## Introduction

Upon exposure to light or ionizing radiation, liquid water may undergo a series of transformations involving electronic excitations, ionization of solvated species, and formation of free radicals and solvated electrons1,2,3,4,5,6,7. These processes occur, for example, in (photo-)electrochemical cells8,9, biological molecules10, as well as in atmospheric water11. Many of them are governed, at least in part, by the nature of the electron transfer and binding energies in aqueous solutions. Hence a detailed understanding of the interaction of light with water, and in general of redox reactions in aqueous systems, requires a precise knowledge of the liquid ionization potential (IP) and electron affinity (EA).

Direct measurements of the IP of liquid water are available from photoelectron spectroscopy, obtained by linear extrapolation to zero energy of the lowest binding-energy peak (1b1). The measurement of Delahay et al.12 yielded 10.06 eV for the water IP, while more recent photoelectron spectroscopy studies led to a slightly smaller value of 9.9 eV13, leading to an overall agreement that the water IP is ~10 eV. On the other hand, direct measurements of the liquid EA, or the energy gain due to the injection of an electron into the liquid, are not available, due to the rapid solvation of electrons in water1,2,14. Current estimates of the EA are based on thermodynamic arguments involving adiabatic and vertical ionization energies of the solvated electron7,15,16, computed reorganization energies, and specific assumptions on the behavior of photoionized electrons in water. As a result, the reported values of EA span a wide range between 0 and 1.2 eV7,15,16,17,18,19.

The only direct measurement of the EA of condensed aqueous systems was performed for the surface of a thin film of amorphous ice (a-ice) by pump-probe femtosecond spectroscopy20,21. Ice films only 2.5-bilayer thick were grown on a copper electrode, hence the measurement probed the energy of the lowest empty state of the aqueous surface. This study revealed a short-lived (~40 fs) spectral feature at −0.8 eV below the vacuum level, which was attributed to an electron residing at the bottom of the conduction band of amorphous ice interfaced with copper20.

Multiple predictions of water IP and EA from quantum calculations7,22,23,24,25,26 produced conflicting results depending on the structural models and electronic-structure methods employed. For example, many-body perturbation theory (MBPT) calculations using wavefunctions derived from density functional theory (DFT) and the PBE approximation22 yielded a value of 0.7 eV for the EA (configurations were generated either with classical potentials or the PBE functional); this value appeared to be consistent with estimates reported in the literature15. The quasiparticle self-consistent GW calculations of Chen et al.25, who generated a water structural model using van der Waals functionals and included nuclear quantum effects, obtained EA = 0.5 eV. A later thermodynamic integration study performed by the same group7 using hybrid functionals23 without nuclear quantum effects, predicted a larger value of 0.97 eV.

Here we present accurate predictions of the EA of liquid water and its surface using a combination of path-integral molecular dynamics with ab initio potentials, and MBPT calculations, thus determining a key missing property of electrons in water. We compared our results for the water surface with experiments on a-ice, thus validating our computational strategy against the only available direct measurement. We then used the same protocol to calculate the EA of the bulk, which we predict to be 0.1–0.3 eV. This value is similar to that estimated by Coe et al.16, although we show below that the agreement is fortuitous and effectively derives from inaccurate assumptions made in ref.16. We also revisit several other experiments and the assumptions that led to current literature values of the EA.

## Results

### Computational model

The structural models of liquid water were generated using path-integral molecular dynamics (MD) simulations with interatomic interactions described with the many-body MB-pol potential energy function27,28,29, which has been shown to accurately predict the properties of water in gaseous and condensed phases30. Electronic-structure properties were then computed for MD snapshots utilizing MBPT at the G0W0 level of theory, starting from eigenvalues and wavefunctions determined from hybrid density functional calculations. The water IP was computed by extrapolation of the (1b1) peak energy, similar to the procedure followed to analyze experimental data13; the EA was computed as the lowest unoccupied single-particle energy level of the liquid22,31,32,33,34,35 (see SI). Within MBPT, the energies −IP and −EA correspond, respectively, to the valence band maximum (VBM) and conduction band minimum (CBM), which were referred to vacuum using a procedure commonly utilized to compute band offsets in semiconductors (see Supplementary Methods).

In order to validate our computational protocol, we first compared the energies computed for the occupied electronic states of water with well-established liquid jet experimental data13,36. Consistent with our previous studies37,38, we found that the most accurate protocol for computing the binding energies of electrons in water is the G0W0 approximation based on dielectric-dependent hybrid (DDH) functionals39,40. In particular, G0W0 calculations with the range-separated hybrid (RSH) yielded a mean absolute error (MAE) with respect to measured spectra of only 0.18 eV for trajectories including nuclear quantum effects (NQE) and 0.38 eV for classical trajectories.

In addition, to assess the performance of the same level of theory for the unoccupied states of the liquid, we relied on the comparison of our results with highly-accurate quantum chemistry calculations of the EA of water clusters41, due to the lack of experimental data for the bulk system. Specifically, we compared G0W0 electron affinities with those obtained with the coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] method for a series of water hexamers, and found that overall, G0W0/RSH quasiparticle energies agree with the CCSD(T) results within 0.05 eV. This accuracy is comparable to the statistical uncertainty of our calculations.

### Energy levels of bulk water and the water surface

Figure 1 (left-hand side) summarizes the computed VBM and CBM of a slab representing the water/vacuum interface, as obtained at the G0W0/RSH level. The computed CBM values of −0.88 eV (G0W0/RSH) and −0.79 eV (G0W0/sc-hybrid) are in agreement with the direct measurement of Stähler et al.20, who reported a conduction band minimum of −0.8 eV at the interface of amorphous ice and a metal electrode. The agreement with experiment found here for the a-ice surface, expected to be representative of that of the liquid as well, further proves that our computational protocol accurately predicts the energy of the empty states of water.

We now turn to discussing the EA of the bulk liquid. As Fig. 1 shows, the position of the VBM of the surface and bulk water are similar (differing by less than 0.1 eV); however the position of the bulk CBM is ~0.6 eV higher than that of the surface. Our results obtained with G0W0/RSH (−0.29 eV) and G0W0/sc-hybrid (−0.17 eV) yield EA of the bulk within 0.1 and 0.3 eV, when including statistical uncertainties of our calculations (0.04–0.05 eV). This range has been obtained using MD trajectories inclusive of nuclear quantum effects; with classical trajectories and the same level of electronic structure theory, we found smaller values: 0 to 0.1 eV.

### Nuclear quantum effects

The results presented in Fig. 1 highlight the influence of nuclear quantum effects on the electronic structure of liquid water. In particular, the inclusion of NQE affects the VBM and CBM position by 0.3 and 0.2 eV, respectively (the energy of the VBM is increased and that of the CBM lowered) yielding a gap 0.5 eV smaller than that calculated for classical trajectories. The decrease in the gap is consistent with the presence of longer O–H bonds (0.99 ± 0.14 Å) when NQE are included (to be compared with 0.97 ± 0.05 Å for classical trajectories): Indeed, ref.22 showed that water models with longer O–H bonds exhibited smaller gaps. Accounting for NQE also results in a wider bond-pair peak in the distribution of maximally localized Wannier centers (Supplementary Fig. 6), consistent with larger O–H bond-length fluctuations.

Our result for the fundamental gap is in agreement with the study of Chen et al.25 reporting a 0.5–0.7 eV reduction of the band gap of water due to nuclear quantum effects. Instead our findings are inconsistent with the density of virtual states reported in ref.34 showing a rather large shift of the CB edge upon inclusion of NQE (of almost 1 eV). We also note that the fluctuations of the VBM and CBM observed in our quantum trajectories are of the order of 0.4 and 0.1 eV for the bulk and 0.6 and 0.4 eV for the surface, indicating substantial fluctuations of band edges depending on the nuclear configuration, which should be taken into account when modeling chemical reactions at aqueous interfaces.

The marked difference in the EA of the bulk and the surface is reflected in the localization properties of their respective conduction band edges. In Fig. 1 we report the isosurface of the wavefunctions corresponding to the CBM of the surface and the bulk, showing the different localization properties of the two states. Consistent with previous reports42,43, we found that the bulk unoccupied edge is delocalized over the entire supercell, while that of the slab is localized at the surface, in proximity of broken hydrogen bonds. The different bonding environment of the surface, with respect to that of the bulk, is reflected in the values of effective polarizabilities44: As expected from our previous work on salts45, we found that the effective molecular polarizabilities in the slab, where more broken hydrogen bonds are present, are smaller than in the bulk. In addition the distributions of polarizabilities and dipole moments are broader in water with NQE than in the liquid treated classically (Supplementary Figs. 5 and 6).

## Discussion

Our prediction for the liquid vertical EA (0.1–0.3 eV) differs from data often quoted as experimental results in the literature, e.g., 0.74 eV from ref.15. However, these data are not obtained from direct measurements; rather, they are derived using both experimental and theoretical results, under specific assumptions which, as we show below, need to be revisited. By combining our theoretical prediction of the vertical EA with the most recent measurements on solvated electrons, we discuss below the construction of an energy diagram for an electron in liquid water.

The vertical (EA) and adiabatic (AEA) electron affinities of water are defined as:

$${\mathrm{EA}} = E_{{\mathrm{water}}} - E_{{\mathrm{e}}^ - }$$
(1)

and

$${\mathrm{AEA}} = E_{{\mathrm{water}}} - E_{{\mathrm{e}}^ - {\mathrm{(aq)}}},$$
(2)

where Ewater is the energy of pristine, neutral water and $$E_{{\mathrm{e}}^ - }$$ that of the liquid with a non-solvated electron; $$E_{{\mathrm{e}}^ - \left( {{\mathrm{aq}}} \right)}$$ is the energy of the liquid with a solvated electron e (aq). The vertical detachment energy (VDE), or energy necessary to remove an e (aq) from water, is defined as

$${\mathrm{VDE}} = E_{{\mathrm{water}}}^{{\mathrm{cavity}}} - E_{{\mathrm{e}}^ - {\mathrm{(aq)}}},$$
(3)

where $$E_{{\mathrm{water}}}^{{\mathrm{cavity}}}$$ refers to the energy of the neutral liquid with a defect (cavity) created by a solvated electron. The energy required to create the defect is the reorganization energy λ:

$$\lambda = E_{{\mathrm{water}}}^{{\mathrm{cavity}}} - E_{{\mathrm{water}}}.$$
(4)

Finally, $$\left[ {E_{{\mathrm{e}}^ - \left( {{\mathrm{aq}}} \right)}} \right]^ \ast$$ is the energy of the solvated electron’s first excited state with excitation energy μ = $$\left[ {E_{{\mathrm{e}}^ - \left( {{\mathrm{aq}}} \right)}} \right]^ \ast$$ − $$E_{{\mathrm{e}}^ - \left( {{\mathrm{aq}}} \right)}$$.

Figure 2 shows the electron energy diagram in liquid water constructed from our EA calculations and the most recent measurements for AEA, VDE, and μ. Specifically, AEA = 1.34 eV was obtained by extrapolation of water cluster data46; VDE = 3.7 eV was measured by photoemission spectroscopy including corrections for surface scattering effects47; and μ = 1.73 eV is a well-established position of the maximum in the measured optical absorption spectrum of the solvated electron48. The reorganization energy λ derived from the data of Fig. 2 is 2.36 eV, larger than the most recent computational estimates of 1.99–2.17 eV7.

The diagram obtained here differs substantially from those obtained using the values reported by Bernas et al.15 and by Coe et al.16, shown on the left-hand and right-hand sides of Fig. 3, respectively. Focusing first on the results of ref.15, the authors considered AEA = 1.47 eV, λ = 1 eV49,50, and μ = 1.73 eV48. This led to a VDE of 2.47 eV, which is significantly different from that reported by recent measurements (3.3–3.8 eV)1,14,47,51,52,53. This discrepancy originates from the incorrect magnitude of the reorganization energy used in ref.15, a value obtained from classical MD simulations49,50, which turned out to be underestimated by more than 1 eV with respect to all best estimates known today. In addition, a crucial assumption of ref.15 was that the first excited state of the solvated electron coincides with the bottom of the conduction band of water, hence $$E_{{\mathrm{e}}^ - }$$ ≡ $$\left[ {E_{{\mathrm{e}}^ - \left( {{\mathrm{aq}}} \right)}} \right]^ \ast$$. This assumption was later disproved and is now known to be incorrect54.

Finally, we note that Bernas et al.15 defined the bottom of the conduction band of water from the energy difference $$V_0 = E_{{\mathrm{e}} - } - E_{{\mathrm{water}}}^{{\mathrm{cavity}}}$$, which, however, is not equal to −EA. Using the energy diagram of Fig. 3, one would obtain a negative value for the EA of liquid water: −0.26 eV.

Coe et al.16 (right-hand-side diagram) used the values VDE = 3.32 eV and AEA = 1.72 eV, both derived from extrapolated experimental data for water clusters55, and obtained a larger reorganization energy (λ = 1.60 eV) than Bernas et al.15 They then computed an upper bound to the EA as (AEA − λ) by assuming that the energy difference ($$E_{{\mathrm{e}}^ - }$$ − $$E_{{\mathrm{e}}^ - \left( {{\mathrm{aq}}} \right)}$$) is always smaller than the reorganization energy. However, the derivation incorrectly assumed that the creation of a cavity in pure water implies an energy gain, rather than a cost, as pointed out in ref.7. If energy gains and costs are correctly taken into account, the only conclusion that can be drawn from the data of ref.16 is EA ≤ 1.72 eV.

We compare our results with other computational studies, focusing on first-principles calculations. Using several water models, previous studies predicted the EA of the bulk between 0.7 and 0.9 eV at the G0W0/PBE level of theory. As shown in Supplementary Table 7, we obtained a similar result (0.99 eV) for classical MB-pol water with the same electronic structure methods. Our best prediction for the bulk water EA is smaller than that reported recently using first-principles MD trajectories with van der Waals functionals and nuclear quantum effects (0.5 eV)25. These calculations used self-consistent GW with approximate vertex corrections and smaller cells (32 water molecules) than those adopted in our study (64–256 molecules).

Knowledge of the water EA is key to understanding mechanisms of redox reactions in aqueous systems that involve either molecular species or solid surfaces53,56,57, yet no direct experimental measurement is available. In this paper we reported predictions of the EA of bulk water and the surface of water, obtained entirely from first principles, by combining path-integral MD simulations with ab initio potentials, and state-of-the-art electronic structure methods based on MBPT. The accuracy of the theoretical methods used in this work was carefully checked against quantum chemistry calculations for virtual states of aqueous systems41, and against well-established experiments for electronic binding energies of the liquid. Our results for the surface were found to be in good agreement with recent pump-probe spectroscopic measurements20, thus further validating the accuracy of our computational framework.

We determined the EA of the liquid to be between 0.1 and 0.3 eV, much smaller than some of the accepted estimates present in the literature, but consistent with the speculations of Coe et al.16,18 and the measurements of the electron ejection lengths in a two-photon ionization process by the Bradforth group58,59 (The authors of refs.58,59 determined that the average ejection length of the electrons in a two-photon ionization process is roughly constant below 9.5 eV but increases rapidly after 9.5–9.8 eV, and associated this increase with the electrons being able to access the conduction band upon excitation. Given that the ionization threshold of water is 9.9 eV13, the energy of 9.5–9.8 eV corresponds to an energy level positioned ~0.1–0.4 eV below vacuum, although refs.58,59 do not explicitly mention these values for the conduction band minimum). We used our theoretical prediction of the vertical EA together with the most recent measurements for the solvated electron, to construct an energy level diagram for an electron in water, which we believe is the most accurate known to date. The diagram proposed here differs substantially from those generated using estimates reported in the literature, for which we provided a detailed, critical reassessment. We suggest that the EA of water may be determined experimentally by extending the pump-probe spectroscopic study of ref.20 to thicker a-ice films and extrapolating the results as a function of size, building, e.g., on work reported by King et al.21 on trapped electrons at interfaces.

Our results showed that while the IP of bulk water and the surface are very similar, their EAs differ by more than 0.6 eV, with the conduction band edge of the surface being much deeper in energy than that of the bulk. We also found that the band gap of water as well as both band edges are substantially affected by proton quantum effects and couplings with nuclear configurations, highlighting the importance of including energy level fluctuations when modeling chemical reactions in water. Work is in progress to investigate the same effects in aqueous solutions.

## Methods

### Molecular dynamics simulations

Classical and path-integral (PI) molecular dynamics (MD) simulations were performed using the ab initio MB-pol potential and a modified version of the DL_POLY code. Bulk water was simulated in the constant volume-constant temperature (NVT) ensemble at 298 K, using a Nosé–Hoover chain thermostat60 with the length of 4, and supercells containing 64 molecules. The equilibrium density (ρ) of the liquid was determined with constant pressure and temperature (NPT) simulations, corresponding to cell sizes of 12.41 Å3 (ρ = 1.001 g cm−3) and 12.42 Å3 (ρ = 0.999 g cm−3) for classical and PI molecular-dynamics simulations, respectively. Path-integral simulations were performed using 32 beads per atoms. The water surface was simulated with a slab containing 108-molecules, in a cell of dimensions 12.74 × 12.74 × 57.35 Å3. After equilibration, we carried out simulations for 0.5–1 ns for the bulk and the slab. In particular, we insured that the electric field across the water slab vanished within 0.0005 eV/Å. We tested size effects on the position of energy levels by performing MD simulations for 256-molecule bulk water and 216- and 384-molecule slabs. The details of the simulations used for finite size-scaling are discussed in the Supplementary Methods. We note that the choice of the system size in our MD simulations was limited by the cost of MBPT calculations on MD trajectories.

### Electronic structure calculations

We carried out calculations of the electronic properties of water and water slabs at the density-functional theory (DFT) and G0W0 level. DFT calculations were carried out using the Quantum Espresso code61, with a plane-wave energy cutoff of 85 Ry, and HSCV pseudopotentials62,63. G0W0 calculations were performed using the the West code64,65, starting from wavefunctions obtained with the PBE66, PBE067, sc-hybrid39, and RSH40 functionals. Quasiparticle energies were determined using the same exchange-correlation potential as the one employed in the self-consistent-field DFT calculations. All of our G0W0 calculations were carried out with 1600 eigenpotentials and extrapolated using the procedure described in the Supplementary Table 5. DFT calculations were done for 128 equally-spaced snapshots along each of the bulk trajectories. G0W0 calculations for bulk water, starting from hybrid functional wavefunctions, were performed for 4 snapshots. The G0W0 results for the slabs were obtained using computed DFT values for the slab and G0W0 corrections for the bulk. Our analysis of the accuracy of this approach and error estimates are given in the Supplementary Methods.

To determine absolute energy levels from DFT and G0W0 calculations performed in plane-wave basis sets, we aligned the plane-average electrostatic potentials in the bulk water and in the slab, and referenced them to the vacuum energy level, following the procedure proposed by van de Walle and Martin68, as explained in the Supplementary Methods.

### Data availability

All data associated with our article are available at https://datahub.uchicago.edu/.