Eur. Phys. J. D (2014) 68: 239DOI: 10.1140/epjd/e2014-40816-1

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

A density functional theory study of Na(H2O)n: an exampleof the impact of self-interaction corrections�

Phuong Mai Dinh1,2,a, Cong Zhang Gao1,2,3, Peter Klupfel4, Paul-Gerhard Reinhard5, Eric Suraud1,2,Marc Vincendon1,2, Jing Wang3, and Feng Shou Zhang3

1 Universite de Toulouse, UPS, Laboratoire de Physique Theorique, IRSAMC, 31062 Toulouse Cedex, France2 CNRS, UMR 5152, 31062 Toulouse Cedex, France3 The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science

and Technology, Beijing Normal University, Beijing 100875, P.R. China4 Universite de Toulouse, UPS, Laboratoire Collisions-Agregats-Reactivite, IRSAMC, 31062 Toulouse Cedex, France5 Institut fur Theoretische Physik, Universitat Erlangen, Staudtstraße 7, 91058 Erlangen, Germany

Received 20 December 2013 / Received in final form 29 April 2014Published online 14 August 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014

Abstract. We present a detailed analysis of ground state and optical response properties of small metalwater complexes. Such complexes represent prototypical systems for analysing chromophore effects inrelation to irradiation in a biological environment. The mixing of a metal atom with organic ones leadsto the coexistence of covalent and metallic bondings which requires an elaborate treatment of the selfinteraction correction (SIC) within density functional theory (DFT). This is a particularly key issue in thecontext of time dependent DFT which represents the natural tool of investigation of irradiation scenariosin such systems. We show that these complexes require a highly elaborate treatment of the SIC which canbe attributed to the mixing of bonding types.

1 Introduction

Understanding the microscopic mechanisms of irradiationof molecules of biological interest is a key issue in radiationdamage studies [1]. A full analysis requires a dynamicaldescription allowing one to treat the various stages of ex-citation and relaxation in detail. Fully correlated quantumdynamical approaches are presently well beyond compu-tational capabilities and this will remain the case for sometime. The need for efficient microscopic theories is obvi-ous. The most robust candidate here is density functionaltheory (DFT) [2,3] and its time-dependent extension [4,5]which have been extremely successful in describing struc-ture and dynamics of many-electron systems in numerousareas of physics, such as molecular physics, cluster physics,quantum chemistry, and surface science. In spite of itssuccess, DFT is not exempt from difficulties, in particularin the dynamical description of ionization (a key aspectof radiation damage studies), which requires accurate sin-gle electron spectra, especially close to threshold emission.This implies the development of elaborate approximations

� Contribution to the Topical Issue “Nano-scale Insights intoIon-beam Cancer Therapy”, edited by Andrey V. Solov’yov,Nigel Mason, Paulo Limao-Vieira and Malgorzata Smialek-Telega.

a e-mail: dinh@irsamc.ups-tlse.fr

in the treatment of the exchange-correlation (xc) func-tional. As the analytical form of the exact functional isstill unknown, one has in practice to employ approximatefunctionals.

An especially demanding case, in this context, con-cerns molecular complexes mixing various types of bind-ing, so that simple approximations may easily turn insuf-ficient. Still, such systems are of crucial importance forunderstanding key mechanisms in irradiation studies. Inparticular, it has been shown fifty years ago that one ofthe products of water radiolysis is the so-called solvatedor hydrated electron [6]. Such a species is a free elec-tron transiently captured in a liquid water by hydrogenbonds. It can also be formed by an ionizing irradiationand its high reactivity in living cells makes it a poten-tial agent in DNA damage [7]. Many theoretical and ex-perimental works have been devoted to unravel the for-mation of the solvated electron [8]. One possible routeconsists in studying alkali-doped water clusters as proto-types of loosely bound electrons in a polar environment.The existence of systems such as Na(H2O)n up to n = 20have been experimentally demonstrated since long [9–11].These complexes, mixing covalent and metallic bondings,can also stand as model systems of radiosensitizers or chro-mophores in a biological context of irradiation. Indeed oneof the promising tools for a better targeting of cancerouscells is the use of metal nanoparticles in concomittance

Page 2 of 8 Eur. Phys. J. D (2014) 68: 239

with an ionizing irradiation, see e.g. [12,13]. The idea hereis to excite a collective mode of the chromophore, e.g. bymeans of a resonant laser pulse. These collective modesusually lie in the infra-red or visible range, while those ina biomolecule or in water clusters are rather in the UV orXUV range. The resonant coupling of a laser pulse withplasmon modes of a metal cluster will therefore producemore secondary electrons which can in turn become pre-solvated, attach DNA and induce dissociation. In this pa-per, we focus on the case of Na(H2O)n for n = 1, 2 and wewill demonstrate that this type of complexes is particu-larly demanding for (TD)DFT studies, and thus requiringthe use of elaborate approximations.

The simplest and widely used approximation in DFTis the local density approximation (LDA). Numerous prac-tical calculations have confirmed the applicability of LDAin systems where the density slowly changes [14]. Even incases of non-vanishing density gradients, LDA can providefair results for static and dynamical properties [14]. How-ever, LDA is plagued by the self-interaction (SI) error, re-sulting from an incomplete cancellation of the SI betweenthe Hartree potential and the xc counterpart. One of theproblems is that the SI error induces a wrong asymptoticsof the electronic potential. As a consequence, the singleelectron states are usually too weakly bound and LDAviolates Koopmans’ theorem [15]. Thus (TD)LDA is notsuited to describing the dynamics of near-threshold ion-ization. Therefore, a lot of efforts has been devoted toelaborate functionals that cure the drawbacks of LDA.

Two options for a solution are possible. One is to climb“Jacob’s ladder” [16,17], which adds more constraintsstep by step to construct new xc functionals, such as hy-brid functionals [18,19], generalized gradient approxima-tion (GGA) [20–23] and meta-GGA [24]. The resultingfunctionals become more and more involved, and they areoften not applicable for time-dependent problems. More-over, the SI is never fully removed. For instance, GGA failsto reproduce the valence-conduction band gap in semicon-ductors because of a remaining SI [25]. The other optionis to cure the SI error explicitly. The first self-interactioncorrection (SIC) scheme along that line has been suggestedlong ago by Perdew and Zunger [26]. Although this orig-inal SIC is conceptually very simple, it is not straigh-forward to apply in practice, especially in the time do-main, since it provides a state-dependent and non-unitaryHamiltonian.

Several approximations and implementations of SICschemes have been proposed since [27–30], for a recent re-view, see [31]. Among these schemes, we consider in thispaper two options: the approximate average-density SIC(ADSIC) [28] and the two set SIC method (2setSIC) asa practical implementation of full SIC [29]. Both can bederived from a variational principle and both are invariantunder unitary transformation amongst occupied states.This thus allows to formulate a correct time-propagationscheme where conservation laws, as e.g. the zero-force the-orem [32], are fulfilled. ADSIC appears as the simplestSIC: the corresponding Hamiltonian is state-independentand local in space, and the time-propagation is as simple

as in pure LDA. It has been successfully applied in molec-ular and cluster physics, see e.g. [33–35]. The 2setSIC ismore involved, formally and numerically. However its ap-plicability has been recently demonstrated in the calcula-tion of structural properties in atoms [36] and of dynam-ical observables (total ionization, photoelectron spectra)in molecules and clusters [37,38]. In principle, the rangeof validity of 2setSIC is much larger than that of ADSIC,especially in dynamical scenarios of molecular dissociationor high ionization. However, the simple ADSIC has beenfound in several cases to work better than expected, atleast at the side of static observables [39]. This may be notso much of a surprise for metallic systems, because ADSIClives from averaging the SI error over all electrons, butit remains questionable in covalent molecules. We have,however, demonstrated on a variety of systems (atoms,small molecules, carbon chains and rings) covering differ-ent binding types and different geometries (1D, 2D,. . . )that ADSIC performs surprisingly well, often even betterthan the Perdew-Zunger SIC [15]. The aim of this paperis to analyze ADSIC and 2setSIC in the more demandingcase of metal seeded covalent complexes which representprototypical chromophore systems.

The paper is organized as follows. In Section 2, wepresent ADSIC and 2setSIC in some detail, and we givesome ingredients for their numerical realizations. In thenext section, we compare static properties and the opti-cal response of four systems, namely the Na atom, as theperfect example of a metallic system, the water molecule,as a prototype of a covalent system, and the two com-plexes NaH2O and Na(H2O)2, which mix metallic and co-valent bondings. We extensively discuss the performancesof LDA, ADSIC and 2setSIC on various observables. Wefinally give some conclusions.

2 Theory

2.1 Basics on SIC

We work in the standard Kohn-Sham (KS) picture ofDFT [3] where the starting point is the total electronicenergy:

E = Ekin[{ψα}] + Eext + ELDA[ρ], (1)

where ψα (α = 1, . . . , N) describe the valence electronstates, Ekin denotes the kinetic energy, Eext stands forthe coupling of the valence electrons with some externalfield (that from the ions, a laser or a charged projectile),and ρ =

∑α |ψα|2. The last term ELDA embraces the elec-

tronic Hartree and xc energy within LDA [26]. Variationwith respect to ψ∗

α, assuming the orthonormality of theψα’s, gives the standard KS one-body equations. For sim-plicity, we ignore the dependence on electron spin in allequations. The extension to the full spin-dependent for-mulation is straightforward. Of course actual computa-tion have been performed with an explicit account of spindegree of freedom.

Eur. Phys. J. D (2014) 68: 239 Page 3 of 8

Each electron adds a contribution to the SI error. Theidea by Perdew and Zunger is to subtract “by hand” thesecontributions from the total energy [26]:

ESIC = Ekin[{ψα}] + Eext + ELDA[ρ] −N∑

α=1

ELDA

[|ψα|2].

(2)

The corresponding one-body Hamiltonian stems fromvariation with respect to ψ∗

α:

hSIC,α = hLDA − Uα, Uα = ULDA[|ψα|2] (3)

where ULDA = δELDA/δψ∗α. The net result is that the

obtained one-body Hamiltonian depends on the orbitalon which it acts. The non-hermiticity of hSIC,α cantherefore induce a violation of orthonormality in a timepropagation.

The essence of ADSIC is to treat the SI contributionof each electron on the same footing by averaging the SICterm in equation (2) over all N valence electrons. The SICenergy then reads [28]:

EADSIC = Ekin[{ψα}] + Eext + ELDA[ρ] −NELDA

[ ρ

N

].

(4)ADSIC can thus be interpreted as an approximation to thePerdew-Zunger one. As already mentioned in Section 1,ADSIC presents numerous formal and numerical advan-tages, one of them being that it renders the Hamiltonianstate-independent. ADSIC however breaks down if the to-tal electron number N is not conserved in time, as e.g. ina dissociation process.

To circumvent this drawback of ADSIC and also tocure the state-dependency of the one-body Hamiltonianobtained from the Perdew-Zunger SIC, we have developedduring the past years another scheme based on the use oftwo complementary sets of orbitals, hence called “2set-SIC”. All the details can be found elsewhere [29,37]. Wehere recall the main ingredients. The idea is to enforceorthonormalization of the ψα’s through Lagrange param-eters λβα. This modifies the variational problem to:

δψ∗α

⎛

⎝ESIC −∑

β,γ

〈ψβ |ψγ〉λγβ⎞

⎠ = 0

in the stationary case, and to:

δ

∫ t

t0

dt′

⎛

⎝ESIC −∑

α

〈ψα|i�∂t|ψα〉 −∑

β,γ

〈ψβ |ψγ〉λγβ⎞

⎠ = 0

in the time domain. The stationary and time-dependentSIC equations satisfied by the ψα’s, respectively, read [37]:

hSIC,α|ψα〉 =∑

β

λβα|ψβ〉, (5)

(hSIC,α − i�∂t) |ψα〉 =∑

β

λβα|ψβ〉. (6)

Note that in the last equation, the case β = α leads to aglobal phase on the ψα’s which does not have any physicalcontent and can thus be ignored in the following. In bothcases, these equations have to be complemented by:

〈ψβ |Uβ − Uα|ψα〉 = 0. (7)

This equation, called the “symmetry condition”, stemsfrom the orthonormality constraint and appears crucial inthis scheme [37]. Fulfillment of this condition restores thehermiticity of hSIC (however, only in the subspace of occu-pied states). Note that equation (7) was originally called“Pederson condition” [40] or “localization condition” inthe seminal work by Pederson et al. [41]. Since then, ithas been shown in many atomic and molecular systemsthat localized wave functions usually minimize the SICenergy in the stationary case, see e.g. [42,43] and veryrecently [40,44]. However, there are few cases where con-dition (7) does not lead to localization [30,45]. This is whywe prefer the notation “symmetry condition”.

We now come back to equations (5) and (6). Note thatneither the first equation presents the usual form of aneigenvalue equation, nor the second one exhibits a clearpicture for a time propagation. We now exploit the left-over freedom of any unitary transformation among theoccupied states to rewrite these equations as:

hSIC|ϕj〉 = εi|ϕj〉, (8)hSIC|ϕj〉 = i�∂t|ϕj〉, (9)

hSIC = hLDA −∑

α

Uα|ψα〉〈ψα|, (10)

ϕj =N∑

α=1

ujαψα. (11)

We have thus introduced a second set of orbitals, the ϕi’s,related to the ψα’s, by the unitary transformation u inequation (11). This set appears as the “canonical” one inthe stationary case, since it diagonalizes the SIC mean-field. Note that the use of a second set of orbitals is notmandatory in the stationary case, since the numerical im-plementation of the convergence of equations (5) and (7) isfeasible as such. However, it makes more sense in the time-dependent SIC equation (9), since the ϕi’s then appear asthe “propagating” set:

|ϕj(t)〉 = exp(

− i�

∫ t

t0

dt′hSIC

)

|ϕj(t0)〉. (12)

In brief, the 2setSIC uses a first set of orbitals, the ψα’s,which satisfies equation (7) and enters the construction ofthe mean-field through equation (10), and a second set,the ϕi’s, which follows the usual time evolution (12) andwhich is related to the symmetryzing set by equation (11).

2.2 Numerical scheme

From the numerical point of view, the electronic wavefunctions are discretized in a 3-dimensional coordinate-space grid. The typical sizes of the numerical box are

Page 4 of 8 Eur. Phys. J. D (2014) 68: 239

Fig. 1. Geometries of H2O, NaH2O, and Na(H2O)2, optimizedat the ADSIC level. The electron valence cloud is depicted oneach structure. The white balls stands for the H atoms, the(red) light ones for the O atoms, and the (blue) dark ones forthe Na atom.

643 for the Na atom and the H2O molecule, and 723 forthe complexes NaH2O and Na(H2O)2. The mesh size istaken as 0.589 a0. We use the xc functional from Perdewand Wang [46], augmented by ADSIC or 2setSIC. The cou-pling of the valence electrons to the ionic cores is treatedby Goedecker-like pseudopotentials [47]. For the calcula-tion of the ground-state, the KS equations are solved bya damped gradient method [33]. The ionic structures areobtained by standard cooling procedures [35].

A word of caution is in order here. For a given chemicalelement, the original Goedecker pseudopotentials use twodifferent Gaussian widths for the local and non-local parts.Moreover, these two widths also differ from one element tothe other. Since we naturally have a length scale imposedby the mesh size of our numerical box, we have refitted theparameters of the pseudopotentials for Na, H and O, to ob-tain a unique width. This refitting has been performed atthe ADSIC level. In this paper, we apply 2setSIC withoutreadjustment of the pseudopotential parameters. There isthus a slight inconsistency here at the side of the pseu-dopotentials. However we preferred to avoid a re-fit with2setSIC and keep the same set of pseudopotentials for allfunctionals. Note also that the ionic structures have beenoptimized with ADSIC, and that the geometries mightdiffer if 2setSIC was used instead. But for the analogousreason mentioned above about the pseudopotentials, weprefered to keep the same ionic structures in all energyfunctionals, to focus the comparison of LDA, ADSIC and2setSIC at the level of the SI error alone.

3 Results and discussion

3.1 Structural properties

The configurations for the considered molecules weretaken from the ab initio calculations of [48,49] and slightlyre-optimized for our present schemes and functional usingADSIC. The resulting structures are shown in Figure 1.This figure alone is not sufficient to disentangle the na-ture of the binding between the Na atom and the watercounterpart. This issue will be addressed in more detailin Section 3.2 and in Figure 6. For H2O, the calculatedbond length r(O-H) and bond angle ∠H-O-H are 0.986 Aand 104.48◦, respectively. Comparing with experimentalvalue [50], the calculated bond length is slightly stretched,

-2.5

-2

-1.5

-1

-0.5

0

2setSIC ADSIC LDA

s.p.

ener

gies

[Ry]

NaH2O

2setSIC ADSIC LDA

Na(H2O)2

Fig. 2. Single-particle energies of the valence electrons of spinup (black solid lines) and spin down (red dashed lines) inNaH2O (left panel) and Na(H2O)2 (right panel), calculatedin 2setSIC, ADSIC, and LDA.

while the bond angle is fully correct as such. When addinga Na atom to an individual water molecule, we found thatthe Na 3s electron is microsolvated in water, which re-sults in electron delocalization of the “sodium” valenceelectron over water, and formation of a weak Na–O bond.The mixture of metallic bond (Na) and covalent bond(H2O) makes NaH2O to be a weakly bound system. Theoptimized structure parameters are r(O-H) = 0.993 A,∠H-O-H = 104.72◦, r(O-Na) = 2.231 A, and ∠Na-O-H=127.64◦. It indicates that the presence of the Na atomcauses the water molecule structure to be “looser”, whichmeans r(O-H) and ∠H-O-H are a bit larger than the onesof the isolated water molecule. Note that both H2O andNaH2O are planar. A more complicated (3D) structureis formed when the Na atom is added into the waterdimer, see in right part of Figure 1. The similar micro-solvation of Na 3s electron still exists, and the struc-ture of (H2O)2 is distorted somewhat. If one makes aclose inspection of the Na(H2O)2 structure, one can findthat it consists in a planar NaH2O substructure, and theadditional H2O molecule extends almost orthogonal tothe NaH2O molecular plane. The calculated parametersare r(O-H) = 0.967 A, ∠H-O-H = 106.2◦, r(O1-Na) =2.355 A, r(O2-Na) = 4.499 A, r(O1-O2) = 2.808 A, ∠Na-O1-O2 = 121◦.

As mentioned in the introduction, the energy of theHOMO, i.e. the ionization potential (IP) is of great im-portance in near-threshold ionization dynamics. Theoret-ically, it is strongly related to the asymptotic behavior ofthe KS potential and thus sensitive to SIC. Figure 2 showssingle-particle (s.p.) energies for NaH2O and Na(H2O)2 atthree different levels of approximation. For NaH2O, thecalculated HOMO energy is −0.318 Ry, −0.411 Ry, and−0.191 Ry for 2setSIC, ADSIC, and LDA, respectively.Compared with the experimental IP of 0.322 (±0.002) Ry,it is apparent that the most satisfactory result is obtainedby 2setSIC with an error of 1.24%. For Na(H2O)2, again2setSIC leads to the best IP value with the smallest error(8.2%), while ADSIC overestimates the IP by ∼0.083 Ry(29.75%) and LDA underestimates the value by ∼0.107 Ry(38.35%).

As expected, LDA yields the largest error for theIP, because the self-energy in the Hartree energy is notcancelled by LDA exchange-correlation energy. The case

Eur. Phys. J. D (2014) 68: 239 Page 5 of 8

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

H2O

NaH

2O

Na(

H2O

) 2

I Δ−

I ε[R

y]

LDAADSIC2setSIC

Fig. 3. Non-Koopmans error for the three stages of SIC. Seetext for details.

is better when ADSIC is used, but the error cannotbe neglected. This suggests that ADSIC could not de-scribe Na(H2O)n correctly. The reason is that the ADSICused the same corrected scheme (the total number ofelectron N) for all s.p. orbitals. For example, for theNaH2O complexe, the HOMO arises from the Na atom(IP = 0.378 Ry) while other s.p. orbitals mostly comefrom H2O (IP = 0.928 Ry). It is thus clear that the den-sity of the HOMO is different from other orbitals, so thatthe SI in such a special electronic structure requires an ex-plicit correction rather than an average-density one. An-other finding is that for SIC, adding a sodium atom toH2O elevates the Na level and deepens the O core level.However it is exactly the opposite for ADSIC, which ishighly suspicious and rules out ADSIC as an approachfor this type of complexes. Generally, the s.p. levels fromLDA are up-shifted when making comparison with bothversions of SIC. That is an obvious and expected effect.In that sense, LDA is a reasonable approach.

The performance of an exchange-correlation functionalcan also be tested through the check of the Koopmans’theorem which states that the negative HOMO energyshould be equal to the (vertical) IP [51]. The lattercan be thus evaluated from the HOMO energy, that isIε = −εHOMO. But in principle, in finite systems, onecan also calculate the vertical IP by the difference of thetotal energy E(N) of the system with N electrons andthat of the system with one electron removed, E(N − 1),the ionic structure remaining frozen. We then define an-other IP, namely IΔ = E(N − 1)−E(N). The fulfillmentof Koopmans’ theorem should give a vanishing value ofIΔ − Iε, called the non-Koopmans (NK) error. The lat-ter quantity is depicted in Figure 3 for H2O and the twocomplexes. There is no doubt that LDA is completely offfor all cases, and the largest NK error is 0.428 Ry forH2O. In contrast 2setSIC is preferable and compulsoryfor Na(H2O)n, where ADSIC is as wrong as LDA with adifferent sign.

3.2 Optical response and polarizabilities

We now consider optical response as a dynamical ob-servable of the studied systems. To calculate the optical

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 3 6 9 12 15 18 21 24

xdi

pole

[a0]

time [fs]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

ydi

pole

[a0]

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

zdi

pole

[a0]

Na

H2ONaH2ONa(H2O)2

Fig. 4. Time evolution of the electronic dipole in x (bottom),y (middle), and z (top) directions, calculated in 2setSIC, forH2O (black lines), NaH2O (dashed red curves), and Na(H2O)2(light or green lines). The dipole of Na is the same in all spatialdirections and is shown only in the middle panel as the bluesolid curve of largest amplitude.

response, we apply techniques of spectral analysis as out-lined in reference [52]: the same instantaneous dipole boostis applied to all wave functions; the time evolution ofdipole momentum D(t) is protocoled; finally, the time sig-nal D(t) is Fourier transformed to the frequency domainfrom which �{D(ω)} emerges as the wanted optical ab-sorption strength. One can equivalently look at

∣∣D(ω)

∣∣2

which exhibits the same frequency peaks. Figure 4 illus-trates the dipole response in the time domain for the caseof 2setSIC. The Na results are the same in all three direc-tions due to a perfect spherical symmetry, and are shownonly in the middle panel. They also show a significantlylarger response amplitude than those for Na bound toH2O. It implies that the vicinity of the water moleculeconstitutes obviously a great hindrance for the the mo-tion of the Na electron. The trend continues when anotherH2O is added: the amplitude for Na(H2O)2 is once moresmaller than that for NaH2O. This happens although the

Page 6 of 8 Eur. Phys. J. D (2014) 68: 239

1

10

100N

a

NaH

2O

Na(

H2O

) 2

H2O

pola

riza

bilit

yα

LDAADSIC2setSIC

Fig. 5. Static polarizability of the indicated systems, calcu-lations in LDA (black), ADSIC (light or green), and 2setSIC(dark or red).

second H2O is not so close to the Na. The signal from pureH2O is, of course, the weakest among all four because thismolecule has the best bound electron cloud in the sam-ple. We also find a similar trend in amplitude of dipoleresponse for ADSIC and LDA (not shown here).

The results for dipole response in Figure 4 need a bet-ter understanding, thus we computed the static dipole po-larizability for the studied systems. Figure 5 depicts thepolarizabilities for the three SIC stages on a logarithmicscale. The results nicely corroborate the previous find-ings from the dipole response amplitudes. The binding toH2O induces a dramatic reduction of the polarizability ofNa. The next water molecule induces a second reduction.Surely, pure water has a very small polarizability as com-pared to Na and Na(H2O)n compounds. It is interestingto note that 2setSIC and LDA are closer together whileADSIC tends to be farther off for the compounds. Thisindicates again that ADSIC is not so appropriate for thiscase.

Figure 5 also motivates to discuss the type of theNa-water binding in the mixed complexes. Let us focus onthe case of NaH2O here. The high polarizability obtainedfor the Na atom (224 a0

3) is consistent with the picture ofa delocalized wave function. This also agrees well with thepolarizability of the Na2 dimer, which is slightly reducedto 187 a0

3. At the opposite side, the covalent binding inH2O delivers a very small polarizability around 1.6 a0

3,consistently with a high localization of the valence elec-trons. Now, the case of Na(H2O)n precisely lies in be-tween, indicating that the binding of the Na atom to thewater molecule(s) is neither fully metallic nor fully ionicor covalent, but just in between. For closer inspection, weshow in Figure 6 the electronic density of the HOMO inNaH2O, integrated over the z coordinate and in the xyplane. According to Figure 2, the HOMO should be a Na-like state, that is a s state localized at the Na ion, sinceits energy stays very close to that of the Na atom alone.However, in ADSIC (left panel), we rather observe a local-ization of the HOMO density around the O atom. This isfar from the expected result and demonstrates again thefailure of ADSIC for this system. More reliable in this case

-6 -3 0 3 6 9 12

x coordinate (a0)

-9

-6

-3

0

3

6

9

yco

ordi

nate

(a0)

0

0.01

0.02

0.03

0.04

0.05

0.06ADSIC

-6 -3 0 3 6 9 12 15

x coordinate (a0)

-9

-6

-3

0

3

6

9

yco

ordi

nate

(a0)

0

0.01

0.02

0.03

0.042setSIC

Fig. 6. Electronic density of the HOMO of the NaH2O com-plex in the xy plane and integrated over the z coordinate, cal-culated in ADSIC (left) and 2setSIC (right). The white circlesindicate the positions of the Na, O and H atoms.

10−5

10−4

10−3

10−2

10−1

100

0 5 10 15 20 25 30 35 40

|D(ω

)|2

frequency ω [eV]

NaH2O

10−5

10−4

10−3

10−2

10−1

100

|D(ω

)|2

Na

0 5 10 15 20 25 30 35 40

10−5

10−4

10−3

10−2

10−1

100

|D(ω

)|2

frequency ω [eV]

Na(H2O)2

10−5

10−4

10−3

10−2

10−1

100

|D(ω

)|2

H2O LDAADSIC2setSIC

Fig. 7. Optical response∣∣D(ω)

∣∣2 of Na, H2O, and Na(H2O)1,2,

calculated in LDA (black solid lines), ADSIC (light or greencurves), and 2setSIC (red dashes).

is a SIC calculation and that delivers indeed an enlighten-ing result: the HOMO density is mainly located around theNa atom and is even repelled by the water molecule. Thisis far from a standard picture of metallic or covalent bind-ing between the Na atom and the O atom. We rather see astrong dipolar binding combined with strong Pauli repul-sion near the H2O molecule. This once again confirms theconclusion we drew previously when discussing the polar-izability of the mixed system: the bond is neither metal-lic nor covalent, thus demonstrating once again a non-standard feature of the Na(H2O)n complexes, which thencall for a specific treatment at the side of the electronicdescription.

Finally, we discuss the optical response of the fourcases, as shown in Figure 7. Note that the same loga-rithmic scale is used for a better resolution of peaks. Foreach system, the spectrum is generally similar for the var-ious SIC, especially below 5 eV in the spectrum of systemscontaining a Na atom. This proves that, in the low energyrange, the spectral structure from Na carries over to someextent to the complexes, so that the Na atom can be con-sidered as a chromophore if a laser with a frequency closeto the Na plasmon peak irradiates the sodium-water com-plexe. Furthermore, a series of peaks are observed above7 eV. Naturally, the Na atom does not contribute in thisrange. The peaks over ionization threshold are covered by

Eur. Phys. J. D (2014) 68: 239 Page 7 of 8

signals “from” the H2O molecule(s). The two complexesgive similar gross results as those for the water moleculealone, but there are differences in the remaining quantumfluctuations. These spectra however lie in the continuumwhere there are many unbound states. Transitions fromdeeper lying shells to these states in the continuum aremost probably the sources for the quantum fluctuations.

4 Conclusions

In this paper, we discussed the application of two self-interaction schemes, namely ADSIC and 2setSIC, in threegeneric cases: the Na atom, the water molecule and the twocomplexes Na(H2O)1,2. The first one stands for a proto-type of a metallic system with its delocalized valence elec-tron, while the second one exhibits (localized) covalentbonding. At the side of the single particle spectra, bothSIC schemes perform equally well. The optical propertiesof Na and H2O (static polarizabilities, optical response)are quite independent of SIC, since LDA, ADSIC and 2set-SIC provide very similar results. This is not surprising asthe relative positions of the energies of the single particlestates are more relevant than the absolute values of theenergies in this kind of observables.

The conclusion is totally different in the complexes.While ADSIC performs even better than 2setSIC as soonas only one type of bonding is concerned, it strongly failsto produce correct energy spectra in Na(H2O)n. This isparticurlary visible in the violation of the Koopmans’ the-orem in these cases. On the contrary, 2setSIC perfectlyrestores it. Therefore the simple ADSIC is not sufficientenough in complex situations where a mixing of metallicand covalent bondings come into play.

Note however that the optical response look very sim-ilar in the Na-water complexes, whatever the SIC scheme.We observed the reminiscence of the Na atom plasmonpeak which dominates the optical spectrum of both com-plexes. One can thus exploit this property by shining anoptical laser on the complexes, excite the plasmon peakof the Na atom, extract part of its valence electron, andfinally study the impact of such an ionization on the watermolecules around. This would represent a model systemfor the chromophore effect observed in a laser irradiationin the therapeutic window (where there is basically nolight absorption by the living tissues) of gold nanoparti-cles attached to DNA, where more secondary electrons areproduced thanks to the electron reservoir provided by themetal clusters. However, in this close to threshold ioniza-tion scenario, the energetics would be totally wrong at thelevel of ADSIC. It is thus compulsory to use 2setSIC inthis complex dynamical process. Work in this direction isin progress.

This work was supported by the Institut Universitairede France. C.Z.G. receives financial support from ChinaScholarship Council (CSC) (No. [2013]3009). J.W. and F.S.Z.acknowledge the support from the National Natural Sci-ence Foundation of China under Grant Nos. 11025524 and

11161130520, National Basic Research Program of China un-der Grant No. 2010CB832903, and the European Commis-sions 7th Framework Programme (FP7-PEOPLE-2010-IRSES)under Grant Agreement Project No. 269131. This work wasgranted access to the HPC resources of IDRIS under the allo-cation 2013–095115 made by GENCI (Grand Equipement Na-tional de Calcul Intensif ), and of CalMiP (Calcul en Midi-Pyrenees) under the allocation P1238.

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