GENERAL STATIC POLARIZABILITY IN SPHERICAL NEUTRAL METAL CLUSTERS AND FULLERENES WITHIN THOMAS-FERMI THEORY

Transcription

1 ATOMIC PHYSICS GENERAL STATIC POLARIZABILITY IN SPHERICAL NEUTRAL METAL CLUSTERS AND FULLERENES WITHIN THOMAS-FERMI THEORY D. I. PALADE 1,2, V. BARAN 2 1 Department of Theoretical Physics, National Institute of Laser, Plasma and Radiation Physics, PO Box MG 36, RO , Măgurele-Bucharest, Romania, EU dragos.i.palade@gmail.com 2 Faculty of Physics, Bucharest virbaran@yahoo.com Received August 28, 2014 We study the static linear response in spherical Thomas-Fermi systems deriving a simple differential equation for the general multipolar moments and the associated polarizabilities. We test the validity of the equation on the sodium clusters having between 20 and 100 atoms and on fullerenes between C 60 and C 240 and propose it for general Thomas-Fermi systems. Our simple method provides results which deviates from experimental data with less then 10%. Key words: Metal clusters, Thomas-Fermi, polarizability, Sodium clusters, C 60, fullerenes. 1. INTRODUCTION The topic of linear response in an external field is a crucial problem in manybody physics and it has been intensively studied through different techniques. For the static case, the problem simplifies the static response functions as polarizability, magnetic susceptibility, etc. For some mesoscopic systems, various classical methods have been applied successfully during the first part of the 20th century, especially Mie s theory for electromagnetic scattering and electrostatic modelling 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 the experimental data. Semi-classical or full quantum models are appropriate to describe the observed features. Methods as Hartree-Fock theory, RPA, Density Functional Theory, mean field theories in general, give good results regarding the stationary (or dynamic) properties of such atomic systems. Nonetheless, they have also a flaw: the computational efforts are huge and may not worth to use such complex methods to derive more transparent physical interpretations for simple quantities as is polarizability, for example. On the other hand, there are classes of systems, like metallic clusters, in which some special properties manifest. For example, in alkali elements it is well known RJP Rom. 60(Nos. Journ. Phys., 7-8), Vol , Nos. 7-8, (2015) P , (c) 2015 Bucharest, - v.1.3a*

2 1020 D. I. Palade, V. Baran 2 the presence of a quasi-free electron on the last unclosed shell which induces a behaviour for the valence electrons similar with the one described by the free homogeneous electron gas model (HEG). This picture is also employed in solid-state physics and can be exploited in the atomic domain through the Thomas-Fermi model [1 3] which approximates the term of kinetic energy in the electron system with the local form of HEG and simplifies the treatment. Extended versions of it can consider additional terms for the exchange-correlation potential (from Kohn-Sham potential) [3] or supplementary gradient correction from the Weizsacker term [4]. Even if such an approximation provides a very simple way to obtain the electron density, the systems must be carefully chosen, since usual errors can be around 10 20%, or larger, for different observables and due to no-binding Teller s theorem [5], in molecular systems appears the phenomenon of instability. Usually, the extended versions of Thomas-Fermi provide better results. We will refer in the present work to a simpler and a new form of the theory since our main goal is to achieve a quantitative description with a reduced computational effort. The best suited cases are those with a large volume extension, metallic or transition character and in which, we are not interested in fine details of the density profile or single particle (pseudo)wave functions, excitation energies, etc. For all these reasons, we shall focus next on the Thomas-Fermi theory and on modelling the linear response to a static external potential in its most general form. The theoretical results will be applied to various medium-sized sodium clusters and to the famous C 60 fullerene and C 240 and show that the method provides reasonable results. 2. FORMAL BACKGROUND 2.1. METALLIC CLUSTERS AND THE JELLIUM MODEL Clusters are mesoscopic systems containing even [6] atoms of the various elements. The theoretical methods of investigation extends from molecular to bulk (solid-state) domain, both classical and quantum. As we mentioned, the clusters formed from metallic elements have the property that the unclosed shell electrons are loosely bound and their behaviour is close to HEG one. We shall treat the problem in the Born-Oppenheimer approximation considering that their dynamics and de-localization are high enough to consider the ionic background effects in an averaged manner. By ionic background we understand the system obtained from the coupling between charged nuclei and the core electrons which determine a net positive charge. So our framework is the jellium model in which the ionic background is approximated by a homogeneous positive charge distribution while the free electrons will have an almost constant density in the

3 3 General static polarizability within Thomas-Fermi theory 1021 bulk region. For clusters, the jellium model it was also applied, the ionic background determining the smooth Coulomb potential with the appropriate symmetries. Then Coulomb potential may be included as an input in different theoretical approaches (Hartree-Fock, DFT, etc.) to derive the electron density. The self-consistent jellium model proved to be a very appropriate approximation predicting quantitative results in good agreement with the experimental data [7], [8]. The simplest geometry is that of a sphere, but can be somehow a troublemaker in numerical simulations since it has a discontinuity at the edge. A more refined model is the soft jellium which works with a Wood-Saxon radial profile and any possible angular dependence, giving access to deformed geometries of the ionic part. The jellium density in its most general form is: ρ jel (r,θ,ϕ) = 3 r R(θ,ϕ) 4πrs 3 [1 + exp( )] 1 (1) σ jel with R(θ,ϕ) = R 0 (1+ l,m α lmy lm (θ,ϕ)). The Wigner-Seitz radius r s is a parameter of the bulk domain interpreted as the volume occupied by a single atom while Z is the difference between the numbers of protons and the number of bound electrons and so Z e is the total charge of the jellium. The jellium density satisfies the condition: ρ jel d 3 r = Z. The limiting case of spherical sharp distribution is obtained when σ jel THOMAS-FERMI MODEL The Thomas-Fermi model (TF) was derived independently by L. Thomas and E. Fermi in 1927, soon after the introduction of Schrödinger equation in The basic approximation of the model is to treat the electron density distribution in the atom using a local approximation for the kinetic term i.e. the HEG approximation: E kin = 3γ 5 R ρ 5/3 ( r)d 3 r, with γ = (3π 2 ) 2/3 ħ 2 /2m. If V ( r) is the external potential, the total energy density functional (EDF) can be 3 written: E[ρ( r)]= 3γ ρ 5/3 ( r)d 3 r+e ρ( r)v ( r)d 3 r+ 1 e 2 ρ( r)ρ( r ) 5 R 3 R 3 2 4πε 0 R 3 r r d 3 r dr 3 (2) The ground-state energy and the corresponding density distribution are obtained within a Ritz variational principle, searching for the minimum of the EDF: E T F = min{e[ρ( r)] ρ L 5/3 (R 3 ), ρ( r)d 3 r = N, ρ( r) 0} (3) R 3 The condition ρ L 5/3 (R 3 ) refers to the fact that density is a 5/3 integrable function over R 3 and so, is a condition for finite kinetic energy to emerge. The constrain associated with the condition R ρ( r)d 3 r = N is introduced through Euler-Lagrange 3

4 1022 D. I. Palade, V. Baran 4 multiplier technique. By solving the variational problem is arrived to the Thomas- Fermi equation: γρ 2/3 = max[0,φ µ] (4) In this equation, Φ( r), Φ : R 3 R is the Coulomb potential contribution while µ is the chemical potential: Φ( r) = e V ( r) e2 4πε 0 R 3 ρ( r ) r r dr 3 (5) µ = ET F (6) N In turn, the Coulomb potential is the generated by the charge distribution and is the solution of the Poisson equation Φ( r) = e V ( r)+e 2 ρ( r)/ε 0. In the case of neutral electric systems, the chemical potential is zero [1], but this feature is maintained only within the Thomas-Fermi method. The additional terms, as Dirac or Wiezsacker contributions, break this property. In the jellium approximation V ( r) is induced by ρ jel and the Thomas-Fermi is expressed in differential form as: Φ( r) = e 2 /ε 0 (γ 3/2 Φ 3/2 ( r) ρ jel ( r)) (7) 3. THEORY 3.1. PERTURBATION THEORY AND DENSITY CHANGES In the absence of external interactions, for the ground-state, the TF equation leads to γρ 2/3 0 (r) = Φ 0 (r) with ρ 0 the ground state density of the free electrons. We shall study in the following the static linear response based on TF approximation by considering a time-independent potential v( r) of arbitrary form. If we denote the coupling strength to the free electrons by λ, then we can expand: v( r) = λ v lm (r) Y lm (θ,ϕ) (8) r With the new contribution to the energy density, ρv( r), the TF equation for the stationary state becomes: γρ 2/3 ( r) = Φ( r) (9) The perturbation will induce a spatial change of the charge distribution which can be expanded as a power series in the coupling constant λ: ρ( r) = ρ 0 (r) + λρ 1 ( r) + λ 2 ρ 2 ( r) +... If we resume to the first order term (λ e 2 /(4πεr 0 )), then ρ 1 satisfy the condition R ρ 1 ( r)dr 3 = 0 since 3 R ρ 3 0 ( r)dr 3 = R ρ( r)dr 3 = Z. In the 3 presence of the external potential the initial spherical symmetry is broken and the densities verify the properties: ρ : R 3 R +, ρ 0 : R 3 R +, ρ 1 : R 3 R.

5 5 General static polarizability within Thomas-Fermi theory 1023 The linearized kinetic density energy term and the potential Φ are: γρ 2/3 ( r) = γρ 2/3 2γ 0 (r) + λ 3ρ 1/3 0 (r) ρ1 ( r) (10) ρ 1 ( r) Φ( r) = Φ 0 ( r) + v( r) λ r r dr 3 (11) and the TF equation for the perturbed part becomes: 2γ ρ 1 ( r) λ 3ρ 1/3 0 ( r) ρ1 ( r) = v( r) λ r r dr 3 (12) In spherical coordinates we adopt the following expansion of ρ 1 : ρ 1 ( r) = ρ 1/3 0 (r) Y lm (θ,ϕ) u lm(r) r lm By using the expressions for the potential (8) and for the density (13) in the equation (12) is obtained the following radial equation for u lm (r): (13) d 2 u lm (r) l(l + 1) dr 2 u lm (r)( r 2 + 6π γ ρ1/3 0 (r)) = d2 v lm (r) l(l + 1) dr 2 r 2 v lm (r) (14) We want to mention that our equation is an approximate version of the equation (23) deduced in [9] in the absence of Weizsacker, exchange and correlation terms. However the present equation is just a differential equation, simpler than the integrodifferential version deduced in [9]. This fact can be seen by setting the β factor to 0 and excluding exchange-correlation effects BOUNDARY CONDITION AND GENERAL POLARIZABILITY In order to have finite perturbed density, ρ 1 (0), in the origin we require that lim u lm(r)/r to be finite. Concerning the behaviour at infinity we shall ask for r 0 the coefficient u lm (r) to follow the behaviour of the perturbation i.e. u lm (r) = 3/(2γ)v lm when r. From TF eq (12), the asymptotic behaviour of u lm is: u lm (r) 3 2γ (v lm 4π q lm 2l + 1 r l ) (15) Here, the q lm term is the multipole moment associated with the induced charge: q lm = 0 u lm (r )ρ 1/3 0 (r)r l+1 dr (16) From numerical point of view we solve the equation (12) with the associated boundary conditions as it follows: first we guess the term q lm (considering the particular

6 1024 D. I. Palade, V. Baran 6 system to be studied, the magnitude can be quite easily deduced) and solve the equation in such a way that the solution satisfies the asymptotic behaviour mentioned above for the selected value of q lm. With the solution constructed in this way we find a new value of q lm and repeat the procedure until we reach convergence condition for the solution. In practice, a variation of the solution norm smaller then 10 3 is reached within 10 iterations. We mention that the method of iteration for finding the polarizability as a parameter of asymptotic behaviour is new at the best of our knowledge and essential for the results. The above mentioned reference does not discuss such aspects DIPOLE CASE In this section we shall apply the method described above to the specific case of dipolar response. The applied field is v( r) = rcosθ and consequently v 10 = r 2. The equation (14) becomes: d 2 u 10 (r) dr 2 u 10 (r)( 2 r 2 + 6π γ ρ1/3 0 (r)) = 0 (17) While this equation can be solved numerically, there are some specific cases with analytic solutions which can be helpful for the boundary conditions. For ρ 0 (r) = 0, which characterize the regions with large r where the density cancel the solution associated to the boundary condition (19) for q 10 is u 10 (r) = Ar 2 + B r. In the general case with ρ 0 (r) = const a more elaborated solution (k 2 = 6πρ 1/3 0 /γ) is: u 10 (r) e kr 1 ( 2k 3 r 1 2k 2 ) 1 e kr ( 2k 3 r + 1 2k 2 ) (18) 4. RESULTS AND DISCUSSION 4.1. Na CLUSTERS Sodium clusters represent a textbook metal cluster due to the nature of the atomic element which has a single electron on the last shell, namely the Na with the atomic number Z = 11 and the electronic configuration 1s 2 2s 2 2p 6 3s 1. This system is considered in this work, due to almost spherical symmetry for medium sized clusters and due to the strong metallic character. We are aware that the condition for spherical symmetry is satisfied only for those clusters with a magic number of electrons, but, as we shall see, the small deformations do not have a significant effect on the polarizability. The classical electromagnetism provides, in the frame of small metal sphere model, a polarizability connected with the radius: α classic = R 3 (19)

7 7 General static polarizability within Thomas-Fermi theory 1025 Experimental data reveals higher poralizabilities for all Na clusters, only in the high radius limit, when the classical value is reached. In our calculations, different clusters were taken into account as having spherical symmetry and a constant density of atoms. The electrons on the first two atomic levels were considered as core electrons and so, the jellium model reduces to a sphere of a radius connected to the number of atoms through the Wigner-Seitz radius. The positive jellium charge of N e has a Wood-Saxon profile like in (1) without any angular dependence, with a sharp fall of density around the radius of the cluster. We adopted a value σ 0.8a 0 usually used [6] and a Wigner-Seitz radius r 0 = 3.93a 0, see Fig. 1a), for the case N = 40. Other parametrization with smaller σ have been explored, but due to sensitivity of the method far from the centre of the cluster, this parametrizations give usually worse results. Taking into account the spherical symmetry, the TF equation reduces to a radial differential equation: d 2 dr 2 (Φ 0(r) ) = 4πr(( Φ 0(r) r rγ )3/2 ρ jel (r)) (20) In Fig. 1a) we have plotted the radial profile for the density of electrons (obtained with TF) vs. jellium density. Both are presented in units of 1/r0 3, with r 0 the radius of the cluster and one can observe the tail of density beyond the jellium volume, associated with the quantum behaviour of the electrons and responsible for the larger than classical values of polarizability. In Fig. 1 b), for the same cluster we show the smooth Coulomb potential, while in Fig. 1 c) the numerical solution of eq. (17) in full line fitting the asymptotic condition (dashed) prescribed by eq. (18). Finally, the induced density variation is presented in Fig. 1 d) in azimuthal profile. The proposed equation (14) has been used in the dipolar case with the analytic limits (15), (18). The values for the polarizability are in a quite good agreement with the experimental results [10]. In tabel 1 and in Fig. 2 the obtained results are compared with experimental values and with those from [10]. We plotted the quantity α/n as a function of N. The solid horizontal line corresponds to the classic solution. For small numbers of atoms, the equation fail to describe quantitatively the polarizability due to the fact that the jellium model and the spherical shape of the cluster do not represent realistic approximations. In the range N = 40, N = 100 the relative errors are bellow 10% (plotted in 2), the deviations being related to the shell effects. We also remark the evolution towards (bulk) classical value when the number of atoms of the cluster increases. From numerical results a possible empirical parametrization of the approximative u 10 (r) solution in the general case as: u 10 (r) = 3 2γ (r2 4π 3r (1 e βr )) (21)

8 1026 D. I. Palade, V. Baran 8 (a) (b) (c) Fig. 1 a) Jellium density (dashed line) and electron density (continuous line) ρ for the N = 40 Na cluster; b) Ground-state electrostatic potential Φ 0 for the Na cluster with N = 40; c) The fitted solution u 10 (r) (continuous) to the asymptotic function (red,dashed) at large distances for the N = 40 Na cluster; d) The ground-state electron density (dashed) and induced charge density (continuous) along the Oz direction. (The induced charge was intentionally enlarged in order to have a visible effect. In reality, its effect is much smaller than ground state so no negative density region can arise). which allows us to formulate a final expression for static dipole polarizability: α 10 (r) = r 3 0 2γ 3 + 4π ρ 1/3 0 (r)r 4 dr (d) ρ 1/3 0 (r)(1 e ( /N 2 )r )rdr. (22) We can compare this expression with the well known sum rule for static dipole polarizability [8] exhibited by spherical metal clusters. In this case the main contribution is given by the spilled out electrons which are considered outside the jellium region and appear proportional to δ : α r0 3 (1 + δ) [11]. From a comparison of our results with those from [9] we observe a similar qualitative behaviour of polarizability with the size of the cluster C 60 FULLERENE The Buckminster fullerene represents one of the most studied molecule in the last decades due to its high symmetry, special features, high stability, etc. Conse-

9 9 General static polarizability within Thomas-Fermi theory 1027 Table 1 The static polarizability of Na clusters with 20 < N < 100 N Exp. Result quently, the associated polarizability has been also studied [12] within many models, the best theoretical results being obtained in the frame of DFT-LCAO. The fullerene is a carbon molecule with the atoms placed on a structure similar with that of a soccer ball. Due to the fact that carbon is not a genuine metal, one could argue that to study it along with true metallic clusters like sodium is not consistent. Nonetheless, even if from electronic structure and band gap point of view our approach can not be justified, it is also known that the optical spectra from fullerene exhibits a large, well localized plasmon. Further, this plasmon is explained as being a surface plasmon [13] and so, it can be reasonably concluded that in the dynamic regime, part of the electrons from fullerene behave close to the ones from a metal. Moreover, we have obtained recently [14] good results in the dynamic regime using TF approximation for C 60, therefore we expect good results regarding the static properties as polarizability, as well. We have employed as jellium model, a Gaussian distribution centered on the radius at which the carbon nuclei are placed but with a small width, described by the equation ρ jel (r) exp( σ(r r 0 ) 2 ). Our jellium model contain as core electrons those from the 1s 2 2s 2 while the other 240 electrons from 2p 2 are considered quasifree and taken into account in the TF calculations. The results are quite sensitive

10 1028 D. I. Palade, V. Baran 10 (a) Fig. 2 a) Theoretical (dashed line) vs. Experimental polarizability (solid line) in Na [10]; b) Relative error of obtained polarizability. (b) Fig. 3 Geometry of C60 fullerene. to the width ( = full width at half maximum) of the jellium Gaussian distribution and for that reason we have performed our calculations with different values for this quantity, between 0.01Å and 0.6Å. This impediment is hard to be avoided since the physical meaning of this width is the radius of the core electrons in which they can be accounted as part of jellium but our choices cover the usual values for this feature. Other approaches to the jellium model considering the spherical homogeneous shell are discussed in several papers presenting investigations for C 60, see [15][16][17]. The distribution is centred at the mean radius of r 0 = 3.54Å and with a width of 1.5Å [15]. The results concerning the electron density are in a good agreement with the experimental values for the inner, 1.8Å and outer radius of the fullerene 5.1Å, see 4a),c). From the calculations of dipolar polarizability, we have obtained the expected volumic shift in density on the direction of the potential gradient (Oz axes) as is shown in Fig. 4 b),d). The value for polarizability is sweepings an interval between 80Å 3 and 85Å 3, depending on the chosen full width at half maximum of the Gaussian jellium. In Fig. 5 we have plotted the results for C 60 polarizability considering different parameterizations of the jellium model. The results are resonable close to the experimental value 78Å 3. While the global features of the ground state electron density are not strongly influenced by, the values of the density far from centre or the cluster

11 11 General static polarizability within Thomas-Fermi theory 1029 (a) (b) (c) Fig. 4 a) Jellium density (Dashed line) and electron density (continuous line) ρ for the C60 cluster in homogeneous shell parametrization; b) Ground-state electron density (dashed) and induced charge density (continuous) along the Oz direction for the homogeneous shell; c) Jellium density (Dashed line) and electron density (continuous line) ρ for the C60 cluster for the narrow Gaussian parametrization; d) Ground-state electron density (dashed) and induced charge density (continuous) on the Oz direction for the narrow Gaussian jellium. (d) are affecting the value of the polarizability. Since in the case of spherical homogeneous shell the obtained polarizability was α = 92Å 3 we conclude that the results from the Gaussian parametrization of the jellium are closer to the experimental value. In order to test further the validity of our approximations we calculated the polarizability of fullerenes with 180 and 240 carbon atoms for a spherical symmetry and with a Gaussian profile. The obtained results are around 260Å 3 for C 180 comparable with the RPA result of 300Å 3 [18] and 340Å 3 for C 240 compared with 432Å 3 from RPA [18]. As the size of fullerene increases, the method starts to fail, one of the reasons being the fact that the spherical symmetry begins to be broken. Nonetheless, the results are still comparable with those from more involved methods [12].

12 1030 D. I. Palade, V. Baran 12 ΑA ΣA Fig. 5 Polarizability vs. width of different parametrization for C 60 fullerene. 5. CONCLUSIONS In this work we exploit the Thomas-Fermi theory to compute the ground-state density distribution of the electrons in a number of Na clusters and C 60 fullerene assuming the spherical symmetry and the jellium model for ionic background. A perturbation approach is used to derive a differential equation for such TF systems in the presence of a general external one-body potential from which the induced change in the density of electrons can be derived for any multipolarity. The equation for multipolar moments was solved in the case of dipole external potential and the dipole polarizabilities were calculated. The errors are below 15% for the Sodium clusters while for fullerene and for certain parametrization of the jellium model, we can obtain results even closer to the experimental value of the polarizability. From comparison with similar but more complex equations derived in [9] we find a nice quantitative agreements for the Na clusters. Moreover our method presents the advantage of being fast for numerical point of view and allows a rapid estimate of the static linear response for arbitrary multipolarities. REFERENCES 1. E. H. Lieb, B. Simon, Advances in Mathematics 23(1), (1977). 2. L. H. Thomas, in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 23, pp , (Cambridge Univ. Press, 1927). 3. G. W. Drake, Springer handbook of atomic, molecular, and optical physics, (Springer, 2006). 4. C. v. Weizsäcker, Zeitschrift für Physik A Hadrons and Nuclei 96(7), (1935). 5. H. A. Jahn, E. Teller, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences pp (1937). 6. P.-G. Reinhard, E. Suraud, Introduction to cluster dynamics, (John Wiley & Sons, 2008). 7. M. Brack, Rev. Mod. Phys. 65, (1993). 8. W. A. de Heer, Rev. Mod. Phys. 65, (1993).

13 13 General static polarizability within Thomas-Fermi theory L. Serra, F. Garcias, M. Barranco, J. Navarro, L. C. Balbas, A. Rubio, A. Mananes, Journal of Physics: Condensed Matter 1(51), (1989). 10. G. Tikhonov, V. Kasperovich, K. Wong, V. V. Kresin, Phys. Rev. A 64, (2001). 11. D. R. Snider, R. S. Sorbello, Phys. Rev. B 28, (1983). 12. G. K. Gueorguiev, J. M. Pacheco, D. Tománek, Phys. Rev. Lett. 92, (2004). 13. S. W. J. Scully, E. D. Emmons, M. F. Gharaibeh, R. A. Phaneuf, A. L. D. Kilcoyne, et al., Phys. Rev. Lett. 94, (2005). 14. D. I. Palade, V. Baran, Journal of Physics B: Atomic, Molecular and Optical Physics 48(18), (2015). 15. A. Rüdel, R. Hentges, U. Becker, H. S. Chakraborty, M. E. Madjet, J. M. Rost, Physical Review Letters 89(12), (2002). 16. M. J. Puska, R. M. Nieminen, Phys. Rev. A 47, (1993). 17. J. H. Weaver, J. L. Martins, T. Komeda, Y. Chen, T. R. Ohno, G. H. Kroll, N. Troullier, R. E. Haufler, R. E. Smalley, Phys. Rev. Lett. 66, (1991). 18. R. R. Zope, T. Baruah, M. R. Pederson, B. I. Dunlap, Phys. Rev. B 77, (2008).