ionization and double ionization of small water clusters · ionization and double ionization of...

Ionization and double ionization of small water clustersImke B. Müllera� and Lorenz S. CederbaumTheoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg,Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany

�Received 8 May 2006; accepted 31 August 2006; published online 27 November 2006�

The valence ionization and double ionization spectra of the water molecule, of the water dimer, andthe cyclic water clusters �H2O�3 and �H2O�4 are calculated by ab initio Green’s function methodsand discussed in some detail. Particular attention is paid to the analysis of the development of thespectra with increasing cluster size. Electronic decay following inner valence ionization is addressedand a crude estimate for the kinetic energy spectrum of the secondary electrons is given for theclusters. © 2006 American Institute of Physics. �DOI: 10.1063/1.2357921�

I. INTRODUCTION

Water is an ubiquitous liquid with particularly interestingproperties induced by the H bonds and strong impact oneveryday life. Naturally, water has attracted major interest inexperimental and theoretical research. As it is expensive tostudy liquid or solid water by quantum chemical calcula-tions, clusters consisting of two to several water monomershave often been studied to understand the properties of inter-monomeric bonding or structural properties, see, for ex-ample, Refs. 1–3 and references therein. Accurate quantummechanical calculations have in this way helped to param-etrize model potentials, which are used to simulate solvationprocesses and liquid water in general.4–6

Among the water clusters, the water dimer, in particular,attracts a lot of experimental7–10 and theoretical effort be-cause it is suspected to play a very important role in atmo-spheric chemistry.11,12 As the water dimer has been addressedover a large time interval and is still a topic of current re-search, it has evolved to a test case for new methods.13,14

With the theoretical prediction 15 of a new electronicrelaxation process following inner valence ionization of mo-lecular clusters leading to electron emission, the study ofwater clusters and their ionization, in particular, has gainedadditional attraction. Even though the new deexcitation pro-cess intermolecular Coulombic decay �ICD� was already pre-dicted to take place following double ionization of waterclusters,15 it was not yet studied following single ionization.In this contribution, we discuss the single and double ioniza-tion spectra of the water clusters �H2O�2, �H2O�3, and�H2O�4 in their ground state geometry employing ab initioGreen’s function methods. The development of the spectrawith growing cluster size is discussed with particular empha-sis on electronic decay processes following water inner va-lence ionization.

Previous studies on ICD following Ne�2s� ionization inneon clusters16–19 have inspired several experiments, whichhave lead to experimental results on ICD in larger clusters aswell as to detailed coincidence measurements of secondaryelectrons and Ne+ fragments which are products of the ICD

decay following Ne�2s� ionization of the neon dimer.20–22

With this work, we hope to stimulate experimental work onICD in water clusters by analyzing the properties of ioniza-tion and double ionization spectra of the clusters in generaland their evaluation with increasing cluster size and by de-fining the energy interval for the ICD electrons.

II. COMPUTATIONAL DETAILS

All ionization and double ionization spectra have beencalculated by propagator methods.23–26 Propagator methodsallow for the direct computation of ionization energies bysearching the poles of an approximation to the full Green’sfunction. The corresponding pole strengths are related tospectral intensities, which are a measure for the intensities inan experimental spectrum. The poles and pole strengths ofthe approximated Green’s function are derived by solving aneigenvalue problem.

The one-particle Green’s function is evaluated accordingto a third order algebraic diagrammatic construction�ADC�3�� scheme. The ADC scheme transforms the problemof determining the poles and the corresponding polestrengths of the approximated Green’s function to a matrixeigenvalue problem utilizing a diagrammatic perturbationtheory based on a Hartree-Fock �HF� ground-stateansatz.27–29

We adopt the following nomenclature throughout thiscontribution: configurations derived from the HF groundstate by removing one electron from an occupied orbital arereferred to as one hole �1h� configurations. Configurationsderived by adding an electron to a virtual orbital of the HFground state are referred to as one particle �1p� configura-tions. 2h1p and 2p1h configurations are accordingly derivedfrom the HF ground state by removing two electrons fromoccupied orbitals and adding one particle to a virtual orbital,or adding two electrons to virtual orbitals and removing oneelectron from an occupied orbital, respectively.

Owing to the Dyson equation23 behind the ADC�3� ap-proach applied here, the ADC�3� matrix includes, in additionto the large ionization block spanned by all 2h1p configura-tions, an even larger affinity block spanned by all 2p1h con-figurations. These two blocks, in particular, the 2p1h block,a�Electronic mail: [email protected]

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are the leading contributions to the matrix dimension. Thereis no direct coupling between the 2h1p and the 2p1h block ofthe ADC�3� matrix. The two blocks are coupled indirectlyvia their respective coupling to the comparingly small 1h and1p blocks of the ADC�3� matrix. Because of the absence ofdirect coupling of the 2p1h block to the 2h1p block, theformer may be replaced by an approximation of small di-mension, which preserves the information of the correctspectral envelope. This reduction is done by a few iterationsof the Block-Lanczos algorithm29–31 to the affinity part. Thereduction of the affinity part leads to a substantial reductionof the largest matrix dimensions per irreducible representa-tion from 23 000 to 2200 for �H2O�2/aug-cc-pVDZ, 150 000to 16 000 for �H2O�3/aug-cc-pVDZ, and 110 000 to 15 000for �H2O�4/aug-cc-pVDZ�O�cc-pVDZ�H�. All matrices, ex-cept the �H2O�2 ADC�3� matrix, which was subjected to anexact diagonalization, were approximately diagonalized byapplying 400 Block-Lanczos �block size=size of 1h+1pblock� iterations, such that the part of the ionization spec-trum discussed in this contribution was converged accordingto a residue criteria.

To characterize the ionized states, we apply a Mulliken-type 1h population analysis.32 It assigns the ionized states tothe contributing atoms of cluster monomers. This informa-tion may be derived from the 1h information of the �approxi-mated� eigenvector of the respective ionized state.

Double ionization spectra are provided by calculatingthe poles and pole strengths of the particle-particle propaga-tor. The particle-particle propagator is approximated accord-ing to a second-order algebraic diagrammatic construction�ADC�2�� scheme based on the HF ground state. The con-figuration space of the ADC�2� method is spanned by all 2hand all 3h1p configurations.33,34 The ADC�2� matrix is ahigh-dimensional matrix because of the large number of3h1p configurations which have to be taken into account. Itis thus favorable or even necessary to avoid the storage ofthe matrix. Consequently, we performed direct ADC�2� cal-culations, which means that the ADC�2� matrix is never ex-plicitly built up and stored on hard disk.35 Matrix elementblocks are calculated from suitably sorted two-electron inte-grals and orbital energies when they are needed for thematrix�vector multiplication during the Block-Lanczos pro-cedure and are directly passed to the Lanczos diagonalizer.The convergence of the relevant part of the double ionizationspectrum was achieved after 400 block iterations, accordingto a residue criterion. The Block-Lanczos algorithm imple-mented allows for the calculation of approximate eigenvec-tors, such that the 2h part of the vectors may be assignedusing a 2h-population analysis.36 The 2h-population analysisindicates the localization of the two holes on cluster mono-mers.

All Green’s function calculations have been performedstarting from integrals and HF molecular orbitals �MOs� andenergies computed by the GAMESS-UK program package.37

Green’s function calculations are performed using an ADC

program package described in Refs. 29, 30, and 36 and ref-erences therein. As already described above, we used theaug-cc-pVDZ Refs. 38 and 39 basis set taken from a basis setlibrary40 on all atoms to calculate the spectra of �H2O�2 and

�H2O�3. The spectra of �H2O�4 are calculated omitting thediffuse functions on hydrogen because of restricted compu-tational capacities. The aug-cc-pVDZ basis sets were chosenbecause they seem to provide a good compromise betweenthe feasibility of the calculations and the accuracy of theresults. In all calculations, the O�1s� orbitals as well as theirhighest-lying virtual counterparts have been excluded fromthe active configuration space.

III. GEOMETRIES OF THE CLUSTERS „H2O…2–4

The geometries of the water molecule, the water dimer,and the cyclic water trimer were optimized on MP2/aug-cc-pVDZ level using the GAMESS-UK program package. The ge-ometry of water was restricted to C2v symmetry, and thegeometry of the water dimer to CS symmetry during the op-timization procedure. The geometry of the water tetramerwas optimized in S4 symmetry on MP2/�aug�-cc-pVDZ level,which means that no diffuse functions on hydrogen wereapplied in order to use the same basis set during the geom-etry optimization and the computation of the ionization anddouble ionization spectra. The optimized geometries areshown in Fig. 1.

The geometries of all these clusters have been studied inthe literature previously. In particular, the geometries of wa-ter and �H2O�2 are well known. Our structure of the waterdimer is not far from the geometries which were derived inprevious calculations. The O–O distance of 2.92 Å is slightlysmaller than the experimental distance of 2.97–2.98 Å�Refs. 1 and 7–10� and slightly smaller than the theoreticallyderived value from Ref. 13, but slightly larger than that re-ported in Ref. 14. The geometries of the trimer and tetramerare close to the geometries which are optimzed in Ref. 2 withthe same basis set, respectively the basis set, including dif-fuse functions on hydrogen. The differences between the Hbond lengths of the water trimer are smaller in the geometrywe use for the calculation of the spectra than in the geometryoptimized in Ref. 2. Our geometry is rather close to D3 sym-metry. The influence of these smaller deviations between thegeometries on the spectra is small. The geometry of �H2O�4

derived here deviates from the results in Ref. 2, in that theO–O distance is slightly increased instead of decreased com-pared to the water trimer. As it was shown in Ref. 14 thatdiffuse functions are rather important for the geometry opti-mization of the water dimer, we conclude that the differencesbetween the geometries of the water tetramer are due to the

FIG. 1. �Color online� Geometries of the clusters �H2O�2 and �H2O�3 opti-mized by MP2/aug-cc-pVDZ methods, and of �H2O�4 optimized by MP2/�aug�-cc-pVDZ methods. The colored frames have been inserted to helpassign spectral lines in other figures to specific H2O molecules of theclusters.

204305-2 I. B. Müller and L. S. Cederbaum J. Chem. Phys. 125, 204305 �2006�


use of a basis set with different number of diffuse functions.Cyclic structures are the global minimum for clusters com-posed of three and four water molecules.2

IV. OUTER VALENCE IONIZATION SPECTRAOF „H2O…1–4

In Fig. 2, the outer valence ionization spectra of the cy-clic clusters �H2O�3 and �H2O�4 are compared to the spectraof H2O and �H2O�2. The water spectrum shows the well-known structure of three lines. The outermost line, whichdescribes the ionization from the lone pair perpendicular tothe molecular plane, is found at 12.8 eV. The second line at15.2 eV describes the ionization of the lone pair in the mo-lecular plane. This MO is involved in the OH bond. The lastline at 19.11 eV is assigned to the ionization out of the MOwhich forms the OH bond. These values are in good agree-ment with ionization energies calculated by Green’sfunction41,42 and configuration interaction methods.43

The ionization of water clusters, in particular, the ioniza-tion of the water dimer, has previously been addressed in theliterature. In 1982, the He I photoelectron spectrum of thewater dimer was published44 and the nature and moleculardynamics of the first two ionized states were discussed in aseries of subsequent publications.44–48 The characeterizationof the second ionized state was controversial, finally it wasassigned as ionization of the H-acceptor molecule,48 which isin agreement with our results which are discussed in thefollowing. Three more recent publications address the com-parison of the first two ionized states of the water dimer tothose of HF–H2O and NH3–H2O,49 and the molecular dy-namics following ionization50 and multiphotonionization.51

Fewer publications have appeared on ionized states on largerwater clusters. The structural changes of �H2O�3 and �H2O�n,n=3–6, have been addressed by geometry optimization52

and trajectory methods,53 respectively. Valence ionizationspectra including the inner valence region are, to the best ofour knowledge, not yet available for the water clusters stud-

ied here. From the publications addressing the relaxation ofthe water clusters following ionization, it is obvious that theionization of water clusters leads to severe changes of thecluster structure, for example, to H transfer. This leads tobroad bands in the experimental ionization spectra, which arenot found in our numerical results, which consist exclusivelyof vertical ionization energies.

The ground state geometry of the �H2O�2 cluster is char-acterized by two H bond water molecules on perpendicularmolecular planes. The outer valence ionization spectrum of�H2O�2 reflects this geometry. It is composed of two sets oflines. The lines composing the outer valence ionization spec-trum of the H donor are approximately 0.8–0.9 eV below thecorresponding lines in the ionization spectrum of the watermolecule. The ionization energies of the H-acceptor watermolecule are 0.6–0.7 eV above the corresponding lines inthe spectrum of the water molecule. The energy gap betweencorresponding lines in the ionization spectrum of theH-donor water molecule and the H-acceptor water moleculeis large and not substantially smaller than the energy split ofthe first two ionization energies of the water molecule.

As the two water molecules of the water dimer are situ-ated on perpendicular molecular planes, corresponding MOsof the two water molecules are also perpendicular, such thatthey do not mix. The lines of the ionization spectrum of thewater dimer are consequently dominated by 1h contributionson one of the two water molecules. The energy gap betweencorresponding states of the two molecules is thus a result ofcharge transfer and not of molecular orbital interaction.

The third spectrum shown in Fig. 2 describes the outervalence ionization spectrum of the cyclic �H2O�3 cluster. Asalready stated above, the cyclic structure discussed here isvery close to a D3-symmetric structure with three equivalentwater molecules. The outer valence ionization spectrum of�H2O�3 consists of three line groups, assigned to linear com-binations from the respective MOs on each water molecule.Each of the groups is composed of three resolved lines. None

FIG. 2. �Color online� Outer valenceionization spectra of H2O, and of theclusters �H2O�2, �H2O�3 and �H2O�4.The colors of the spectra indicate con-tributions of holes on different watermolecules to the ionized states. Thecolors correspond to those of theframes in figure VII. The orange linesinterconnect ionized states with simi-lar characteristics.

204305-3 Ionization of small water clusters J. Chem. Phys. 125, 204305 �2006�


of the lines is dominated by 1h contributions on one of thewater molecules only. This means that, in contrast to thewater dimer, the three water molecules strongly mix. Themixing of the three water molecules composing the watertrimer is due to the fact that their molecular planes are notperpendicular in the cyclic structure of �H2O�3 leading to anearly D3-symmetric structure.

The three groups of outer valence lines of �H2O�3, eachcontaining three individual lines, are composed of 1h contri-butions according to the following principles. In each group,one line is characterized by approximately equal 1h contri-butions from each of the water monomers, one line is char-acterized by equal 1h contributions from two of the mol-ecules, whereas the remaining water molecule contributesapproximately two-thirds to the third line. One-third of thislast line is divided into 1h contributions on the other twowater molecules. As a rule, the lines with nonequal 1h con-tributions of the three water molecules are close in energy,whereas the gap to the “equally” distributed line is larger.This pattern reflects the proximity of the �H2O�3 structure toD3 symmetry. A pattern of one line with equal contributionsand two degenerate lines with 1/2, 2 /3, 1 /6, 1 /6 contribu-tions is characteristic for D3 symmetry. The deviations fromthis pattern are thus a measure for the deviation of the clustergeometry from the ideal D3 symmetric one. The deviation ofthe cyclic water trimer from D3 symmetry is most promi-nently reflected by the line group of the lone pairs perpen-dicular to the respective molecular plane. Here, the third lineshows little contributions from “green” water, whereas greenwater is overrepresented in the second line to make up forthe character of the third line. In addition to the compositionof the lines, also the energy gaps between the lines deviatefrom the pattern discussed above. Instead of finding a smalland a considerably larger energy gap between the lines asexpected, the energy gaps between the three lines are rathersimilar.

It is a characteristic of D3 symmetric clusters containingthree water molecules that the position of the line character-ized by equal contributions of all three water molecules withrespect to the two energetically degenerate lines reflects therelative orientation of the corresponding MOs. In case thatthe orientation of the MOs of the three water molecules fa-vors a bonding interaction, the line with equal contributionsis found at higher ionization energies. In the case of �H2O�3,the orientation of the water MOs is such that a bonding com-bination with equal contributions from all water moleculesmay be formed for the lone pairs perpendicular to the mo-lecular planes and for the OH levels, but not for the remain-ing lone pairs in the molecular plane. The equally distributedline of the latter thus shows up at lower ionization energiesthan the remaining two close-lying lines.

It is noteworthy that the lines in each group in the spec-trum of �H2O�3 are spread over a smaller energy intervalthan the pairs in the spectrum of the water dimer. Obviously,the three water molecules are rather equivalent in �H2O�3,such that all of them act both as H-donor and H-acceptormolecules. The MO interaction described above, whichcauses the energy gaps between the lines in the spectrum of�H2O�3, is weaker than the effect of the charge transfer

between H donor and H acceptor in the spectrum of �H2O�2,resulting in smaller energy gaps between correspondinglines.

The outer valence ionization spectrum of cyclic �H2O�4

consists of three resolved lines for each of the three outervalence line groups. The second line of each group hides anenergetically degenerate line. The first and the last line ofeach group are characterized by equal 1h contributions ofeach of the four water molecules. The degenerate line pairshows equal contributions of each water molecule in total,but not to the individual lines. Contributions of opposite wa-ter molecules are equivalent for all lines, though. This linepattern is achieved due to the S4 symmetry of the cluster,which enforces equal contributions from opposite water mol-ecules. Furthermore, equivalent 1h contributions from allwater molecules are found for those lines representing ion-ization from cluster MOs with zero or two additional nodalplanes due to cluster formation with respect to the number ofnodal planes in the composing water MOs. Typically, theso-formed line groups are energetically more widespreadthan corresponding line groups in the spectrum of compa-rable clusters with three water molecules. This is because therelative orientation of four water molecules in a circle avoid-ing distortion of the HOH angles allows for a more efficientinteraction between the MOs of the molecules than the ori-entation of three water molecules in a circle avoiding distor-tion of the H2O geometry.

Interestingly, the first ionization potential of the waterclusters shows no systematic development with increasingnumber of water molecules in the cluster. As we have alreadydiscussed above, different influences are mapped in the spec-tra, at least comparing that of �H2O�2 to the other spectra. Itis therefore interesting to intercompare mean values of thegroups’ ionization energies in a first step to analyze the in-fluence of cluster formation and address the development ofthe ionization potential afterwards. The mean value of theionization energy of the lone pairs perpendicular to the re-spective water molecular plane slightly decreases from 12.85for water to 12.68 for the water dimer and 12.58, and 12.53for the trimer and tetramer, respectively. The mean value ofthe ionization energy of the lone pair in the respective waterplane also shows only small changes, 15.16 is found forwater, 14.95, 14.96, and 14.89 for the dimer, trimer, andtetramer. The mean ionization energy of the OH bond levelsdeceases slowly from 19.11 for water to 19.01, 18.86, andfinally 18.85 for the tetramer. Obviously, cluster formationfrom neutral monomers does not change the mean ionizationenergy of corresponding groups significantly. Nevertheless,the ionization potential monotonically decreases with in-creasing number of water molecules in the cluster, disregard-ing the noncyclic water dimer. The decrease of the ionizationenergies with increasing number of water molecules in cycliccluster is achieved by stronger interaction between corre-sponding levels of particular water molecules with increasingcluster size. The impact of MO interaction increases alsobecause the relative orientation of the water molecules be-comes more favorable for interaction. This means that a fur-ther decrease of the ionization potential of correspondinglevels is thus expected with increasing cluster size of cyclic



water clusters composed of �next to� equivalent monomers aslong as the molecules are oriented favorable for interaction.As long as the the ring strain is small, the ionization energyof cyclic water cluster can be expected to decrease slowlywith increasing cluster size. The mean ionization energy isexpected to remain rather constant.

V. INNER VALENCE IONIZATIONAND INTERMOLECULAR ELECTRONIC DECAY

A. Spectra and their analysis

The inner valence ionization spectra of the cyclic �H2O�3

and �H2O�4 clusters and those of the water molecule and thewater dimer are shown in Fig. 3. Below each of the innervalence ionization spectra, the lowest states in the doubleionization spectrum of the respective cluster are depicted.The inner valence ionization spectrum of the water moleculeis characterized by a “breakdown of the MO” picture,54

which means that the intensity originating from the MO isdistributed over several ionic states and no main line, repre-senting the majority of the spectral intensity, is present. Theinner valence ionized state of water is not at all well repre-sented by its 1h contribution alone.

The breakdown characteristics already present in the iso-lated water molecule are even more pronounced in the innervalence ionization spectrum of �H2O�2, and particularly inthose of the larger clusters. The inner valence ionizationspectrum of �H2O�2 is composed of two groups of severallines which gain small spectral intensity. The group belong-ing to the spectrum of the H donor appears at lower ioniza-tion energies than that of the H acceptor. The mean innervalence ionization energy is rather similar to that of the watermolecule.

The line groups describing the inner valence ionizationof �H2O�3 and �H2O�4 are also found at about 33 eV. As allwater molecules are equivalent in �H2O�4 and nearly equiva-lent in �H2O�3, only one group consisting of many lines isfound in the spectrum. Individual line groups representingthe three and four inner valence MOs are not resolved.

It is interesting to compare the shape of the inner valenceionization spectra of the clusters somewhat more quantita-tively. The spectra of water, the water dimer, and the trimerhave been calculated using the aug-cc-pVDZ basis set for allatoms. The ionization spectrum of �H2O�4 was calculatedwithout diffuse functions on hydrogen, such that the differ-ence between the basis sets has to be taken into account inthe discussion. The shape of the inner valence ionizationspectrum changes from water to �H2O�3 in that the numberof low-intensity lines increases and the spectral intensity ofthe individual lines decreases. The intensity is obviouslyspread over an increasing multitude of lines. The spectralintensity is a measure of the 1h contribution to a line, suchthat the difference to “1” describes the total impact of the2h1p contributions. The impact of 2h1p contributions in-creases with increasing number of water molecules in thecluster. Visibly, also the �H2O�4 spectrum is characterized bya large number of lines with very small spectral intensity, butthere exist lines with 1h contributions larger than those in thespectrum of �H2O�3. A priori, one would expect the intensi-ties of the individual lines in the spectrum of �H2O�4 to de-crease or at least to remain comparable to those of �H2O�3.The numerical finding that this is not the case can be attrib-uted to the lack of diffuse functions on hydrogen.

To analyze the composition of the lines in detail, wehave performed a 1h+2h1p population analysis of the eigen-vectors of the inner valence ionized states of water and of

FIG. 3. �Color online� Inner valenceionization spectra of H2O, and of theclusters �H2O�2, �H2O�3 and �H2O�4.The colors of the spectra indicate con-tributions of holes on different watermolecules to the ionized states. Thecolors correspond to those in Fig. 2. Atthe bottom of each figure, a small fig-ure is attached depicting the energeti-cally lowest states of the double ion-ization spectrum. Here, green colorssymbolize “two-site states” with 2hcontributions distributed over at leasttwo water molecules, and black colorsstand for “one-site states” with 2h con-tribution with both holes localized onthe same water molecule. Red colorsappear only for �H2O�2 and indicateone-site states of the H-donor watermolecule while black colors indicate“one-site” states of the H-acceptormolecule of this cluster. Orange colorsappear only for �H2O�4 and indicatesholes on adjacent molecules whilegreen indicates holes on opposite mol-ecules in this cluster.



�H2O�2. Full eigenvectors of the ionized states of �H2O�3 andof �H2O�4 are not available because a full diagonalization ofthe ADC�3� matrix was not feasible, and a1h+2h1p-population analysis is not accessible. The 1h+2h1p population analysis assigns the contributions of theADC eigenvectors to particular basis functions applying aMulliken-type algorithm. In the analysis the contributions ofdiffuse functions on each water molecule are summed upseparately from the contributions of nondiffuse functions foreach water molecule and ionic state. The resulting excesscharges with respect to the ground state charge distributionare shown in Fig. 4. The excess charge in nondiffuse func-tions of each state is plotted by circles, whereas the trianglessymbolize excess charge in diffuse functions. For the isolatedH2O, the states between 30 and 34 eV are characterized byone positive charge in nondiffuse functions and no excesscharge in diffuse functions. Above 35 eV, highly excitedstates with up to two holes in nondiffuse functions and up toone particle in diffuse functions appear due to the breakdownof the molecular orbital picture.

The excess charge of the H-donor water molecule of�H2O�2 is plotted in the right upper part of the Fig. 4 and theexcess charge of the H-acceptor molecule is depicted in thelower part of this figure. Obviously, the number of states inthe energy region of water inner valence ionization is muchlarger for the water dimer than for the water molecule. Somestates with more than a single positive excess charge in non-diffuse functions and up to one negative charge in diffusefunctions are found in the charge distributions. The majorityof the states, however, exhibit a different behavior. Here, onepositive charge is localized on each of the two water mol-ecules, and the negative excess charge is distributed over thediffuse functions of both molecules. This kind of states is a

clear signature of intermolecular electronic decay �ICD�, amechanism leading to the ultrafast emission of secondaryelectrons as a consequence of the initial inner valence ion-ization. The ICD electron is simulated in the calculationby particles in diffuse functions, because continuum electronfunctions are not included in the calculation. The finaldicationic states of the cluster are characterized by two va-cancies, each localized on a different H2O molecule of thecluster.

B. On ICD

When is ICD following inner valence ionization of waterclusters possible? The emission of secondary electrons as aconsequence of inner valence ionization is possible if theinner valence ionized state is energetically above at least thethreshold of double ionization of the system. We thus have tocompare the inner valence ionization spectra of the watermolecule and the water clusters to the energetically mostfavorable doubly ionized states. This comparison is done inFig. 3. Double ionization potentials and inner valence ioniza-tion energies are also listed in Table I. For the isolated watermolecule, the energetically lowest states in the double ion-ization spectrum are slightly above the higher-lying satellitesof the inner valence ionization spectrum and approximately5 eV above the line carrying the largest spectral intensity.Relaxation of the inner valence ionized states by electronemission is energetically forbidden.

The onset of the double ionization spectrum of �H2O�2 isbelow the inner valence ionization energy of the H-acceptorand also of the H-donor water molecules. Electronic relax-ation to double ionized states is thus possible. The ICD cantake place because the double ionization potential of �H2O�2

FIG. 4. �Color online� Distribution ofexcess charge �relative to the HFground state� over the various ionicstates of H2O and its dimer �H2O�2 inthe inner-valence binding energy rangebetween 28 and 40 eV. On the left:Excess charge in diffuse functions �tri-angles� and in nondiffuse functions�circles� of water are shown. On theright: Data on �H2O�2 separately foreach of the two water molecules �up-per panel: H-donor water molecule,lower panel: H-acceptor watermolecule�.



is considerably decreased compared to that of water. Theenergetically low-lying states of the dication are character-ized by delocalized holes, where one hole charge is localizedon each of the water molecules. These states are referred toas “two-site states” in the following. Due to the hole delo-calization, the Coulombic hole-hole repulsion is muchsmaller for these states than for the states where both holesare on the same monomer. The latter are referred to as “one-site states.” The onset of the “one-site state” double ioniza-tion spectrum of �H2O�2 is indeed above the inner valenceionization energies of �H2O�2. Relaxation by electron emis-sion following inner valence ionization of �H2O�2 thus leadsto final states with distributed hole charges. The majority ofthe states in the �H2O�2 inner valence ionization spectrumhas indeed been found by our analysis discussed above tosimulate two-site states and an electron in the continuum.The presence of such states is a consequence of ICD. Themechanism of the ICD process has already been described indetail elsewhere.16,55–57 The ICD mechanism can be brieflydescribed as follows. The initial inner valence hole on mol-ecule 1 �M1� is refilled by an electron of the same monomer�M1�. The excess energy gained by refilling the hole is thentransferred to a second molecule �M2� of the cluster where asecondary electron is set free.

Certainly, an increasing number of accessible decaychannels increases the efficiency of this process. Conse-quently, ICD processes become more and more efficient withincreasing cluster size, because more and more decay chan-nels open. This means that the lifetimes of the initial statesusually decrease with increasing cluster size. Typically, theline group in the ionization spectrum, which describes theelectronically decaying state, is characterized by several low-intensity lines and, if the basis set is sufficiently large, bypronounced line bundles. If the final states are relatively wellrepresented in the basis set chosen despite the fact that thefree electron cannot be described, the envelope of the line

bundle describes a Lorentzian. The half-width of the Lorent-zian is a measure for the decaying state’s lifelime. The innervalence spectra of the water clusters, in particular, in those of�H2O�3 and �H2O�4, certainly show line bundles, but as thecontributions of different inner valence MOs are not ener-getically resolved, the profile of the envelop is not a Lorent-zian. It is thus not possible to derive lifetimes from the spec-tra alone. It is nevertheless probable that the lifetimesdecrease with increasing cluster size as can be envisioned bycomparing the line groups and counting the accessible finalstates in the double ionization spectrum. It should be favor-able to compute the lifetimes by calculating the decay widthsexplicitly, for example, by propagator/complex-absorbingpotential methods

To crudely estimate the distribution of the ICD electronsas function of their kinetic energy, which is of interest forexperimental purposes, we make use of the envelopes of theinner valence ionization spectra and of the double ionizationspectra of the clusters. The spectral envelopes were calcu-lated by convoluting each individual line by a Lorentian witha half-width of 0.5 eV. The kinetic energy distributionsshown in Fig. 5 are derived by assuming that the probabilityof the transitions is independent of the energy differencesbetween the initial and the final state. The kinetic energydistribution was accordingly derived by summing up all pos-sible products of the envelope of the inner valence spectrumand that of the double ionization spectrum for which theenergy difference corresponds to the kinetic energy of thesecondary electron considered.

The resulting ICD electron distributions for �H2O�3 and�H2O�4 are similar in shape. The majority of the ICD elec-trons emitted has kinetic energies below 8 eV, and the dis-tributions exhibit local maxima at kinetic energies of 3 and5 eV. The ICD electron distribution of �H2O�2 is less struc-tured.

TABLE I. Ionization energies and double ionization potentials of water clusters. First column: Outer valenceionization energies Eov. Second column: The energy�ies� Eiv of the most intense inner valence ionized state.Third column: The double ionization potential of two-site state formation �ionization of two different watermolecules�. For �H2O�4, the values for opposite molecules and for adjacent molecules are given separately.Fourth column: The double ionization potential of one-site state formation �double ionization of one watermolecule�. All energies are given in eV.

System Eov Eiv DIPtwo-site DIPone-site

H2O 12.85 33.37 ¯ 38.6315.1619.11

�H2O�2 11.91, 13.45 32.59, 34.10 27.97 36.45, 39.9214.07, 15.8418.28, 19.74

�H2O�3 12.40, 12.54, 12.80 33.21 27.89 38.1614.32, 15.22, 15.3318.53, 18.59, 19.46

�H2O�4 12.18, 12.54, 12.54, 12.88 33.79 26.28opp., 27.72adj. 37.9814.09, 14.84, 14.84, 15.7918.20, 18.86, 18.86, 19.50



VI. THE COMPOSITION OF THE DOUBLE IONIZATIONSPECTRA OF THE WATER CLUSTERS

The double ionization spectra of clusters are particularlyinteresting because they consist of two parts. The first partcomprises the double ionization spectra of the composingmonomers and the second part the spectrum of the mixeddouble ionized states, describing the ionization of two differ-ent monomers of the cluster. Cluster double ionization spec-tra are therefore more complex and much richer than mo-lecular double ionization spectra, and their assignment isoften impossible without the help of suitable quantumchemical calculations. In this section, we analyze the doubleionization spectra of the clusters �H2O�2–4 and compare themto the double ionization spectrum of water.

The double ionization spectra of water and of the three

clusters �H2O�2, �H2O�3, and �H2O�4 are shown in Fig. 6. Asexpected, the complexity of the cluster spectra exceeds thatof the water spectrum by far. Being the reference system forthe one-site spectra of the clusters, we discuss first the com-position of the double ionization spectrum of water.

The double ionization energies of the water moleculemay be estimated from the HF-MO energies by adding arepulsion/exchange contribution to the respective sum of theMO energies. As a rule, this energy contribution is approxi-mately 13 eV for the singlet states and about 10 eV for thetriplet states. The lowering of the energy of the triplet statesis due to exchange energy of the two holes and is approxi-mately 3 eV for the water molecule.

The water double ionization spectrum is accordinglycomposed as follows. The energetically lowest doubly ion-

FIG. 5. Envelops of the inner-valence ionization spectra �A� and of the double ionization spectra �B� of the water clusters �H2O�n, n=2–4. Each line in thespectra has been folded by a Lorentzian. For each spectrum two envelops are depicted, one folded with a Lorentzian with a half-width of 0.5 eV and one witha half-width of 0.1 eV. �C� shows an estimated distribution of the ICD electrons as a function of their kinetic energy. The kinetic energies are derived fromthe two 0.5 half-width curves. It has been assumed that the cross section for the transition from an ionized to a doubly ionized state is independent of thekinetic energy of the emitted electron and of the states involved. More details are given in the text. All intensities are not normalized and thus given in arbitraryunits.



ized state is a triplet state with one hole charge in each of thetwo highest-lying outer valence levels. The next state is thesinglet state with two holes in the energetically highest levelwhich preceeds the singlet state corresponding to the tripletstate. The fourth state is the triplet state describing doubleionization with one hole charge in the highest level and onehole charge in the energetically lowest outer valence leveland is followed by a group of three states with rather similarenergies. These are the singlet state corresponding to the lat-ter triplet state, the singlet state with both holes in the secondouter valence level, and the triplet state with one hole in thesecond and one in the lowest outer valence level. The corre-sponding singlet state of the latter is the next state and ispursued by the singlet state with two holes in the lowest-lying outer valence level.

The first line of the double ionization spectrum where avacancy in the inner-valence orbital is present is found at57 eV. It represents a triplet state with an outer valence holein the outermost orbital. The two other triplet states made ofone hole in inner-valence and one hole in one of the remain-ing outer-valence orbitals follow. The next two individuallines and the line group depicted in Fig. 6 are assigned to theouter valence/inner valence singlet states corresponding tothe latter three triplets, and the last line group is related to theinner valence/inner valence singlet state. The latter twogroups of lines are not well characterized by their 2h contri-butions alone, and one may speak of a breakdown of the MOpicture for these states.

As already mentioned above, the double ionization spec-trum of �H2O�2 is more complex than that of the water mol-ecule as two-site states with delocalized holes appear anddecrease the double ionization potential considerably. We

start our discussion of the double ionization spectra of thewater clusters by assigning the one-site states. In Fig. 6, theone-site double ionization spectrum of the H-donor watermolecule is depicted in red colors and that of the H acceptoris depicted in black. The structure of each of these spectra isanalogous to the double ionization spectrum of H2O dis-cussed above. The H-donor one-site spectrum is subjected toa shift of approximately 2 eV towards lower energies com-pared to the water spectrum, that of the H-acceptor watermolecules is shifted by approximately 1 eV to higher ener-gies. Interestingly, these shifts are smaller than those foundin the singly ionized states.

Above 55 eV, the states in the one-site double ionizationspectrum of the dimer are no longer well represented by their2h contributions alone. Interestingly, the onset of the innervalence/outer valence one-site spectrum coincides with theonset of important 3h1p contributions to the doubly ionizedstates. Line bundling and clear breakdown of the molecularorbital picture are thus observed at lower energies than forthe isolated water molecule. Indeed, ICD of doubly ionizedstates of the dimer takes place and is not present in the watermolecule.15

The one-site double ionization spectra of the cyclic clus-ters �H2O�3 and �H2O�4 are assigned more easily by com-parison to the water spectrum than for the water dimer be-cause all water molecules are �nearly� equivalent and theinteraction between them weaker. Consequently, their doubleionization spectra are characterized by groups of nearly de-generate lines. The energies of corresponding line groups inthe spectra of �H2O�3 and �H2O�4 are very similar to eachother. The one-site double ionization spectrum of the water

FIG. 6. �Color online� Double ionization spectra of H2O, and of the clusters �H2O�2, �H2O�3 and �H2O�4. The colors of the spectra stand for different 2hcontributions to the double ionized states. Green colors symbolize “two-site states” with 2h contributions distributed over at least two water molecules, blackcolors symbolize “one-site states” with 2h contributions characterized by both holes being localized on the same water molecule. Red colors appear only for�H2O�2 and indicate one-site states on the H-donor water molecule, black colors indicate those on the H-acceptor molecule in this spectrum. Orange colorsappear only for �H2O�4 and indicates holes on adjacent molecule while green indicates holes on opposite molecules in this cluster. Light blue and magentalines connect states with similar characteristics in the spectra of �H2O�2 and �H2O�3 and in those of �H2O�3 and �H2O�4.



dimer shows the most complex composition among the one-site spectra discussed here because the cluster is composedof nonequivalent monomers with different electrostatic prop-erties

Let us now discuss the two-site spectra and begin withthe water dimer. The outer valence of the water moleculeconsists of three energy levels with energy gaps of 2 and4 eV between them. These levels are assigned as “low �l�,”“medium �m�,” and “high �h�” in the following. Furthermore,each water molecule contributes an inner valence level, at anionization energy of 14 eV above the highest outer valencelevel “h.” The two-site spectrum of the water dimer and thecyclic clusters consists of all combinations between the “l,”“m,” “h,” and inner-valence levels on different water mol-ecules. Energetically, the outer-valence part of the spectrumbegins with l/l lines followed by l/m lines, which precedem/m lines. The latter are followed by l/h lines, which precedem/h and h/h lines. Singlet and triplet states of the two-sitespectrum are not resolved in Fig. 6 because the exchangeinteraction between electrons on different monomers is verysmall. Specifically, for the water dimer in Fig. 6, the gapbetween the l/l line and the two l/m lines is approximately2.5 eV, that between the two l/m lines is 0.5 eV. The gapbetween the l/m lines and the m/m line is again 2.5 eV. Thetwo unresolved l/h lines in Fig. 6 are 1.5 eV above the m/mline. The m/h lines are 3 eV above the l/h line and 3 eVbelow the h/h line. For �H2O�2, the energetical gap betweenthe outer valence/outer valence part of the spectrum and theonset of the inner valence/outer valence part is 7 eV. Theiv/ov two-site spectrum is composed in an obvious manner:“l / iv” � “m/ iv” � “h/ iv” �iv=inner valence�. The energygaps are in the range of 3 eV. The “iv/iv” two-site statesappear approximately 14 eV above the highest outer valence/inner valence two-site state.

The two-site spectrum of �H2O�3 may readily be as-signed by comparison to that of �H2O�2. The energies oflines with iv hole contributions, in particular, are similar forboth clusters. The outer valence spectra appear rather differ-ent at first sight due to the larger number of states with simi-lar characteristics in the spectrum of �H2O�3. The energies ofcorresponding states in the spectra are nevertheless rathersimilar. The energy difference between the H-donor and theH-acceptor water molecule of �H2O�2 is averaged out in thetwo-site spectrum of the dimer because it is necessary toionize both H donor and H acceptor to create a two-site state.The average two-site ionization energy is consequently simi-lar for �H2O�2 and �H2O�3 for states with similar hole char-acteristics.

The two-site double ionization spectrum of �H2O�4 issomewhat more complex than the respective spectra of�H2O�2 and �H2O�3. As a consequence of the cyclic structureof the cluster, each water molecule has two direct neighbors�adjacent molecules� and one water molecule opposite to it�opposite molecule�. The O–O distance between oppositewater molecules is 4.05 Å, that between adjacent monomersis only 2.86 Å. The hole-hole repulsion between adjacentand opposite pairs of water molecules differs consequentlyby roughly 1.5 eV. The two-site spectrum of adjacent holelocalization is plotted in orange in Fig. 6, that of opposite

two-site states in green. The energies of the orange linesobviously coincide with the energies of corresponding statesin the two-site spectrum of �H2O�3 for iv/ov and iv/iv states,i.e., at ionization energies above 45 eV. The ov/ov energiesof the adjacent two-site spectrum are also very similar tothose of the corresponding states in the spectrum of �H2O�3.Because of the richness of the spectrum, this similarity isharder to recognize, though.

In the iv/iv and ov/iv parts of the spectrum, one observesthat each orange �adjacent� line group is indeed preceded bya green �opposite� one. The energy gap is approximately1.5 eV, according to the different hole-hole repulsion esti-mated above. The same pattern is found, though less obviousbecause of overlapping line groups, in the ov/ov spectrum.The extra lines in figure VII connecting spectra of differentclusters, colored in cyan and magenta, help to identify cor-responding line groups. The presence of two-site states withholes on opposite water molecules is responsible for thecomparingly low double ionization potential of �H2O�4,which leads to a slightly broader kinetic energy distributionsof ICD electrons than for the other clusters as discussed inSec. V B.

Above 50 eV for �H2O�2 and 45 eV for �H2O�3–4, thetwo-site spectrum of �H2O�3 and �H2O�4 is characterized bydistributions of lines with particularly low spectral intensi-ties. These line groups, and also those of the one-site spec-trum above 50 eV, show pronounced line bundling. We havecalculated the triple ionization potential of �H2O�2, �H2O�3,and �H2O�4 to find out whether this line bundling may berelated to electronic decay processes of the doubly ionizedstates by quadratic configuration interaction single double�QCISD� methods. We find that the triple ionization thresh-old is about 60 eV for �H2O�2, whereas it is only about50 eV for �H2O�3 and �H2O�4. Although computed by differ-ent ab initio methods, the possible error of the relative ener-gies between the triple ionization threshold and the energiesof the double ionization spectrum is too small to affect ourresults qualitatively. We conclude that for the larger clusters�H2O�3 and �H2O�4 most structures showing pronounced linebundling are above the triple ionization threshold. The linebundles located above 50 eV certainly correspond to dica-tionic states which are subjected to electronic decay pro-cesses. We further note that the onset of line bundling in thedouble ionization spectrum of �H2O�2 is in the energy regionof the triple ionization threshold. Clearly, electronic decay isalso relevant to understand the higher energy part of thisspectrum.

VII. SUMMARY AND CONCLUSION

We have analyzed the ionization and double ionizationspectra of the water molecule, the water dimer, and the cyclicwater trimer and tetramer calculated by Green’s functionmethods. The mean valence ionization energies of all relatedstates of all water clusters studied here are close to the va-lence ionization energies of water. Differences of individualenergies to those of the water molecule are found in thecluster spectra due to water-water interactions. The valenceionization spectrum of the water dimer shows the most



prominent deviations from the ionization spectrum of water.In contrast to the cyclic clusters, the water dimer is com-posed of two nonequivalent water molecules, the H acceptorand the H donor of the H bond connecting the two watermonomers. The ionization energies of the H-donor watermolecule are considerably smaller than the ionization ener-gies of the H-acceptor water molecule.

In the cyclic clusters, the energy splittings of similarlevels of different water molecules which would be degener-ate in the absence of interactions, are rather the result ofMO-MO interactions and not of the H bonds interconnectingthe cluster monomers. These energy splittings are conse-quently smaller than the energy differences between theH-donor and the H-acceptor units, found in the spectrum of�H2O�2. This leads to the interesting finding that the ioniza-tion potential of the water dimer is smaller than that of thecyclic water clusters. The total impact of the MO-MO inter-actions increases with increasing number of water moleculesin the cyclic clusters and this is also favored by the changingrelative orientation of the water molecules. Consequently, theionization potential of the cyclic clusters decreases with in-creasing number of water molecules and the outer valenceionized states evolve from being localized on one water mol-ecule to being delocalized over several water molecules.

Already the inner valence ionization spectrum of the iso-lated water molecule shows a breakdown of the molecularorbital picture. The characteristics of a breakdwon of the MOpicture are, however, much more prominent in the inner va-lence spectra of the clusters. The number of lines increasesdisproportionate to the number of water molecules in thecluster. The spectral intensity of individual lines clearly de-creases from H2O to �H2O�2 and from �H2O�2 to the largerclusters. In contrast to the inner valence ionized state of wa-ter, the respective states of the water clusters are above thedouble ionization threshold and may thus undergo electronicrelaxation processes via electron emission. We have shownby means of a population analysis that ionized states withtwo holes, one on each of the two water monomers, and anelectron in diffuse virtual orbitals, contribute significantly tothe inner valence ionization of the water dimer. This is char-acteristic for ionized states decaying by ICD. Clearly, theinner valence ionized states of the cyclic water clusters un-dergo the same kind of electronic decay processes as theinner valence ionized states of the water dimer.

We have crudely estimated the kinetic energy spectrumof the ICD electrons emitted following inner valence ioniza-tion of the investigated water clusters. Owing to the manycontributing states and open channels and the approxima-tions �equal decay probability of each computed state to eachcomputed final state� used, the kinetic energy spectra of theICD electrons are rather similar for all three clusters. Themajority of these electrons acquire kinetic energies below8 eV and the distributions exhibit an oscillatory behavior.These estimated distributions of the kinetic energies of theICD electrons emitted following inner valence ionization ofsmall water clusters should be at least useful for the setup ofmeasurements of intermolecular electronic decay processes.

The double ionization spectra of the water clusters arevery rich in structures and phenomena. Each spectrum con-

tains several subspectra of different physical origin. Onekind of subspectra is due to the “one-site” doubly ionizedstates describing the double ionization of one of the clusters’molecules. The other kind is due to the “two-site” states withtwo holes, each on a different molecule of the cluster. Thisdelocalization of the holes reduces substantially the doubleionization potential of water clusters placing it below thewater inner valence ionization energy and opening in thisway the channel for ICD. The two-site double ionizationenergies are similar for �H2O�2 and �H2O�3 and for thosestates describing ionization from two adjacent water mol-ecules of �H2O�4. The two-site states with one hole charge oneach of the two opposite water molecules of �H2O�4 appearat approximately 1.5 eV lower energies than the correspond-ing states with holes on adjacent molecules. Consequently,the two-site double ionization spectrum of the tetramer ismore complex than those of the water dimer and trimer. Theone-site double ionization spectra of the clusters may bereadily assigned by comparison to the spectrum of the watermolecule. In contrast to the case of the two-site spectrum, theone-site spectrum of �H2O�2 is more complex than the otherspectra because the two water molecules are not equivalent.

Like for the single ionization, also the double ionizationspectra of all systems investigated here show characteristicsof a breakdown of the MO picture. In addition, the latterspectra of the clusters also exhibit the fingerprints of ICD.Interestingly, the triple ionization threshold of H2O is 85 eV,that of �H2O�2 is approximately 60 eV, and that of the cyclicclusters approximately 50 eV. As a consequence, most of thedoubly ionized states of the clusters in the energy range ofthe breakdown of the MO picture are subjected to relaxationby secondary electron emission. We conclude that electronicdecay must be taken into account to unterstand both valenceionization and double ionization processes of water clustersand hope that the present contribution will thus stimulatefurther studies on intermolecular electronic decay processesin water clusters.

ACKNOWLEDGMENT

Financial support by the DFG is gratefully acknowl-edged.

1 K. S. Kim, B. J. Mhin, U.-S. Choi, and K. Lee, J. Chem. Phys. 97, 6649�1992�.

2 S. S. Xantheas and T. H. Dunning, Jr., J. Chem. Phys. 99, 8774 �1993�.3 M. Yang, P. Senet, and C. van Alsenoy, Int. J. Quantum Chem. 101, 535�2005�.

4 W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L.Klein, J. Chem. Phys. 79, 926 �1983�.

5 J. Caldwell, L. X. Dang, and P. A. Kollmann, J. Am. Chem. Soc. 112,9145 �1990�.

6 L. Perera and M. L. Berkowitz, J. Chem. Phys. 95, 7556 �1991�.7 T. R. Dyke and J. S. Muenter, J. Chem. Phys. 60, 2929 �1974�.8 T. R. Dyke and J. S. Muenter, J. Chem. Phys. 57, 5011 �1972�.9 T. R. Dyke, K. M. Mack, and J. S. Muenter, J. Chem. Phys. 66, 498�1977�.

10 J. A. Odutola and T. R. Dyke, J. Chem. Phys. 72, 5062 �1980�.11 N. Goldman, R. S. Felers, C. Leforestier, and R. J. Saykally, J. Phys.

Chem. A 105, 515 �2001�.12 N. Goldman, C. Leforestier, and R. J. Saykally, J. Phys. Chem. A 108,

787 �2004�.13 J. G. C. M. van Duijneveldt-van de Rijdt and F. B. van Duijneveldt, J.

Chem. Phys. 97, 5019 �1992�.



14 D. Feller, J. Chem. Phys. 96, 6104 �1992�.15 L. S. Cederbaum, J. Zobeley, and F. Tarantelli, Phys. Rev. Lett. 79, 4778

�1997�.16 R. Santra, J. Zobeley, and L. S. Cederbaum, Phys. Rev. B 64, 245104

�2001�.17 N. Moiseyev, R. Santra, J. Zobeley, and L. S. Cederbaum, J. Chem. Phys.

114, 7351 �2001�.18 S. Scheit, L. S. Cederbaum, and H.-D. Meyer, J. Chem. Phys. 118, 2092

�2003�.19 S. Scheit, V. Averbukh, H.-D. Meyer, N. Moiseyev, R. Santra, T. Som-

merfeld, J. Zobeley, and L. S. Cederbaum, J. Chem. Phys. 121, 8393�2004�.

20 S. Marburger, O. Kugeler, U. Hergenhahn, and T. Möller, Phys. Rev.Lett. 90, 203401 �2003�.

21 T. Jahnke, A. Czasch, M. S. Schöffler et al., Phys. Rev. Lett. 93, 163401�2004�.

22 G. Öhrwall, M. Tchaplyguine, M. Lundwall et al., Phys. Rev. Lett. 93,173401 �2004�.

23 A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Sys-tems �McGraw-Hill, New York, 1971�.

24 L. S. Cederbaum and W. Domcke, Adv. Chem. Phys. 36, 205 �1977�.25 J. Linderberg and Y. Öhrn, Propagators in Quantum Chemistry �Aca-

demic, London, 1973�.26 J. V. Leszcyinski, Computational Chemistry: Reviews of Current Trends

�World Scientific, Singapore, 1997�, Vol. 2, p. 1.27 J. Schirmer, L. S. Cederbaum, and O. Walter, Phys. Rev. A 28, 1237

�1983�.28 J. Schirmer and G. Angonoa, J. Chem. Phys. 91, 1754 �1989�.29 H.-G. Weikert, H.-D. Meyer, L. S. Cederbaum, and F. Tarantelli, J. Chem.

Phys. 104, 7122 �1996�.30 F. O. Gottfried, L. S. Cederbaum, and F. Tarantelli, Phys. Rev. A 53,

2118 �1996�.31 H.-D. Meyer and S. Pal, J. Chem. Phys. 91, 6195 �1989�.32 J. Zobeley, L. S. Cederbaum, and F. Tarantelli, J. Phys. Chem. A 103,

11145 �1999�.33 J. Schirmer and A. Barth, Z. Phys. A 317, 267 �1984�.34 A. Tarantelli and L. S. Cederbaum, Phys. Rev. A 39, 1656 �1989�.35 F. Tarantelli �unpublished�.36 F. Tarantelli, A. Sgamellotti, and L. S. Cederbaum, J. Chem. Phys. 94,

523 �1991�.

37 M. F. Guest, J. H. van Lenthe, J. Kendrick et al., GAMESS-UK, a package ofab initio programs. The package is derived from the originel GAMESScode due to M. Duduis, D. Spangle and J. Wendoloski, NRCC SoftwareCatalog, Vol. 1, Program No. QG01 GAMESS, 1980.

38 T. H. Dunning, Jr., J. Chem. Phys. 90, 12007 �1989�.39 R. A. Kendall and T. H. Dunning, Jr., J. Chem. Phys. 96, 6769 �1992�.40 D. Feller, and K. Schuchardt, basis sets were obtained from the Exten-

sible Computational Chemistry Environment Basis Set Database, Version2/12/03, as developed and distributed by the Molecular Science Comput-ing Facility, Environmental and Molecular Sciences Laboratory which ispart of the Pacific Northwest Laboratory, P.O. Box 999, Richland, Wash-ington 99352, USA. Contact David Feller or Karen Schuchardt for furtherinformation.

41 G. M. Seabra, I. G. Kaplan, V. G. Zakrzewski, and J. V. Ortiz, J. Chem.Phys. 121, 4143 �2004�.

42 A. B. Trofimov and J. Schirmer, J. Chem. Phys. 123, 144115 �2005�.43 M. Ehara, M. Ishida, and H. Nakatsuji, J. Chem. Phys. 114, 8990 �2001�.44 S. Tomoda, Y. Achiba, and K. Kimura, Chem. Phys. Lett. 87, 197 �1982�.45 L. A. Curtiss, Chem. Phys. Lett. 96, 442 �1983�.46 K. Sato, S. Tomoda, K. Kimura, and S. Iwata, Chem. Phys. Lett. 95, 579

�1983�.47 S. Tomoda and K. Kimura, Chem. Phys. Lett. 102, 560 �1983�.48 L. A. Curtiss, Chem. Phys. Lett. 112, 409 �1984�.49 M. Sodupe, A. Oliva, and J. Bertran, J. Am. Chem. Soc. 116, 8249

�1994�.50 H. Tachikawa, J. Phys. Chem. A 106, 6915 �2002�.51 R. M. Sosa, K. Irving, and O. N. Ventura, J. Mol. Struct.: THEOCHEM

254, 453 �1992�.52 S. J. Knak Jensen and I. G. Csizmadia, J. Mol. Struct.: THEOCHEM

488, 263 �1999�.53 H. Tachikawa, J. Phys. Chem. A 108, 7853 �2004�.54 L. S. Cederbaum, W. Domcke, J. Schirmer, and W. von Niessen, Adv.

Chem. Phys. 65, 115 �1986�.55 J. Zobeley, L. S. Cederbaum, and F. Tarantelli, J. Chem. Phys. 108, 9737

�1998�.56 J. Zobeley, L. S. Cederbaum, and F. Tarantelli, J. Phys. Chem. A 103,

11145 �1999�.57 R. Santra, J. Zobeley, L. S. Cederbaum, and F. Tarantelli, J. Electron

Spectrosc. Relat. Phenom. 114–116, 41 �2001�.



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