Optical absorption spectroscopy probes water wire and its ordering in a hydrogen-bond network

We present our calculated optical absorption spectra for liquid water at 300 K and ice at 80 K in 1 (a) and (b), respectively. The calculations are performed using the $ab$ $initio$ GW plus Bethe Salpeter equation (GW-BSE) method [39, 40, 41]. Details of the calculation may be found in Section I and II of the Supplemental Material. For comparison, the recent experimental measurements from Refs. [58, 57] are also shown in the same figure. The theoretical water and ice spectra are blue-shifted by $\sim$0.6 eV and $\sim$0.8 eV, respectively, and both broadened by 0.4 eV with a Gaussian function to match the energy and width of the first peak in experiment. Clearly, an accurate agreement can be seen between experiments and theoretical predictions. In both water and ice, the absorption onset is dominated by a major absorption peak centered at $\sim$8 eV followed by a more broadly distributed spectral feature at $\sim$15 eV. Despite the similarities, ice is distinct from water in its spectrum. Compared to that in water, the spectrum of ice shows a much stronger absorption peak at $\sim$8 eV together with more pronounced spectral features for the incident photon energy between $\sim$8 eV and $\sim$15 eV.

In the condensed phase, intermolecular interactions broaden molecular orbital energy levels into bands of finite width. Optical absorption spectroscopy probes dipole-allowed transitions between these bands as shown in Fig. 2 (a). To understand the nature of the spectral features, we break down the absorption spectrum by contributions from specific band to band transitions. In water, the highest occupied bands are valence band states with $1b_{1}$, $3a_{1}$, $1b_{2}$, and $2a_{1}$ character, respectively, and the lowest unoccupied bands have characters $4a_{1}$ and $2b_{2}$, respectively. The schematic of the orbital character of water molecules is shown in Fig. 2(a). In the full calculation, all electron-hole pairs contribute to the spectra. We decompose the spectra into contributions from specific transitions by solving the BSE on a subspace with only selected orbitals, we find that the full spectrum is well approximated by the solution on a subspace containing only the $p$ orbital electrons near the Fermi level (i.e. the $1b_{1}$, $3a_{1}$, $1b_{2}$ bands), see the details in the Section IV of the Supplemental Material.

In Fig. 2(a), we label the transitions between the valence and conduction bands that contribute to the optical absorption. Then, in Fig. 2(b), we show how each transition contributes to the overall optical absorption by restricting the calculation of the optical absorption spectrum to specific valence and conduction bands. From this, we see that in both liquid water and ice the first absorption peak around $\sim$8 eV comes from transitions from valence states of $1b_{1}$ (or oxygen $p$) character. The broad feature at $\sim$15 eV comes from the broadening of several distinct exciton peaks arising from transitions from $1b_{1}$ and $3a_{1}$ orbitals, because of the dispersion of the oxygen $p$ band in Fig. 2 (a). The low-lying oxygen $2s$ valence band of $2a_{1}$ character does not contribute to the optical absorption spectrum in the current energy range.

The first absorption peak at $\sim$8 eV, as shown in Fig. 3(a), exhibits charge-transfer characteristics, as noted in previous studies [28, 29, 30]. Therefore, it can be classified as a bound exciton of the charge-transfer type. In the condensed phase, water molecules construct a tetrahedral structure via H-bonding. The H-bonding can be described by a Coulombic attraction between the electronegative oxygen on an acceptor molecule and the electropositive proton on a donor molecule, as illustrated in Fig. 3 (a). During the process of optical absorption, electron-hole pair excitation also occurs between the donor and acceptor molecules as shown in the Fig. 3 (a). An electron of $4a_{1}$ characteristic is excited in the lowest conduction band, which is an antibonding orbital centered on the protons of a water molecule, and at the same time, a hole of $1b_{1}$ character is excited in the highest valence band, which consists of lone pair electrons located in close vicinity of the oxygen atom. Not surprisingly, the resulting exciton strongly couples to the H-bonding, which applies an electrostatic field pointing from the donor to the acceptor. In the photo-absorption process, the electron in the lone pair orbitals on the H-bond acceptor molecule is excited onto the antibonding orbital on the H-bond donor molecule. Driven by the Coulombic attraction, the process of electron-hole pair excitation takes place along the H-bonding direction, and a charge-transfer exciton is stabilized on the H-bonded water molecules. As a bipolar molecule, a water molecule serves both as an H-bond donor and an H-bond acceptor, and therefore both electron and hole states can be simultaneously found on a single water molecule during an excitation. As a result, large excitonic effects can be seen in Fig. 3 (b), which are manifested in the pronounced absorption features and large exciton binding energies of $\sim$2.2 eV in water and $\sim$3.4 eV in hexagonal ice.

In water, the H-bond constantly breaks and reforms on a time scale of picoseconds [36, 37, 38]. In its liquid structure, the strength of H-bonding undergoes a significant fluctuation as well. The strength of H-bonding and its variance in water can be seen in Fig. 3 (c) from the broad distribution function of the proton transfer coordinate $\nu=d_{\rm OH}-d_{\rm O\cdots H}$ [59, 60], where $d_{\rm OH}$ denotes the length of the OH covalent bond, $d_{\rm O\cdots H}$ denotes the distance between the H atom and the O atom in the H-bond acceptor. By convention, $\nu$ has been adopted in Fig. 3 (a) as a descriptor to represent the H-bonding strength, where a more positive (negative) $\nu$ means a physically stronger (weaker) H-bonding strength between two neighboring water molecules. Because of its coupling to the H-bonding, it is expected that the charge-transfer exciton and its spectral signals are sensitive to its H-bonding environment in liquid. In Fig. 3 (c), we present the electron-hole overlap function ($\rho_{h}\times\rho_{e}$) as a function of H-bonding strength in terms of the proton transfer coordinate $\nu$ for each pair of neighboring water molecules averaged by using many configurations which are extracted from molecular dynamics simulation trajectories. The electron-hole overlap function is computed from the sum of the product of electron and hole densities obtained from BSE eigenstates whose excitation energies are located within the first absorption peak at $\sim$8 eV, and the electron-hole overlap function has been averaged over the pair of neighboring water molecules. The details of the calculations of the electron and hole densities can be found in the Section III of the Supplemental Materials. In Fig. 3 (c), a strong correlation between a charge-transfer exciton and its H-bonding strength can be clearly identified. When the H-bonding between two water molecules is relatively weak in Fig. 3 (c), the hole and electron barely overlap with each other and therefore contributes little to the observed absorption spectra. As the H-bonding become stronger, the polarization field greatly facilitates the electron-hole excitation and their overlap on the H-bonded water molecule yielding a significant increase in optical oscillator strength. Our analysis shows that the first spectral peak is dominated by charge-transfer excitons on pairs of water molecules with nearly ideal H-bonding strength comparable to the H-bonding strengths in hexagonal ice.

Under ambient pressure, the crystalline structure of regular ice Ih can be viewed as an assembly of corrugated oxygen bilayers stacked on top of each other as shown in Fig. 4 (a). In ice Ih, the oxygen atoms are located on lattice sites satisfying hexagonal symmetry, while, the protons take nearly random positions that are allowed by the ice rule [51, 52], which is referred to as proton disorder in the field [61, 62]. In ice, all neighboring water molecules are H-bonded, and the bonding strength of $\nu\simeq-0.70$ in ice is much stronger than the average bonding strength of $\nu\simeq-0.85$ in liquid water. These intact H-bonds not only construct an extended tetrahedral structure but also become interconnected among themselves and form persistent water wires on their underlying H-bond network. As shown in Fig. 4 (a), we present a typical water wire in regular ice. Along this water wire, five water molecules on the top three bilayers are threaded by four consecutive H-bonds in a zigzag chain. Because of the highly directional nature of H-bonding, a water wire can be defined to be a proton ordered chain if all the participating H-bonds adopt the same bonding direction pointing from donor molecule to acceptor molecule along the wire, or vice versa. As such, the above prototypical water wire can be assigned with an ordering length ($l=4$) from the number of constituent H-bonds and with an ordering direction which is along the $c$-axis of the hexagonal lattice. The ordering of the above water wire is interrupted by a proton disorder occurring at the bottom bilayer as denoted by the black circle in Fig. 4 (a). Therefore, only finite sized proton ordered water wires exist, whose lengths and locations are randomly distributed in ice Ih.

In ice Ih, the presence of water wires greatly enhances excitonic effects at the main peak in the optical absorption spectrum. On a proton ordered water wire, the H-bonds are connected head to tail along its ordering direction, and each water molecule in ice carries an electric dipole $\sim$3.09 $D$ along a direction roughly bisecting its covalent bond angle. In compliance with the ordering, the water molecules on a wire also position themselves in a structured configuration. As a result, the axes of the individual electric dipoles of water molecules on the wire are aligned as well, yielding a polar ordering along the $c$-axis as shown in Fig. 4 (b). Because of the proton disorder, the polar ordering occurs on water wires of random sizes and random direction in regular ice. Therefore, no ferroelectricity exists [63, 1, 7], and ice Ih remains paraelectric. Nevertheless, along an ordered water wire, a local polarization field arises. In ice Ih, the charge transfer exciton is not only promoted by its intact H-bonds but also further stabilized by the local polarization field along the wire. As a result, the main peak of the optical absorption spectrum in ice Ih is mainly characterized by a collective exciton excitation of charge transfer type on a proton ordered water wire as shown in Fig. 4 (c). In this collective excitation, a relay of electron-hole pairs takes place. For a water molecule along the wire, a hole state is generated at the $1b_{1}$ valence orbital centered on oxygen atom by passing electrons onto the $4a_{1}$ empty orbital at the hydrogen atom of one of the neighboring molecules, and simultaneously the empty state of the same molecule receives electrons from the other neighboring molecule as shown in Fig. 4 (b). Here, the proton ordered water wire plays a key role in stabilizing the collective charge-transfer exciton, since its polarization field tilts the energy landscape in favor of formation of an electron and a hole on the H-bond acceptor and donor molecules, respectively, and creates a potential that reduces recombination. In the liquid water, the H-bond constantly breaks and reforms. Therefore, it cannot contain any long water wire ($l>2$) , and the number of water wires stays roughly constant with respect to temperature, since they are dominated by pairs of molecules forming H-bonds at all temperatures. Indeed, as shown in Fig. S4 of the Supplemental Material, the first peak of the optical spectra in liquid water at different temperatures is almost identical. Since, the polarization field increases with the ordering length of a water wire before reaching a limit (see Section VIII of the Supplemental Material), the collective exciton is expected to be more stabilized on a water wire with longer ordering length. Indeed, the overlap density of electron-hole pairs clearly increases when ordering length ($l$) increases as plotted in Fig. 4 (d), which correlates with an enhanced exciton oscillator strength. Consistently, the collective excitation on water wires in ice Ih results in a stronger binding energy of $\sim$3.4 eV compared with that in water of $\sim$2.2 eV as well as a more pronounced absorption intensity as shown in Fig. 4 (c).

At temperatures below 80 K, ice XI, a proton ordered hexagonal phase [63, 7], becomes energerically more stable relative to ice Ih. In the absence of proton disorder, the water wires in ice XI develop a long-range ordering length ($l=\infty$) along the $c$ axis. On this ordered water wire with ($l=\infty$), the aligned electric dipoles maximize the polarization field and yield a ferroelectricity with $P\simeq 21{\rm\mu C/cm^{2}}$ along the $c$ axis of the hexagonal crystal. Not surprisingly, excitonic effects are also maximized, which is evidenced by the large increase in the magnitude of the electron-hole overlap density at $l=\infty$ in Fig. 4 (d). As expected, the binding energy of the main peak increases to $\sim$3.6 eV relative to $\sim$3.4 eV in ice Ih, and its intensity is also the most pronounced among all the solid and liquid phases of water under current investigation as shown in Fig. 4 (c).