That is, as the electron charge of the sodium atom decreases (e.g.approaching 0.6e), the electron charge of the water molecule (labeled with index 1 and called as Wat 1 ) also decreases.On the other hand, the charge fluctuations of the other water molecule (labeled with index 2 and called as Wat 2 ) behave in the complete opposite way, i.e. when the electron charge of the sodium atom decreases, the electron charge of the water molecule (Wat 2 ) increases.It is particularly interesting that in these charge fluctuations there are short time moments when the electron charge of Wat 2 exceeds 0.7e.As for the atomic distances d(Na • • • O i ), i = 1,2, two different behaviours are also observed.In the case of d(Na • • • O 1 ), after an initial larger oscillation, the value of the distance seems to stabilise around 2.4 Å which means that sodium and Wat 1 remain relatively compact.For the other distance d(Na • • • O 2 ), however, one or two much larger excursions can be observed at every 5 ps or so.These fluctuations ( > 4.5 Å) are closely related to the charge fluctuations at Wat 2 , since the maxima of the distance deviations are also the maxima of the electron charge fluctuations of the Wat 2 .

Na • • • (H 2 O) 3 cluster
The starting geometry for cluster structures containing three water molecules and one sodium atom is shown in the third row of Table S1 in the SM file.Is built by a hydrogen-bonded water dimer and by a separate, third water molecule, but the oxygen in all three water molecules is turned towards the sodium atom.20 ps long time resolved molecular dynamics of the fourth and fifth electronic excited states of the Na • • • (H 2 O) 3 mixed cluster show a relatively higher amplitude fluctuating S 5 state, while the S 4 state still behaves with only small fluctuations compared to the < S 4 > value as already shown in the previous, Na • • • (H 2 O) 2 case (See Figure 1(c)).However, it can also be observed that the energy levels of S 5 and S 4 are closer in value than was seen in the previous case.The charge distribution on the sodium atom shows an increasing value over time, from 0.15e to 0.6e (See the first row form Table S7 in the SM file).If one looks at the dynamic behaviour of water molecules, it can be observed that the electron charge of the Wat 1 molecule fluctuates between −0.25 e and 0.05e for the first 12 ps and stays close to the sodium atom, then after 5 ps it moves away from the sodium atom and its charge jumps to −0.5 e.The situation is not so clear for the other two water molecules.The distance graphs show that in the 7-10 ps interval, for example, Wat 2 -Wat 3 forms a dimer configuration, while in the 14-17 ps interval the Wat 1 -Wat 2 complex is formed.What is clear, however, is that there is no single water molecule that stays tightly attached to the sodium atom during the 20 ps dynamics, but rather one of the three water molecules alternates with the sodium, while the other two form a hydrogen-bonded water dimer.At the same time, the charge distribution of the S 5 excited state is looks like an excess effective electron charge (between 0.4e and 0.6e) is observed on the water molecule that is at least 4 Å away from the sodium atom.

Na • • • (H 2 O) 4 cluster
For the n = 4 water molecule, the dynamics starts from an initial geometry where the four waters form a pyramidlike structure (See the fourth row of Table S1 in the SM file), where the three water molecules which form the pyramid base also interact with the sodium atom, and thus, the fourth water molecule (in this case indexed as Wat 2 ) in the cluster will be a little further away from the sodium atom.As for the dynamics of the 4th and 5th excited electron states, on first sight, the same behaviour is observed as for the n = 3 case, i.e. the two energies are close to each other (See Figure 1(d)).But at the same time there is a significant difference, i.e. at around t = 14 ps the two energy states come very close to each other and can even cross each other.However, this does not exclude the possibility that the fifth excited state can turn into the fourth excited state through the internal conversion phenomenon [57][58][59] at well-defined geometry conformations (also called conical intersection geometries) and then the system can continue its dynamics in this lower state, i.e. the non-radiative relaxation of the excited states can start.It should also be noted that for isolated molecules, such as DNA bases, the relaxation from higher excited states to the S 1 state tends to occur in the 100 femtosecond time interval [60,61], which is much faster than the ≈ 15 ps seen in the present case, which can be explained by the intermolecular nature of the cluster.The fluctuation of the effective charge attached to the sodium atom shows during the whole simulation time an oscillating behaviour between 0.2e and 0.6e electron charge (See the first row form Table S8 in the SM file).As for the charge of the water molecules, in the first half (≈ 10 ps) of the simulation time, excess charge is seen partly on the Wat 1 -Wat 2 dimer and partly on the Wat 4 monomer, and in the second half of the simulation time, it appears first on the Wat 3 and then on the Wat 2 monomer.From distance graphs can be seen that increased charges on waters also mean increased distances d(Na • • • O i ), i = 1,..,4 between sodium and oxygen atoms.In general, it can be seen that throughout the simulation, the Wat 1 , Wat 3 and Wat 4 water molecules alternately stay together with the sodium atom for shorter periods of time (the d(Na • • • O) value is around 2.5 Å), while Wat 2 only approaches the sodium for short moments.

Na • • • (H 2 O) 5 cluster
In the present work, n = 5 is the largest system whose time-scale dynamics has been studied.The initial geometry is very similar to n = 4, with the pyramidal configuration of the four water molecules on one side of the sodium atom and the fifth water on the other side (See the fifth row of Table S1 in the SM file).The energies of the fourth and fifth excited states are even closer together than seen in the case of the lower dimensional clusters (See Figure 1(e)).Thus, by consequence, the probability of crossing the energy surfaces and of non-radiative relaxation is even higher than it was observed in the previous cases, as such an event can occur in less than 10 ps.As far as the charge migration is concerned, a similar behaviour is observed as for the n = 3,4 cases.That is, during the first 10 ps of the dynamics, the charge of the sodium atom changes between -0.1e and +0.2e, followed by a larger jump during which the sodium charge stabilises around 0.5e -0.6e during the second 10 ps of the dynamics.As for the distances from sodium or charges of water molecules, it can be seen that in this case there is also at least one water molecule which moves away from sodium and has a negative charge (For example, see the behaviour of Wat 4 in Table S9 in the SM file).

Conclusions
The equilibrium geometries of the neutral and ionised species of the Na • • • (H 2 O) n (n = 1, .., 8) mixed clusters were computed at MN15/def2-TZVPD level of density functional theory (DFT).Based on these geometries the vertical-(VIE) and adiabatic ionisation energies (AIE) as well as the adiabatic ionisation enthalpies (AIH) of the clusters were computed at the same MN15/def2-TZVPD theory level.Significant energy differences between the vertical and adiabatic values (larger than 0.5 eV) are obtained suggesting that substantial rearrangements in the geometries may occur during adiabatic relaxation.Our previous experimental analysis [24] also showed that vertical ionisation of clusters is not necessary, it is enough to ionise them to higher electron states, i.e.Rydberg fields, the adiabatic ionisation will occur in this case as well.A detailed analysis of the Rydberg-type excitations, computed at TD-DFT/ωB2PLYP/def2-TZVPD level of theory, shows that the 4th and 5th excited electron states can lead to adiabatic ionisation.When the aug-cc-pVQZ basis set with larger diffuse function is used, the excited energy level reached by the laser irradiation could be the 16th order or even higher.The nature of the excitations shows that electrons move from the highly delocalised Rydberg orbitals bound to sodium to the less delocalised Rydberg orbitals close to water molecules.In order to follow the adiabatic relaxation more thoroughly, ab initio molecular dynamics calculations were performed for 20 picosecond time intervals for the n = 1,..,5 cluster cases.After the first 10 ps dynamics, the results based on the charge and the sodium-oxygen atomic distances predict that, on the one hand, the 3s 1 electron of the sodium atom are transferred to the Rydberg orbitals around the water molecules and the sodium atom becomes positively charged with around 0.6e.On the other hand, some of the water molecules will not stay permanently near the sodium, they can move away up to 5 Å from the sodium with a significant negative charge on them.Moreover, it has also been shown that non-radiative relaxation cannot be ruled out, although relaxation of higher excited states occurs more slowly than is the case for organic molecules.This kind of radiation-free relaxation has been observed mainly for the n 4 cases.In this way, the results confirm the idea that the phenomenon of adiabatic photoionisation occur on the basis of cluster disintegration [35], where the larger part of the cluster is a positively charged core together with the sodium atom, and a smaller part is a negatively charged water molecule or water dimer.However, it is also important to stress that the current model needs improvement in several areas.In addition to taking into account the longer time interval of the dynamics and more accurate electron structure methods, it is important, for example, to treat the radiationless relaxation, which allows a more accurate description of the relaxation of excited states, and to also take into account basis sets with larger diffusive functions in order to achieve the basis set saturation for the population analysis.
• • • (H 2 O) n (n = 1, .., 8) mixed clusters were also computed with ωB2PLYP range-separated double-hybrid XC functional by considering the same def2-TZVPD basis set.The VIE, AIE and AIH values are shown in the second part of Table 1.It can be observed that the AIE and AIH values obtained with the ωB2PLYP method give on average values 0.2 eV lower than those obtained with MN15.