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Field-Induced Electron Generation in Water: Solvation Dynamics and Many-Body Interactions

The solvated electron represents an elementary quantum system in a liquid environment. Electrons solvated in water have raised strong interest because of their prototypical properties, their role in radiation chemistry, and their relevance for charge separation and transport. Nonequilibrium dynamics of photogenerated electrons in water occur on ultrafast time scales and include charge transfer, localization, and energy dissipation processes. We present new insight in the role of fluctuating electric fields of the liquid for generating electrons in presence of an external terahertz field and address polaronic many-body properties of solvated electrons. The article combines a review of recent results from experiment and theory with a discussion of basic electric interactions of electrons in water.


Introduction
Electrons and protons solvated in liquid water are among the most elementary chemical species. They represent prototypic quantum systems embedded in a fluctuating environment without long-range structural order. Interactions between the charged particles and their solvation shell include electric as well as other attractive or repulsive forces from noncovalent couplings. Vice versa, the presence of electrons and/or protons affects the local structure of the liquid and, upon nonequilibrium excitation, induces ultrafast solvation and energy dissipation processes. Such properties make solvated charges a direct probe of liquid dynamics. Beyond such basic aspects, solvated electrons and protons are relevant for the self-dissociation of water, charge transport in and radiolysis of aqueous systems, e.g., of hydrated DNA and RNA structures.
Solvated electrons have attracted strong and continued interest from both theory and experiment. [1][2][3][4] The so-called cavity model describes an electron in an excluded volume, surrounded by water molecules which form mainly electrostatic hydrogen bonds with the charged particle. The orientation of solvent molecules by interaction with the electron results in a reduction of free energy and, thus, a binding electron potential of a depth of some 5 eV.
The wavefunction of the electron ground state is essentially localized within the cavity but displays some overlap with the first few water shells. This description has been challenged by a non-cavity picture according to which the electron's spin density is delocalized over several water molecules and induces an enhancement of water density near the center of the electron wavefunction. 5 Disputing this view, it was argued that the absence of a cavity was the result of a problem with the pseudo-potential parametrization. [6][7][8] Following a vigorous debate, the cavity model is now widely accepted. [9][10][11][12][13][14][15] Multiphoton ionization of water by femtosecond pulses has allowed for mapping ultrafast nonequilibrium dynamics of solvated electrons in pump-probe and photon-echo experiments. [16][17][18][19][20][21] The transition from initially delocalized electronic states to a localized bound quantum state in a self-consistent potential has been probed via the related spectral shift 2 of optical absorption of the electron from the terahertz (THz) to the near-infrared spectral range. Localization occurs on a subpicosecond time scale, in parallel to the dissipation of excess energy which extends up to some 10 ps. After excitation of pre-existing solvated electrons from their electronic ground to higher electronic states, a sequence of subpicosecond relaxation processes between excited electronic states has been observed.
Recently, a novel generation mechanism of electrons based on tunneling ionization of water molecules in presence of an external THz field has been demonstrated. 22 The extremely high fluctuating electric field from water molecules in thermal motion spontaneously induces tunneling ionization of individual water molecules while irreversibility of the process is achieved via the separation of the electron from its parent ion in the THz field. The dynamics of such processes and the relevant molecular interactions are discussed in this article.
The many-body character of Coulomb interactions between the electron and its fluctuating surrounding and between polar solvent molecules leads to local-field effects with a direct impact on the character of excitations of the solvated electron. As a consequence, nuclear and electronic degrees of freedom are being coupled and polaronic properties arise, beyond what is captured by existing one-electron models. Insight in this important aspect has remained very limited, in particular from the experimental side. Most recently, first experimental evidence for polaron excitations of solvated electrons has been presented. 23 In this article, we combine a short review of recent advances in understanding nonequilibrium dynamics of solvated electrons with a discussion of basic electric interactions of electrons in water. The role of fluctuating electric fields for generating electrons in an external THz field is addressed and new insight in Coulomb many-body interactions and polaronic properties of solvated electrons presented. The content is organized as follows. Section 2 gives a summary of theoretical and experimental knowledge on structural properties and nonequilibrium dynamics of bulk water and solvated electrons. In Section 3, we discuss how the very high local electric fields in water induce tunneling ionization of water molecules and the separation of the electron from its parent ion in presence of an external THz field. Section 4 3 focuses on many-body aspects of Coulomb forces and addresses the polaronic properties of solvated electrons. Conclusions and an outlook are given in Section 5.
2 Electrons solvated in water: Structure and Spectroscopy 2

.1 Electric interactions and dynamics of bulk water
Water molecules in the liquid phase display a permanent electric dipole moment of approximately 3 Debye which is substantially larger than the dipole moment of an isolated water molecule of 1.86 Debye. This dipole enhancement is due to polarization by the electric field the other molecules in the liquid impose on a particular molecule. 24 In addition to long-range In Fig. 1(d), the time correlation function C(t) = δE(t)δE(0) / δE(0)δE(0) of the fluctuating electric field is shown, where E(t) stands for the amplitude of the electric field projected on the axis of the 3a 1 orbital [ Fig. 1(c)] and δE(t) for its excursion from the average valueĒ. The brackets represent the ensemble average. The correlation function exhibits an initial decay on a 50-fs time scale and slower subpicosecond decay components.
The initial decay is governed by thermally excited librations in the frequency range between 500 and 1000 cm −1 . 30,31 The fluctuating electric field acting on a polarizable water molecule distorts its electronic structure and leads to time-dependent changes of electronic and vibrational transition frequencies, i.e., spectral diffusion and decoherence of quantum-coherent excitations and the resulting macroscopic polarization.
The combination of two-dimensional infrared (2D-IR) spectroscopy with theory and MD simulations has established a quantitative picture of the ultrafast dynamics of neat water, aqueous solutions of ions, and hydrated biomolecules. 25,26,32 Spectral diffusion has been followed in time via changes of 2D-IR line shapes of O-H stretching excitations which are governed by the frequency fluctuation correlation function (FFCF) δν(t)δν(0) of the transition frequency ν. 33 In turn, the FFCF is determined by the correlation function of the fluctuating electric field C(t) and, thus, the underlying structural dynamics of the liquid. A link between the momentary electric field amplitude E(t) and δν(t) is established with the help of vibrational frequency maps ν(E).
The FFCF of neat water exhibits an initial decay on a 50-fs time scale, due to highfrequency librational motions and followed by a decay on a time scale of several hundred femtoseconds. The latter is caused by molecular motions at lower frequency and, for O-H stretching excitations, resonant energy transfer between water molecules. In an even longer picosecond time range, hydrogen bond breaking and reformation contribute as well.
In case the water structure is subject to particular steric boundary conditions, such as in the solvation shell of ions or at the surface of a biomolecule, a moderate lengthening of the shortest correlation decay arises. 32  The molecular level understanding of the properties of the hydrated electron has been covered in recent comprehensive reviews, e.g., Refs. 1,2, while the basic structural motif relevant for hydrated electron stabilization by surrounding water molecules was outlined already in 1953: 34 the trapping of the electron in a potential well arises from a polarization of the surrounding water molecules via their orientation under the influence of the negative charge ( Fig. 2). The thermalized electron thus occupies an excluded volume in the structure of liquid water and coordination of water molecules occurs via the directed, hydrogen bond-like interaction with a single hydroxyl group of water molecules in the first solvation shell. At room temperature, the instantaneous coordination number is fluctuating and the hydration structure notably deviates from an idealized Kevan structure which consists of an octahedral arrangement of six water molecules with one OH group each pointing to the electron. 35 The predicted average number of hydrogen bonds varies between 4 and 6, depending on the particular theoretical model and the employed level of theory. Accordingly, the solvation pattern of the hydrated electron in the canonical cavity model 36,37 resembles that of a pseudohalide Fig. 2b,d). Different theoretical methods, ranging from single active electron quantum mechanical -molecular mechanical (QM/MM) methods in the Turi-Borgis (TB) model, 37 density functional theory (DFT), 13 and many body perturbation theory (MP2), 14 predict the hydration structure in the first solvation shell (Fig. 3a) with high consistency. For all methods, a characteristic of the cavity model is the breaking of water-water hydrogen bonds in the solvation shell surrounding the hydrated electron.
The formation of the cavity by a reorganization of the first few water shells leads to a stabilization of the electron ground state in a trap potential, characterized by an elec- cavity-forming models, 41,43,44,53,54 and various technical aspects of the LGS model have been criticized. 47,64 Here, we report new simulations using the LGS pseudopotential model and compare the results to two alternative, cavity-forming, one-electron models. 41,44 The latter are found to afford much better agreement with experimental data for the vertical electron binding energy (VEBE), the electronic absorption spectrum, and the electron's radius of gyration. In addition, we find that the agreement between experiment and LGS predictions deteriorates, for certain observables, when Ewald summation is used to evaluate the long-range Coulomb interactions, rather than the minimum-image convention that has been used in all previous LGS simulations.
We also report time-dependent density functional theory (TD-DFT) calculations of the e À (aq) absorption spectrum, using a carefully calibrated quantum mechanics/molecular mechanics (QM/MM) methodology based on solvent configurations extracted from the one-electron pseudopotential simulations. Quantitative or semiquantitative agreement with the experimental calculations are discussed in section 4. marizes the current state of affairs with r for one-electron pseudopotential mode

SIMULATIONS
Several different one-electron pseu examined herein, including the LGS mod above); the TuriÀBorgis (TB) model, 41 in some respects but employs a different potential; and also a polarizable mode developed. 44 The mathematical struct briefly reviewed in section 2.1, and then details regarding how the condensed-p were performed.

One-Electron Pseudopotentia
TB models of the hydrated electron bea with one another, even though their ph markedly. In particular, both models em "simple point charge" (SPC) force field a one-electron Hamiltonian of the form The symbol R B J denotes a collective set o water molecule. These coordinates ser one-electron Schr€ odinger equation,Ĥ propagate classical molecular dynamics along the ground-state eigenvalue, E = E( Feynman forces. 66 The electronÀwater interaction poten several components, Here, V Coulomb denotes the Coulomb electron and the SPC atomic point char is an ad hoc electronÀwater polarizati used in many previous studies. 67,68 We u in eq 3 to denote the distance between th water site is chosen as the origin for the spherically averaged polarizability function, 67 and C is a constant whose more sophisticated polarization potent previously. 66 [Note that C = 0 for the L the TB model; eq 3 is used for generali The final component of V eÀwater , and Representative images of the ground-state and first excitedstate wave functions for e À (aq), obtained from the cavity-forming "PEWP-2" pseudopotential model 44 and from the non-cavity-forming LarsenÀGloverÀSchwartz (LGS) pseudopotential model. 46 (Wave functions obtained from the cavity-forming TuriÀBorgis model 41 and the non-cavity-forming LGS-mPol model 47 are not shown, as they are qualitatively similar to the PEWP-2 and LGS wave functions, respectively.) Isosurfaces depicted here encapsulate 70% of |ψ| 2 . In panels (a) and (b), all water molecules within 4.5 Å of the centroid of ψ are shown, whereas panels (c) and (d) depict the entire simulation cell, which contains 600 water molecules in (c) and 499 molecules in (d).
The Journal of Physical Chemistry A stretches out approximately up to two water shells (Fig. 3a). 10 The cavity model has been invoked to explain a range of properties of hydrated electrons 8 co-workers (TB) ran MQC molecular dynamics (MD) simulations that also suggested that the hydrated electron occupied a cavity in the water, although the particular structure (with only roughly four first-shell water molecules, Figure 1b) was different than that suggested by SR. 10,11 More recently, the effects of dynamic water polarization on the hydrated electron's structure and energetics were explored by Jacobson and Herbert. 12,13 Although electronic polarization is included in most hydrated electron models in a mean-field manner, 14 Jacobson and Herbert extended the TB model to treat the water polarization explicitly and self-consistently. These authors found that the explicit inclusion of water polarizability is important both in obtaining meaningful ionization potentials and in correctly capturing the blue tail of the optical absorption spectra. However, these authors also observed that including polarizability only slightly modified the structure of the hydrated electron from that seen by TB: even with polarizability, the hydrated electron still resides in a  co-workers (TB) ran MQC molecular dynamics (MD) simulations that also suggested that the hydrated electron occupied a cavity in the water, although the particular structure (with only roughly four first-shell water molecules, Figure 1b) was different than that suggested by SR. 10,11 More recently, the effects of dynamic water polarization on the hydrated electron's structure and energetics were explored by Jacobson and Herbert. 12,13 Although electronic polarization is included in most hydrated electron models in a mean-field manner, 14 Jacobson and Herbert extended the TB model to treat the water polarization explicitly and self-consistently. These authors found that the explicit inclusion of water polarizability is important both in obtaining meaningful ionization potentials and in correctly capturing the blue tail of the optical absorption spectra. However, these authors also observed that including polarizability only slightly modified the structure of the hydrated electron from that seen by TB: even with polarizability, the hydrated electron still resides in a properties of the hydrated elec nature of the pseudopotential c TB and LGS derived their Phillips−Kleinman (PK) fo prescription for calculating a the wave function of the exces wave functions of all the elec calculated at the Hartree−Foc though they are constructed for is important to note that these the quantum chemistry of an e a single, gas-phase molecule molecule, however, does not b when constructing such pseudo the quantum chemistry of the by adding a confinement poten in the vicinity of the water m basis set with finite range. It i approximations only alter the range, so that the short-range p condensed-phase simulation is r any detailed exploration of approximations in the literature Regardless of the details beh pseudopotential, it is not comp full numerical PK potential standard procedure has been analytic function. TB decid determined PK pseudoorbital They adjusted the parameters o eigenenergy would be correct, chose missed many of the phy the pseudopotential. For examp for an excess electron interactin attractive region behind the w atoms) and a strongly repulsive ( Figure 2a) that are missed by t on the other hand, decid pseudopotential in such a wa features but at the cost of chan eV ( Figure 2b). Perhaps even in the shape of the fitting funct the simulated hydrated electron back again. 16,18 Thus, starting w co-workers (TB) ran MQC molecular dynamics (MD) simulations that also suggested that the hydrated electron occupied a cavity in the water, although the particular structure (with only roughly four first-shell water molecules, Figure 1b) was different than that suggested by SR. 10,11 More recently, the effects of dynamic water polarization on the hydrated electron's structure and energetics were explored by Jacobson and Herbert. 12,13 Although electronic polarization is included in most hydrated electron models in a mean-field manner, 14 Jacobson and Herbert extended the TB model to treat the water polarization explicitly and self-consistently. These authors found that the explicit inclusion of water polarizability is important both in obtaining meaningful ionization potentials and in correctly capturing the blue tail of the optical absorption spectra. However, these authors also observed that including polarizability only slightly modified the structure of the hydrated electron from that seen by TB: even with polarizability, the hydrated electron still resides in a  The radius of gyration, the root-mean-square distance of electronic charge from its center of gravity, is frequently used to quantify the spatial extent of the electron wave function. It shows a strong inverse correlation with the electronic excitation energy. 2 Such correlation of the electronic excitation energy and gyration radius provides access to the electron localization dynamics following generation of nonequilibrium excess electrons. The transition from nonequilibrium delocalized states to trap states confined in the cavity is connected with a shrinking radius of gyration and a concomitant shift of electronic absorption from the THz to the NIR spectral region, 21 in qualitative agreement with the estimate of the initial spatial wavefunction delocalization over some 40Å.
An alternative view to the cavity model was proposed where the electron occupies a region of enhanced water density of some ∼10Å diameter (Fig. 2c,e). 5 Within the first minimum of the electron-oxygen radial distribution function, ≈37 water molecules are contained, compared to only ∼30 water molecules expected for neat water, corresponding to a locally increased water density of ρ ≈ 1.2 g/cm 3 within the spatial extent of the electron. 12 In this so-called Larsen-Glover-Schwartz (LGS) model a cavity is absent and the electron has a rather delocalized character. Within the electron volume, a net orientation of water molecules is induced with a preference of the water O-H bonds pointing to the center of charge of the hydrated electron, but the hydrogen bond network within this volume is largely preserved. The charge density of the electron shows appreciable spatial overlap with the density of water molecules and the hydrated electron extends beyond the first hydrogen and oxygen peaks of the radial distribution function. The region of enhanced water density extends to some ∼ 6Å from where on bulk behavior is observed (Fig. 3b). Almost immediately after publication of the non-cavity model criticism of the employed pseudo-potential parametrization was raised 6-8 (see also Refs. 2,9-12,41 on the cavity vs. non-cavity model debate). Recently, short-range correlation effects were suggested to be particularly important for stabilizing non-cavity solvation structures of the condensed-phase hydrated electron. 15,42 Nevertheless, due to the largely consistent predictions of the electron hydration structure from ab-initio and DFT simulations 13,14

Resonance Raman spectrum of the hydrated electron
The resonance Raman spectra of the hydrated electron 53-55 suggest a predominant homogenous broadening of the optical absorption. 54 They further suggest that water vibrations, presumably of molecules in direct interaction with the hydrated electron, are resonantly enhanced upon electronic excitation of the hydrated electron and that the stretching, bending and librational modes of water associated with the hydrated electron are red-shifted com- 3 Electrons as a probe of fluctuating electric fields

Field-induced ionization and charge separation
The highest sub-50 fs peaks of the fluctuating electric field in bulk water reach an amplitude on the order of 200 MV/cm (cf. Fig. 1), which is sufficient for ionizing a water molecule by electron tunneling. The basic scenario is sketched in Fig. 4(a,b). The external electric field distorts the electronic potential of the molecule in a nonperturbative way, thus creating a tunneling barrier of finite width and height for electrons in the highest occupied molecular orbitals 1b 1 (HOMO) and 3a 1 (HOMO-1). Electrons can tunnel from the bound orbital states into continuum states outside the barrier, leaving an H 2 O + parent ion behind. The released electron represents a wavepacket made up of a superposition of continuum states and propagating along the spatial electric field gradient. Upon propagation, the wavepacket spreads in space and, in parallel, loses its quantum-coherent character under the action of the fluctuating electric field from the water environment. As a result of such decoherence, propagation comes to an end and the electron localizes at a distance from the parent ion.
For water in thermal equilibrium, this charge separation is not persistent but recombination probability has a value around 5 × 10 −12 for a peak electric field E peak = 150 MV/cm and rises to 10 −6 for E peak ≈ 300 MV/cm. Averaging over the 200-ps period of the MD trajectory in Fig. 1(b) gives an average tunneling probability of 3.4 × 10 −7 from the 3a 1 orbital. The tunneling probability from the 1b 1 orbital is much lower because of the smaller projected electric field (Fig. 1a).
• The wavepacket spreads to a several 10-Å width within the first few femtoseconds of propagation and reaches a distance of some 40Å from the parent ion after 2-3 fs propagation time. It is important to note that the model describes an electron wavepacket which propagates ballistically in an external electric field. In real liquid water, this field is screened on a length scale comparable to the estimated propagation length and additional random electric forces arise from the dipolar water molecules.
Thus, the model only covers the initial few femtoseconds of transport under the action of the electric-field spike. Propagation distances on a longer time scale can be derived from an analysis of experimental results and will be discussed in section 3.2.
• Using the Caldeira-Leggett path integral approach, 67 one estimates a decoherence rate Γ decoh = 2.6 × 10 14 s −1 ≈ 5Γ f luc where Γ f luc = 1/(20fs) = 5 × 10 13 s −1 is the fluctuation rate of the electric field in the liquid as derived from the fast decay of the field correlation function (Fig. 1d).
• The recombination process is described as a radiationless decay from the charge- The strength of the local directed electric field needs to be sufficient to accelerate the released electron to a ponderomotive energy U p = U I . U p is given by

Experimental results
Tunneling ionization of water molecules in presence of a strong THz electric field has recently been studied in ultrafast THz pump/ optical probe experiments and by 2D-THz spectroscopy. 22,73 Single-cycle THz pulses with an electric field amplitude up to 2 MV/cm and a 1-kHz repetition rate were generated by tilted-wavefront optical rectification of nearinfrared pulses in a LiNbO 3 crystal. 74 The THz transients were detected by phase-resolving free-space electrooptic sampling in a 10-µm thick ZnTe crystal. 73 This detection method gives the THz electric field as a function of real time t, as shown in Fig. 4(c).  A basic spectroscopic probe for the generation of solvated electrons is their optical absorption. The absorption spectrum of equilibrated electrons solvated in water is plotted as a solid line in Fig. 5(a). 76 The symbols in Fig. 5(a)  Electron motions driven by the THz field represent a time-dependent electric current in the water sample. This current gives rise to the emission of an electric field E em which is , the difference between the incoming THz field E i (t) and the total transmitted field E T (t). The current density j(t) can be derived from the measured

Transverse and longitudinal elementary excitations
Most studies of the dielectric properties of liquid water have considered a frequency dependent dielectric function (ω) (ω = 2πν) but neglected any spatial dispersion. Spatial dispersion introduces a q-vector dependence of the dielectric response, i.e., (q, ω), which accounts for the discrete short-range molecular structure of the liquid and, in particular, allows for distinguishing transverse (T) and longitudinal (L) elementary excitations. In general, the dielectric function is a tensorial quantity 78,79 which can be decomposed in a T-and L-part with respect to the direction of an external electric field E(r, t): with the transverse T (q, ω) and longitudinal dielectric function L (q, ω) and the spatial directions i, j = x, y, or z. The longitudinal L (q, ω) describes the linear response to planewave driving fields with E q, whereas T (q, ω) that to fields with E ⊥ q. The T-and L-components of ij (q, ω) of liquid water have different imaginary parts in the limit q → 0, i.e., on a macroscopic length scale.
We now derive the T-and L-components of the dielectric response of water from the MD simulation described in section 3 and Ref. 22 with the help of the fluctuation-dissipation theorem. 80 Instead of the complex dielectric function (eq. 2), we consider the complex conductiv- The current density j(q, ω) is given by j(q, ω) = σ(q, ω)E(q, ω) and in real space connected to the macroscopic polarization by j(r, t) = dP(r, t)/dt. The conductivity tensor is written as Applying the fluctuation-dissipation theorem, the transverse, e.g., j x ⊥ q y , and longitudinal conductivity, e.g., j x q x , are determined by the current-density-fluctuation-correlation functions defined as follows: 81,82 We calculated the q-vector and time-dependent current densities j x (q x , q y , q z , t), j y (q x , q y , q z , t), Results are presented in Fig. 6 The blue curve in Fig. 6(a) shows the real part of the transverse conductivity (eq. 4) which is measured in the linear optical response. The MD simulation reproduces the librational absorption band of water in good agreement with experiment. 83 The red solid curve shows the real part Re[σ L (q min , ν)] of the longitudinal conductivity (eq. 5). This quantity is not accessible in the linear optical response using transverse electromagnetic waves only.
The spectrum of Re[σ L (q min , ν)] differs from its transverse counterpart in several respects:

24
(i) The longitudinal conductivity resonance (eq. 5) is blue shifted relative to its transverse counterpart (eq. 4) and displays a smaller spectral width.
(ii) At low frequencies ν → 0 shown in Fig. 6(b), the L-and T-components exhibit a markedly different limiting behavior. While the transverse conductivity (blue curve) approaches a finite value, its longitudinal counterpart decays much faster for ν → 0 and vanishes at ν = 0. Since the real part of conductivity is connected with dissipative processes, any elementary longitudinal excitation of liquid water is largely underdamped at frequencies ν < 2 THz. The dielectric displacement field D(r, t) due to a gaussian electron wavepacket ψ(r) = (η/π) 3/2 exp[−η r 2 ], which has a time-dependent width [determined by η(t)] centered around the time-dependent position r 0 (t), is given by 88

The solvated electron in a polaron picture
The semiclassical force field acting back on the electron wavepacket when propagating and shrinking in the medium is given by the electric field distribution E(r, t) with E(q, ω) and D(q, ω) is the four-dimensional Fourier transform of E(r, t) and D(r, t), respectively. During the solvation process of the initially free electron, the force-field distri-bution (8) organizes both the deceleration in the medium via a friction force 88 The kinetic energy contained in the zero-point motion of the electron wavepacket is: The corresponding radial density r 2 |Ψ(r)| 2 is shown as black line in panel (a). In Fig. 7(c) we compare the radial densities r 2 |Ψ(r)| 2 predicted by the Turi-Borgis (TB), Larsen-Glover-Schwartz (LGS), and the polaron model. Interestingly, the three models give almost identical electron wave packets.
The polaron model allows for analyzing the different contributions to its self-induced potential (red curve in Fig. 7a). Such analysis shows that the electronic polarizability of water molecules, their intra-molecular vibrations and the librational contribution to the polarizability are mainly responsible for the size of the electron wavepacket. In contrast, contributions to V (r) caused by the orientational alignment of water molecules in the solvation shell which occurs at frequencies below some 10 THz, have no influence on the wavepacket size, but on the depth and detailed shape of the self-induced potential (cyan curve in Fig. 7a). Obviously, this contribution is predominantly located at radial distances outside the electron wavepacket.
The calculated depth of the electron potential depends on the particular water model The dielectric continuum model 88,89 intrinsically contains the dynamic response of the combined system, i.e., the electron plus its solvation shell, via the linear relation (8). In

Polaron oscillations of electrons solvated in water
with Here, electrons with a polarizability α el (ν) are added to the neat water dielectric response The red lines in Fig. 9(d) to (f) repesent spectra −Im[1/ L (ν, c e )] calculated from eq. (12) together with their spectral widths ∆ν (FWHM) for electron concentrations c e . In panel (g) the calculated resonance frequencies are shown as a solid line together with the experimental frequencies (symbols) as a function of the electron concentration c e . In all cases, the calculated spectra and frequency positions are in excellent agreement with experiment. The good agreement demonstrates that the oscillatory signals are due to impulsively excited coherent longitudinal oscillations of the polaron. As expected from the extremely small damping of the longitudinal conductivity (eq. 5) at frequencies ν < 2 THz, the polaron oscillations are highly underdamped and last for up to 20 ps and beyond at low electron concentrations c e .
In a system of charged and/or polar particles, the time scale of damping of longitudinal excitations is set by the charge-density-fluctuation-correlation function, which is, via the continuity equation of electric charge, connected to the longitudinal current-density-fluctuationcorrelation function (eq. 5). 79,81,82,91 In contrast, the dephasing of transverse excitations is governed by the transverse current-density-fluctuation-correlation function (eq. 4). The latter is much more susceptible to fluctuations and scattering processes than the correlation function of charge density. For instance, elastic scattering may decrease the electric current, but does not change the charge density. Accordingly, transverse excitations are much more strongly damped than longitudinal excitations.

Conclusions
The results discussed in this article demonstrate the multifaceted character of Coulomb in-