It is clear that in order to investigate the energetics of the itinerant hydronium radical, the band structure must be clearly established. However, this issue is still a matter of debate. Most of the confusion has been generated by researchers who have used experimental energy differences related to what they believed to the "hydrated electron". In our view, since this latter interpretation is incorrect, it not suprising that it is difficult to reconcile these model-dependent data with other measurements. In order to avoid any circular argument, we are going to try to establish water energy levels, in a most rigourous way, without relying on experiments related to the hydrated electron.

In condensed phase, one must take into account the formation of the "conduction" band which results from the overlap of the wavefunctions of each individual water molecules. These new electronic states specific to the condensed state are allowed because, through the creation of smooth delocalized states, electron may obtain a lowering of it kinetic energy [Zallen 83].

if we consider a perfect crystal, the conduction band constitutes a specific set of delocalized electronic states which pertains to the whole sample. If we consider a liquid, the overlap between individual water wavefunctions is going to fluctuate, and instead of the whole sample, I think we must consider regions or "flickering clusters" of water molecules the wavefunctions of which, are more strongly overlapping, depending on their spatial separation. The gap between various allowed energy levels, depends on the spatial location, and the band structure diagram displays energy intervals which are in reality (averaged) spatial minima of these gaps.

Band theory is a one-electron theory and as such neglects electron correlation. One must pay attention [Zallen 83] that the correlation cost (Fermi hole, Coulomb hole), associated with the probability of two delocalized electrons being seated in close proximity, may exceed the kinetic energy gain associated with delocalization, and therefore induces a localization ( Mott insulator). In most theoretical treatments, the extra electron is assumed to be alone, but this does not correspond to most experimental conditions (Radiolysis, laser photon pumping), where swarms of extra electrons are produced altogether.

In supersonic expansion clusters, the emergence of a conduction "band" is going to depend on the cluster size.

The competition between localization and delocalization has been analyzed by Anderson [Zallen 83] who introduced a disorder parameter. When the medium is disordered enough, no delocalized set of states may exist anymore.

One theoretical approach to understand the electronic structure of water, as a condensed medium, is to use the theory develloped for amorphous semiconductors [Williams 76].

It is quite important to distinguish between vertical and adiabatic energies, and also to underline the difference between threshold and average quantities.

Vertical energies are in fact optical data.
• 1/The energy deposition is fast. The nuclear geometries have no time to adapt, electronic clouds, however, may have the time to relax ( electronic polarization ).
• 2/The energy deposition is unilateral. The liquid cannot give back energy to the optical source. It is more like an excitation energy or a transition barrier but it is different however, because the TST theory assumes equilibrium with the solvent,along the reaction pathway, which is not the case.

Thermodynamic quantities are balances of average adiabatic energies. Threshold energies are constituted of the intramolecular energy required plus the threshold hydration energy which is a fluctuating quantity, depending on the liquid local geometries. According to the direction of the threshold, the threshold hydration energy would the maximun or the minimun avalaible. So one must be extremely careful not to add quantities of a different nature.

Few theoretical computations concerning the band structure of water or ice have been achieved, and they are somewhat old. Because of its symmetry, cubic ice is much amenable to calculations. While the amount of orbital overlap which creates the band structure might be different in water and in various forms of ice, it allows to separate out the instrinsic structure from the "extrinsic" structure caused by native defects. A first ab initio computation was performed in 1973 [Parravicini 73] within the tight-binding approximation. The water basis set comprises Slater-Type Orbitals( STOs). The interactions of an excited electron with its hole is taken into account,so we must consider more exactly of the excited band (EB) which comprises the conduction band (CB) and the adjacent Mott exciton band. The valence band (VB) is constituted from the 1B1 and 1B2 MOs of the single molecule. Within a more localized MO picture, the lone-pair MOs contribute mostly to this band, we expect therefore that the value of this electronic level should be quite sensible to an adequate description of the hydrogen bridges. The VB is quite narrow and lies at -10.6 eV/Vacuum. This latter latter value seems to quite in agreement with UPS values for cubic ice [Shibaguchi 77]. The next inner VB is constituted with 2A1 MOs and its upper edge lies around -12 eV/Vacuum. The lower excited band extends between -2.76 and -0.84 eV. An upper EB is quite narrow around 0 eV. In another ab initio computation in 1977 [Resca 77], with the same minimal STO basis set, but with a different procedure. The VB is located -10.83 eV/Vacuum. Surprisingly the CB does not appear to correspond to bound states, but Resca and Resta [Resca 77] underline that the procedure employed is not reliable for estimating the CB parameters. More recently [Choe 88], within a MNDO tight-binding approach, the band structure of an infinite one-dimensional chain was investigated. The VB lies around 11.7 eV, while the chain was found to be an insulator. The MNDO method is of course not well suited to treat excited states.

We are now going to try to establish the intrinsic band structure of liquid water with experimental data.

The top of the valence band can be determined by photoelectron emission spectroscopy [Delahay 82,Delahay 70,Baron 69]. In this technique, the photocurrent is recorded while varying the wavelength of the UV source. In a careful experiment, by Delahay and Von Burg [Delahay 81], the threshold photoemission energy was determined to be 10.06 eV/Vacuum. Remarkably enough there is no Urbach tail. This tend to indicate that the recorded photoejection process did not took place via the ionization of impurities or hydrated electrons. The yield of hydrated electron is probably too low, anyway, because of the lack of sufficient intensity of the light source [Hart 70,p10]. With a different photoelectron technique [Ghosh 83], the UV (UPS) or X-ray (XPS or ESCA) source is maintained at a fixed frequency, but the energies of ejected photoelectron are analyzed yielding an energy distribution curve (EDC). With proper calibration and taking account work functions, it is possible to assimilate the recorded electron kinetic energies with ionization potentials and, to a good approximation, with the molecular orbital or band energies [Ghosh 83]. Using UPS data [Pache 89], Michaud,Sanche, and coworkers [Michaud 91] computed a value of 10 eV for the photoelectric threshold. From the EDCs of UPS spectroscopy, at eight different wavelengths from 13.5 to 19.2 eV, of heavy ice deposited on a gold substrate, Shibaguchi and coworkers derived a value of 10.5 +/- 0.5 eV from the top of the valence band to the vacuum. Although the threshold was determined as an onset point [Shibaguchi 77], therefore taking into account the energy loss during the emission process, Baron and coworkers [Baron 78] argued that the value of 10.5 eV should be lowered to approximately 8.5 eV by a correction for work functions [Baron 76], in order to be in agreement with their own UPS results [Baron 78]. Although the photoelectron threshold is related to the outer valence band, this latter UPS work [Shibaguchi 77] has been also critized [Abbati 79] because background electron scattering prevents from discerning inner valence structures.

Extreme purification of water appears indeed to be required. In a previous photoelectron emission experiment by Watanabe,Flanagan and Delahay [Watanabe 80], a value of 9.3 eV was found, which appeared like an Urbach tail threshold. The error in this earlier experiment was probably due to water contamination by organic or inorganic impurities [Delahay 81]. As a matter of fact, it is their very own results related to the photoemission spectroscopy of inorganic anions in aqueous solutions [Von Burg 81] which forced Delahay and Von Burg to redo their experiments. In passing this series of experiments, shows that,in that case, the Urbach tail was caused simply by impurities and not by sideband phonons or excitons.

In a paper containing an interesting discussion, Watanabe and Gerischer [Watanabe 81b] conducted photoelectrochemical experiments of photohole injection [Sass 78b,78a] into the water solvent at a gold electrode interface. By exploiting the linear relationship between the photon energy and the electrode potential, and extrapolating to a zero value electrode potential, a value of 9.3 eV/Vacuum was estimated.

Besides contamination issues, depending on the wavelength of the monoenergetic source, UPS and XPS spectroscopy techniques may involve the formation of native defects and therefore are possibly not well suited for estimating the the top of the intrinsic valence band as well as the band gap. In UPS [120 to 200nm] by Baron and coworkers [Baron 78] estimated the photoelectric threshold as 8.7 eV. It is possible that the large photocurrent coming from the gold substrate and the smaller photocurrent from the thin ice layer have not been well separated out in the data analysis. More likely,like in their XPS experiments [Baron 76], gold impurities and native defects may have been promoted at the gold/ice interface or water was comtaminated. Assuming a band gap of around 7.8 eV determined from optical absorption, Baron estimated the CB width as low as 0.9 eV. This suspiciously low CB width value has been employed by Bowen and coworkers [Lee 91,Coe 90] in a comparison with the extrapolated VDE (Vertical Detachment Energy) of their cluster water anions. UPS experiments by Abbati [Abbati 79] do not allow a quantitative estimation of the photoelectric threshold.

XPS experiments have been conducted by Siegbahn and coworkers [Siegbahn 69] both on water vapor and on ice. The valence band edge of ice apparently has been lowered by approximately 6 eV to 5 eV by comparison to a free molecule. However, since the experiments [Siegbahn 69] were made with different instruments, Baron and Williams [Baron 76] warned that a zero binding energy reference level must be consistently defined. XPS experiments of ice [Shibaguchi 77] indicate an internal valence structure which correlate qualitatively with vapor XPS experiments. Although it appears hazardous to extract quantitative values from their reported EDC figure [Shibaguchi 77], it seems that the edge of the 1B1 band has been lowered by approximatly 2 eV upon condensation. Baron and Williams [Baron 76] performed a XPS of both water vapor and amorphous ice deposited on gold films and found that the peak of the valence band is only 0.7 eV lower than the 1B1 gaz phase peak. It is difficult to interpret this value of "change of binding energy upon condensation", since we notice the lack of resolution of the 1B2,3A1 and 1B1 bands, which may be clearly discerned in other XPS spectra [Shibaguchi 77,Siegbahn 69]. Baron and Williams estimated the edge of the valence band around 9 eV/Vacuum. We suggest that impurities may have been promoted at the gold/water interface or some contamination took place, since an apparent Urbach tail is discernible on their EDCs.

The band gap is the minimun energy difference between band (mobility) edges [Zallen 83]. It is possible to relate the mean interband energy difference [Grevendonk 84] with the dispersion of the refractive index of water [Wemple 73]. Dividing by an empirical factor of 1.5, the mean energy difference, Grevendonk and coworkers [Grevendonk 84] obtain a band gap value of 8.28 eV,

By measuring the magnetic field required to rotate the plane of polarization of an incoming polarized light (Faraday rotation effect), it is possible to determine the Verdet constant of water [Balbin 79,Grevendonk 84]. The Verdet constant can also be related to the optical dispersion[Balbin 79]. From the Verdet constant it is possible with the KLN equation [Kolodziejczak 62] to compute a band gap of 8.33 eV. For an amorphous material, yet another equation [Mort 71] which takes into account some degree of disorder, yields a value of 8.55 eV for the band gap. The Faraday rotation effect is mostly caused [Kireev 75] either by intrinsic absorption (interband rotation) or by exciton absorption (excitonic rotation). Experimentally it has been difficult [Kireev 75] to find any evidence of Faraday rotation caused by impurity absorption. This may be intuitively be understood that magneto-optical effects are related to collective oscillations, while impurities, defects or localized excitations represent a random superposition of oscillators.

It is therefore interesting to determine if the computed Faraday rotation band gap is related to interband transitions or to exciton transitions. It is possible to distinguish [Painter 68] in the real and more easily in the imaginary part of the dielectric constant two bands with respective peaks at 8.3 eV and 9.6 eV. The band centered around 9.6 eV, with an extrapolated threshold around 8.7 eV, is assigned [Painter 68] to an interband transition. The band centered around 8.3 eV, with an extrapolated threshold around 7 eV is interpreted tentatively to be an exciton band. If this latter band were to correspond to excitons, then it would have induced excitonic Faraday rotation which is not obsverved. Therefore the 8.7 eV centered band should correspond to impurities or local excitations. On the other hand, the interband assignement seems to be coherent with the band gap value computed from the Verdet constant. In UPS experiments [Shibaguchi 77], from the quantum efficiency decrease, a band gap is estimated around 8.7 eV .

A recent study of the one photon absorption spectrum of liquid ethanol, methanol and water [Jung 01] confirms that, in water, the absorption threshold starts around 6 eV and that the band gap is around 8.5 eV.

If we retain a value of 10.05 +/- 0.05 eV as the intrinsic conduction band upper edge and a value of 8.6 +/- 0.2 eV as the intrinsic band gap , we obtain indirectly an intrinsic conduction band width of 1.45 +/- 0.25 eV .

The value of photoemission threshold/vaccum of electron injection from metal electrodes into water has been determined by Pleskov and Rotenberg [Pleskov 69] to be -1.26 +/- 0.1 eV Sass and Gerischer [Sass 78a p 480] found a value of -1.3 eV . Grand and Bernas [Grand 79] quote a number of other experimental results ranging from -1.14 to -1.3 eV [Yamashita 77]. It seems that all these threshold were deduced while neglecting the adsorption potential drop at the water/electrode surface [Pleskov 69].

It is possible to relate intuitively the electron injection threshold with the lower edge of the intrinsic conduction band but these two concepts belong, in fact, to two different systems. The former is a charged system ( H₂O + an extra electron), while the latter is neutral. Within the band diagram, the CB corresponds to a possible one-electron energy level of neutral water molecules. The injection threshold is the lowest bound state of an extra electron injected in a condensed assembly of water molecules. This lowest bound state is not a localized discrete state but pertains to a pseudo-continuum.

On the other hand, the consideration of the photoinjection threshold implies the assumption [Barker 66,Pleskov 78, Sass 78a] that the electron is first injected in the CB, and that there no direct electronic transition to the "hydrated electron" state. We think that this assumption is essentially correct but physical justification is rather delicate.

In an electrochemical experiment [Szklarczyk 89], the dielectric breakdown of water under high electric field was related to the passage of electrons in the conduction band. The bottom of the conduction band was estimated to lie at -1.3 eV .

Absorption spectra reveal both optical allowed intramolecular transitions as well as interband transitions. Within the UV absorption spectrum [Watanabe 53] of water vapor, a first broad continuun can be discerned in the region from 186nm(6.6 eV) to 145nm(8.5eV) with a flat maximum around 170nm(7.3eV). A second continuum superimposed with small bands exists in the region 125nm(9.9eV) to 145nm. There have been a large number of reports of absorption spectra of liquid water and ice, with no general agreement. Onaka and Takahashi [Onaka 67] observe in liquid water a first continuum from 5.8 eV with a maximum about 6.2 eV which constitute the shoulder of much larger continuum peaking around 8.3 eV. In the spectra of ices at various temperature, an band at 8.7 eV can be distinguished. Verrall and Senior [Verall 69] observed a first contiuum starting at 175 nm(7.08eV) to approximately 135nm(9.2eV). The vapor/solid blue shift is estimated [Verall 69] around 6000-8000 cm-1( 0.7-1.0 eV). The first continuum maximum is located around 146nm(8.5eV). It was noted that the anomalously low variation of this maximum with temperature could be explained by an increase of its repulsive character in comparison to gaz phase. This does make sense if the diffuse character of the first excited state is repressed by the surrounding medium in favor of a more localized dissociative antibonding component.

The optical absorption of water displays a minor absorption band, shaped like an Urbach tail, starting around 6.2 eV [Quickendem 80]. Another larger absorption band starts around 7.5 eV [Heller 77,Heller 74].

In VUV experiments [Shibaguchi 77], an absorption band was found on thin films of heavy ice on LiF substrates with a peak around 8.75 eV. The peak and the shape was almost independent from temperature and was assigned to intramolecular transitions.