The ionization potentials of electrolyte solutions provide important information about the electronic structure of liquids and solute-solvent interactions. We analyzed the positions of solute and solvent bands of aqueous hydroxide and the influence of the solvent environment on the ionization potential of hydroxide ions. We used the concept of a computational hydrogen electrode to define absolute band positions with respect to vacuum. We found that many-body perturbation theory in the G0 W0 approximation substantially improves the relative and absolute positions of the band edges of solute and solvent with respect to those obtained within Density Functional Theory, using semi-local functionals, yielding results in satisfactory agreement with recent experiments.
I. INTRODUCTION
Water represents the most important liquid in chemistry and biology. Despite the vast number of studies devoted to elucidate the structure, solvation properties, and electronic states of liquid water,1–4 the theoretical description of the water band structure including its autoprotolysis products is still far from complete. Recently developed techniques of liquid-microjet photoemission spectroscopy (PES) can directly probe the electronic structure of the valence band (VB) of aqueous solutions.5 In particular, experiments by Winter et al.6,7 revealed an energy gap of 0.9–1.5 eV between the electronic states of solvated hydroxide ions and the energy of the VB maximum of liquid water. PES experiments further demonstrated that aqueous hydroxide ions show insignificant or no surface affinity,7 indicating that molecular dynamics (MD) simulations of hydroxide in water using a periodically repeated unit cell may represent an accurate model.
Theoretical studies of solvated hydroxide often focused on the structure of the solvent environment of the ion and its incorporation into the hydrogen-bond network of bulk water. Marx et al.8 ruled out several proposed solvation models using both computer simulations and experimental data and introduced the concept of the “dynamical hypercoordination” mechanism to describe the fluctuating solvation shell of OH−.
While structural properties using ab initio MD simulations based on Density Functional Theory (DFT) have received widespread attention, the electronic structure of hydrated ions has been studied only recently.4 DFT with semi-local functionals is known to underestimate band gap energies. Many-body perturbation theory (MBPT), on the other hand, provides a consistent theoretical description of quasi-particle energies corresponding to ionization potentials (IPs) and electron affinities. It has recently been shown that even with approximate formulations of MBPT, the description of the band gap, IP, and electron affinities of water are significantly improved and in good agreement with experiment.3 Swartz and Wu9 carried out electronic-structure calculations of proton transfer reactions between solvated OH− and water molecules. In order to compute IPs as measured in PES experiments, they performed ab initio calculations for hydrated OH− using MBPT within the static Coulomb hole and screened exchange (COHSEX) approximation.9 The quasi-particle energy of the highest occupied molecular orbital (HOMO) of the solute, averaged over snapshots including proton-transfer reaction intermediates, was determined at 9.9 eV, 0.7 eV above the experimental value. Zhang et al.10 applied MBPT to hydrated chloride, taking into account the frequency dependence of the self-energy within the G0 W0 approximation, showing a clear improvement over DFT results with semi-local functionals, although discrepancies with experiments persisted. Overall, the results of both Refs. 9 and 10 indicate that the description of IPs and band gaps of electrolyte solutions are substantially improved by MBPT corrections to the DFT energy levels, which underestimate the water band gap by more than 50% in the generalized-gradient approximation (GGA).
In addition to the band gap, spectroscopic and electrochemical properties of electrolyte solutions are determined by the absolute position of the band edges of the solvent and the energy levels of the solute. Adriaanse et al.11 calculated the positions of the electronic energy levels of solvated hydroxide and the conduction and valence band of bulk water using DFT and the GGA, and the computational standard hydrogen electrode (SHE) concept to determine absolute positions with respect to vacuum. This concept originated from calculations of acidity constants in aqueous solutions and is based on free-energy perturbation theory.12 The key to the use of the SHE is that within ab initio MD using periodic boundary conditions (PBC), the offset (or arbitrary constant) of the average potential for removal of an electron is minus the offset for the removal of a proton. So the offset cancels. There is one central approximation in this argument, namely, that the offset is determined by the solvent, i.e., is independent of the solute. This is strictly true only in the limit of infinite dilution. Adriaanse et al.11,13 found that the 31 H2O molecule system with one small solute such as OH− is close to the infinite dilution limit. The positions of the valence band maximum (VBM) and conduction band minimum (CBM) of water, determined using the SHE and DFT with the BLYP functional, are −6.41 V and −1.84 V, respectively,11,14 to be compared with the values of −6.1 V and −1.9 V obtained in Ref. 3 within PBE and calculations of the IP and electron affinity of liquid water, using slabs.
In the present work we adopt the SHE concept for simulations with MBPT in the G0 W0 approximation. The article is organized as follows: In Sec. II we describe the computational details of our simulations, Sec. III contains the results and a discussion of the band alignment. In Sec. IV the results are summarized and discussed.
II. METHODOLOGY
The potential-energy surface of a simulation box comprising 31 water molecules and one hydroxide ion has been sampled using ab initio MD simulations. Additionally, a system of 32 water molecules has been simulated with the same protocol for comparison. We carried out simulations in the NVT ensemble for about 20 ps at a temperature of 330 K, maintained with a loosely coupled Nosé-Hoover thermostat. We used the GGA with Becke's exchange and the Lee-Yang-Parr correlation functional15,16 and a plane-wave basis set, with a 70 Ry cut-off and norm-conserving pseudo-potentials.17
The density of states (DOS) of the liquid has been computed averaging over 40 snapshots, chosen equally spaced, every 0.5 ps. For all single-point calculations a plane-wave cut-off of 80 Ry has been used. Moreover, the DOS calculations were carried out on a 4 × 4 × 4 Monkhorst-Pack grid of k-points to correctly sample the conduction band of the system.18 The KS orbitals of the same snapshots have been used to compute six quasi-particle energies per snapshot, using MBPT in the G0 W0 approximation, including the lowest two unoccupied and the highest four occupied orbitals. The head of the dielectric matrix was computed on a 2 × 2 × 2 Monkhorst-Pack grid of k-points. The dielectric matrix has been evaluated by the projective dielectric eigenpotential method19,20 including 400 eigenvectors in its spectral decomposition. All G0 W0 calculations were performed at the Γ-point only. All DFT results in the present work were computed with the QUANTUM ESPRESSO software distribution.21 The G0 W0 calculations were carried out with the methods and codes developed in Refs. 19 and 20.
III. RESULTS
The DOS as obtained from the average of 40 snapshots along the MD trajectory shows the characteristic features of liquid water simulated with Kohn-Sham DFT and the GGA.18 In Fig. 1 the highest peak centered around −2.5 eV corresponds to the well-known 1b1 band of the water molecules. The high-energy shoulder of the 1b1 band can be identified from the projected DOS as an electronic state localized on the OH− ion. At the GGA DFT level, the HOMO of the solution can therefore be assigned to the 2pπ orbitals of the solvent. PES experiments by Winter et al.7 with NaOH solutions of varying concentrations showed a qualitatively identical feature from the OH− ions with an onset at 9.2 eV, 0.7 eV below the photoemission threshold of liquid water.
Figure 2 illustrates the computed energies of the highest four occupied and the lowest two unoccupied KS orbitals. The HOMO and the second highest occupied orbital (orange lines) correspond to the nearly degenerate 2pπ orbitals of the dissolved OH− ion. The solute orbitals appear above the top of the valence band of liquid water throughout the trajectory, but on average they are separated from the band edge by only ≈0.31 eV—significantly less than the reported experimental value.6,7 Figure 3 shows the quasi-particle energies at the G0 W0 level of theory, computed for the same geometries as in Fig. 2. In addition to increasing the fundamental band gap, as expected, the G0 W0 corrections shift the relative positions of the KS orbitals of water and the hydroxide ion. As in Fig. 2, the two highest occupied solute molecular orbitals are plotted in color. Contrary to DFT/GGA calculations, crossings between solvent and solute energy levels are observed at the G0 W0 level. It remains to be seen if these intersections would remain, should one chose a different starting point (e.g., PBE0) for the wavefunctions used in the G0 W0 calculations. The computed unoccupied bands correspond to solvent states.
The G0 W0 calculations give a fundamental gap of the solution of ≈7.5 eV compared to 4.1 eV obtained within the GGA. The band gap of the solvent computed from the differences between the third HOMO and the LUMO quasi-particle energies of the system was determined to be 8.2 eV, close to the fundamental gap of a pure water system (8.5 eV) that we simulated at the same conditions. These values are in good agreement with previously reported values of 8.4 eV, 8.7 eV, and 8.1 eV from G0 W0 calculations for pure water1–3 and the proposed experimental energy gap of 8.7 ± 0.5 eV.22,23 Pham et al.3 recently showed that the effect of the potential or the functional used to model the water structure is of the order of 0.3–0.5 eV. In particular the average temperature in the simulation cell is known to affect the water structure.24,25 Hence the different sampling methods and density functionals used in previous works may explain the differences among the reported theoretical values for the band gap of liquid water. Finite size effects, however, have been found to be negligible for the average structural properties of water at the conditions of our simulations.24,25
Table I displays the numerical values of the energy levels computed in this study, with respect to vacuum. All energies in Table I are expressed with respect to vacuum. The ionization threshold of a pure water system has been determined as 8.98 eV with a conduction band minimum at −0.63 eV, the latter being unaffected by the presence of the solute. Our results which have been obtained by alignment vs SHE are in agreement with previous work of Pham et al. who aligned the bands of liquid water based on the average electrostatic potential and reported values of −8.8 eV and −0.7 eV with respect to vacuum for the VBM and CBM, respectively.3 We found that the average energy gap between the lowest IP of the solute and solvent is ≈0.7 eV in good agreement with experiment. Moreover, the average splitting of the 2pπ orbitals of the solvent is approximately 0.15 eV which suggests that the arrangement of the solvent molecules around the OH− ion is rather symmetrical.
. | EKS (eV) . | $E^{\text{G_{\text{0}}W_{\text{0}}}}$ (eV)
. | ΔE (eV) . |
---|---|---|---|
LUMO | −1.93 | −0.67 | 1.26 |
HOMO | −6.04 | −8.15 | −2.11 |
HOMO-1 | −6.16 | −8.22 | −2.06 |
HOMO-2 | −6.41 | −8.86 | −2.45 |
LUMO (H2O) | −1.93 | −0.63 | 1.30 |
HOMO (H2O) | −6.41 | −8.98 | −2.57 |
. | EKS (eV) . | $E^{\text{G_{\text{0}}W_{\text{0}}}}$ (eV)
. | ΔE (eV) . |
---|---|---|---|
LUMO | −1.93 | −0.67 | 1.26 |
HOMO | −6.04 | −8.15 | −2.11 |
HOMO-1 | −6.16 | −8.22 | −2.06 |
HOMO-2 | −6.41 | −8.86 | −2.45 |
LUMO (H2O) | −1.93 | −0.63 | 1.30 |
HOMO (H2O) | −6.41 | −8.98 | −2.57 |
As reported in Ref. 7, the fitting of the hydroxide component of the measured peaks depends weakly on the concentration. For 0.1 mol/l the OH− (aq) binding energy (BE) was measured to be 8.8 eV with 1.3 eV full width at half maximum (fwhm), while for 0.2 mol/l the BE is 9.0 eV with 1.6 eV fwhm. No further change in the peak width was observed up to the highest concentration but BEs further increased to 9.2 eV for concentrations above 1.5 mol/l. Hence, the measured ionization potential of OH− is 9.2 ± 0.8 eV, putting the threshold energy at 8.4 eV. This value is in good agreement with the threshold energy of 8.45 eV reported by von Burg and Delahay26 as discussed by Winter et al.6
Our computed G0 W0 value of 8.15 eV for the ionization potential of OH− is to be compared with the measured threshold energy and it is in good agreement with experiments, with a moderate upward shift of only 0.3 eV. As mentioned above, the calculated threshold energy of liquid water is 8.86 eV, also in reasonable, qualitative agreement with experimental values ranging from 9.3 eV to 9.9 eV.5,23 Hence the computed energy separation between OH− and water is 0.7 eV (within G0 W0) consistent with experimental results ranging between 0.9 and 1.5 eV (see Fig. 4). On the contrary the separation is only 0.3 eV within GGA. Overall our results illustrate that the relative and total energies obtained within DFT/GGA are seriously affected by the quality of the exchange-correlation energy and that MBPT is necessary to obtain at least a qualitatively correct description of the electronic properties.
IV. SUMMARY
We showed that many body perturbation theory within the G0 W0 approximation yields a reasonably accurate description of the electronic properties of a aqueous hydroxide solution in the limit of infinite dilution, much improved compared with DFT/GGA. We found that the separation between the OH− band and the VBM of water in the solution is ≈0.7 eV, in qualitative agreement with measured values ranging from 0.9 to 1.5 eV. DFT/GGA gives instead a smaller separation of only 0.3 eV. In addition, the computed onset of ionization of OH− with respect to vacuum (8.15 eV) is in agreement with the measured ionization threshold of 8.4 eV in Refs. 26 and 6. We note that our results for the IP of OH− differ from those computed by Swartz and Wu9 for a proton-transfer reaction involving solvated hydroxide. This difference might stem from the neglect of the explicit frequency dependence in the COHSEX approximation used in their work.
Finally, we note that the method used here to align levels to vacuum using a SHE yielded results in agreement with those reported in Ref. 3 for pure water, using slab calculations.
ACKNOWLEDGMENTS
This work was supported by a research grant of the Deutsche Forschungsgemeinschaft (D.O.) and by NSF-CHE-0802907 (G.G.). Computing resources provided by the Leibniz Rechenzentrum of the Bavarian Academy of Sciences are gratefully acknowledged.