Density functional calculation of the electronic ...

Density functional calculation of the electronic absorption spectrum of Cu+ and Ag+ aqua ions Leonardo Bernasconi, Jochen Blumberger, Michiel Sprik, and Rodolphe Vuilleumier Citation: J. Chem. Phys. 121, 11885 (2004); doi: 10.1063/1.1818676 View online: http://dx.doi.org/10.1063/1.1818676 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v121/i23 Published by the AIP Publishing LLC.

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JOURNAL OF CHEMICAL PHYSICS

VOLUME 121, NUMBER 23

15 DECEMBER 2004

Density functional calculation of the electronic absorption spectrum of Cu¿ and Ag¿ aqua ions Leonardo Bernasconi, Jochen Blumberger, and Michiel Sprik Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Rodolphe Vuilleumier Laboratoire de Physique Theórique des Liquides, Universite´ P. et M. Curie, 4 Place Jussieu, 75005 Paris, France

共Received 12 August 2004; accepted 23 September 2004兲 The UV absorption of aqueous Cu⫹ and Ag⫹ has been studied using Time Dependent Density Functional Theory 共TDDFT兲 response techniques. The TDDFT electronic spectrum was computed from finite temperature dynamical trajectories in solution generated using the Density Functional Theory 共DFT兲 based Ab Initio Molecular Dynamics 共AIMD兲 method. The absorption of the two ions is shown to arise from similar excitation mechanisms, namely transitions from d orbitals localized on the metal center to a rather delocalized state originating from hybridization of the metal s orbital to the conduction band edge of the solvent. The ions differ in the way the spectral profile builds up as a consequence of solvent thermal motion. The Cu⫹ absorption is widely modulated, both in transition energies and intensities by fluctuations in the coordination environment which is characterized by the formation of strong coordination bonds to two water molecules in an approximately linear geometry. Though, on average, absorption intensities are typical of symmetry forbidden transitions of metal ions in the solid state, occasionally very short 共⬍100 fs兲 bursts in intensity are observed, associated with anomalous Cu–H interactions. Absorption by the Ag⫹ complex is in comparison relatively stable in time, and can be interpreted in terms of the energy splitting of the metal 4d manifold in an average crystal field corresponding to a fourfold coordination in a distorted tetrahedral arrangement. Whereas the spectral profile of the Ag⫹ aqua ion is in good agreement with experiment, the overall position of the band is underestimated by 2 eV in the BLYP approximation to DFT. The discrepancy with experiment is reduced to 1.3 eV when a hybrid functional 共PBE0兲 is used. The remaining inaccuracy of TDDFT in this situation is related to the delocalized character of the target state in d→s transitions. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1818676兴

I. INTRODUCTION

The merits and limitations of approximate schemes for electronic structure calculations only become clear when these techniques are applied to a variety of different physical regimes. Time dependent density functional theory1 based methods for the computation of electronic absorption spectra2,3 are no exception. Accurate calculations of optical transition energies and intensities between valence states of molecules can be achieved at negligible computational cost in TDDFT compared to other techniques, such as quantum many-body approaches.4 These savings are largely due to the use of relatively simple adiabatic 共frequency independent兲 kernels derived from LDA or GGA ground state functionals. This approximation is usually referred to in the literature by the acronym ALDA.3 Its shortcomings become apparent when excitations involve rearrangement of the electron charge distribution over large spatial regions. Well known examples are the excitations from a compact valence state to a Rydberg state which is substantially more spread out in space.5,6 Similarly, GGA extensions of ALDA-TDDFT achieve but little improvement over Kohn–Sham 共KS兲 eigenvalue energy differences when applied to condensed-phase systems.4 This is also true for the energy of excitations trans0021-9606/2004/121(23)/11885/15/$22.00

ferring charge between valence states localized in different regions of space, as was recently pointed out.7–10 Molecular solids and liquids at finite temperature are of particular interest in this context. Owing to the relatively weak nonbonding nature of intermolecular interactions, and the related inherent disorder in molecular configurations, the electronic structure of extended molecular systems consists of a mixture of localized and delocalized states. As a result, the full variety of excitations mentioned above can occur. Our recent study of the electronic absorption of aqueous acetone11 provides a good illustration of this point. TDDFT proved to be able to yield an excellent estimate of the absolute value of the intramolecular n→ ␲ * transition in solution and of its solvatochromic shift. Analysis of the optically active KS states confirmed that the n→ ␲ * transition essentially retains the character of valence excitation in solution, thus supporting the model of a predominantly electrostatic coupling between the solvated molecule and the solvent 共as implicit, e.g., in the QM/MM approach described in Ref. 12兲. Our calculation, treating solute and solvent at strictly the same level of theory 关BLYP 共Refs. 13, 14兲兴, did however also expose several intrinsic limitations of this approach. The n→ ␲ * band was obscured by a series of intense spurious

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n→solvent and solvent→␲* charge transfer 共CT兲 transitions. Similarly, the onset of solvent-to-solvent excitations was found to be severely underestimated, as compared to experimental data. As a consequence, the high frequency end of the multiplet of transitions 共false and real兲 between solute and solvent and the solvent-to-solvent excitation region turned out to overlap in the calculated spectrum. A procedure for assigning and isolating the n→ ␲ * band of interest from the latter unphysical spectral features was devised. This procedure was verified and supported by a subsequent calculation15 employing hybrid exchange-correlation functionals 关B3LYP 共Ref. 16兲, PBE0 共Ref. 17兲兴. In accordance with the prediction of Ref. 8 we found that exact exchange shifts the CT transitions to higher frequencies, separating them from the carbonyl valence excitation. The present work reports on a similar computational investigation of the electronic absorption spectrum of another type of model solute, namely the d 10 transition metal aqua ions Ag⫹ and Cu⫹ . The excitation responsible for the absorption is a d→s transition. As was the case for the n orbital of acetone, the occupied d states are localized on the metal ion and the H2 O molecules directly coordinated to it.18 A further parallel to n→ ␲ * is that the 共pure兲 d→s transition is symmetry forbidden, though a gain in intensity may be induced by thermal fluctuations and interaction with the solvent. However, while the ␲* of acetone is a compact valence state, the empty s orbital of free group IB monocations has the character of a Rydberg state. Its spatial extension in vacuo is sufficiently large to reach well beyond the first solvation shell. An important question is therefore how such a state interacts with the conduction band of the solvent. As discussed in a previous publication on the redox chemistry of Ag⫹ and Cu⫹ ions,18 the answer obtained from the ab initio MD simulation is that the metal s state is incorporated in the band of delocalized empty states of the solvent and effectively spread out over the metal center and all solvent molecules. The energy of this collective state was found to be, compared to occupied d levels, rather insensitive to solvent fluctuations 共an account of what happens if the s level is filled with one electron to form a neutral silver atom can be found in Ref. 19兲. As a result, the s level can be regarded as a reference for the solvent modulation of energy of the d states, which greatly simplifies the correlation of the shape of the d→s absorption band to structural features of the coordination environment in the liquid. This type of analysis is familiar from the interpretation of the optical response of open d shell transition metal complexes, where it yields direct information on the crystal field splitting between d orbitals. A similar analysis led experimentalists to the conclusion that the Ag⫹ aqua ion has on average a tetrahedral coordination.20 Exploiting the microscopic data available from the simulation we prove in this paper that this spectroscopic structure must be refined to a dihedrally distorted tetrahedral coordination shell. However, as also indicated in Ref. 18, such an interpretation in terms of an average crystal field fails for the UV spectrum of the Cu⫹ ion which is dominated by rare but intense absorption events. This observation will be substantiated in the present paper by a more detailed analysis. This assignment of the absorption

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profile of aqueous Ag⫹ and Cu⫹ occupies the main part of the present paper. In addition, after having established an understanding of the electronic structure behind the optical spectrum, we return to the question of the nature of the empty s state and use the experimental UV spectra to obtain an estimate of the location of the edge of empty states in water with respect to the water valence band. The paper is organized as follows. After a review of some of the relevant aspects of Ag⫹ and Cu⫹ coordination chemistry 共Sec. II兲 and a short overview of recent developments in TDDFT 共Sec. III兲 we outline in Sec. IV the ab initio MD implementation used in this work and give further practical details of our calculations. In Sec. V the main results regarding the electronic structure of the Cu⫹ and Ag⫹ solutions are presented, together with a description of the computed absorption spectra. The interpretation of line shape in terms of cation solvation environment is the subject of Sec. VI while in Sec. VII we comment on our result for the average position of the absorption band. These findings are briefly discussed and summarized in Sec. VIII. The Appendix contains details regarding the calculation of multipole moments and charge density deformation effects on the basis of a localized Wannier function analysis of the ground state wave function, as used in Sec. VI. II. BACKGROUND ON Ag„I… AND Cu„I… COORDINATION CHEMISTRY

For a better appreciation of the structural and dynamical properties of the Cu共I兲 and Ag共I兲 aqua ions, some of which are rather peculiar for closed shell species, we start with a brief review of some of the relevant background information on the chemistry of these ions.21 The monocations will be designated by M ⫹ or M (I), M ⫽Ag, Cu. Their characteristic behavior arises from the elements being located at the very end of the transition metal series. The closed d shell guarantees that only relatively weak interactions are established with the ligands. The presence of nd levels at suitable energy with respect to (n⫹1)(s⫹ p) orbitals allows for effective s – d orbital mixing. In the solid state this leads to a remarkable and unique coordination chemistry. For example, in Ag共I兲 and Cu共I兲 chalcogenides and halides the energy barriers separating two, three, and four coordination are sufficiently low to allow for coexistence of competitive coordination sites22 and facile onset of ionic23 and superionic24 conduction. Similar structural features are observed in Cu共I兲 sulfides,25–31 whereas the linear two coordinate structure is dominant in Cu共I兲 oxides.32 Two, three, and four coordination geometries are also found in Ag共I兲 chemistry.33 All these geometries can be rationalized in terms of distortions of the ideal tetrahedral environment with the cation in the center 共resulting in fourfold coordination兲 or rather offset toward one of the faces or edges of the tetrahedron 共threefold and twofold coordination, respectively兲.34 This extraordinary variety of coordination motives should be contrasted with the rather inconspicuous structural solid state chemistry of the isoelectronic Zn共II兲, which is almost completely dominated by fourfold coordination.



The accepted explanation in inorganic chemistry for this structural flexibility is that appropriate changes in the ideal tetrahedral coordination maximize the degree of HOMOLUMO mixing by decreasing the energy gap between one nd orbital and the empty (n⫹1)s orbital with respect to the ideal t 2 manifold.34 The balance between the tetrahedral distortion and the gain in energy originating from the s – d mixing, usually referred to as second-order 共or pseudo兲 JahnTeller effect,35–37 explains the lack of preferential coordination sites for the cations and their large mobility at finite temperature. In aqueous solution the same effects lead to the highly fluxional hydration shells. Differences between first and second row Cu共I兲 and Ag共I兲 are even more noticeable than in the solid state.21 Ag⫹ is a stable if poorly soluble ion. A coordination number of 4 with a slightly distorted tetrahedral geometry has been established for hydrated Ag⫹ . 20,38,39 Cu共I兲 is very unstable with respect to disproportionation, 2Cu⫹ Cu2⫹ ⫹Cu(s), and can only be produced in vanishingly small concentrations. The short life of Cu⫹ in solution has so far hampered a direct determination of its solvation structure. The effects of the s – d hybridization are also reflected in gas-phase hydration energies in the form of a marked discontinuity in the stepwise formation constant of Cu⫹ (H2 O) n complexes when n is increased from two to three.40– 43 A qualitatively similar decrease in the differential hydration energy is observed for Ag⫹ (H2 O) n , but the size of the discontinuity is much smaller compared to gas-phase copper hydrates. III. TDDFT FOR CONDENSED MOLECULAR SYSTEMS

In the Introduction we made a very brief reference to some of the difficulties encountered in TDDFT. In this section we will expand on these remarks in order to place our calculations in the framework of recent developments in the field. A crucial step was the recognition that in response calculations the one-electron 共KS兲 potential, and not the response kernel, is often the major source of error. The KS potential has a direct effect on excitation energies as it determines the energy difference between empty and occupied KS orbitals. The discrepancies in Rydberg excitations are a good example. As it was realized in the context of static response calculations,44 LDA and GGA potentials fail to reproduce the correct 1/r long range behavior.44 As shown by a number of studies, virtually exact values for Rydberg energies can be obtained by restoring the 1/r tail by appropriate asymptotic corrections.5,6 The same effect can be achieved by hybrid functionals such as B3LYP or PBE0. The improvements are however not as impressive in the latter case, unless the fraction of exact exchange is allowed to approach 100% beyond a certain radial distance.45 Asymptotic corrections not only affect high-energy excitations, but also shift the energy of the HOMO to lower values, bringing them in better agreement with the ionization potential. From these and other observations the Amsterdam group of Baerends,46,47 concluded that improvement in the shape of the KS potential should be a priority in the development of time dependent methods in DFT.

Electronic absorption aqueous Cu⫹ and Ag⫹

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A solution to the problem of underestimation of CT excitation energies,7,11 however, must involve the time dependent response at a fundamental level. This was convincingly demonstrated in recent contributions by Dreuw et al.,7,8 Tozer,9 and Gritsenko and Baerends.10 These authors pointed out that for large spatial separation between donor and acceptor the cost of transferring an electron approaches the difference in the ionization potential of the donor and the electron affinity of the acceptor. There are fundamental reasons why this redox energy should substantially exceed the HOMO-LUMO energy difference even in the limit of the elusive exact energy functional. This is in contrast to Hartree–Fock where the redox energy and the orbital energy gap are similar 共in fact identical according to Koopman’s theorem兲. This important point is clearly explained in Ref. 10. The energy deficit in TDDFT must be recovered through the kernel which, as a result, must contain a discontinuity to counter the vanishing of the overlap between donor and acceptor orbitals. Gritsenko and Baerends10 made a proposal for an asymptotic correction to the kernel meeting these requirements. The approach to the CT problem followed by Dreuw et al. in their latest work on photo-induced electron transfer in biomolecules8 is rather different. For large donor– acceptor separation they essentially return to wave function based methods using CIS 共corrected with dynamical correlation derived from DFT兲 while at shorter distances, where donor and acceptor levels overlap, they switch back to regular TDDFT. Their earlier B3LYP based work7 can be considered a precursor of the new CIS/TDDFT hybrid approach, because the exact exchange part of B3LYP introduces a CIStype term in the time dependent response, with a weight determined by the fraction of exact exchange in the hybrid functional. Our calculations of Ref. 15 applied this older scheme. Turning now to the band gap problem in condensed molecular systems, LDA is known to underestimate the optical gap in semiconductors of up to 40%, and little improvement is achieved through the introduction of nonlocal gradient corrections in the exchange-correlation functional. The vanishing contribution of the response term is usually blamed on the absence of a long range (⬃1/q 2 ) term from the exchange-correlation kernel responsible for the attraction between 共screened兲 particles and holes.4 It is however not clear how relevant this observation is for excitations in molecular crystals. The origin of the emphasis on the long range behavior of the kernel for semiconductors lies in the delocalized nature of excitons in these systems. Particle–hole pairs can be separated by large distances and are, in this respect, similar to CT excitations. However, excitations in molecular crystals are localized 共Frenkel excitons兲 and have predominant single molecule character with only a minor CT component. Recalling the success of suitably adjusted oneelectron potentials for high energy excitations of molecules in vacuo, one would therefore expect that opening of KS energy gaps could also bring significant progress in the case of condensed molecular systems. This probably also holds true for band gaps in semiconductors, because shortcomings in the kernel are most likely not the main reason why ALDA


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band gaps are too small. This line of thought is supported by the results of exact exchange calculations in the OEP implementation giving KS gaps in reasonable agreement with experiment,48,49 even if interactions between excitons are ignored. All of these considerations are helpful to understand and interpret the results of our calculations of the spectrum of aqueous acetone11,15 and we will come back to them in the discussion of the results of the present study of the optical response of Ag⫹ and Cu⫹ in Sec. VII. As emphasized in Ref. 11, it is important to distinguish between the carbonyl n→solvent and solvent→␲* transitions. The latter are genuine CT transitions between an occupied state localized on one or a small number of solvent molecules and an empty state localized on the solute 共the ␲* orbital兲. These occupied states consist of n orbitals of water molecules and are similar to the n orbital of the solute. The arguments of Refs. 7–10 concerning the intrinsic difficulties of TDDFT in describing CT transitions apply to these solvent→␲* excitations, and explain why their energies end up too low in our calculation. Indeed, in accordance with the predictions of Ref. 7, when a functional with a fraction of exact exchange 共B3LYP,PBE0兲 is used, the solvent→␲* transitions are shifted to higher frequencies, while the solute n→ ␲ * band is largely unaffected.15 The n→solvent excitations are equally found to move toward higher energies. These transitions have delocalized states in the conduction band as target. CT effects probably play only a secondary role in this situation, and the excitation mechanism is more likely to resemble a valence→conduction band transition in a truly periodic condensed system. IV. AIMD AND MODEL SYSTEM

The TDDFT computations make use of the same methodology as applied in our previous studies.11,15 The method is based on a novel implementation of TDDFT within the plane wave pseudopotential framework developed by Hutter,50 augmented by a scheme for computing transition intensities in condensed phase systems within periodic Born–von Karman boundary conditions. Rather than the usual current representation, the coupling to the radiation probe is treated in the dipole representation, which seems more natural for molecular systems. The dipole representation is adapted to periodic boundary conditions by a timedependent extension of the Berry phase formulation of the bulk theory of polarization.51–53 共for a related approach to time variations of polarization, see Ref. 54兲. Details regarding the calculation of transition energies and intensities can be found in Refs. 50 and 11, respectively. Kohn–Sham DFT calculations were performed within the BLYP 共Refs. 13, 14兲 approximation to the exchangecorrelation energy. Norm-conserving fully separable55 ab initio pseudopotentials of the Troullier–Martins-type56 were employed, with s and p nonlocal projectors for O, and a local s potential for H. Gaussian–Hermite integration was used for Cu and Ag, with Cu 3d and Ag 4d states treated explicitly as valence electrons. All pseudopotentials were extensively tested in previous simulations18,57 and their accuracy assessed by comparing to all-electron ab initio calculations.18

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A cubic supercell of size a⫽9.86 Å was used, containing one metal atom and 32 water molecules, corresponding to 266 valence electrons per unit cell. The supercell size was chosen so as to guarantee vanishing mean total pressure, as determined from classical simulation using a parametrized force field. Charge neutrality was enforced through the introduction of a uniform neutralizing negative background, replacing a counterion at infinite dilution conditions.18 The Kohn– Sham orbitals were expanded in plane waves at the ⌫ point of the Brillouin zone up to a kinetic energy cutoff of 70 Ry. Constant-volume AIMD simulations were performed using the Car–Parrinello method58 – 60 as implemented in the 61 CPMD package. A fictitious mass of 600 a.u. and a time step of 5 a.u. were used in the integration of the equations of motion. The simulation temperature was set to 300 K and controlled through a Nose´ –Hoover thermostat with a thermostat parameter of 400 a.u. Excitation energies were estimated within the Tamm–Dancoff approximation.62– 64 The time dependent density functional response equations50 were solved using Davidson’s iterative subspace method.65– 67 For each instantaneous ionic configuration considered, the ground state Kohn–Sham potential was computed at the BLYP level of theory, while the ALDA 共Ref. 3兲共based on BLYP兲 was adopted for the exchange-correlation kernel. V. RESULTS FOR ELECTRONIC STATES AND OPTICAL SPECTRA A. Finite-temperature Kohn–Sham electron densities of states

We show in Fig. 1 the time evolution of a selected number of Kohn–Sham energy levels and the corresponding electron density of states 共EDOS兲, computed by diagonalizing the Kohn–Sham Hamiltonian 共including a few empty states兲 for several instantaneous ion configurations extracted from the room temperature AIMD simulations for the two ions. This analysis was restricted to 24 configurations 共with a separation of 240 fs兲 for Ag⫹ , spanning an overall simulation time of 4.6 ps. A much longer analysis turned out to be necessary for Cu⫹ , as will be shown in due course. 200 configurations were examined in this case, with a separation of 50 fs and an overall simulation time of 10 ps. The overall structure of the valence-electron energy bands 共indicated in Fig. 1 using the conventional molecular symmetry labels for the water molecule兲 is essentially unchanged with respect to clean liquid water11,68 in both solutions. Five Kohn–Sham orbitals with predominant d character and largely localized on the metal center 共Fig. 2兲 are found to concentrate within a rather narrow energy window immediately above the top of the valence band of water. The small delocalization over ligand water molecules resembles the familiar antibonding states of open shell transition metal complexes, which in the case of a d 10 configuration are all occupied. In the Cu⫹ solution, the clear cut separation between the d band and the water 1b 1 band allows a total dispersion to be estimated of ⬃2.2 eV, which is equally distributed between a purely electronic contribution 共ligandfield-like orbital splitting within the d manifold, ⬃1.2 eV兲 and the fluctuations in the orbital energies induced by the



FIG. 1. Time variation of higher energy Kohn–Sham energy levels 共left panels兲 and finite-temperature EDOS plots for aqueous Cu⫹ 共upper panels兲 and Ag⫹ . Dashed lines correspond to empty states. All energies are referred to the value of the energy for the lowest empty level, averaged over time. In the left panels, dotted lines indicate Kohn–Sham eigenvalues with d-orbital character which are localized on the metal ion. Bands corresponding to solvent states have been labeled following the standard nomenclature for the molecular orbitals of the water molecule in which the first valence orbital 共containing O 2s) is indicated as 1a 1 共this band has not been included in the plot兲. The arrows mark the approximate position of the d band in the EDOS.

thermal relaxation of the solvent molecules. Also, the energy splitting of the d orbitals remains more or less constant throughout the AIMD, and thermal effects manifest themselves as a slow modulation in the energy of the whole d manifold. The situation is not as clear for Ag⫹ , owing to some of the d orbitals on the cation being most of the time degenerate with states near the top of the valence band of water. Indeed the overall shape of the 1b 1 band is slightly perturbed near the band edge by the admixture of cation d states. The total 共thermal⫹electronic兲 dispersion of the d bands may nonetheless be estimated to be larger than ⬃1.4 eV. One of the states within the d manifold 共HOMO兲 is well separated in energy from the remaining ones 共up to ⬃0.7 eV higher兲, except for a few configurations during the trajectory 共e.g., at ⬃2.1 ps兲 where the d orbital degeneracy is approximately restored. The lowest unoccupied state, corresponding to the edge of the conduction band of water, is partly localized on the metal center in both cation solutions, with significant admixture of character from states belonging to water molecules extending well beyond the first solvation shell. In clean liquid water, semilocalized states detached in energy from the rest of the conduction band have previously been described69 共see also Ref. 18兲. Their origin can be rationalized in terms of constructive overlap between ␴* orbitals 共more precisely 3a 1 orbitals兲 extending over a cluster of water molecules in the liquid. This is promoted by the relative ease with which a large number of water molecules in the liquid environment can achieve a suitable reciprocal orientation for optimal inphase matching of orbitals. In the minimal size samples used


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FIG. 2. 共Color兲 Isosurfaces corresponding to the highest occupied 共A,C兲 and lowest unoccupied 共B,D兲 orbitals of aqueous Cu⫹ 共upper panels兲 and Ag⫹ . The plots are roughly centred around the metal ion and different colors are used to identify the phase of the wave function. Note the large degree of dispersion in space of the unoccupied states. Due the comparatively stronger tendency for d – s hybridization of Cu⫹ the metal component of the LUMO for this aqua-ion has enhanced d character.

here the lowest of these empty states 共the water LUMO兲 is clearly separated from the states higher in energy by a gap of the order of ⬃1 eV. In the presence of a Ag⫹ or Cu⫹ this state acquires metal s orbital character. A similar state was also observed in our study of the acetone solution.11 In the latter case, however, this state is indistinguishable from the LUMO of water, owing to negligible hybridization between acetone ␲* and water LUMO, notwithstanding the quasidegeneracy of the two states 共this degeneracy is accidental as we showed in Ref. 15兲. This behavior is in sharp contrast with the present case of the metal-aqua ions, where only a single mixed state is present. The qualitative model emerging from the analysis of the Kohn–Sham orbitals and of their distribution in energy is therefore that of a closed shell metal ion whose d energy levels are split by the ligand field of nearby solvent molecules. Admixture of orbital character from an empty metal s state is favored by the highly mobile nature of the local coordination environment in the liquid, which can readily distort and explore in turn different energy minima corresponding to slightly different ligand arrangements. With reference to the density of states of clean water, the 3d orbitals of Cu⫹ form a rather narrow intra-gap band, oscillating due to thermal relaxation well above the top of the valence band. In the Ag⫹ solution, the 4d states lay much closer to the valence band edge, with which they can occasionally merge. The degree of s – d mixing is therefore expected to be more pronounced in the case of Cu⫹ . As will be shown in Sec. VI C,


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FIG. 3. Calculated TDDFT absorption spectra at 300 K. Thick lines are total spectra for the 10 lowest-energy electronic excitations. Thin lines in the positive half of each panel represent composite partial spectra obtained by including only subsets of low-energy transitions, ␻ 1 ⫺ ␻ n (n⫽1,5), from the metal d orbital manifold to the lowest energy empty state 共so the thin solid line represents the spectrum of only the lowest excitations, ␻ 1 , the dotted line consists of the contributions of ␻ 1 and the next line ␻ 2 , etc.兲 In the negative half the corresponding single-excitation partial spectra are reported. A Gaussian smearing of 0.03 eV has been introduced in each plot. Note the difference in the intensity scale between the two panels.

the large deformation in the cation charge density induced by the orbital interaction is the sole responsible for the optical activity of this ion in solution. B. Optical spectra

The total finite-temperature absorption spectrum for each of the two cation solutions, corresponding to the 10 lowestenergy electronic transitions 共thick lines in Fig. 3兲 was computed according to J

f 共 ␻ 兲⫽

兺

I⫽1

F I共 ␻ 兲 ,

共 J⫽10兲 ,

共1兲

where F I ( ␻ ) are single-excitation spectra, given by F I共 ␻ 兲 ⫽

1 T

冕

T

dt f I 共 t 兲 ␦ 共 ␻ I 共 t 兲 ⫺ ␻ 兲

N

1 R R ⫽ f ␦ 共 ␻ IR ⫺ ␻ 兲 . N R R⫽1 I

兺

共2兲

In this formula, T is the total simulation time, ␻ IR (t) is the value of ␻ I for the instantaneous ionic configuration R(t), with f IR the corresponding instantaneous oscillator strength. The total number of ion configurations N R was, as for the EDOS plots, 200 for Cu⫹ and 24 for Ag⫹ . In both absorption spectra a rather structured low-energy region 共2.1– 4 eV in Cu⫹ , 3.2– 4.5 eV in Ag⫹ ), characterized by modest transition intensities, is followed by an abrupt upturn in absorption. The latter marks the onset of transition with predominant solvent→solvent character.11 We remark that these excitations involve transfer of electrons from pure water states 共occasionally with small admixture of metal character兲 to states at the bottom of the conduction band 共lowest unoccupied orbital or second lowest兲 which carry a considerable amount of metal character, and the label

‘‘solvent→solvent’’ 共STS兲 should therefore be considered largely a convention. The underlying band structure of the solvent is thus still fully recognizable in the general shape of the TDDFT spectrum. As observed elsewhere,11 the KS gap of clean water 关 E DFT g ⫽4.6 eV 共Ref. 70兲兴 is carried over, essentially unchanged to the TDDFT calculations of the absorption spectrum, strongly narrowing the optical gap compared to the experimental values which are in the range 8.40–10.06 eV.71–76 This remains true even if some metal character is now mixed with states at the bottom of the conduction band. The intra-gap states in the EDOS are however alone responsible for the low-energy features absent from the pure water spectrum. These all correspond to transitions from d states of the metal ion to the lowest energy state in the conduction band or, extremely rarely, to one of the next few low-lying states at its bottom. They will be indicated as ‘‘metal→metal’’ 共MTM兲 transitions. Important differences between the spectra of the two cations are apparent in the MTM region. One order of magnitude separates the average value of the transition intensities for the two ions, with Ag⫹ reaching intensities typical of allowed d→d excitations in the solid state. Resolution into partial spectra, obtained from Eq. 共1兲 by restricting J over the five lowest excitations, makes it possible to isolate the MTM tail of the spectrum from the low-energy portion of the STS transition region and to estimate its spread in energy: up to 2.3 eV for Cu⫹ and ⬃1.1 eV for Ag⫹ . These values are both comparable to the estimates obtained from the KS energies in the previous section. In addition, the Ag⫹ spectrum shows a well defined structure, with three main peaks at ⬃3.66, ⬃4.09, and ⬃4.30 eV 共with relative intensities at peak maxima increasing as 1, 5, 7兲, while the Cu⫹ tail evidences a more complicated sequence of partially merged peaks with only minor variations in oscillator strengths throughout the whole range of MTM transition energies. Ultraviolet absorption studies of aqueous Ag共I兲 tosylate, sulfate, perchlorate, and tetrafluoroborate salts20 show three bands attributable to isolated Ag⫹ ions, located at 5.51, 5.90, and 6.46 eV, with relative absorbances increasing as 1, 2, 4. The calculated difference in transition energy between the two lowest peaks 共0.43 eV兲 is thus in excellent agreement with the experimental value 共0.39 eV兲, whereas a mismatch of ⬃0.4 eV between theory and experiment is observed for the two highest peaks. Although the overall MTM spectral profile is reasonably well reproduced, a large underestimate of the computed transition energies 共⬃1.8 eV兲 is apparent, which remain nonetheless much smaller than the DFT error in the optical gap of clean water 共⬃4 eV兲. We are aware of only a very limited number of experimental studies of Cu⫹ in aqueous solution.77,78 Optical absorption measurements of transient intermediate species generated in pulse radiolysis experiments on aqueous Cu共ClO4 ) – HCOONa 共Ref. 77兲 yields a featureless absorption peak at 4.7 eV attributed in Ref. 77 to short lived Cu⫹ ions. This is ⬇1 eV higher in energy compared to the center of the computed Cu⫹ band. If this assignment is correct, the agreement with experiment for Cu⫹ is similar to or even better than for Ag⫹ . The experimental spectrum shows no structure, however, and our conclusions on the rather pecu-




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TABLE I. Mean transition energies ( ␻ ¯ I , eV兲, standard deviations ( ␴ I , eV兲 and frequency-integrated oscillator strengths ( f I⬘ , a.u.兲 for MTM (I ⫽1 – 5) and solvent→solvent excitations. Cu⫹

1 2 3 4 5 6 7 8 9 10

f I⬘

␴I

␻ ¯I

I

3.07 3.20 3.29 3.45 3.70 4.25 4.38 4.46 4.52 4.57

Ag⫹

⫺5

0.33 0.31 0.28 0.31 0.28 0.17 0.16 0.15 0.14 0.13

1.92⫻10 1.95⫻10⫺5 2.53⫻10⫺5 6.90⫻10⫺5 3.49⫻10⫺5 3.62⫻10⫺4 3.91⫻10⫺4 3.90⫻10⫺4 4.73⫻10⫺4 7.21⫻10⫺4

␻ ¯I

␴I

f I⬘

3.68 4.06 4.20 4.29 4.35 4.40 4.44 4.48 4.53 4.57

0.17 0.08 0.10 0.09 0.10 0.07 0.07 0.06 0.07 0.07

2.82⫻10⫺4 4.51⫻10⫺4 3.99⫻10⫺4 4.82⫻10⫺4 5.36⫻10⫺4 4.97⫻10⫺4 4.32⫻10⫺4 6.44⫻10⫺4 5.41⫻10⫺4 5.35⫻10⫺4

liar properties of the Cu⫹ aqua-ion 共see Sec. VI C兲 have to remain largely hypothetical. VI. ANALYSIS AND CORRELATION TO SOLVATION A. Fit to a Gaussian model

Tracing a particular feature in the spectrum back to a well defined electronic transition is not straightforward, owing to different transitions contributing at different times to the same region of the spectrum. We compare in Table I the mean transition energies

␻ ¯ I⫽

冕

d␻ f I 共 ␻ 兲 ␻

共3兲

and frequency-integrated oscillator strengths f I⬘ ⫽

冕

d␻ f I 共 ␻ 兲 ,

共4兲

for the 10 lowest excitations. In addition to carrying modest absorption intensities compared to both MTM Ag⫹ and STS excitations, the lowest Cu⫹ transition energies oscillate over ranges which are up to three times as wide as those of Ag⫹ . Standard deviations for Ag⫹ MTM transitions are comparable to those of STS excitations, with the possible exception ¯ 1. of ␻ From the spectral resolution of the MTM tail given by Eqs. 共3兲 and 共4兲 a theoretical absorption profile can be constructed, for instance through the function f⬘

5

f 共 ␻ 兲⫽ G

I e ⫺ 共 ␻ ⫺ ␻¯ 兲 /2␴ 兺 I⫽1 ␴ 冑2 ␲ I

2

2 I

FIG. 4. Gaussian fit of metal→metal transitions 共MTM, see text for details兲. Dashed lines correspond to MTM spectra computed from AIMD simulation, thick solid lines are MTM spectra obtained through Eq. 共5兲. In the lower part of each panel the elementary Gaussian functions contributing to Eq. 共5兲 are shown. Note different scales in transition energies 共given in eV兲 and intensities in 共a.u.兲.

is inadequate for Cu⫹ . The optical activity of the two ions is thus likely to involve processes for electronic excitations which are quite dissimilar from each other. B. Interpretation in terms of an average crystal field

On the basis of the results of the previous sections, it is possible to account for the absorption of Ag⫹ in terms of one particle transitions from a ligand-field split manifold of d states to the lowest virtual state 共the water LUMO—metal s hybrid兲. Analysis of AIMD atomic configurations demonstrates the persistence of a tetragonally distorted tetrahedral coordination shell throughout the trajectory 共Fig. 5兲. The deviation from the ideal tetrahedral geometry can be described through the contraction parameter ␰ 共Fig. 6兲, corresponding to the ratio between the shortest and largest projection of the four shortest Ag–O distances onto two perpendicular im-

共5兲

I

共which is normalized to the integral of the MTM absorption, 5 兺 I⫽1 f I⬘ ). Using the values for ␻ ¯ I , f I⬘ , and ␴ I given in Table I, Eq. 共5兲 is found to reproduce rather accurately the MTM spectrum of Ag⫹ 共Fig. 4兲. The large energy dispersion of the modes contributing to the Cu⫹ spectrum does however result in a featureless theoretical line profile which hardly matches the absorption computed from the AIMD simulation 共though it happens to be in better agreement with the experimental absorption profile77兲. Indeed dynamical analysis of the intensity fluctuations in the next section shows that the simple model of superimposed Gaussian lines underpinning Eq. 共5兲

FIG. 5. OAg d⫹ O angle distribution function for the four oxygen atoms closest to Ag⫹ , averaged over 4000 configurations taken from the room temperature AIMD. Notice the presence of two maxima at ⬃90° and ⬃160°, indicating a deviation of the coordination shell structure from the ideal tetrahedral geometry ( ␪ id⫽109°).


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FIG. 6. 共Left兲 Contraction parameter ␰ in a distorted tetrahedral coordination geometry. Black circles represent the O atom of the ligand H2 O molecule, 丣 is the cation center. 共Right兲: Crystal field splitting of Kohn–Sham orbitals with d character in gas phase Ag⫹ in the presence of the static external field given by Eq. 共6兲. See main text for details. Orbital energies E KS were computed at values of ␰ indicated by circles, and they are plotted relative to the 共unsplit兲 Ag⫹ manifold of d states in the absence of external fields 共energy E KS ). Lines are guides to the eye. Conventional symmetry labels for ␰⫽0 共square-planar兲 and ␰⫽1 共regular tetrahedron兲 are used.

proper fourfold rotation axes of the tetrahedron. An average over time value 具␰典⯝0.6 was computed from the AIMD trajectory. The effect of varying ␰ onto the relative energies of the split manifold of Ag⫹ d states 共in the gas phase兲 was studied using a simple ion-ligand potential model of the form V共 r 兲⫽v0

兺i e ⫺ ␣兩r⫺r 兩, i

共6兲

where v 0 and ␣ are adjustable parameters and the sum is over the ligand-field source positions r1 ⫽(a,0,0), r2 ⫽(0,a,0), r3 ⫽(0,0,b), r4 ⫽(a,a,b). The cation is located at rAg⫹ :a/2, a/2, b/2. The tetrahedron axis a and b are related to the contraction parameter by a⫽

2d 共 2⫹ ␰ 2 兲 1/2

,

represent the thermal broadening. For this choice of potential parameters, the TDDFT and the model absorption profiles agree qualitatively within the range ␰⫽0.8 –0.4. Despite the simplistic representation of the cation-ligand interaction of Eq. 共6兲共bare Pauli repulsion between unpolarizable neutral ligand and cation-centered one particle densities兲, this result is indicative of a direct correlation between the main features in the absorption profile and the ligand field splitting of cation d states. In particular, this analysis indicates that the more pronounced cation absorption intensity in the energy range 4.0– 4.4 eV is likely to originate from the coincidence of up to 4 absorption lines, rather than to an intrinsic intensity enhancement mechanism. Thermal effects would mainly be responsible for the peak broadening, though they could also affect non-negligibly the relative height of the peaks.

共7兲

and b⫽ ␰ a.

共8兲

d is the ion-ligand distance for undistorted tetrahedral coordination 共␰⫽1兲. The KS equations for the cation were solved self-consistently in the presence of the static potential 共6兲 for values of the contraction parameter ranging from ␰⫽0 共square-planar coordination兲 to ␰⫽1 共ideal tetrahedral coordination兲. The KS eigenvalue energies ⑀ i ( ␰ ) corresponding to the Ag⫹ 4d manifold are shown in Fig. 6 for the parameter set 共in atomic units兲: d⫽2.0, v 0 ⫽0.5, ␣⫽1. From this correlation diagram, theoretical Ag⫹ absorption profiles f ( ␻ ) were computed using ␻ i ( ␰ )⬅⫺ ⑀ i ( ␰ ), i.e., we ignore any dispersion of the virtual state. This simplification is justified by the relative stability of this level in Fig. 1. We further assume that each absorption line contributes an overall oscillator strength f i ⫽1.79 Results are shown in Fig. 7. Gaussian smearings ␴ G⫽0.03 and 0.06 have been added to

FIG. 7. Model theoretical absorption profiles as a function of the dihedral distortion parameter ␰ 共see Fig. 6 and Sec. VI A兲. Reasonable agreement with experiment is achieved for values of ␰ in the interval ␰⫽0.4 –0.8.



FIG. 8. Time variation of transition energies 共upper panels兲 and intensities for the 10 lowest energy transition of the Cu⫹ and Ag⫹ solutions. The metal→metal transitions are indicated by thick curves.

It might be interesting to observe that the lowest energy excitation 共⬃3.7 eV兲 is well separated from the upper ones and therefore reasonably well accounted for by Eq. 共5兲. This excitation is associated with a transition from the highest occupied Kohn–Sham state, which was found to lay most of the time well within the solvent gap 共Fig. 1兲. Its abnormally high spread in energy is comparable to that of the MTM transitions of Cu⫹ , all originating from pure intra-gap states. This latter feature is indicative of a larger sensitivity to changes in the ligand field, compared to the remaining absorption lines. This is consistent with the large dispersion of the b 2g (t 2 ) orbital energy as a function of ␰ 共Fig. 6兲. By contrast, Eq. 共5兲 is much less reliable in the 4.0– 4.4 eV region. Fluctuations in the instantaneous ligand field may in this case be crucial in promoting mixing between quasidegenerate oscillation modes within this energy range. More care is required in examining the Cu⫹ MTM spectrum. We compare in Fig. 8 the time series of the 10 lowest transition energies for both cations. Similar to Ag⫹ the slow modulation in the Cu⫹ MTM transition energies reproduces the variation in time of the Kohn–Sham energy levels of localized on the cation 共Fig. 1兲. However the effect on the intensity is more severe. Occasional near degeneracy of MTM and STS transitions 共typically lasting for less than 100 fs兲 is observed. While relatively rare, these events are associated with large enhancements in the MTM intensities. These may at times increase by more than one order of magnitude 共e.g., at ⬃5.5 and 7.0 ps兲, though moderate bursts of 2–3 times the average oscillator strength occur roughly every 0.5 ps 共Fig. 9兲. A similar behavior, on a much more modest scale, can also be detected in the Ag⫹ plots 共e.g., at ⬃3.5 ps兲, though the too short simulation time prevents the frequency of occurrence of these events from being estimated with certainty. A rise in Cu⫹ absorption intensity is always associated with high MTM transition energies 共Fig. 8兲. The absorption line shape 共Fig. 3兲 is therefore the result of few intense lines concentrating in the high-energy region


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FIG. 9. Time variation of the total Cu⫹ absorption, computed from the sum of the oscillator strengths of the corresponding MTM transitions 共five lowest excitations兲. The dashed line indicate the mean value of the absorption intensity.

of the MTM tail 共3.5– 4.4 eV兲 and a large number of weak features in the lowest-energy portion. C. Details of cation coordination

As argued in the previous section, the absorption spectrum of Cu⫹ cannot be understood without taking the fluctuations of the liquid environment into account. We will therefore subject the coordination to a closer structural analysis focusing on the difference between Ag⫹ and Cu⫹ . The radial distribution functions 共RDFs兲 for the two solutions were given in Ref. 18. We will summarize here the main features. Four water molecules are on average with their oxygen atoms within a sphere of radius 3.0 Å centered on Ag⫹ . Only two water molecules are however found to contribute to the full height of the first peak in the Cu–O and Cu–H RDFs, with the two oxygen atoms within 2.00 Å and the four hydrogens within 2.46 Å. Two additional oxygen atoms are within 3.1 Å. Effective fourfold coordination of Cu⫹ is therefore restored at distances comparable to the Ag⫹ . The mean distances of the two closest water molecules 共which never exchange with other solvent molecules throughout the ˆ –O trajectory兲 are 1.81⫾0.18 and 1.67⫾0.17 Å. The O–Cu angle oscillates over a range of ⬃60°, with a minimum value of 150.34°. This avoids strict linearity in the O–Cu–O geometry for most of the time. The second coordination shell is much more mobile, with loosely bound molecules frequently exchanging with the bulk solvent. The peculiar twofold coordination of Cu⫹ can be rationalized as a distortion induced by s – d hybridization similar to the examples of low coordination structures in the solid mentioned in Sec. II. Interaction with the empty s orbital allows the ionic charge cloud to contract in one direction and extend in the perpendicular direction.18 This enables the oxygen atoms of two water molecules to approach from opposite sides of the ion center greatly increasing their electrostatic binding energy. This creates a highly stable 关 H2 O–Cu–OH2 兴 ⫹ moiety with a geometry similar to the gas-phase dihydrate as was verified by performing a 6 ps room-temperature AIMD simulation on the gas-phase 关 H2 O–Cu–OH2 兴 ⫹ . Metal–oxygen coordination along the


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FIG. 10. Detail of the metal-hydrogen radial distribution functions showing the increased affinity of Cu⫹ for solvent hydrogen atoms compared to Ag⫹ . The position of the first peak in both functions corresponds to r/r max⫽1 and the height has been rescaled so that g(r/r max)⫽1. A spatial resolution of 0.01 Å has been used, and a Gaussian broadening of ␴⫽0.02 Å has been introduced in the final plots.

perpendicular direction is however disfavored by the protruding electronic charge, as illustrated in a density difference plot in Ref. 18. The building up of charge density in the equatorial plane is sufficient to prevent additional molecules from establishing stable oxygen coordination with the cation. A similar deformation is in principle also possible for Ag⫹ and is in fact observed in gas-phase hydration.40– 43 In solution the energy gain for Ag⫹ is not large enough to overcome the more uniform tetrahedral coordination. For Cu⫹ , however, the charge deformation is sufficiently strong to have a noticeable destabilizing effect on the second coordination shell, which makes a wide range of solvent configurations energetically accessible. These include fluctuations in which water molecules belonging to the second coordination shell orient one of their O–H bonds toward the cation. Evidence of this tendency, already discussed in Ref. 18, can be seen in Fig. 10 where we compare the shape of the Cu–H and Ag–H RDFs in the region immediately preceding the first peak. The rise toward the first maximum is markedly nonmonotonic in Cu⫹ , with at least one shoulder at ⬃r/r max⫽0.85 共corresponding to a distance from Cu⫹ of 2.09 Å兲. This confirms the anomalous accumulation of hydrogen atoms in the close vicinity of the cation. No such feature can be identified in the Ag⫹ plot. D. Metal ion shape deformations and moment analysis

In order to quantify the deviation from a spherical ion picture we have carried out a moment analysis of the electron density distribution of the Cu⫹ and Ag⫹ ions, as obtained using the maximally localized Wannier function method.80,81 Technical details of our approach can be found in the Appendix. The resulting dipole and quadrupole distribution functions averaged over the AIMD trajectories are shown in Fig. 11. The total dipole moment of Ag⫹ remains rather small throughout the simulation and spread over no more than ⬃0.2 a.u. This is consistent with the expected response of the charge density to a slightly distorted tetrahedral coordination

Bernasconi et al.

FIG. 11. Left panels: distribution functions for the total cation dipole moments ␮. A resolution of 0.01 a.u. in the dipole moment has been used, and a Gaussian smearing has been introduced in the final plots of ␴⫽0.01 a.u. Right panels: distribution functions for the eigenvalues of the cation quadrupole tensor (⌰ xx (t)⫹⌰ y y (t)⫹⌰ zz (t)⫽0, with 兩 ⌰ xx 兩 ⬎ 兩 ⌰ y y 兩 ⬎ 兩 ⌰ zz 兩 ). The resolution is the same as in the dipole plots, but a value ␴⫽0.1 a.u. has been used in the Gaussian smearing.

environment, with moderate fluctuations introduced by the thermal motion of the ligands. The dipole moment of Cu⫹ is distributed over a much wider range 共⬃0.7 a.u.兲 and the plot lacks a well defined main peak. Compared to Ag⫹ , the Cu⫹ dipole polarizability is therefore dramatically enhanced by solvation, and a stronger coupling is established between the cation dipole and the thermal fluctuations of the solvent molecules. The eigenvalues of the quadrupole tensor 共Fig. 11, right panels兲 of Ag⫹ are concentrated around ⌰ ␣␣ ⫽0, with values varying between ⫺1 and 1 a.u. Any given eigenvalue changes sign several times during the simulation, which is indicative of the absence, on average, of a preferential elongation axis for the charge density ellipsoid. ⌰ xx do however oscillate between two main peaks, symmetrical wrt ⌰ ␣␣ ⫽0, with a marked preference for the negative value, ⌰ xx ⯝⫺1.2 a.u. A negative eigenvalue at a given time, larger in absolute value than the two positive ones, corresponds to a prolate distribution of the negative electron charge density along the x axis.82 Ag⫹ does therefore retain moderate quadrupole distortion throughout the AIMD, with the two more frequent situations corresponding to the two peaks in Fig. 11. By contrast, one of the eigenvalues of Cu⫹ maintains a very large and positive value ⌰ xx ⯝2.2 a.u., corresponding to an oblate electron density distribution. The eigenvector corresponding to ⌰ xx was found to orient approximately along the O–Cu–O direction in all the situations in which the O–Cu–O angle was close to 180°, and the dipole moment vector roughly perpendicular to it. The absolute value of the cation dipole moment is therefore related to the difference between ⌰ y y and ⌰ zz , which yields an estimate of the deviation from a symmetrical distribution of the electron density in any plane perpendicular to ⌰ xx . Fluctuations in the relative magnitude of the eigenvectors of the quadrupole tensor perpendicular to the



FIG. 12. 共Color兲 Left panels: time dependence of Cu⫹ dipole moment 共bottom兲, transition energies for the ten lowest excitations 共solid lines are MTM transitions兲 and total cation absorption, obtained from the sum of the oscillator strengths of the MTM transitions 共top兲. All the plots have been smoothed using a Gaussian convolution of ␴⫽0.2 ps, in order to emphasize dependencies on the slow solvent dynamics over effects due to fast intramolecular fluctuations. Right panels: instantaneous ion configurations corresponding to the times labeled in the upper left panel: t A⫽2.0 ps, t B ⫽3.4 ps, t C⫽6.8 ps, representative of low intensity, polarized 共A兲, and high intensity depolarized 共B,C兲 states. Cu is at the plot center and I indicates the strongly bound water molecules.

关 H2 O–Cu–OH2 兴 ⫹ axis may at times succeed in temporarily polarizing the excess charge density. This results in the cation carrying a net temporary dipole, in addition to the permanent quadrupole moment. Water molecules do themselves carry large dipole and quadrupole moments 关DFT estimates for clean liquid water being 兩␮兩⫽1.16 a.u. and ⌰ ␣␣ ⫽ ⫺2.35, 2.51, ⫺0.16 a.u. 共Ref. 83兲 and suitably oriented bulk solvent molecules can therefore establish direct attractive interactions with the cation for short periods of time. Dipole– dipole attraction would bring the positive end of the water molecule to point toward the cation, allowing one hydrogen atom to penetrate closer toward the ion center. Overall, this mechanism can therefore contribute to explain the anomalous concentration of hydrogen atoms near Cu⫹ . Such effect is not observed in Ag⫹ , despite the non-negligible induced cation moments, mainly owing to steric interference and shielding on part of the four coordinated water molecules. E. Correlation to optical absorption

How are these ion-shape fluctuations correlated with the time variation of optical absorption? We show in Fig. 12 a comparison between the time dependence of the induced cation dipole moment, the transition energy of the MTM transitions 共and of the five lowest solvent→solvent ones兲 and the total cation absorption. All three properties oscillate with similar frequencies 共minima occurring on average every ⬃1 ps兲. Periods of large absorption intensity and transition energy 共e.g., at 3.2, 5.8, 7.1 ps兲 are always associated with local minima in the 兩 ␮ (t) 兩 plot. The cation thus exhibits maximum optical activity when its dipole moment is lowest. In these situations, the 3d orbitals are selectively stabilized with respect to both solvent valence and conduction band, as is clear from the relative stability in time of the STS 共essentially


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valence→conduction band兲 transition energies. The absence of induced cation dipole moment is associated with a nearly linear O–Cu–O geometry 共Fig. 12, left panels B, corresponding to the instantaneous ion configuration at t⫽3.4 ps and C, t⫽6.8 ps) and with up to two loosely-bound water molecules approximately pointing one O–H bond toward the cation 共with Cu–H distances as short as ⬃1.8 Å, well within the first Cu–H RDF peak兲. Periods of poor optical activity are associated with bent and dipole polarized O–Cu–O geometries 共Fig. 12, panel A, t⫽2.0 ps), and at most one water molecule ‘‘hydrogen-bonded’’ to the cation, with shortest Cu–H distances of ⬃1.9 Å. The temporary binding of water molecules to 关 H2 O–Cu–OH2 兴 ⫹ and, ultimately, the transient optical activity of Cu⫹ are therefore mainly to be associated with quadrupole (Cu⫹ )-dipole, and quadrupole–quadrupole interactions, rather than to direct dipole–dipole attraction. Dipole–dipole interactions do however contribute to stabilizing ‘‘anomalous’’ orientations of water molecules w.r.t. the positively charged ion, overcoming unfavorable chargedipole interactions and bringing hydrogen atoms within the first coordination shell. The results displayed in Fig. 12 are indicative of a strong correlation between fluctuations in the effective dipolemoment and the optical absorption of the complex. This raises the question how these fluctuations affect the transition dipole controlling the intensity. Answering this question will require more detailed analysis which was deemed outside the scope of the investigation reported here. VII. ASSESSMENT OF THE TDDFT METHODOLOGY

As remarked in Sec. VI C a large mismatch is observed between the energy region associated with electronic transitions from the metal center in our TDDFT theoretical absorption profile 共3.7– 4.3 eV兲 and in experimental ultraviolet spectra 共5.5– 6.5 eV兲.20 The time dependent correction to the DFT energy gap at the ALDA level of theory does not appear to approach the desired value which would be necessary to bring the energy difference between the highest 共cation兲 occupied state and the lowest 共cation⫹water兲 empty state into agreement with experiment. Moreover the TDDFT correction is virtually the same for all lowest-energy transitions, i.e., no distinction is made between the energy of transitions involving 共partially兲 localized states 共MTM兲 and collective 共STS兲 excitations. In Sec. III we have argued that the degree of localization of optically active states is an important factor for the accuracy of TDDFT methods. We introduced the distinction into 共a兲 excitations between valence states localized on the same molecule 共solute兲, 共b兲 transitions between states localized on different molecules 共proper CT excitations兲, and 共c兲 excitations of an electron from a localized to a delocalized state. This was one of the conclusions deduced from the results of the calculations of Ref. 15, where we examined the effect of exact exchange on the electronic excitations of aqueous acetone. The d→s excitations of the metal ions solutions studied here belong the last of the above categories. Based on the results of acetone calculations, we expect that an exact exchange hybrid scheme should reduce the error in the energy difference between cation 共occupied兲 d states and the lowest


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FIG. 13. Absorption profiles of solvated Ag⫹ computed from TDDFT at the BLYP 共32 solvent molecules per unit cell, solid curve兲 and PBE0 共16 solvent molecules, dashed curve兲 levels of theory. The experimental spectrum 共from Ref. 20兲 is shown for comparison 共dotted-dashed curve兲.

unoccupied state already at the KS level. Furthermore, increased selectivity in the time dependent hybrid correction to MTM and STS transitions could result in a sharper separation between spectral features attributable to the two kinds of excitations. Accordingly, an absorption spectrum was computed from TDDFT at the PBE0 level of theory from a Car– Parrinello simulation at 298 K. The system size was reduced to one Ag⫹ ion and 16 water molecules. Calculations on larger samples using exact-exchange functionals and a planewave basis set are at present prohibitive. The unit cell size was rescaled to yield the same density as for the 32 water molecule system. BLYP pseudopotentials were used throughout. The system was equilibrated for ⬃1 ps. The absorption spectrum was obtained as previously described from a set of 16 ion configurations spanning a total simulation time of 2.0 ps. We compare in Fig. 13 the absorption profile corresponding to the MTM transitions obtained from BLYP 共32 water molecule sample兲 and PBE0 to the experimental spectrum. The absolute position of the major cation absorption peak is blueshifted by some 1 eV in PBE0, bringing the computed absorption towards the experimental region. This confirms that the d→s transition is indeed not a valence excitation of the type found in the example of acetone. A large 共⬃1.5 eV兲 underestimate of the peak positions is nonetheless still quite evident. Rather remarkably, the overall PBE0 cation absorption profile remains largely unchanged w.r.t. to BLYP and in good qualitative agreement with experiment. This fact lends support to the reliability of the cation absorption analysis described in the previous sections. System-size effects and the short simulation time may be responsible for the more developed peak structure at low energies. The assignment of absorption bands played a crucial role in the analysis of the computed spectra and the comparison to experiment. It was on the basis of this assignment that we were able to separate the metal→LUMO from the solvent→LUMO transitions 共see Fig. 3兲. This interpretation was entirely based on the orbital character of the Kohn– Sham states with the highest contributions to the oscillator strength. In general assignment in terms of component transitions between KS states is not sufficient to discriminate

between optical excitations, as some of the bands may be the result of orbital term splitting. The electronic spectra of openshell transition metal complexes are a good example of this phenomenon. In the case of the d→s transitions discussed here, however, the danger of degeneracy between occupied and empty Kohn–Sham orbitals is minimal and we may assume that analysis purely in terms of transitions between one-electron orbitals is justified. However at the GGA level of theory applied here, KS energies can be particularly poor approximations to excitation energies, to the point that some of the virtual states must be considered spurious 共see, for example, Ref. 47兲. Our interpretation of the excitations of the Ag⫹ and Cu⫹ aqua-ion on the basis of KS states only must therefore treated with some caution. VIII. SUMMARY AND CONCLUSIONS

We have described a mixed DFT/TDDFT AIMD study of the electronic and the optical properties of Cu⫹ and Ag⫹ in aqueous solution at room temperature. Using our calculations as a tool to assign and interpret the experimental UV spectrum, we have obtained information about the states involved, the occupied d and empty s states. The d states are of interest in the context of the coordination chemistry of these closed shell transition metal aqua ions. The s state, because of its delocalized character, is of more general interest as a probe of the low lying empty states in liquid water which play a role in radiation chemistry and electrochemistry. Both cations were found to give origin to a rather narrow electron band oscillating in time well (Cu⫹ ) or only slightly (Ag⫹ ) above the valence band of the solvent, and spread in energy by both electronic 共crystal-like field splitting兲 and thermal broadening. We conclude that the standard picture of the occupied states as metal orbitals with some antibonding admixture of ligand orbitals and energies situated well above the bonding ligand states remains valid in solution, with the manifold of optically active occupied d states ending up in the gap of the collective states of the solvent. The empty s state was found to merge with the lowest unoccupied state of the pure solvent and to be comparatively insensitive to solvent fluctuations. This allowed us to treat this state as a reference. As target in the electronic excitation, it was found to determine the overall position of the d→s band, without itself contributing to the band broadening mechanism. Electronic excitations from the metal center were isolated from higher energy features arising from transition between states with large solvent character and analyzed in terms of the local coordination environment and cation charge density deformation effects. Consistent with the interpretation of UV absorption experiments,20 we found Ag⫹ to be stably coordinated by four water molecules in a distorted tetrahedral arrangement, which also explains its relatively large transition intensities ( f ⯝10⫺3 a.u.). The overall absorption profile is reasonably well reproduced by the TDDFT calculations and was shown to be consistent with splitting by an average crystal field of dihedral geometry. Cu⫹ is tightly bound to two water molecules in a linear 共though widely oscillating兲 geometry. Transition intensities are roughly one order of magnitude lower than in Ag⫹ and the cation spectrum could not be resolved into elementary




oscillators associated with an average ligand field. In addition, all transition energies and intensities are strongly modulated in time, with short bursts 共characterized by high transition energies兲 alternating to longer quiescence periods. High transition intensities were always found to be associated to a nearly linear 关O–Cu–O兴 geometry, with up to two water molecules pointing one of their O–H bonds toward Cu⫹ . Cation polarization, driven by the s – d hybridization, was shown to be responsible for these interactions. These rare metal ion states with dual hydrogen bonding affinity 共for both oxygens and hydrogens兲 were found to make up the high frequency end of the spectrum. This led to the conclusion that, contrary the Ag⫹ case, fluctuations dominate the shape of the Cu⫹ absorption band, and a static ligand-field picture does not apply. Absolute transition energies as measured by the frequency of maximum absorption in the Ag⫹ spectrum are underestimated by 2 eV in our TDDFT approach based on BLYP. This discrepancy is reduced to 1.3 eV when the PBE0 functional is used. We have argued that the significant shift in response to the introduction of exact 共Hartree–Fock兲 exchange is consistent with the delocalized character of the s orbital, resembling the semilocalized states which are thought to occur at the bottom of the conduction band of pure water 共Urbach tail兲.76 On a technical level, this somewhat negative result can be seen as an indication that TDDFT methods need further improvements before they can be reliably applied to study delocalized states in condensed molecular systems. In a more positive sense, it gives us confidence in our spectral assignment. An interesting implication is that the experimental UV spectrum of Ag⫹ can be interpreted as a lower bound for the energy difference between the valence states and the semilocalized empty states of liquid water. Hybridization with the metal orbitals strongly enhances the intensitiy of transitions to these levels which, appearing below the main optical absorption edge in the clean water, are normally too weak to be seen. This leads to a gap of ⬇6 eV, which should be compared to the 8.5 eV optical gap in pure water. This estimate for the lower boundary of semilocalized states is in fairly good agreement with values deduced from various experimental techniques.76 ACKNOWLEDGMENTS

This work is part of the 2001–2004 CCP1 flagship project supported by EPSRC. We thank E. K. U. Gross and Juerg Hutter for useful discussions. Computer resources were provided by the CLRC at Daresbury Laboratories and the UKCP which gave us access to the HPCx facility at Daresbury. APPENDIX: MULTIPOLE MOMENTS FROM WANNIER FUNCTION INTEGRATION

Given a set of orthonormalized doubly-occupied singleparticle Bloch 共Kohn–Sham兲 orbitals 兵兩 ␺ n 典其 at the ⌫ point of the Brillouin zone of the simulation supercell, maximally localized Wannier functions 兵兩 w n 典其 are obtained through the unitary transformation80,81

兩 w n典 ⫽

兺m U mn兩 ␺ m 典 ,

11897

共A1兲

where all the indices run over the total number of occupied states, and the matrix U is chosen which minimizes the total Wannier function spread ⍀⫽

兺n 共具 w n兩 r 2兩 w n 典 ⫺ 具 w n兩 r兩 w n 典 2 兲

共A2兲

with respect to all unitary rotations among the Bloch orbitals. As the Bloch functions are assumed59,60 to obey Born–von Karman boundary conditions84 ( 具 r⫹R兩 ␺ n 典 ⫽ 具 r兩 ␺ n 典 , with R any real space lattice vector of the supercell兲, the Wannier orbitals are in this case also lattice periodic, as Eq. 共A1兲 amounts to a simple linear combination of 共real兲 periodic functions. In ionic insulators and molecular liquids, subsets of Wannier functions within the supercell can be associated unambiguously with single ionic centers or groups of ions by monitoring the relative position of the respective centroids, defined for a cell of general symmetry by85 3

r n␣ ⫽⫺

兺 m⫽1

M nm Im ln关 U† S共 m 兲 U兴 nn , bm

共A3兲

is where ␣ labels Cartesian directions, M nm ⫽(bn "um )b ⫺1 n the normalized projection of the nth reciprocal lattice vector on the mth Cartesian vector, and 兲 ⫺ibm "r S 共i m 兩␺ j典 j ⫽ 具 ␺ i兩 e

共A4兲

represents the overlap between 兩 ␺ i 典 and 兩 ␺ j 典 translated by the reciprocal lattice vector bm . The system can in this way be partitioned into independent, spatially separated distributions of charge. The total charge density ␳ S associated with each of these subsystems is given by the sum of the electronic charge density ␳ e and the ionic contribution ␳ I :

␳ S ⫽ ␳ e ⫹ ␳ I ⫽⫺2e

兺

n苸S

冕

V

dr兩 w n 共 r兲兩 2 ⫹e

兺

I苸S

ZI ,

共A5兲

where ⫺e is the electron charge, w n (r)⫽ 具 r兩 w n 典 , Z I is the formal charge of the ion I, and the integral is over the supercell volume. Owing to Eq. 共A1兲 and the normalization of the Bloch orbitals over the supercell, the integral on the r.h.s. always yields a positive integer number, and ␳ S is therefore always an integer multiple of e. In the case of clean liquid water in normal conditions83 this decomposition yields a set of neutral moleculelike units, whose densities overlap only negligibly, similar to the classical Clausius–Mossotti model.52,84 This fact motivated and justified the approach followed in Ref. 83 for the calculation of dipole and quadrupole moments of liquid water as expectation values over restricted sets of Wannier functions. In solid ionic systems, like crystalline MgO,86 a separation into anions and cations is again naturally achieved. Though still tightly associated with an ion center the optimally localized Wannier orbitals do however retain a more developed nodal structure, which leaves part of the single-particle density 具 w n 兩 r典具 r兩 w n 典 extending far away from the main localization region. This may be due in part to small inaccuracies in the


11898

determination of the matrix U, which has to be carried out numerically:80,81 no analytic solutions of the minimization problem of Eq. 共A2兲 are in fact known at present for a general three-dimensional periodic system.80,87 The reason for the remaining delocalization is however mainly related to the orthonormality condition between Wannier functions implicit in Eq. 共A1兲, which requires the overlap between Wannier functions with different indices to be strictly zero when integrated over the volume V. This results in practice in the appearance of an orthogonalization tail when a Wannier function is nonvanishing in regions of space corresponding to localization peaks of other functions. The relative importance of these oscillations increases with the degree of covalency in the cation–anion interaction.88,89 The relaxation of the orthonormality constraint 共as proposed, e.g., in the context of linear scaling electronic structure methods90,91兲 do bring about greater localization, but it also violates the condition that ␳ S be an integer, and the connection with the adiabatic change in the total electronic polarization 共the central physical observable in the bulk theory of polarization52,51兲 is thus unpleasantly lost. We remark that multipole moments of separate subunits within a quantum system should not be regarded as genuine physical observables, and differing equally consistent definitions may therefore be adopted in various situations. In the present study we follow a formalism analogous to the one presented in Ref. 86 which was developed in view of the construction of highly accurate and transferable models for many-body interaction potentials for ions in oxide materials.92–94 Potential parameters were chosen by insisting that forces, stresses, and single-ion dipole and quadrupole moments, as obtained from ab initio 共DFT兲 calculations for systems of the same size and in the same conditions, should all be reproduced, simultaneously, by the same classical model. Our choice of definitions is therefore justified a posteriori by the fact that the multipoles thus computed are fully consistent with other single ion properties, as embodied in the classical interaction potential, and is therefore similar in spirit 共though slightly different in practical implementation兲 to the one of Ref. 83. Cu⫹ and Ag⫹ multipoles were computed from the maximally localized Wannier orbitals whose centers were within a threshold of 1 Å. In practice, five centers 共corresponding to the ten valence 3d and 4d electrons兲 were always found to be closer than 0.1 Å to the ion. The total electronic contribution to the ionic density ␳ eM , was then estimated as for ␳ c in Eq. 共A5兲. A spherical cutoff radius of integration r cut was defined such that

兺

n苸M

Bernasconi et al.


冕

兩 r ⬘ 兩 ⫽r cut

兩 r ⬘ 兩 ⫽0

dr⬘ 兩 w n 共 r⬘ 兲兩 2 ⫽0.99755␳ eM ,

共A6兲

where r⬘ ⫽r⫺RM , and RM is the ion position. Truncation of the real space integrals was found to be necessary according to the results of Ref. 86 in order to minimize spurious contributions from the long range oscillation of the Wannier functions far away from the nuclear position, which are amplified by the dependence of the dipole and quadrupole operators on the distance and its square, respectively. The large value of the density fraction guarantees the accurate evalua-

tion of the integrals. By exploiting the localization of the Wannier functions inside the periodic cell, we therefore computed ion dipole and quadrupole components by integrating ˆ ⫽(3r r the standard multipole operators rˆ ␣ ⫽r ␣ and ⌰ ␣␤ ␣ ␤ ⫺r 2 ␦ ␣␤ )/2, i.e.,

␮ ␣M ⫽⫺2e

冕兺冕

兺

n苸M

M ⌰ ␣␤ ⫽⫺2e

V

n苸M

dr⬘ r⬘ 兩 w n 共 r⬘ 兲兩 2 f 共 r⬘ 兲 ,

V

共A7兲

dr⬘ 共 23 r ␣ r ␤ ⫺ 21 r 2 ␦ ␣␤ 兲兩 w n 共 r⬘ 兲兩 2 f 共 r⬘ 兲 , 共A8兲

with the function f 共 r⬘ 兲 ⫽

再

1 0

if r ⬘ ⭐r cut

, if r ⬘ ⬎r cut

共A9兲

representing the effect of the truncation on the integrand at r cut . Our approach deviates from the procedure of Ref. 83 in that integrals over multipole operators do reflect the real space structure of the charge density distribution and make explicit use of the strict localization properties of the Wannier functions. The two methods should however yield very similar results in the limit of large simulation supercells, owing to the multipole operators used in Ref. 83 converging to the form adopted in Eqs. 共A7兲 and 共A8兲. We verified this by computing the eigenvalues of the quadrupole moment components for a water molecule in a cubic supercell of size a ⯝10 Å. The electronic contribution was estimated using Eq. 共A8兲 computed w.r.t. the center of mass of the molecule and ˆ by replacadded to the ionic contribution obtained from ⌰ ␣␤ ing the electronic coordinates r ␣ with those of the ions and the electron charge by the formal ion charges. This gave 共in 10⫺26 esu cm2 ) ⌰ xx ⫽⫺2.48, ⌰ y y ⫽2.61, and ⌰ zz ⫽⫺0.13, to be compared to the results of Ref. 83 共⫺2.45,2.58,⫺0.13兲 and to the experimental value for the gas-phase molecule 共⫺2.50,2.63,⫺0.13兲.95 E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 共1984兲. M. Petersilka, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett. 76, 1212 共1996兲. 3 M. E. Casida, in Recent Developments and Applications of Modern Density Functional Theory, Theoretical and Computational Chemistry, edited by J. M. Seminario 共Elsevier, Amsterdam, 1996兲, Vol. 4. 4 G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 共2002兲. 5 M. E. Casida, C. Jamorski, K. C. Casida, and D. R. Salahub, J. Chem. Phys. 108, 4439 共1998兲. 6 D. J. Tozer and N. C. Handy, J. Chem. Phys. 109, 10180 共1998兲. 7 A. Dreuw, J. L. Weisman, and M. Head-Gordon, J. Chem. Phys. 119, 2943 共2003兲. 8 A. Dreuw and M. Head-Gordon, J. Am. Chem. Soc. 126, 4007 共2004兲. 9 D. J. Tozer, J. Chem. Phys. 119, 12697 共2003兲. 10 O. Gritsenko and E. J. Baerends, J. Chem. Phys. 121, 655 共2004兲. 11 L. Bernasconi, M. Sprik, and J. Hutter, J. Chem. Phys. 119, 12417 共2003兲. 12 U. F. Ro¨hrig, I. Frank, J. Hutter, A. Laio, J. VandeVondele, and U. Ro¨thlisberger, Chem. Phys. Chem. 4, 1177 共2003兲. 13 A. Becke, Phys. Rev. A 38, 3098 共1988兲. 14 C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 共1988兲. 15 L. Bernasconi, M. Sprik, and J. Hutter, Chem. Phys. Lett. 394, 141 共2004兲. 16 A. D. Becke, J. Chem. Phys. 98, 5648 共1993兲. 17 C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 共1999兲. 18 J. Blumberger, L. Bernasconi, I. Tavernelli, R. Vuilleumier, and M. Sprik, J. Am. Chem. Soc. 126, 3928 共2004兲. 19 R. Spezia, N. C. Boutin, and R. Vuilleumier, Phys. Rev. Lett. 91, 208304 共2003兲. 1 2



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20

64

21

65
