Absolute energy levels of liquid water from many-body perturbation theory with effective vertex corrections

Significance

The knowledge of the electronic structure of liquid water is essential for fundamental science and technology. However, state-of-the-art electronic-structure schemes have so far been unable to match experimental energy levels, by which severe ambiguities persist for the ionization potential and the electron affinity of liquid water. Here, it is shown that the consideration of a vertex function within many-body perturbation theory succeeds in producing photoemission and absorption spectra in excellent agreement with experiment on the absolute scale, overcoming this long standing issue.

Abstract

We demonstrate the importance of addressing the Γ vertex and thus going beyond the GW approximation for achieving the energy levels of liquid water in many-body perturbation theory. In particular, we consider an effective vertex function in both the polarizability and the self-energy, which does not produce any computational overhead compared with the GW approximation. We yield the band gap, the ionization potential, and the electron affinity in good agreement with experiment and with a hybrid functional description. The achieved electronic structure and dielectric screening further lead to a good description of the optical absorption spectrum, as obtained through the solution of the Bethe–Salpeter equation. In particular, the experimental peak position of the exciton is accurately reproduced.

Liquid water is a fundamentally important system for science and technology. For processes such as solvation, thermochemical water-splitting, and catalytic reactions, an accurate description of the electronic structure of liquid water is essential and has been the object of numerous experimental (1–6) and theoretical (7–11) studies. Nevertheless, the experimental uncertainty in the band-edge levels and in the band gap remains large, and theoretical calculations have so far not reached a consensus regarding these values (5, 12). For instance, recent theoretical estimates for the electron affinity (EA) range from 0.2 to 1.0 eV (9, 10, 13). In order to resolve the ambiguity related to the values of the EA and the ionization potential (IP), Bischoff, Reshetnyak, and Pasquarello performed a thorough analysis of the available experimental data and significantly reduced the uncertainty, leading to reference values of $10.0\pm 0.1$ eV for the IP, of $1.0\pm 0.2$ eV for the EA, and of $9.0\pm 0.2$ eV for the band gap (12). In addition, these values allow for a critical assessment of ab initio methods for determining the electronic structure of liquid water.

The most widely used ab initio approaches based on density functional theory (DFT) with (semi)local exchange-correlation functionals, such as the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation, generally give underestimated IPs and band gaps for most semiconductors and insulators, including liquid water (11). Hybrid functionals combining local expressions for the exchange and correlation with Fock exchange have the potential to overcome such limitations, but their dependence on adjustable parameters makes such approaches unsatisfactory, despite the significant progress achieved recently (9, 11, 14–18). In principle, the many-body perturbation theory (MBPT) in the GW approximation is considered to be the most reliable method for an accurate description of the electronic structure of weakly correlated materials (19–22). The most commonly used one-shot GW approximation ($G_0{W}_0$) performs only one iteration of Hedin’s equations and neglects the vertex function, thereby significantly speeding up the calculations. However, when using the PBE electronic structure as starting point, the fundamental band gap of liquid water determined with this approach was found to be severely underestimated, while the absolute levels of the band edges did not match their experimental counterparts (7, 8, 23). The case of liquid water is therefore more complex than many semiconductors, for which $G_0{W}_0$ yields reasonably accurate band gaps, presumably due to the partial cancelation between the effects due to the self-consistency and the lack of vertex corrections (24). When the starting electronic structure is determined through a hybrid functional, $G_0{W}_0$ was shown to be dependent on the fraction of Fock exchange (10, 25). The dependence on the starting point can be overcome by solving Hedin’s equations self-consistently (20, 24). However, the band gap and the IP of liquid water obtained via the quasiparticle self-consistent GW method are severely overestimated when the vertex function in the dielectric function is not accounted for (8, 23, 26).

The electron–hole interaction introduced via vertex corrections in the polarizability has been shown to correct the band gaps of semiconductors and insulators in general (21, 22, 27) and the band gap of liquid water in particular (8). However, despite the improved band gaps, the self-consistent GW approach generally fails in producing accurate absolute band levels, which has been attributed to the neglect of vertex corrections in the self-energy (28, 29). This analysis indicates that the vertex function is essential for an accurate determination of quasiparticle energies in MBPT, but the diagrammatic approach is computationally too expensive to be used for systems such as liquid water. Hence, effective schemes for including the effect of the vertex function are required.

Recently, a quasiparticle self-consistent scheme going beyond the GW approximation has been introduced including an effective vertex function in both the polarizability and the self-energy (QS$G\widehat{W}$). In this approach, the vertex function is separated into short-range and long-range parts, which are described separately. In the long range the vertex complies with the Ward identity (30), while the adiabatic local density approximation (ALDA) (31) is applied in the short range. The QS$G\widehat{W}$ scheme has been found to yield band gaps, IPs, and dielectric constants in excellent agreement with experiment for a diversified set of materials (27).

In this article, we demonstrate that many-body perturbation theory yields an accurate description of the electronic structure of liquid water. In particular, we use an effective vertex function $f_{\mathrm{xc}}$ in both the polarizability and the self-energy and find that the QS$G\widehat{W}$ approach yields the band gap, the IP, and the EA of liquid water in close agreement with experiment. Furthermore, the energy levels and the wave functions obtained with our advanced many-body perturbation theory are found to be well approximated by hybrid functionals, thereby lending support to the latter for describing the electronic structure of liquid water. Next, we calculate the optical absorption spectrum relying on the electronic structure achieved with QS$G\widehat{W}$ and solving the Bethe–Salpeter equation. We find overall good agreement with experiment, including the position of the excitonic peak. Hence, our work provides a consistent description for the absolute energy levels of liquid water, which reconciles state-of-the-art electronic structure theory and experiment.

To investigate the role of the vertex function for an accurate description of the electronic structure of liquid water in MBPT, we carry out quasi-particle self-consistent calculations (20) with two types of effective vertex functions. In the QS$G\tilde{W}$ scheme (22), the effective vertex function consists of the bootstrap kernel (32, 33) and only occurs in the polarizability. To facilitate the comparison with previous results, we here only use the head of the kernel (8, 12). In the QS$G\widehat{W}$ calculations, the vertex function is included in both the polarizability (27)

$$\begin{array}{c}\hfill \tilde{\chi }=\chi +\chi f_{\mathrm{xc}}\tilde{\chi }\end{array}$$ [1]

and the self-energy

$$\begin{array}{c}\hfill \mathrm{\Sigma }=i{G}_0(1+Z{f}_{\mathrm{xc}}^{\mathrm{SR}}\tilde{\chi })W.\end{array}$$ [2]

In Eq. 2, $G_0$ is the noninteracting Green’s function, Z is the derivative of the self-energy $\mathrm{\Sigma }$ w.r.t. the real frequency ω, i.e., $Z={(1-\partial \mathrm{\Sigma }/\partial \omega )}^{-1}$, and $f_{\mathrm{xc}}^{\mathrm{SR}}$ is the effective vertex function in the short-range. More specifically, the diagonal part of the corresponding exchange-correlation kernel $f_{\mathrm{xc}}$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_{\mathrm{xc},\mathbf{G},G}(\mathbf{q},\omega )& =\frac{1-Z(\mathbf{q}=0,\omega =0)}{{\chi }_0^{00}(\mathbf{q},\omega =0)}{e}^{{-|\mathbf{q}+G|}^2/k_{\mathrm{TF}}^2}\hfill \\ \hfill & +f_{\mathrm{xc},\mathbf{G},G^{\prime }}^{\mathrm{LDA}}(1-e^{{-|\mathbf{q}+G|}^2/k_{\mathrm{TF}}^2}),\hfill \end{array}\end{array}$$ [3]

and the off-diagonal terms are found via

$$\begin{array}{c}\hfill f_{\mathrm{xc},\mathbf{G},G^{\prime }}(\mathbf{q},\omega )=f_{\mathrm{xc},\mathbf{G},G^{\prime }}^{\mathrm{LDA}}(1-{\delta }_{\mathbf{G},\mathbf{0}})(1-{\delta }_{0,{\mathbf{G}}^{\prime }}).\end{array}$$ [4]

Here, ${\chi }_0^{00}$ is the head of the non-interacting polarizability and $k_{\mathrm{TF}}$ the Thomas-Fermi wave vector. In Eqs. 3 and 4, q stands for a wave vector in the first Brillouin zone, while G indicates a reciprocal lattice vector.

As structural model of liquid water, we use atomic configurations from the ab initio path-integral molecular dynamics simulation carried out at ambient temperature by Chen et al. (8). To achieve energy levels on the absolute scale, the determination of the average electrostatic potential relative to the vacuum level is required. Ambrosio et al. established an accurate model for the water–vacuum interface and made available a representative set of configurations obtained through ab initio molecular dynamics in the same DFT approximation as for bulk liquid water (9). To determine the average electrostatic potential, we use these configurations within our own computational set-up for consistency with our calculations of bulk liquid water.

In Fig. 1, we show the average DOS calculated with the QS$G\widehat{W}$ method in comparison with the experimental photoemission spectrum (6). We highlight that the calculated DOS in Fig. 1 is presented on the absolute scale without including any ad hoc alignment. The absolute binding energies obtained in the UV photoemission experiment, namely $-11.2$ eV for 1$b_1$, $-13.5$ eV for 3$a_1$, $-17.3$ eV for 1$b_2$, and $-30.9$ eV for 2$a_1$, are accurately reproduced by our QS$G\widehat{W}$ calculations, which yield respective values of $-11.3$, $-13.5$, $-17.3$, and $-30.7$ eV. The broad background of 2$a_1$ has been attributed to the energy loss of the 1$b_1$ electron (6, 34), and is not accounted for in our calculations.

Fig. 1.

Density of states (DOS) calculated with QS$G\widehat{W}$ and compared on the absolute scale with the photoemission spectrum measured by Winter et al. (6).

In Fig. 2, we assess the accuracy of the QS$G\widehat{W}$ approach for the energy levels in comparison with experiment and other theoretical schemes. The present scheme yields a band gap of 9.2 eV, which is in a good agreement with the range of experimental values ($9.0\pm 0.2$ eV) (12) and with a recent theoretical estimate resulting from an improved treatment of the electron screening (9.3 eV) (13). As far as the measured absolute energy levels, the calculated IP falls within the range of the experimental references and the calculated EA deviates by at most 0.1 eV. Hence, the present theoretical scheme yields an excellent description of both relative and absolute energy levels of liquid water.

Fig. 2.

Energy levels of the valence band maximum (VBM) and conduction band minimum (CBM) with respect to the vacuum level, as calculated through QS$G\widehat{W}$, QS$G\tilde{W}$, and a nonempirical hybrid functional (HF) developed in ref. 12 that satisfies both the condition of piecewise linearity and the asymptotic limit of the potential. The shaded areas depict the experimental values with their corresponding uncertainties (12).

To achieve accurate results within MBPT for liquid water, it is important to remark that the vertex function must be taken into account in both the polarizability and the self-energy. Indeed, as shown previously, $G_0{W}_0$ strongly underestimated the IP (8, 10, 26, 35), self-consistent GW calculations without vertex corrections lead to an overestimation of both the band gap and the IP (8, 12, 35), and the vertex function in the sole polarizability improves the band gap (8, 12, 13) but not necessarily the IP and EA (8). In this respect, we compare the QS$G\widehat{W}$ results with those obtained with QS$G\tilde{W}$. The alignment in QS$G\tilde{W}$ has been obtained for classical nuclei and corrected for NQE (8). This comparison indicates that the short-range vertex corrections in the self-energy are responsible for shifting the quasiparticle energies upward by approximately 0.5 eV. The comparison between the QS$G\widehat{W}$ and QS$G\tilde{W}$ schemes can be extended to the dielectric constant ${\epsilon }_{\infty }$ (Table 1), for which we find 1.69 and 1.96, respectively. In particular, ${\epsilon }_{\infty }$ obtained with QS$G\widehat{W}$ slightly underestimates the experimental value of 1.77 (36, 37), consistent with the general tendency of QS$G\widehat{W}$ applied to insulators (27).

Table 1.

Band gap, IP, and EA obtained with QS$\mathit{G}\stackrel{\mathbf{~}}{\mathit{W}}$ and QS$\mathit{G}\widehat{\mathit{W}}$ calculations for liquid water, compared with the results obtained with a nonempirical hybrid functional (HF) and with experimental references (12)

Method	Band gap (eV)	IP (eV)	EA (eV)	${\epsilon }_{\infty }$
QS$G\widehat{W}$	9.20	9.86	0.66	1.69
QS$G\tilde{W}$	9.21	10.44	1.23	1.96
HF	9.20	9.97	0.77
Expt.	9.0 ± 0.2	10.0 ± 0.1	1.0 ± 0.2	1.77

The hybrid functional has been defined in ref. 12. In the last column, the dielectric constants ${\mathit{\epsilon }}_{\mathit{\infty }}$ obtained with the $\mathit{GW}$ methods are compared with the experimental value from ref. 36.

The description achieved with QS$G\widehat{W}$ can also be compared with the energy levels obtained from a nonempirical range-separated hybrid functional (12) (Fig. 2). The latter functional reproduces two properties of the exact generalized Kohn–Sham functional, namely the asymptotic potential in the long-range and the piecewise linearity condition (38). We see that the energy levels obtained with QS$G\widehat{W}$ closely match the results found with the hybrid functional. The band gap at 9.2 eV, the IP at 9.9 to 10 eV, and the EA at 0.7 to 0.8 eV are all in agreement within 0.1 eV (cf. Table 1) corresponding to the estimated error for the QS$G\widehat{W}$ calculations (SI Appendix). This agreement is strengthened by inspecting the self-consistent wave functions. The hybrid-functional, QS$G\tilde{W}$, and QS$G\widehat{W}$ schemes all yield very similar wave functions, while the PBE wave functions differ more dramatically, as we illustrate in SI Appendix, Fig. S7 for conduction and valence band states lying close to the band edges. This provides additional support to the hybrid-functional description of the electronic structure of liquid water. Hence, besides matching the experimental data, the QS$G\widehat{W}$ results yield a global agreement between different theoretical approaches for both the energy levels and the wave functions of liquid water.

Next, we calculate the absorption spectrum to further investigate the performance of the QS$G\widehat{W}$ method. We solve the Bethe–Salpeter equation using the quasiparticle energies and wave functions as well as the dielectric matrix taken from the preceding QS$G\widehat{W}$ calculations. The spectrum is obtained by averaging over 18 molecular-dynamics configurations in order to achieve a proper statistical description of the liquid. The imaginary part of the calculated dielectric function is compared in Fig. 3 with the experimental optical absorption spectra obtained by Hayashi and Hiraoka (37) and by Heller et al. (36). The experimental spectra slightly differ between each other in the absolute intensities as well as in the position of the first excitonic peak. Here, we use the more recent spectrum from Hayashi and Hiraoka for comparison with our results. Overall, the agreement between calculated and measured spectra is excellent near the absorption edge, and only a slight deviation occurs at higher energies. In particular, the excitonic peak in the calculated spectrum occurs at 8.2 eV, very close to its experimental counterpart at 8.4 eV. Accounting for the finite size effect of 0.2 eV (13) brings the excitonic peak on top of the experimental peak.

Fig. 3.

Imaginary part of the dielectric function ${\epsilon }_2$ of liquid water. The calculated spectrum is obtained by solving the Bethe–Salpeter equation and using quasiparticle energies and wave functions from QS$G\widehat{W}$ calculation (BSE@QS$G\widehat{W}$). An average over 18 configurations of the liquid is carried out. For the calculated spectrum a Lorentzian broadening of 0.1 eV is used. The experimental spectra are taken from Hayashi and Hiraoka (37) and Heller et al. (36).

Our calculated spectrum can also be compared with the result of other schemes applied previously. The comparison of our spectrum with the Bethe–Salpeter equation spectrum based on the QS$G\tilde{W}$ electronic structure (13) shows good agreement, due to the similar description of the band gap in the two schemes (SI Appendix, Fig. S6). In particular, we determine an exciton binding energy of 2.1 eV (SI Appendix, Fig. S3), which coincides with the value obtained in ref. 13. Our absorption spectrum based on the QS$G\widehat{W}$ electronic structure also shows close agreement with the shape of the spectrum obtained with $G_0{W}_0$ (39). However, as the $G_0{W}_0$ approximation strongly underestimates the band gap of liquid water, the energy scale had to be adjusted to match the experimental spectrum (39). A comparison with an unshifted spectrum obtained within the $G_0{W}_0$ approximation is given in SI Appendix, Fig. S6. At variance, the present QS$G\widehat{W}$ approach does not require any alignment as the band gap is accurately reproduced. Hence, the good agreement with experiment for the resulting absorption spectrum stems from the reliable description of the energy levels.

Conclusions

To conclude, we have shown that the vertex function is essential for an accurate description of the electronic structure of liquid water within MBPT. Due to the formidable computational cost of the diagrammatic approaches for this system, we have used an effective vertex function that has recently successfully been applied to semiconductors and insulators (27). In this work, we show that such an effective vertex function applied to liquid water not only reproduces the band gap, but also accurately describes the ionization potential and the electron affinity. Further support for the determined energy levels comes from the good agreement with experiment found for the calculated absorption spectrum. The present results obtained within many-body perturbation theory support the description achieved with nonempirical hybrid functionals, both as far the energy levels and the wave functions are concerned. This global agreement among different theoretical methods and with experiment resolves the longstanding issue associated with the actual values for the band gap, the ionization potential, and the electron affinity of liquid water.

Materials and Methods

All ab initio calculations are performed within the projector-augmented-wave (PAW) scheme (40) implemented in the VASP software suite (41, 42). Approximatively norm-conserving PAW potentials (a small norm violation of up to 10% was allowed) are used in the calculations for both H ($\mathrm{H}\_\mathrm{h}\_\mathrm{GW}$) and O ($\mathrm{O}\_\mathrm{h}\_\mathrm{GW}$). To achieve converged quasiparticle energies, an energy cutoff of 800 eV is used in the ground-state calculations. The polarizability matrix is computed with 60 frequency grid points and a reduced basis set defined by an energy cutoff of 250 eV. We explicitly account for 3,072 bands, 256 of which are updated in every iteration of the QSGW calculations, while all other bands are implicitly taken into account via linear extrapolation to infinite number of bands. A correction of 0.2 eV is found to account for the fact of keeping the other orbitals fixed in the self-consistent iterations. Eight self-consistent iterations are found to be sufficient for achieving converged quasiparticle energies. The derivative of the cell-periodic part of the orbitals w.r.t. the $\mathbf{k}$ vectors is calculated with the PEAD method (43). We sample the Brillouin zone of the 32-water-molecule supercell at the sole $\mathrm{\Gamma }$ point, which has previously been shown to yield energy levels accurate within 0.1 eV (8). Thus, the estimated accuracy of the absolute quasiparticle energies is 0.1 eV. The detailed analysis of the estimated errors is provided in SI Appendix.

To account for the liquid nature of the system, we carry out calculations for 18 configurations, equally spaced in time, taken from the ab initio path-integral molecular dynamics simulation described in ref. 8. In this simulation, the electronic structure was described with the PBE functional supplemented with nonlocal van der Waals interactions via the revisited Vydrov and Van Voorhis (rVV10) density functional (8, 44–46), and the nuclei were treated as quantum particles. The atomic configurations consist of 32 water molecules in a cubic box with a side of $9.81$ Å, corresponding to a density of $1.01$ $\text{g}/{\text{cm}}^3$ (8). Nuclear quantum effects (NQE) on the electronic structure of liquid water resulted in an upward shift of the valence band edge by 0.46 eV and a downward shift of the conduction band edge by 0.25 eV, leading to an overall band gap reduction of 0.7 eV (8).

The structure of the water–vacuum interface was modeled with a supercell consisting of 384 water molecules at the experimental density and with the vacuum taking up 40 Å of the supercell (9). In the present work, we use 540 configurations from ref. 9 for determining the average electrostatic potential.

In solving the Bethe–Salpeter equation, we use the Tamm–Dancoff approximation (47–51) and take into account all the 128 occupied and 256 additional unoccupied states at the $\mathrm{\Gamma }$ point. The application of the Tamm–Dancoff approximation to liquid water is validated in SI Appendix, Fig. S5. The exciton binding energy is determined as the difference between the onsets in the spectra calculated in the random-phase approximation and solving the BSE. More details concerning the computational set-up are given in SI Appendix.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

The data associated with this work are available on Materials Cloud (52). All other data are included in the manuscript and/or SI Appendix.