General static polarizability in spherical neutral metal clusters and fullerenes within Thomas-Fermi theory
aa r X i v : . [ phy s i c s . a t m - c l u s ] J un General static polarizability in spherical neutral metal clusters andfullereneswithin Thomas-Fermi theory
D. I. Palade ∗ and V.Baran † National Institute of Laser, Plasma and Radiation Physics,PO Box MG 36, RO-077125 Magurele, Bucharest, Romania Faculty of Physics, Bucharest
Abstract
We study the static linear response in spherical Thomas-Fermi systems deriving a simple differen-tial equation for general multipolar moments and associated polarizabilities. We test the equationon sodium clusters between and atoms and on fullerenes between C and C and proposeit for general Thomas-Fermi systems. Our simple method provides results which deviates fromexperimental data with less then . ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The problem of linear response to an external field is a crucial problem in many-bodyphysics and it has been intesively studied through different techniques. For the static case,the problem simplifies itself to static response functions as polarizability and magnetic sus-ceptibility, crucial quantities in the description of any classical or quantum system.In the range of mesoscopic systems as metallic clusters are, various classical methods havebeen applied successfully during the first part of the 20th century, especially Mie’s theoryfor electromagnetic scattering and electrostatic modeling for the polarizability. Nonetheless,going down on the length scale, the classical models start to fail such that in the atomic,molecular and mesoscopic domains, the predicted results are no longer consistent with theexperimental data.Semi-classical or fully quantum models are appropriate to describe the observed features.Methods as Hartree-Fock theory, RPA, Density Functional Theory, etc., in general, meanfield theories, give very good results regarding the stationary (or dynamic) properties of suchatomic systems. Nonetheless, they have a single flaw: the computational efforts are hugeand may not worth to use such complex methods to derive simple quantities as polarizability,especially when the rigor of result is not necessary.On the other hand, there are large classes of systems, like metallic clusters, in whichsome special properties are exhibited. For example, in alkali elements it is well known thepresence of a quasi-free electron on the last unclosed shell which is close to the model offree homogeneous electron gas (HEG). This type of thinking it is used in solid-state physicsand can be exploited in the atomic domain through the Thomas-Fermi model [3, 6, 13]which approximates the problematic term of kinetic energy in the electron system with thelocal form of HEG and simplifies the treatment. Extended versions of it can employ alsoadditional terms for the exchange-correlation potential (from Kohn-Sham potential)[3] orsupplementary gradient correction from the Weizsacker term [16].Even if such an approximation provides a very simple way to obtain the electron density,the systems in which we apply it must be carefully chosen, since usual errors can be around − , or larger, for different observables and due to "no-binding" Teller’s theorem [5] inmolecules appear the phenomenon of instability. Even though the extended versions providebetter results, we will refer in the present work just on the simple, original form of the theory2ince our main goal is to achieve quantitative description with minimum of computationaleffort. That is why, the best suited cases are those with a large volume extension, metallic ortransition character and in which, we are not interested in fine details of the density profileor single particle (pseudo)wave functions, shell effects, excitation energies, etc.For all those reasons, we shall focus next on the Thomas-Fermi theory and on modeling thelinear response to a static external potential in its most general form. The theoretical resultswill be tested on various medium-sized sodium clusters and on the famous C fullerene and C and show that the method provides resonable results. II. FORMAL BACKGROUNDA. Metallic clusters and the jellium model
Clusters are by definition, mesoscopic systems formed from − [8] atoms of variouselements. The theoretical methods of investigation are going from molecular to bulk (solid-state) domain, both quantum and classical approaches.As we have mentioned before, the clusters formed from metallic elements have the prop-erty that the electrons from the unclosed shell are loosely bounded and their behavior isclose to HEG one. Also, we treat the problem in the Born-Oppenheimer approximation.This means that their dynamics and de-localization are high enough to consider that they see the ionic background in a averaged manner. By ionic background we understand thesystem obtained from the coupling between charged nuclei and the core electrons whichdetermine a net positive charge.Our working framework is the so called jellium model in which the ionic backgroundis approximated to a homogeneous positive charge distribution while the free electrons willalso have an almost constant density in the bulk region. In clusters the jellium modelit is also applied and the ionic background determines a smooth Coulomb potential withthe appropriate symmetries. Once the Coulomb potential is generated it may be includedas an input in different theoretical approaches (Hartree-Fock, DFT, etc.) to derive theelectron density. The self-consistent jellium model proved to be a very appropriate methodpredicting quantitative results in good agreement with the experimental data [1], [2]. Thesimplest geometry possible is that of a sphere, but can be somehow a troublemaker in3umerical simulations since it has a discontinuity at the edge. A more refined model isthe so called sof t jellium which works with a Wood-Saxon radial profile coupled with anypossible angular dependence, giving access to any possible geometry of the ionic part. Thejellium density is written in its most general form as: ρ jel ( r, θ, φ ) = 34 πr s [1 + exp ( | r | − R ( θ, φ ) σ jel )] − (1)with R ( θ, φ ) = R (1 + P l,m α lm Y lm ( θ, φ )) . The Weitzecker-Seitzs radius r s is a parameterof the bulk domain interpreted as the volume occupied by a single atom while Z is thedifference between the numbers of protons and the number of bound electrons and so Z | e | is the charge of jellium. Therefore, the jellium density satisfies the condition: R ρ jel dr = Z .The limiting case of spherical sharp distribution is obtained when σ jel → . B. Thomas-Fermi model
Thomas-Fermi model (TF) was derived independently by L. Thomas and E. Fermi in1927, soon after Schroedinger equation, 1926. The basic approximation of the model is totreat the electron density distribution in the atom using a local approximation for the kineticterm i.e. the HEG approximation: E kin = γ R R ρ / ( ~r )d r , with γ = (3 π ) / ~ / m . If V ( ~r ) is the external potential , the total energy density functional can be written: E [ ρ ( ~r )] = 3 γ Z R ρ / ( ~r )d r + e Z R ρ ( ~r ) V ( ~r )d r + 12 e πε Z R ρ ( ~r ) ρ ( ~r ′ ) | ~r − ~r ′ | d r ′ d r (2)The ground-state energy and the corresponding density distribution are obtained withina Ritz variational principle, searching for the minimum of this quantity: E T F = min { E [ ρ ( ~r )] | ρ ∈ L / ( R ) , Z R ρ ( ~r )d r = N, ρ ( ~r ) ≥ } (3)The condition ρ ∈ L / ( R ) refers to the fact that density is a / integrable function over R and so, is a condition for finite kinetic energy to emerge. The constrain associated withthe condition R R ρ ( ~r )d r = N is introduced through Euler-Lagrange multiplier techniqueand solving the variational problem, the resulting Thomas-Fermi equation is be derived: γρ / = max [0 , Φ − µ ] (4)4n this equation, Φ( ~r ) , Φ : R → R is the energetic Coulomb potential while µ is thechemical potential: Φ( ~r ) = | e | V ( ~r ) − e πε Z R ρ ( ~r ′ ) | ~r − ~r ′ | d r ′ (5) µ = − ∂E T F ∂N (6)In turn, the Coulomb potential is the generated by the charge distribution and is thesolution of the Poisson’s equation ∆Φ( ~r ) = | e | ∆ V ( ~r ) + e ρ ( ~r ) /ε . In the case of neutralelectric systems, the chemical potential is null [6], but this feature is maintained only withinthe Thomas-Fermi method. The additional terms, as Dirac or Wietzacker contributions,break this property. In the jellium approximation V ( ~r ) is induced by ρ jel and the Thomas-Fermi becomes in differential form: ∆Φ( ~r ) = e /ε ( γ − / Φ / ( ~r ) − ρ jel ( ~r )) (7) III. THEORYA. Perturbation theory and density changes
In the absence of external interactions, for the ground-state, the TF equation leads to γρ / ( r ) = Φ ( r ) with ρ the ground state density of the free electrons. We shall study inthe following the static linear response in TF approximation considering a time-independentpotential of an arbitrary form. If the coupling strength to the free electrons is λ : v ( ~r ) = λ X v lm ( r ) r Y lm ( θ, φ ) (8)And the TF equation for the new stationary state, becomes: γρ / ( ~r ) = Φ( ~r ) (9)Then the interaction energy density is ρv ( ~r ) . This perturbation will induce a spatialchange of the charge which can be treated in a power expansion in the coupling strength λ : ρ ( ~r ) = ρ ( r ) + λρ ( ~r ) + λ ρ ( ~r ) + ... . 5f we resume to first order term ( λ ≪ e / (4 πεr )) , then ρ should satisfy R R ρ ( ~r )d r = 0 since R R ρ ( ~r )d r = R R ρ ( ~r )d r = Z .In the presence of the external potential the initial spherical symmetry is broken and thedensities varify the properties: ρ : R → R + (10a) ρ : R → R + (10b) ρ : R → R (10c)The linearized kinetic density energy term and the potential Φ are: γρ / ( ~r ) = γρ / ( r ) + λ γ ρ / ( r ) ρ ( ~r ) (11) Φ( ~r ) = Φ ( ~r ) + v ( ~r ) − λ Z ρ ( ~r ) | ~r − ~r ′ | dr ′ (12)Then the TF equation for the perturbed part becomes: λ γ ρ / ( ~r ) ρ ( ~r ) = v ( ~r ) − λ Z ρ ( ~r ) | ~r − ~r ′ | dr ′ (13)Working in spherical coordinates we can consider the following expansion of ρ ρ ( ~r ) = ρ / ( r ) X lm Y lm ( θ, φ ) u lm ( r ) r (14)Using the expansions for potential (8) and for density (14) in equation (13) we deducethat u lm ( r ) functions will satisfy the radial equation: d u lm ( r ) dr − u lm ( r )( l ( l + 1) r + 6 πγ ρ / ( r )) = d v lm ( r ) dr − l ( l + 1) r v lm ( r ) (15)Our equation is an approximate version of the equation (23) deduced in [11] in the absenceof Weizsacker, exchange and correlation terms. This fact can be seen by setting the beta factor to and excluding exchange-correlation effects. Nonetheless, their derivation is doneon the basis of variational method, while our is not. We stress in this paper the fact thatthe complexity of the equation proposed in [11] is much more involved being an integro-differential equation and the presence of those supplementary terms is not necessary for6emi-quantitative results. In fact, the results have the same level of accuracy with ours andsince the present equation is pure differential we can take advantage of its simplicity in orderto study the main effects which contribute to polarizability in metal clusters. Moreover, whilein the reference, the Weiszacker correction is used with different constants and is consideredto be the essential reason for which the differential equation is derived, our derivation hasno such condition. B. Boundary condition and general polarizability
Concerning the boundary conditions, in the origin, in order to have finite ρ (0) we requirethat lim r → u lm ( r ) /r to be finite. Concerning the behavior at infinity we shall ask for thecoefficient u lm ( r ) to follow the behavior of the perturbation i.e. u lm ( r ) = 3 / (2 γ ) v lm when r → ∞ . From TF eq (13), the asymptotic behavior of u lm is: u lm ( r ) → γ ( v lm − π l + 1 q lm r l ) (16)Here, the q lm term is the multipole moment associated with the induced charge: q lm = Z ∞ u lm ( r ′ ) ρ / ( r ) r ′ l +1 dr ′ (17)From numerical point of view we solve the equation (13) with the associated boundaryconditions as it follows: first we guess the term q lm (considering the particular system tobe studied, the magnitude can be easily guessed) and solve the equation in such a waythat the solution satisfies the asymptotic behavior mentioned above for the selected value of q lm . With the solution constructed in these way we find a new value of q lm and repeat theprocedure until we reach convergence condition of the solution.Even though can seem to be a long road to the convergence, in practice, this is reachedwithin 10 iterations.Again, in comparison with [11] we have different asymptotic boundary condition, simplyfrom the form of our equation and the meaning of the unknown. The entire method ofiteration for finding the polarizability as a parameter of asymptotic behavior is original, atthe best of our knowledge and essential for the results. The above mentioned reference doesnot discuss such matters. 7 . Dipole case In this section we shall apply the method described above to the specific case of dipolarresponse. The applied field is v ( ~r ) = rcosθ and consequently v = r . The equation (15)becomes: d u ( r ) dr − u ( r )( 2 r + 6 πγ ρ / ( r )) = 0 (18)Of course, this can and must be solved numerically, but there are some specific cases withanalytic solutions which can be helpful for the boundary conditions.For ρ ( r ) = 0 , which usually describes the regions with large r where the density mustbe null we have the solution which respects the boundary condition (19) for q , u ( r ) = Ar + Br . In the more general case of ρ ( r ) = const we have a more elaborate solution( k = 6 πρ / /γ ): u ( r ) ∝ ( e kr ( 12 k r − k ) − ( e − kr ( 12 k r + 12 k )) (19) IV. RESULTS AND DISCUSSIONA. Na clusters
Sodium clusters represent a textbook metal cluster due to the nature of the element whichhas a single electron on the last shell, namely the
N a with the atomic number Z = 11 andthe electronic configuration s s p s . This element has been taken into account in thiswork, due to their close to spherical symmetry for medium sized clusters and due to thestrong metallic character. The classical electromagnetism provides, in the frame of smallmetal sphere model, a polarizability connected with the radius by: α classic = R (20)Experimental data reveals higher poralizabilities for all N a clusters, only in the highradius limit, the classical value is reached.In our calculations, different clusters were taken into account as having spherical sym-metry and a constant density of atoms. The electrons on the first two atomic levels were8onsidered as core electrons and so, the jellium model reduces to a sphere of a radius con-nected to the number of atoms through the Weitzecker-Seitzs radius. The positive jelliumcharge of N | e | distributed by a Wood-Saxon profile like in (1) (but no angular dependence),with a sharp fall of density around the radius of the cluster ( σ ≃ . a usually used [8]) andSeitz radius r = 3 . a , atomic units of length, see Fig. 1a), for the case N = 40 . Otherparametrization of smaller σ have been explored, but due to sensitivity of the method farfrom the center of the cluster, this parametrizations give worse results.Taking into account the spherical symmetry, the TF equation reduces to radial differentialequation: d dr ( Φ ( r ) r ) = 4 πr (( Φ ( r ) rγ ) / − ρ jel ( r )) (21)For the same number of atoms, in Fig 1a) (continuous line) is plotted the electrondensity ρ as obtained from above equation (21). In practice, clusters with the number ofatoms between and were investigated, but in Fig. 1 just a generic plot of the jelliumdensity and the electron density is presented in units of /r , with r the radius of thecluster.This distribution manifests a tail beyond the jellium volume associated with the quantumbehavior of the electrons. The ground state electrostatic potential Φ is plotted in Fig. 1b),while the radial dependence of the induced charge density is represented in Fig 1c) (blue-filled). In figure Fig. 1c) we’re drawn for comparison the asymptotic function as describedby (16) and actual solution.The proposed equation (15) has been used in the dipolar case with the analytic limits(16),(19) and the polarizability was obtain in a quite good agreement with the experimentalresults[14]. The results can be seen and compared with reference [14] in tabel I and in 2where the quantity α/N as a function of N is plotted. The solid horizontal line is associatedwith the classic solution.For low numbers of atoms, the equation fail to describe quantitatively the polarizabilitydue to the fact that the jellium model and the spherical shape of the cluster are no longerrealistic approximations while in the range N = 40 , N = 100 we can see an error bellow (plotted in 2), the only deviations being a consequence of shell effects. Also, the tendencyof decreasing to a constant (bulk) value with the number of atoms involved, explained as a9 .5 1.0 1.5 2.0 r H r L Ρ H r - L r H r L - - - - - F H e (cid:144) ΠΕ r L - - - r H r L - Ρ , Ρ H r - L FIG. 1: a)Jellium density (Dashed line) and electron density (continuous line) ρ for the N a cluster with N = 40 ; b) Ground-state electrostatic potential Φ for the N a cluster with N = 40 ; c)Fitting the solution u ( r ) (continuous) to the asymptotic function (red,dashed) at large distances for the N a cluster with N = 40 ; d) Ground-state electron density (dashed) and induced charge density (continuous) on the Oz direction.( The induced charge was intentionally raised up in order to have a visible effect. in reality, itseffect is much smaller than ground state so no negative density region can arise) classical limit of our semi-classical treatment of the electron system. ææ æ æ æ æææ æ æ æ ææ æ æ æ æææ æàà à à à ààà à à à àà à à à àààà N Α (cid:144) N H A L FIG. 2: Theoretical vs Experimental polarizability in
N a [14]From numerical solution obtained with the above method, we have observed that it is10ABLE I: Static polarizability of
N a clusters with < N < N Exp. Result19 16.66 19.6620 16.86 19.626 16.16 17.94430 17.6 16.7634 16.7 15.839 17.3 15.640 14.7 15.340 16.16 15.346 18. 15.50 16. 14.955 16.76 14.757 13.46 14.758 14.46 14.668 13.86 14.377 15.76 14.184 14.26 14.91 13.66 13.992 13.26 13.993 15.36 13.993 15.36 13.87 possible an empirical parametrization of the approximative u ( r ) solution in the generalcase as: u ( r ) = 32 γ ( r − π r (1 − e − βr )) (22)Which allows us to formulate a final empirical sum rule-like expression for static dipolepolarizability. The potential of this observation is that links the linear response only to theground state properties of the system as in the usual moments of the response function:11 æ æ æ æ æææ æ æ æææ æ æ æ æææ æ
40 60 80 100 N abs. error H % L FIG. 3: Relative error of obtained polarizability α ( r ) = r ∞ R ρ / ( r ) r dr γ + π ∞ R ρ / ( r )(1 − e − (0 . − . /N ) r ) rdr (23)We make comparison with the well known sum rule for static dipole polarizability [[2]]exhibited by spherical metal clusters in which the main contribution is given through theso called spilled − out electron which are considered outside the jellium region and appearproportional to δ in the approximation : α ≃ r (1 + δ ) [12]. B. C fullerene The Buckminster fullerene represents one of the most studied molecule in the last decadesdue to its high symmetry, special features, high stability, etc. Consequently, the polariz-ability has been studied [4] in many models, the best theoretical results being obtainedin the frame of DFT-LCAO while other theories obtained large errors in respect with theexperimental data.Essentially, fullerene is a carbon molecule with the atoms placed on a structure similarwith that of a soccer ball. Due to the fact that carbon is not a genuine metal, one couldargue that to study it along with true metallic clusters like sodium, it is a bad mistake.Nonetheless, even if from electronic structure and band gap point of view our approach cannot be justified, it is a known fact that the optical spectra from fullerene exhibit a large, welllocalized plasmon. Further, this plasmon it is explained as being a surface plasmon [10] and12IG. 4:
Geometry of C60 fullerene so, it can be reasonably concluded that in the dynamic regime, the electrons from fullerenebehave close to the ones from a metal. For this reason we have applied Thomas-Fermiequation for the semi-delocalized electrons and obtain a good description for polarizability,Mie’s plasmon centroid and the density distribution for the ground-state.We have used as jellium model, a gaussian distribution centered on the radius on whichthe carbon nuclei are placed but with a small width, described by the equation ρ jel ( r ) ∝ exp ( − σ ( r − r ) ) . Regarding the charge, our jellium model contain the core electrons fromthe s s while the other 240 electrons from p are considered quasi-free and taken intoaccount in the TF calculations. The results are quite sensitive to the width ( ∆ = fullwidth at half maximum) of the jellium gaussian distribution and for that reason we haveperformed our calculations with different values for this quantity between .
01 ˚ A and . A .This impediment is hard to be avoided since the physical meaning of this width is theradius of the core electrons in which they can be accounted as part of jellium, but in anapproximative physical way, our values cover the usual atomic values for this feature andprovide a set of very close values to the experimental data.Also, another approach to the jellium model was considered the spherical homogeneousshell (discussed in almost all papers using jellium model in C investigations [9][7][15])centered on the mean radius of r = 3 .
54 ˚ A and with a width of . A [9].The results in the electron density are physical and in good agreement with the exper-imental values for the inner, ≃ . A and outer radius of the fullerene ≃ . A , see 5a),c).From the calculations of dipolar polarizability, we have obtained the expected volumic shiftin density on the direction of the potential gradient ( Oz axes) shown in Fig 5b),d) whilethe value for polarizability is sweepings an interval between
80 ˚ A and
85 ˚ A , depending onthe chosen full width at half maximum of the gaussian jellium.In Fig 6 we have plotted the results for C polarizability for different parametrizations13 .5 1.0 1.5 2.0 r H r L Ρ H r - L - - r H r L Ρ , Ρ H r - L r H r L Ρ H r - L - - r H r L Ρ , Ρ H r - L FIG. 5: a)Jellium density (Dashed line) and electron density (continuous line) ρ for the C cluster inhomogeneous shell parametrization; b)) Ground-state electron density (dashed) and induced chargedensity (continuous) on the Oz direction for homogeneous shell; c))Jellium density (Dashed line) andelectron density (continuous line) ρ for the C cluster in narrow gaussian parametrization; d)Ground-state electron density (dashed) and induced charge density (continuous) on the Oz direction fornarrow gaussian jellium of the jellium model. The results are unexpectedly close to the experimental value of A .While the global aspect of ground state electron density has no essential dependence on ∆ ,the values of the density far from center or the cluster influence the value of the polarizability,fact which explains the spectrum of obtained values. æææææææææææ Σ H A L Α H A L FIG. 6: Polarizability vs width of different parametrization for C fullereneIn the case of spherical homogeneous shell the obtained polarizability was α = 92 ˚ A .14he fact that the results from the gaussian parametrization of the jellium were closer tothe experimental value raise the question of weather this aspect is a mathematical propertyof the equations involved, or simply the gaussian case is more realistic as discussed due toasymptotic tail of the core electrons from carbon atoms which must be taken into accountin the geometry of jellium.In order to test further the power of our equation and the validity of the approximationsinvolved we calculate the polarizability of fullerenes with and carbon atoms also ina spherical symmetry and with a gaussian profile. The obtained results are around
260 ˚ A for C comparable with the RPA result of
300 ˚ A [17] and
340 ˚ A for C compared with
432 ˚ A from RPA [17].As the size of fullerene increases, the method starts to fail, one of the reasons being thefact that the spherical symmetry begins to be broken. Nonetheless, the results are stillcomparable with those from more involved methods [4] from computational point of view. Conclusions
We exploit the Thomas-Fermi theory to compute the ground-state density of the electronsystem in a various number of Na clusters and C fullerene using the anzatz of sphericalsymmetry and the jellium model for ionic background. Further the perturbation theory itis used to derive a differential equation in such TF systems for a general external one-bodypotential from which the induced change in the density of electrons can be derived andconsequently the static linear response for any angular dependence or multipolarity.This equation for multipolar moments it is solved for the same metallic clusters in thecase of dipole external potential and the dipole polarizabilities are obtained. The errors areunder for the Sodium clusters while for fullerene, in a certain parametrization of thejellium model, we can obtain even the experimental value of the polarizability.From all the semi-quantitative results, we conclude that our method is fast numericallyand a good replacement for all the ab initio method which allow to compute the static linear15esponse. [1] Matthias Brack. The physics of simple metal clusters: self-consistent jellium model and semi-classical approaches. Reviews of Modern Physics , 65(3):677, 1993.[2] Walt A de Heer. The physics of simple metal clusters: experimental aspects and simple models.
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