On the Position of the Highest OccupiedMolecular Orbital in Aqueous Solutions ofSimple IonsPatricia Hunt[a] and Michiel Sprik*[a]

1. Introduction

Aqueous chemistry is an almost ideal subject for density func-tional ab initio molecular dynamics (“Car–Parrinello”)[1] simula-tion. Perhaps in no other area of condensed-phase chemistrythe coupling between electronic states and dynamical finitetemperature fluctuations is as important as for reactions be-tween aqueous solutes. However, the application of Car–Parri-nello molecular dynamics (MD) to aqueous systems had towait for the development of gradient-corrected density func-tionals capable of describing hydrogen bonding.[2,3] Once ithad been established that structure and dynamics of pureliquid water could be described with reasonable accuracy[4] (foran up-to-date assessment see refs. [5, 6]), the method was soonapplied to the small aqua ions. Limiting the examples to ionssimilar to the systems discussed here we mention Car–Parrinel-lo (CP) simulations of solvation of alkali cations (Li+ ,[7] Na+ ,[8]

K+ [9]) and halide anions (Cl� ,[10,11] Br�[12]). These studies havebeen extended to group-2 and 3 cations carrying multiplecharges (Be2+ ,[13] Mg2+ ,[14] Ca2+ ,[15] Al3+ [11]) and even transitionmetal aqua ions in variable oxidation states (Ag2+/Ag+ ,[8, 16, 17]

Cu2+/Cu+ ,[16–17] Ru2+/Ru3+ [19]).A frequently recurring question, particularly relevant for the

understanding of chemical reactivity and electronic absorptionspectra,[17,20] concerns the position of the highest occupied mo-lecular orbital (HOMO) of the solute and the correspondinglowest unoccupied orbital (LUMO). Many of these phenomenacan be explained using (self-consistent) perturbation methodswhich take only relative values of molecular energy levels intoaccount and the absolute position plays no role. The referenceenergy of one-electron levels is however crucial for electronicprocesses involving ionization (vertical or adiabatic). The abso-lute position of a level is measured with respect to vacuumand can be determined by photoelectron spectroscopy experi-ments. Alternatively it can be deduced via more indirect elec-trochemical methods such as injection of electrons from a suit-able electrode. For recent examples of such experiments forpure liquid water we refer to refs. [21,22] . All these experi-ments take place in inhomogeneous systems consisting of two

bulk regions separated by an interface. This is a most seriouscomplication for plane wave basis set schemes such as theCar–Parrinello method which inherently replicate a system inall three directions. The problem can be resolved, in principle,using a (large) supercell with a slab geometry.[23] In homogene-ous systems under periodic boundary conditions, however, thereference of the Kohn–Sham (KS) energies is completely artifi-cial and has no physical meaning.

For periodic systems consisting of supercells containing asmall set of molecules confined in space, there is no suchproblem. As long as the periodic cell is sufficiently large com-pared to the dimension of the molecule or cluster the spuriousinteractions between periodic images can be eliminated usingspecial screening methods.[24] This suggests to determine theposition of the valence band edge of the liquid by comparingthe energy levels of increasingly larger water clusters invacuum. An X-ray absorption spectroscopy experiment onthese lines is reported in ref. [25]. Here, rather than pure waterclusters, we will focus on microsolvated simple ions, such asalkali or halide ions. Single ion–water clusters are particularlysuitable for our purpose as it has been established by experi-ment and computational (force field) model studies, that hy-dration of these ions in vacuum leads to formation of compactand stable clusters. However, the energy levels of microsolvat-ed ions depend on the ionic species, in particular on thecharge. This raises the question whether these differences arecarried over to solution, and, hence, to what extent the HOMOof a solution differs from the HOMO of the pure solvent. Forexample, one could intuitively expect that in limit of low con-centration the HOMO of the solution is determined by the sol-

The energies of the highest occupied molecular orbital (HOMO)of four simple microsolvated aqua ion clusters (Na+ , Ag+ , Cl� ,CN�) are computed for varying numbers of water molecules. Ex-trapolating to infinite hydration numbers we find that these ener-gies approach a value of �6 eV. This limiting one-electron energyis within a margin of �1 eV independent of the character of the

ion and is 4 eV lower compared to the estimate obtained for theHOMO energy of the ions in aqueous solution under periodicboundary conditions. We argue that this discrepancy must the at-tributed to a shift in the reference of the one-electron potentialof the periodic solvent model.

[a] P. Hunt,+ Dr. M. SprikDepartment of Chemistry, University of CambridgeCambridge CB2 1EW (UK)Fax: (+44)1223-336362E-mail : ms284@cam.ac.uk

[+] Current address:Department of Chemistry, Imperial College, London (UK)

ChemPhysChem 2005, 6, 1805 – 1808 DOI: 10.1002/cphc.200500006 C 2005 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim 1805

vent, unless the solute inserts a level in the solvent energygap.

These and other considerations were behind the particularchoice of model systems we have studied, which consisted, oftwo aqueous cations, Na+ , Ag+ and two anions, Cl� and CN� .Na+ and Cl� are generally considered representative of hardaqua ions and Ag+ and CN� of soft ions. Anticipating our re-sults we find that extrapolation to infinite cluster size indeedgives a rough estimate of the elusive value of the upper edgeof the valence band in liquid water, which is rather differentfrom the value obtained for periodic models of the liquid.

2. Single Ions in Periodic Cells

In molecular dynamics simulation the electrostatic interactionenergy of periodically replicated charge distributions is usuallycomputed using reciprocal space methods. The best exampleof this type of calculations is the Ewald summation method forthe evaluation of the lattice energy of a periodic array of pointcharges.[26,27] Infinite lattice sums of Coulomb interactions areonly conditionally convergent and the Ewald sum representsone special recipe for summation convenient for solid statesystems. The average electrostatic potential f(r) over a unitcell in this scheme vanishes as shown in Equation (1):[26]

Zcelldr�ðrÞ ¼ 0 ð1Þ

Equation (1) has a number of implications. It enables us tomodel the solvation of single ions as mentioned in the Intro-duction. The net charge of the corresponding supercell is auto-matically neutralized by a homogeneous charge distributionplaying the role of counterion. The question that concerns ushere is not the effect on the total energies, which of coursecan be significant,[26,27] but the effect on the one-electron KSenergies. As will be clear from Equation (1) the levels in a peri-odic cell are offset by a uniform shift v0. For finite systems(molecules, clusters) in an otherwise empty periodic box ofsize L this bias vanishes for sufficiently large values of L. In ex-tended condensed systems v0 converges to a finite value inthe limit of large supercells.

It is not difficult to obtain a rough estimate of v0 for solidsconsisting of apolar closed shell molecules. Taking solid argonas example v0 is approximately equal to the average of the po-tential of the isolated atom over a volume equal to thevolume per atom in the solid. This number is not at all smalland depending on the pseudo-potential used it can be severaleV. For polar systems such as water this simple scheme to esti-mate v0 cannot be applied. A general method for computingv0 for periodic systems is, to our knowledge still wanting. Infact it is not clear whether such a method exists even in princi-ple.

Computational Section

As explained in the Introduction the calculation presented here isan attempt to estimate the reference of the energy levels in the

liquid from the change in the HOMO energy with the number ofH2O molecules n hydrating a single ion in vacuum. The size L ofthe cubic cell is kept fixed. The electronic structure is computedusing the same plane wave pseudo-potential apparatus as used forsimulation of the liquid, augmented with a special method forscreening interactions between periodic images.[24] The referenceof the KS potentials in these methods is consistent with a vacuumcalculation with the one-electron potential vanishing at infinity.This has been verified by numerous calculations carried out, for ex-ample, for the parametrization and validation of pseudo-potentialsfor use with the CPMD package,[28] which is also the electronicstructure code employed herein.The model systems we have studied consist of a series microsolvat-ed Na+ , Ag+ , Cl�and CN� ions hydrated by n=1,…10 water mole-cules in a cubic cell of size L=12 K. The energy levels were com-puted for stable (zero temperature) structures generated by a com-bination of annealing and geometry optimization, except for thelargest cluster (n=10) for which the KS levels have been obtainedby averaging over a low temperature (T=20 K) MD run of �5 psduration. These energies will be compared to the time averagedHOMO of model solutions consisting of a single solute ion and nwater molecules in a periodic cubic box of length L=9.85 K. Inorder to adjust the pressure to a uniform value for all four systemsthe number of water molecules varied from n=33, 32, 31, 28 forthe solvated Na+ respectively Ag+ , CN� and Cl� systems. The den-sity functional used is BLYP.[2,3] For O, N, Cl and H we have used thestandard norm-conserving Troullier–Martins pseudo-potentials[29]

validated in numerous CPMD calculations. Na+ and Ag+ ions weretreated according to the same Troullier–Martins scheme. The pa-rameters are given in ref. [8] . The reciprocal space cutoff for theplane wave basis set for the orbitals is again 70 Ry.

3. Results and Discussion

The HOMO energies of the clusters are shown in Figure 1 plot-ted as a function of n�1/3. By representing the size of the clus-ters in terms of this function of the number of water moleculesn we assume that the energy levels scale proportional to theinverse radius of the cluster as is suggested by the Born ex-pression for solvation energy.[27] As a first observation we notethat the energies for the two cations are very similar as are theenergies for the anions. Cation and anion show an opposite re-

Figure 1. Variation of the HOMO energy of microsolvated Na+ (*), Ag+ (*),Cl� (&) and CN� (&) clusters with the number n of water molecules. Thedashed lines represent linear fits extrapolated to a infinite cluster size(n�1/3=0). The minimum in the CN� series is due to the exceptional stabilityof the n=5 cluster (magic number effect).

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M. Sprik and P. Hunt

sponse to addition of more H2O molecules. The increase of theHOMO level of Na+ and Ag+ can be explained by the prefer-ence of cations to coordinate with the electron-rich O atoms.While lowering the total energy, it leads to increased repulsionbetween the electrons of solute and solvent. Similarly theenergy levels of an anion complex are pulled down by theelectrostatic field set up by the solvation. Now the H atomspoint towards the ion optimizing interaction with a negativelycharged solute. The result is stabilization of the one-electronorbitals of anions.

The narrowing of the gap between HOMO levels of the clus-ters in Figure 1 is therefore a direct reflection of the inversionof the orientation of the H2O molecules making up the hydra-tion shells of cations compared to anions. However, as can beseen in Figure 1 the gap effectively closes in the limit of bulksolvation. The four lines all intersect near n=1 at a commonlimiting energy e1F =�6�1 eV (see also Table 1). Moreover, the

discrepancy in convergence is consistent with the small varia-tion in HOMO energies observed for (low concentration) ofbulk solutions under periodic boundary conditions (Table 1).Returning to the question posed in the Introduction, these re-sults confirm that the HOMO level in a low-concentration solu-tion is, in first approximation, determined by the bulk solvent.

These observations are supported by closer analysis of thecharacter of the highest occupied orbitals. Graphical inspectionreveals that these orbitals have predominantly solvent charac-ter comparable to the 1b1 states at the band edge of the puresolvent. Using localization techniques similar to ref. [30] it canbe demonstrated that the HOMO of the Cl� and CN� anionsmerge with the 1b1 band of the solvent and are thus part ofthe HOMO manifold. The levels of the Na+ appear, as expect-ed, way below the solvent HOMO band. The exception is theAg+ aqua ion, which, produces a HOMO in the solvent gap ap-proximately 0.5 eV above the top of the solvent band.[16,17] Thisstate is entirely localized on the metal ion and its four ligandH2O molecules.[16,17] The overall conclusion is, however, thatthe HOMO energy of the model solutions all fall in narrowenergy interval epbc

F =�2�1 eV. This interval is moreover cen-tered around the valence band edge of the pure solvent.[31]

As anticipated by the argument in Section 2 the apparentHOMO level under periodic boundary conditions epbc

F is consid-erably higher compared to the extrapolated cluster result e1Fdiffering by �4 eV. While this 4 eV upward shift is an artefactand has no physical meaning, our results indicate that it is

largely determined by the extended solvent, justifying compar-isons of densities of states of single ions in periodic solventsamples as small as 32 water molecules. Furthermore, takingthe value e1F =�6�1 eV obtained by extrapolation of the clus-ter energies as the estimate of the location of the valenceband edge of liquid water as predicted by BLYP, we find a fur-ther 4 eV discrepancy with the eexp

F =�10 eV which is the cur-rently agreed experimental value.[21,22] This difference must beinterpreted as a generalized gradient approximation (GGA) re-lated error, although also more fundamental theoretical issuesconcerning the experimental interpretation of Kohn–Shamlevels in terms of an (vertical) ionization energy of an extendedsystem may also play a role.

Acknowledgements

We thank Joost VandeVondele and Jochen Blumberger for manyhelpful discussions about this somewhat controversial subject.P.H. is grateful to the Leverhulme Trust for financial support inthe form of a postdoctoral fellowship.

Keywords: cluster compounds · density functionalcalculations · energy levels · solvation

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7070.

Table 1. Energy [eV] of the highest occupied molecular orbital (Fermienergy) for aqueous solutions of four simple ions. e1F is the estimate ob-tained from the cluster data of Figure 1 by extrapolation to an infinitenumber of water molecules (n!1). epbcF is the HOMO computed for amodel solution of a single ion in a periodic cubic cell with solvent (n�30).

Na+ Ag+ Cl� CN�

e1F �5.7 �5.4 �6.6 �7.1epbcF �2.0 �1.5 �2.4 �2.1

ChemPhysChem 2005, 6, 1805 – 1808 www.chemphyschem.org C 2005 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim 1807

HOMO of Simple Ion–Water Clusters

[28] CPMD Version 3.x, The CPMD consortium, http://www.cpmd.org, MPIfRr Festkçrperforschung and the IBM Zurich Research Laboratory.

[29] N. Troullie, J. Martins, Phys. Rev. B 1991, 43, 1993.[30] P. Hunt, M. Sprik, R. Vuilleumier, Chem. Phys. Lett. 2003, 376, 68.[31] We have verified that the density of occupied states of the pure liquid

water is essentially invariant under an increase of the number of water

molecules by comparing the results of solvent systems consisting of upto 1024 molecules (VandeVondele, in preparation).

Received: January 4, 2005Revised: May 18, 2005Published online on August 1, 2005

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M. Sprik and P. Hunt