Electronic Energy Levels and Band Alignment for Aqueous Phenol

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Electronic Energy Levels and Band Alignment for Aqueous Phenol and Phenolate from First Principles Daniel Opalka, Tuan Anh Pham, Giulia Galli, and Michiel Sprik J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b04189 • Publication Date (Web): 01 Jul 2015 Downloaded from http://pubs.acs.org on July 7, 2015

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The Journal of Physical Chemistry

Electronic Energy Levels and Band Alignment for Aqueous Phenol and Phenolate from First Principles Daniel Opalka,∗,† Tuan Anh Pham,‡ Michiel Sprik,¶ and Giulia Galli§ Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany, Lawrence Livermore National Laboratory, Livermore, California 94551, USA, Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom, and The Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA E-mail: [email protected]

Abstract Electronic energy levels in phenol and phenolate solutions have been computed using density functional theory and many-body perturbation theory. The valence and conduction bands of the solvent and the ionization energies of the solutes have been aligned with respect to the vacuum level based on the concept of a computational standard hydrogen electrode. We have found significant quantitative differences between the generalized-gradient approximation, calculations with the HSE hybrid functional ∗

To whom correspondence should be addressed Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany ‡ Lawrence Livermore National Laboratory, Livermore, California 94551, USA ¶ Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom § The Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA †

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and many-body perturbation theory in the G0 W0 approximation. For phenol, two ionization energies below the photoionization threshold of bulk water have been assigned in the spectrum of Kohn-Sham eigenvalues of the solution. Deprotonation to phenolate was found to lift a third occupied energy level above the valence band maximum of the solvent which is characterized by an electronic lone pair at the hydroxyl group. The second and third ionization energies of phenolate were found to be very similar and explain the intensity pattern observed in recent experiments using liquid-microjet photoemission spectroscopy.

Introduction Phenol (PhOH) and molecules with phenyl groups are subject to a rich chemistry of photochemical reactions. The PhOH molecule and its conjugated base, the phenolate anion (PhO– ), are important model chromophores for photochemical and redox processes in the photosystem II of green plants. In proton-coupled electron-transfer process, the phenyl group of the amino acid tyrosine is oxidized and subsequently deprotonated. Both solvated tyrosine and PhOH show very similar photochemical characteristics and the positions of their absorption maxima in the UV spectrum differ only by about 5 cm-1 . 1 Hence it is often assumed that PhOH and PhO– represent realistic models for many systems involving tyrosine or a phenyl group. 2,3 Both in the gas phase and in microsolvated clusters, PhOH has been subject of extensive theoretical and experimental work that examined the electronic structure and relaxation processes after absorption of a photon. 4,5 The first spectroscopic experiments on PhO– ions in aqueous solution were carried out by Jortner et al who observed a significant yield of solvated electrons after absorption of a UV photon. 6 Contrary to solutions of simple ions such as aqueous hydroxide or halide ions, 7 the formation of solvated electrons from photoexcited PhO– or PhOH does not proceed through a charge transfer to solvent (CTTS) state. 6,8,9 Consequently, the first optically excited electronic state of solvated PhO– and PhOH is ex2


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pected to lie above the conduction band minimum of liquid water. In particular, a solvated PhOH molecule in its first excited electronic state may eject an electron into the conduction band of liquid water. This process would then be accompanied by subsequent deprotonation due to a tremendous increase of acidity. 9 Experiments using liquid-microjet photoemission spectroscopy found two photoemission bands for solvated PhOH and PhO– below the photoemission threshold of liquid water, 10 indicating vertical ionization energies (IEs) of 7.8 eV and 8.6 eV for PhOH in a 0.75 mol/l aqueous solution. For the conjugated PhO– anion two IEs of 7.1 eV and 8.5 eV were reported from experiments. 10 While many theoretical investigations focused on isolated PhO– and PhOH, as well as on small PhO– /PhOH-water clusters, 4 only few theoretical studies were carried out in the condensed phase. Recently, Ghosh and coworkers investigated the IEs of solvated PhOH and PhO– in bulk solution. 10 They applied a quantum/molecular mechanics (QM/MM) approach using equation-of-motion ionization potential coupled-cluster theory and effective fragment potential methods, where a single solute molecule was represented in the QM region. Despite the relatively small QM region they obtained results in reasonable agreement with experiments, although quantitative accord required rescaling of the intensities and, for the charged PhO– system, an additional shift of 0.8 eV. The electronic structure of the solvent molecules and the interaction of solute and solvent were not considered. Additionally, the origin of the approximately double intensity of the second photoemission band of solvated PhO– was not identified. The relative positions of the solute energy levels and the valence band (VB) and conduction band (CB) of liquid water are of fundamental importance to understand electrochemical properties and photo-induced charge-transfer in aqueous environments. Clearly, a rigorous QM description of both the solute and the solvent is required to address these issues. Pham et al recently reported first principles calculations of the absolute positions of the valence (VB) and conduction bands (CB) of liquid water, with respect to vacuum. 11 They used the projective dielectric eigenpotential method. 12,13,24 which includes the frequency dependence

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of the self-energy; the same technique was used to investigate aqueous solutions of chloride ions. 14 The relevance of explicit frequency dependence of the self-energy was also demonstrated in a recent study by Opalka et al on solvated hydroxide where the band alignment of solute and solvent levels was addressed within a quasi-particle (QP) picture. 15 In the present work we considered aqueous solutions of PhOH and PhO– using first principles calculations. Based on the model of a computational standard hydrogen electrode (SHE) which mimics the reference potential in an electrochemical experiment, 16–18 we determined the absolute position of the molecular energy levels of each solute and of the valence band maximum (VBM) and conduction band minimum (CBM) of the solvent. The major deficiency of the generalized-gradient approximation (GGA), namely the delocalization error which leads to strongly underestimated band gaps, was corrected by MBPT calculations in the G0 W0 approximation. Additionally, we compared our results to those obtained with electronic-structure calculations using a hybrid functional. The rest of the paper is organized as follows: In the first section the computational methods and simulation parameters are described. Section II is primarily dedicated to the discussion of the electronic structure calculations and solute levels with respect to the solvent bands. In section III we describe the alignment of solute levels and solvent bands with respect to the SHE and vacuum reference potential. The final section provides a summary of our results and conclusions.

Computational details The potential-energy (PE) surface of a periodic cell containing 68 water molecules and one PhOH or PhO– species was sampled by ab initio MD simulations. The volume of the cubic cell was adjusted to match the experimental density of a 0.75 mol/l H2 O/PhOH mixture at a temperature of 330 K, which was determined from experimental density data 19 by interpolation. After an equilibration period of more than 2 ps, the temperature in the simulation

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cell was maintained through a Nosé-Hoover thermostat. Using Born-Oppenheimer MD, the atomic forces were obtained from DFT in the GGA with a mixed plane-wave (PW) and triple-ζ Gaussian basis set. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional 20,21 was used in all MD simulations which were performed with the CP2K software and Goedecker-Teter-Hutter pseudo potentials. 22 In order to analyze the electronic structure of solute and solvent, 20 snapshots were selected from a 5 ps long section of the trajectories at regular time intervals, every 0.25 ps and energy levels were computed at different levels of theory. All electronic structure calculations of the 20 snapshots were performed with the Quantum Espresso software distribution 23 and, in the case of G0 W0 calculations, with the WEST code (http://www.west-code.org). 12,13,24 For these calculations we have used a plane-wave basis set with a cut-off energy of 80 Ry in combination with norm-conserving Troullier-Martins pseudo potentials. 25 The density of states (DOS) for the PhOH and PhO– solutions were computed using the PBE functional on a 2×2×2 Monkhorst-Pack grid of k-points and it was averaged over all snapshots. In addition, we computed the Kohn-Sham (KS) energy levels with a hybrid functional, namely the Heyd-Scuseria-Ernzerhof (HSE) density functional, 26,27 at the supercell Γ-point. Quasi-particle energies were computed in the G0 W0 approximation for six occupied and five unoccupied KS energy levels, for each snapshot. The G0 W0 calculations were carried out with the projective dielectric eigenpotential method. 12,13 Following previous work we computed the head of the dielectric matrix separately using the same 2×2×2 grid of k-points that was employed for the DOS calculations in the GGA. In the spectral decomposition of the dielectric matrix we included 800 eigenvectors and all G0 W0 calculations were performed at the Γ-point only.

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Electronic energy levels in solution Our analysis of the structural properties of the PhOH and PhO– solutions showed that the first maximum and minimum of the oxygen-oxygen radial distribution functions are nearly identical in the two solutions, thus showing little effect on the average O-O structure of the additional charge on PhO– . However, as expected the solute O/water H correlation functions differ in the two solutions, with a strong and well defined peak present in the case of PhO– .

Figure 1: Density of states of a aqueous solution of phenol (left) and phenolate (right), computed using the GGA and averaged over 20 snapshots extracted from an initio MD trajectories. Figure 1 and 2 show the electronic density of states (DOS) computed using the GGA and the HSE hybrid functional, respectively, for the two solutions. The electronic DOS projected on the solute is represented by the colored line and indicates the positions of localized solute states within the band structure of the solution. The data were obtained as an average over the DOS of 20 snapshots extracted from ab initio MD simulations and illustrate the electronic band gap present in the simulated PhOH (left) and PhO– (right) solutions. Electronic structure calculations using the GGA (Fig. 1) predict a water band gap

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Figure 2: Density of states of an aqueous solution of phenol (left) and phenolate (right), computed using the HSE hybrid functional and averaged over 20 snapshots extracted from an initio MD trajectories. of 4.51 eV and 4.45 eV for the PhOH and PhO– solution, respectively, which is much smaller than the experimentally determined band gap of 8.7±0.5 eV. 28,29 Including a contribution of Hartree-Fock exchange in the density functional substantially improves the description of the relative positions of VB and CB with a band gap of 6.27 eV (PhOH) and 6.19 eV (PhO– ), as illustrated in Fig. 2. Both GGA and HSE calculations indicate the presence of at least two solute states within the band gap of liquid water. We note that compared to simple ions such as Cl− and OH− , 14,15 the aromatic PhOH and PhO– species are of considerable complexity. In particular an assignment of the solute states, which is a prerequisite to carrying out G0 W0 calculations, is significantly more difficult, due to a larger number of states and energy levels within the energy range of the water bands. In order to identify the electronic states of the solutes in solution we computed a highresolution projected DOS at the solute. Figure 3 shows the projected solute states for one snapshot computed in the GGA with a broadening factor of 0.005 Ry and the averaged total DOS with a broadening of 0.02 Ry. The black vertical lines in Fig. 3 correspond to the KS 7



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Figure 3: Average total density of states over 20 snapshots (dashed line) and rescaled highresolution projected density of states at the solute phenol (left) and phenolate (right) for one snapshot. energy levels of the solute in the aqueous solution. From the positions of the solute states we identified four KS electronic energy levels of PhOH and five energy levels of PhO– . In the case of PhOH the two highest occupied molecular orbitals (HOMOs) appear above the VBM of the water molecules. For PhO– , an additional occupied electronic energy level was identified, which appears as HOMO-1, resulting from deprotonation. Figure 3 illustrates the positions of two and three solute states at the top of the VB for PhOH and PhO– solutions, respectively. The positions of the lowest unoccupied molecular orbitals (LUMOs) in both the PhOH and PhO– solutions are generally located above the CBM of liquid water and in what follows we consider only the two LUMOs of PhOH and the LUMO of PhO– . Figure 4 and 5 show the KS energy levels of the solute and solvent states together with the peak positions of the high-resolution projected DOS of the simulated solutions. The energy levels of the solute states obtained from the high-resolution projected DOS are represented by the + symbols and illustrate the fluctuations of the solute energy levels along the chosen trajectory. Matching the electronic energies for each snapshot as obtained from the projected 8


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Figure 4: Energy levels of the Kohn-Sham orbitals for a phenol/water (left) and a phenolate/water (right) solution, computed using the GGA. The solute energy levels (orange) have been assigned using the projected density of states and the + symbols show the positions of maxima in the projected density of states at the solute. DOS (cf. Fig. 3) with the KS eigenvalues of the electrolyte solution also reveals that some solute states appear below the (HOMO-5) of the solution; the latter state was the lowest orbital considered in our MBPT calculations. Thus data points which refer to solute states energetically below the top five occupied states of the solution have been discarded in our subsequent analysis. Comparing the results for the GGA and hybrid functionals, we found a significantly increased gap between occupied and unoccupied energy levels for the latter. This effectively increases the energy difference between the HOMO of the solute and the CBM of the solvent by about 1.1 eV for PhOH and 1.0 eV for PhO– when using the HSE functional. The fluctuation (root-mean-square deviation, RMSD) of the energy levels along the trajectories is very similar for both functionals. The HOMO energy level of PhOH is subject to a RMSD of 0.16 and 0.17 eV for the PBE and HSE functional, respectively. For PhO– we found corresponding RMSD values of 0.18 and 0.20 eV. Assuming a Gaussian distribution, an estimated full width at half maximum (FWHM) of about 0.4-0.5 eV can be

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Figure 5: Energy levels of the Kohn-Sham orbitals for a phenol/water (left) and a phenolate/water (right) solution, computed with the HSE hybrid functional. The solute energy levels (orange) have been assigned from the projected density of states and the blue symbols show the positions of maxima in the projected density of states for the solute. estimated for the IEs of aqueous PhOH and PhO– , which is significantly smaller then the kinetic energy distribution of photoelectrons measured in experiments (FWHM≈ 1 eV). 10 The band edge of liquid water is subject to fluctuations of similar magnitude (0.13 eV for PBE and 0.16 eV for HSE). However the separation of the solute states from the VB of water is increased with HSE, with respect to GGA results. Similar observations were made in previous work on electrolyte solutions of simple ions such as hydroxide and chloride. 13,15,30 Figure 4 and 5 also illustrate an important difference between the PhOH and PhO– solutions. Within the thermal fluctuations along the MD trajectory, the PhOH molecules contribute two electronic energy levels just above the VB edge of liquid water. In the PhO– system, on the other hand, three solute levels exist between the VBM and CBM of water. This qualitative result is the same within the GGA and with hybrid functional calculations. The relevant molecular orbitals of the solute are presented in Table 1, where the electronic densities for the solutions were obtained from one snapshot and computed in the GGA.

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Orbitals of PhOH and PhO– in the gas-phase were computed using PBE calculations and a schematic illustration of their symmetry and nodal structure is presented in Table 1. The orbitals were labeled according to the irreducible representations of the molecular point groups Cs and C2v for PhOH and PhO– , respectively. Table 1 illustrates the close correspondence of the orbitals of PhOH and PhO– in solution and in the gas phase. In particular, we found that the solute-solvent interactions do not induce a delocalized charge-transfer state within the considered energy range. The availability of a localized empty orbital of π ∗ character represents a fundamental difference with respect to aqueous halide and hydroxide ions, and it is responsible for the presence of solvated electrons after photoexcitation. In summary, KS orbitals of the solute remain localized in aqueous solution and closely resemble the MOs of the isolated species. The most significant difference between the PhOH and PhO– solution is the HOMO-1 of PhO– which is not present in the water band gap of the PhOH system. Deprotonation lifts an additional energy level of 1a′ symmetry above the VBM of liquid water, giving rise to the 1b1 orbital of PhO– shown in the right column of Table 1. As a consequence, also the energy of the HOMO of the PhO– solute (1b2 ) is shifted to higher energy and effectively reduces the first IE of the solution by 0.80 eV in the GGA and 0.87 eV for the HSE hybrid functional. Liquid-microjet photoemission experiments indicate a shift in the lowest IE due to deprotonation of 0.7 eV 10 in remarkably good agreement with our calculations, considering the approximations involved in our ab initio simulations. The second IE computed in the GGA is rather unaffected by deprotonation although the character of the orbital where the electron is detached from changes from 1a′′ to 1b1 . This result is in excellent agreement with experiment which recorded the second IE in aqueous PhO– solution at 8.5 eV and for PhOH at 8.6 eV. Moreover, our calculations indicate that the second and third IE of a PhO– are very close in energy, which likely would appear as a single feature in the measured photoemission spectrum. Compared to the inclusion of some fraction of Hartree-Fock exchange in the density

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Figure 6: Quasi-particle energies of phenol/water (left) and phenolate/water (right) computed with MBPT and the G0 W0 approximation. The energy levels of the solute have been computed from the assigned electronic Kohn-Sham orbitals obtained in the GGA. functional, MBPT provides a higher level theoretical framework to describe the electronic self-energy and to alleviate the delocalization error which is present in DFT calculations. In this work we used the G0 W0 approximation with GGA orbitals as input. The G0 W0 scheme provides a first-order correction to the single particle energies, with no change to the single particle GGA wavefunctions. Hence our previous assignment of solute and solvent states from the projected DOS (Fig. 4) can be used in the case of G0 W0 calculations as well. Figure 6 shows the QP energy levels of 20 snapshots taken from the trajectories of the simulated PhOH and PhO– solutions. The energy gap between occupied and unoccupied electronic states in the solution is increased by 2.52 eV/1.52 eV for PhOH and 2.65 eV/1.63 eV for PhO– with respect to the results from GGA/hybrid functional calculations, respectively. The fluctuations of the solute energy levels and the water VBM computed with MBPT are typically increased in our calculations compared to KS DFT. We found RMSDs of 0.30 and 0.20 eV for the solute IE of PhOH and PhO– , respectively. The VBM fluctuations of bulk water are increased even further, to 0.30 eV for the PhOH solution and to 0.40 eV in the PhO– 13



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system. Likewise, the band gap of bulk water is much more strongly affected by the level of theory used and it increases for the PhOH system from 4.51 eV for the GGA and 6.27 eV for the hybrid functional to 8.1 eV if computed with MBPT. The PhO– system shows very similar behavior, i. e. a band gap of the bulk solvent of 4.45 eV (GGA), 6.19 eV (hybrid) and 8.2 eV (MBPT). This strongly supports our assumption that our model systems are sufficiently dilute and that the electronic structure of the solvent is mostly unaffected by the solute, irrespective of the presence of an excess charge in the simulation cell. Moreover, the band gap of liquid water computed with MBPT and the G0 W0 approximation is consistent with previous work which reported values of 8.4 eV, 31 8.7 eV, 32 8.1 eV 11 and the experimental band gap of 8.7±0.5 eV. 28,29 The experimental values for the two lowest IEs are 2.1 eV and 1.3 eV (PhOH) and 2.8 eV and 1.4 eV (PhO– ) above the photoionization threshold of liquid water. In our simulations, the difference between the ionization threshold (i. e. the VB edge) and the two lowest binding energies of the solute (the energy of the HOMO and HOMO-1) is underestimated by 0.3–0.4 eV in the PhOH solution. For the PhO– system, on the other hand, our simulations are in good agreement with the experimental values: We found IEs of 2.57 eV, 1.3 eV and 1.2 eV above the photoemission threshold of bulk water for the first, second and third IE, respectively. Since the second and third IE are almost degenerate, the detachment of an electron may occur either from the 1b1 or 1a2 orbital of PhO– . We therefore attribute the experimentally observed intensity pattern, 10 which shows, for the second IE, approximately twice the emission intensity than for the lowest IE, to photoemission processes from both the HOMO-1 and HOMO-2 of PhO– .

Molecular energy levels and band alignment Electro- and photochemical reactions generally depend on the relative positions of the energy levels involved in the process. In solution, these are usually the VBM and CBM of a condensed phase and the energy levels of the frontier orbitals (i. e. HOMO and LUMO)

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Table 2: Experimental 10,29 and computed energy levels of PhOH and positions of the VBM and CBM of water in a PhOH solution relative to the vacuum level. el. state LUMO+1 LUMO HOMO HOMO-1 CBM (H2 O) VBM (H2 O)

Exp. (eV)

E PBE (eV)

E HSE (eV)

E G0 W0 (eV)

E G0 W0 − E PBE (eV)

-7.8 -8.6 -1.2 -9.9

-1.18 -1.53 -5.72 -6.35 -1.91 -6.42

-0.73 -1.09 -6.43 -7.13 -1.52 -7.79

0.29 0.01 -7.06 -7.93 -0.73 -8.83

1.47 1.54 -1.34 -1.58 1.18 -2.41

Table 3: Experimental 10,29 and computed energy levels of PhO– and positions of the VBM and CBM of water in a PhO– solution relative to the vacuum level. el. state LUMO HOMO HOMO-1 HOMO-2 CBM (H2 O) VBM (H2 O)

Exp. (eV)

E PBE (eV)

E HSE (eV)

E G0 W0 (eV)

E G0 W0 − E PBE (eV)

-7.1 -8.5 -1.2 -9.9

-1.11 -4.92 -5.76 -6.05 -1.94 -6.39

-0.67 -5.56 -6.70 -6.96 -1.56 -7.75

0.44 -6.25 -7.52 -7.62 -0.62 -8.82

1.55 1.33 -1.76 -1.57 1.32 -2.43

of molecular species. Since only relative energy levels are accessible in both theory and experiments, alignment with respect to a standard reference level is required to compare different systems. The most important reference levels are the vacuum level and the standard hydrogen electrode (SHE) which are typically used in spectroscopy and electrochemical measurements, respectively. Both reference potentials are related by a shift of 4.44 V which is the experimentally determined potential of the SHE vs vacuum. 33 In computational studies involving periodic boundary conditions, the alignment of energy levels with respect to a reference potential is a challenging task. This applies in particular to dynamic and diffusing systems such as liquids, where the lack of a well-defined surface exacerbates the implementation of slab calculations to determine the vacuum potential. As an alternative, Costanzo and coworkers developed a “computational SHE” which circumvents the need for computationally expensive slab calculations. 16,18,34 The details of this methodology were extensively described in the literature 16,30,35,36 and here we limit our discussion to the basic principles 15



vacuum

0

-4

-6

2

-8

4

-10

6 PBE

G0 W0

HSE

EXP

8 vacuum

-12

0

-4

-2

-2

-4

0

-6

2

-8

4

U / V (vacuum)

U / V (SHE)

-4

0

U / V (vacuum)

-2

-2

U / V (SHE)

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-10

6 PBE

G0 W0

HSE

8

EXP

-12

Figure 7: Aligned electronic energy levels of a phenol/water (top) and phenolate/water (bottom) solution vs SHE and vacuum. of the alignment method used in our work. The potential of the SHE is defined by the reaction 1 − H+ (aq) + e −−→ H2(g) . 2

(1)

According to the fundamental laws of thermodynamics, reaction (1) can be decomposed into

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a sequence of reversible reaction steps 1 I III · + − II − H+ (aq) + e −−→ H(g) + e −−→ H(g) −−→ H2(g) 2

(2)

The reaction difficult to describe in periodic ab initio calculations is reaction step I which involves the reversible insertion of a proton into liquid water. Besides the technical difficulties associated to the insertion of an additional particle into a condensed phase, the proton work function computed within periodic boundary conditions is defined within an arbitrary additive constant. Such constant is precisely the same as that entering the definition of work functions of electrons (ionization energies) computed within periodic boundary conditions and the same cell. Reaction steps II and III are instead well defined and can be determined experimentally and by ab initio calculations without any ambiguity. Based on reaction (2) and the definition of the SHE potential is 33

UH◦ + /H2 (abs) =

1 ◦ , ∆at G◦ + ∆ion G◦ + αH + F

(3)

where F is the Faraday constant, ∆at G◦ denotes the molar dissociation energy of molecu◦ lar hydrogen, ∆ion G◦ the molar ionization energy of a hydrogen atom and αH + the molar

real potential of the proton in aqueous solution; computationally the potential of a SHE is ◦ obtained by substitution of the experimental work function αH + in definition (3) with the

computed work function. We previously computed the free energy of reaction step I by thermodynamic integration using the CP2K software in a periodic simulation cell. In order to avoid difficulties with the insertion of an unbound proton into bulk water, we attached a dummy atom to one of the water molecules which was gradually converted to a hydronium ion H3 O+ according to ∆A =

Z

1 0

∆EH3 O+

17

λ

dλ.


(4)


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In Eq. (4) the symbols ∆A and ∆EH3 O+ λ refer to the computed free energy and the ensemble-averaged energy difference between the protonated system and the uncharged dummy atom. With the resulting potential, adiabatic ionization energies computed with periodic boundary conditions can be aligned with respect to the SHE reference level according to the following equation:

U ◦ (she) =

AIP(pbc) − UH◦ + /H2 (pbc) F

(5)

where U ◦ (she) refers to the redox potential vs SHE, AIP represents the molar adiabatic ionization potential and UH◦ + /H (pbc) the potential of the computational SHE. The latter 2

can be determined by UH◦ + /H2 (pbc) =

g,◦ µH + − WH+ (pbc)

F

(6)

as the difference between the standard chemical potential of the proton in the gas phase g,◦ (µH + ) and the work function of the proton WH+ (pbc).

We found no significant finite size effects for simulation cells larger than 32 water molecules and negligible differences between GGA and hybrid functionals. 35 However, all these calculations were carried out with the CP2K software which uses different simulation parameters, in particular different basis sets, than the QE code used in this work. The use of different basis sets (PW in QE and mixed Gaussian plus PW in CP2K) and possibly different PPs leads to different zeros of total and free energies computed with the two codes. Hence, in order to properly align the energy values obtained with the two different software packages, we aligned the characteristic 1b1 peak in the DOS of liquid water computed in the two cases. In addition to the shift due to the different simulation parameters in CP2K and QE, a correction for the zero-point energy of the bound proton must be applied to the calculated work function to be consistent with Eq. 1. Including all these contributions we determined a potential of 0.41 V for the computational SHE in periodic boundary conditions, by substituting the computed proton work function, the experimental ionization energy of atomic hydrogen 18


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and the experimental dissociation energy of H2 in Eq. (3). We recently showed that the band alignment of liquid water obtained with such a computational SHE is in agreement with the well established approach based on slab calculations and the average electrostatic potential. 11,15 We therefore adopted the computational SHE to carry out band alignments in the present work. Figure 7 illustrates the aligned electronic energy levels of the simulated PhOH (top) and PhO– (bottom) solution computed at different levels of theory along with the experimental data. The discrepancy between experiment and our simulations decreases with higher level of theory for both systems. The aligned VBM, which defines the photoionization threshold of water in the solution, appears in both systems at about -6.4 V, -7.8 V and -8.8 V vs the vacuum level if computed with the GGA, the HSE hybrid functional and MBPT in the G0 W0 approximation, respectively. The CBM of the PhOH solution is shifted by 1.18 V to higher energy when the MBPT/G0 W0 correction is applied to the KS states in the GGA. Again we observed intermediate values for the energy levels in the HSE calculations between the GGA and the MBPT values. The energy levels of PhO– in solution are qualitatively distinct from those of PhOH. As discussed in the previous section, our calculations showed that there exist three occupied energy levels above the VBM of bulk water. The HOMO of PhO– is shifted to higher potential, effectively decreasing the computed first vertical IE to 4.93 eV in the GGA, 5.56 eV for the hybrid calculations and 6.25 eV for the MBPT calculations. For a PhO– solution, the second IE appears as a broad feature in the experimental photoemission spectrum, with approximately twice the intensity of the first IE. 10 Our computational results indicate that the second photoemission band in the experimental spectrum originates from two energetically close electronic states of solvated PhO– . The HOMO of PhOH and PhO– are of the same character but the insertion of the additional electronic level during deprotonation pushes the HOMO in PhO– to higher energy. Experimentally, the difference between the HOMO of PhOH and its anion has been determined as 0.7 eV. This is very close to the deprotonation shift of the hydroxyl ion, i. e. the shift of the OH− HOMO above the VBM

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of liquid water. Table 2 and 3 display numerical values for the aligned energy levels with respect to the vacuum level.

Summary and conclusions The electronic structure of two 0.75 mol/l PhOH and PhO– solutions have been computed from first principles using DFT and MBPT. Ab initio MD trajectories computed in the GGA were used to sample the PE surface and 20 instantaneous geometries were extracted for each solution to carry out electronic structure calculations. In our analysis we considered 11 electronic states, including the six highest occupied and five lowest unoccupied states. In order to identify localized solute states within the electronic band structure of the solution, we developed an assignment scheme based on the projected electronic DOS at the solute molecules. Our assignment technique is generally applicable to solvated species which exhibit localized and energetically separated KS orbitals. Within the considered range of eleven states, we identified four electronic energy levels which are present in almost all of the analyzed snapshots. In the PhOH solution these four states were assigned to two occupied and two unoccupied energy levels, while we found three occupied and one unoccupied level in the PhO– solution. The KS orbitals of the solvated species resemble the MOs in the gas phase and illustrate the effect of deprotonation on the electronic band structure which is analogous to the formation of a hydroxide ion in liquid water. Similar to the deprotonation of a water molecule, which induces a localized electronic energy level about 0.7 eV above the VBM, 15,37,38 the removal of the proton in the hydroxyl group of PhOH was found to induce an additional solute state about 1.3 eV higher than the VBM of bulk water (computed with MBPT). The fundamental gap of bulk water is essentially unaffected by the solute and the excess charge of the PhO– anion. Using different levels of theory including the GGA, hybrid-DFT and MBPT, we found increasingly better agreement with experiment. The most accurate data, obtained from

20


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MBPT, are 8.1 and 8.2 eV for the band gap of water in the presence of PhOH and PhO– , respectively. An implementation of a computational SHE was used to align the electronic molecular energy levels and the VBM and CBM of bulk water. With the experimental absolute potential of the SHE, the aligned bands can be directly compared to experimental data obtained from electrochemical and spectroscopic measurements. The lowest two vertical IEs of PhOH in aqueous solution, as computed with MBPT in the G0 W0 approximation, are 7.06 eV and 7.93 eV. A recent experimental study reported values of 7.8 eV and 8.6 eV, about 0.7–0.8 eV higher than those found in our simulations. 10 This discrepancy may stem from the error in the description of the VB of liquid water when using G0 W0 with GGA wavefunctions as input; 11,15 the difference between the first and second IE was found to be in good agreement with experiment. According to our simulations, the PhO– solution, which can be considered as an infinitely dilute system without counter ions, contains three electronic energy levels above the VBM of water. Our most accurate calculations using MBPT show average vertical IEs from solute states of 6.25 eV, 7.52 eV and 7.62 eV. These results deviate from experiment which reported 7.1 eV and 8.5 eV for the first and second IEs. 10 We note that our results are qualitatively different from those of previous work, which assumed only two IEs above the photoionization threshold of water. The second and third IE occur from photoionization of two almost degenerate solute states and such a degeneracy explains the approximately double intensity of the corresponding peak in the experimental photoemission spectrum.

Acknowledgement 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.

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Electronic Energy Levels and Band Alignment for Aqueous Phenol

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