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ELSEVIER Nuclear Physics A731 (2004) 347-354 www.elsevier.comilocatelnpe

Variational RPA for the dipole surface plasmon in metal clusters

K. Hagino”, G.F. Bertschb and C. Guet ’

“Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

“Institute for Nuclear Theory and Department of Physics, University of Washington, Seattle, WA 98195

“DCpartement de Physique ThCorique et AppliquCe, CEA-Ile de France, Boite Postal 12, 91680 Bruyhres le Chgtel, France

The deviation of the ionic background potential in simple metal clusters from the harmonic shape leads to a red shift of the surface plasmon from the Mie frequency, that is considerably larger than the spill-out correction. In order to estimate this effect, here we develop a variational approach to the RPA collective excitations. Using a simple trial form, we obtain analytic expressions for the energy shift beyond the spill-out contribution. We find that the additional red shift is proportional to the spill-out correction and has the same order of magnitude.

1. INTRODUCTION

The Kohn theorem [l-3] states that for a system with interacting particles in a confining external harmonic oscillator potential, a single state contains the whole dipole strength. The frequency of this collective state is equal to that of the confining potential, inde- pendently of the interaction among the particles and of the number of particles. This results from the fact that for a harmonic oscillator potential, the center of mass motion decouples exactly from the intrinsic motion, and that the interaction between particles is translationally invariant. Any deviation of the confining potential from the harmonic shape leads to an energy shift of the resonance energy as well as a redistribution of the oscillator strength into closely lying dipole states.

In the jellium approximation to simple metal clusters, the ionic background is approxi- mated by an uniformly charged sphere, thus the electron-ion potential inside the sphere is a harmonic oscillator with a frequency given by the classical Mie resonance formula[4,5],

(1)

where n is the density of a homogeneous electron gas, while outside the sphere it is the Coulomb potential, -Ze2/r. The measured resonance peak is considerably red-shifted from the Mie frequency, which can be attributed to a large extent to the finite size, i.e. to the deviation of the ion-electron potential outside the jellium sphere from the harmonic oscillator.

0375-9474/$ - see front matter 0 2004 Elsevier B.V. All rights reserved. doi:i0.1016/j.nuc1physa.2003.11.047

348 K. Hagino et al. /Nuclear Physics A731 (2004) 347-354

I I I 4 I 3 6000 - - - Harmonic Potential - - JelliumPotential N%J -

$5000 - I -; 4000- i s g 3000-

II II

5 - & 2000 II II

1000 -

$ 3508- I

$3000-

E, 2500-

4 2000- &! .s 1500 - s B 1000 -

8 t5 500-

Ol I 2 3 4 5 6

w W)

Figure 1. Strength function of Nazo in the jellium model. Upper panel shows the dipole strength function, broadened by a artifical width. Lower panel shows the integerated strength function. Dashed line is the results of the computation in which the jellium background potential is replaced by a harmonic oscillator.

Aw,, = WMie(l - dm)~ (2)

where AN/N is the fraction of electrons in the ground state that is outside the jellium sphere radius, is shown as w,,. One sees that the strength function is fragmented into two large components that are considerably red-shifted from the Mie frequency, and smaller contributions at higher frequencies. The corresponding spectrum with the jellium background potential replaced by a pure harmonic potential is shown by the dashed line. The integrated strength function is also shown in the lower panel of the figure. -

Recently, Gerchikov, et al. studied anharmonic effects in metallic clusters making use of a coordinate transformation to separate center of mass (cm.) and intrinsic motion [12]. The authors show that in absence of coupling between c.m. motion and intrinsic excitations the surface plasmon associated with a jellium sphere has a single peak which is red-shifted with respect to the Mie frequency by the spill-out electrons, Eq. (2). Turning on the coupling yields a further red shift which indeed is larger in magnitude than the spill-out contribution. Concomitantly, there is a partial transfer of strength into states of higher energy preserving the Thomas Reiche Kuhn sum rule. The approach requires the spectrum of excitations in the intrinsic coordinates, which were obtained by projection on the computed wave functions of the numerical RPAE. A similar approach was employed by Kurasawa et al. to discuss the size dependence of the width of the surface plasmon[l3].

In the present contribution, we wish to find an analytic estimate of the red shift, keeping

It is well established that the photoabsorp- tion spectra for clusters with a “magic” number of valence electrons, which behave as close shell jellium spheres, are properly de- scribed within the linear response theory using either the time-dependent local-density approximation (TDLDA)[6-81 or the ran- dom phase approximation with exact exchange (RPAE)[S,lO]. In order to illus- trate the discussion, we show in fig. 1 the dipole strength function of Naao in the jellium model, obtained with the computer program JellyRpa[ll]. The strength function in- cludes an artificial width of r = 0.1 eV for display purposes. The Mie frequency, Eq. (1)) is indicated by wo, while the prediction of the well known spill-out formula,

K. Hagino et al. /Nuclear Physics A 731 (2004) 347-354 349

as far as possible the ordinary formulation of RPA, and not singling out a collective state in the Hamiltonian [14]. 0 ur approach will be a variational RPA theory, which we present in the next section. The rest of the paper is organized as follows. In Section 3 we apply the formalism to a system of interacting electrons. The model Hamiltonian describes interacting electrons confined in a pure harmonic potential, whereas the perturbation corrects for the jellium confinement. The model RPA solution is derived analytically and first and second order corrections of the frequency shift are given.

2. VARIATIONAL RPA

In this section, we establish our notation for the RPA theory of excitations and develop a variational expression for perturbations to the collective excitation frequency. The perturbation behaves somewhat differently in RPA than in conventional matrix Hamiltonians because the RPA operator is not Hermitean.

As usual, the starting point is a mean field theory whose ground state is represented by an orbital set & satisfying the orbital equations

qpo14i = w$Ji (3)

where pa = Ci ]&(r)]“. The RPA equations are obtained by considering small deviations from the ground state,

(hi --t 4; + qzpt + yp). (4

Here zi, y; are vectors in whatever space (r-space,orbital occupation number,...) is used to represent &. The RPA equations can be expressed as

(G%- 4% + 6p* E * (bi = WCC,,

-(f4Pol - %)Yi - 6p * $ * (fJi = WY,

where the transition density Sp is defined by

and the symbol * denotes an operator or matrix multiplication. Eqs. (5) and (6) represent linear eigenvalue problem for a nonhermitean operator R and the vector 1~) = (~1, y1,22, yz, . ..). We will write the equations compactly as

Rlz) = w/z). (8)

For a nonhermitean operator, the adjoint vector (~1 is defined as the eigenvector of the adjoint equation, (zIR = +I. Fr om the symmetry of R it is easy to see that it is given by (~1 = (XI,-Y~,Q-Y~, . ..)+.

We now ask how to construct a perturbation theory starting from the zero-order wave function 1~~) that is the solution of an unperturbed Ro with eigenfrequency wg. If we had


the complete spectrum of Ro, the perturbation series for R = Ro + AR could be written down in the usual way,

etc. This is in fact what is done in ref. [12]. H owever, this requires diagonalizing Ro

which in general can only be done numerically. Instead we shall estimate the energy perturbation using a variational expression for the

frequency,

w = min (~0 + XwlRlzo 3- Xw)

w (zo+xwlzo+xw) ’ (10)

where ]w) is a vector to be specified later and X is to be varied to minimize the expression. Carrying out the variation and assuming that the perturbation is small, the value of X at the minimum is given by

x- (zolRlw) - wobolw) - - (WIRW} - Wl(WIW)

and the energy shift is

w = wo + (z,,lARIzo) - (+olRw) - wd4w))” (wlRw) - w(wlw) (12)

Here, WI = (zolRjzo) = wo + (z~IARIz~).

3. COLLECTIVE LIMIT OF THE SURFACE PLASMON

We apply the RPA variational perturbation theory derived in the previous section to the surface plasmon of small metal clusters. We write the single particle Hamiltonian as

h = ho + AV(r),

ho h2

= -2mv2 + ;mw;,P + u * PO,

where u * pa is the mean field potential,

u*po= .I

U(T, 7-‘)po(4) d37-‘.

(13)

(14

(15)

Here 21 is the electron-electron interaction, which may contain an exchange-correlation contribution from density functional theory. In this paper, we throughout use the jellium model for the ionic background, and also assume that the ion and the electron densities are both spherical. wo and AV(r) are then given by wg = Ze2/mR3 and

AV(r)= [+- (-4-G) ?$] @(r-R); (16)

respectively, R being the sharp-cutoff radius for the ion distribution.

K. Hagino et al. /Nuclear Physics A731 (2004) 347-354 351

The RPA equations can be solved exactly for the dipole resonance if AV is neglected. The solution is

bo)- (;) =-~(:$)+&g$$ (17)

associated with the eigenfrequency WO. See Ref. [14] f or a proof. Notice that the eigenfrequency wg is the same as the harmonic oscillation frequency in Eq. (14), agreeing with the Kohn’s theorem.

The familiar formula relating the red-shift to the electron spill-out probability can be recovered from the expectation value of the original RPA matrix,

(18)

However, the wave function zo must be taken with the collective ansatz applied to the Hamiltonian h. This is different from the ~0 defined in Eq. (17), which was based on the Hamiltonian ho. In the following, we have no further use for the original zo and we will use the same name here. Applying the RPA operator R to ~0, we find

Rlzo) = wobo) + 14,

where u is given by

(19)

The expectation value eq. (18) then reduces to

AN Aw = (z&L) = -wo -,

2N

with

AN= s ,” 47rr2dr /q)(T).

(20)

Eq. (21) is just the well-known spill-out formula, Eq.(2), to the first order in AN/N.

4. SECOND ORDER ENERGY SHIFT

We now consider the frequency shift in the second order perturbation. With ordinary Hamiltonians, one can construct a two-state perturbation theory using the vector obtained by applying AR to the unperturbed vector, 1~) = ARIzO). Thus, obvious possibilities for the perturbation are w = (y, Z) and U, but we find that neither produces a significant energy shift. The problem with u is that the z component is tied to the y component in Eq. (20). In fact, the energetics are such the y perturbation is much less than the z perturbation. In order to avoid this undesirable feature, we simply take the 2 component of u for the perturbation. That is, we use

(23)


for the 1~) in the variational formula (10). With this perturbed wave function, after performing the angular integration, we find the integrals in the formula to be [14]

A simple analytic formula for the energy shift can be obtained if we estimate Eqs. (24), @5h (261, and (27) assuming that the density po in the surface region is given by

PO(r) N /4-24-R)

(r 2 RI, (28)

with K2/2m = E, where E is the ionization energy. In order to simplify the algebra, we also expand AV(r) and take the first term,

dAV - N -3mwi(r - R).

dr

The result is [14],

3 AN w = w1 - 16 - 8. wO/e

. wo-. N

Note that the perturbation theory breaks down at E = w0/2. In realistic situations, E is always close to ws, and the perturbation theory should work in principle.

5. APPLICATION TO SODIUM CLUSTERS

Let us now apply the variational shifts to the optical response of Na clusters. No- tice that a precise definition of the red shift problematic as the strength is fragmented, particularly in large clusters. We therefore have simplified the jellium model in our numerical computations in order to artificially prevent any fragmentation of strength. To this end we put all the electrons in the lowest s-orbital, treating them as bosons. Other- wise, the model is the same as the usual sherical jellium model, with the electron orbitals determined self-consistently.

The results of the numerical calculation with the full effect of the surface are shown in Fig. 2 as the solid line. The collective spill-out correction from Eq. (2) is also shown as the dotted line. One sees that the additional shift due to the wave function perturbation is comparable to the spill-out correction, and has a similar N-dependence. The shift given by the variational formula Eq. (12) is s h own by the dashed line. The functional dependence predicted by the formula is confirmed by the numerical calculations, but the coefficient of N is too small by a factor of two or so.

K. Hagino et al. /Nuclear Physics A731 (2004) 347-354 353

0.9

3” ;

0.8 -RPA @. 0 Spill out L- -A Perturbation

I I I I I , I I I I I 10 20 30 40 50 60

N

Figure 2. Collective excitation frequency in the s-wave jellium model as a function of N. The solid line is the result of the numerical calculation. This is compared with the spill-out formula eq. (2) and the perturbation formula eq. (12) as the dotted and dashed lines, respectively.

6. CONCLUDING REMARKS

We have developed a variational approach to treat perturbations to the collective RPA wave functions, and have applied it to the surface plasmon in small metal clusters. Our zeroth order solution is the same as that used by Gerchikov et al. [la] and Kurasawa et al. [13]. It corresponds to the center of mass motion, and is the exact RPA solution when the ionic background potential is a harmonic oscillator. The deviation of the background potential from the harmonic shape is responsible for the perturbation. The first order perturbation yields the well-known spill-out formula for the plasmon frequency, as was also shown in Refs. [12,13]. The higher order corrections lead to the additional energy shift of the frequency [12], th e anharmonicity of the spectrum [la], and the fragmentation of the strength [13]. Th ose effects were studied in Refs. [12,13] by considering explicitly the couplings between the center of mass and the intrinsic motions. In this paper, we assumed some analytic form for the perturbation and determined its coefficient variationally. We found that this approach qualitatively accounts for the red shift of the collective frequency, but its magnitude came out too small by about a factor of two.

The method developed in this paper is general, and is not restricted to the surface plasmon in micro clusters. One interesting application might be to the giant dipole resonance in atomic nuclei. In heavy nuclei, the mass dependence of the isovector dipole frequency deviates from the prediction of the Goldhaber-Teller model, that is based on a simple c.m. motion[l5,16]. The shift of collective frequency can be attributed to the effect of deviation of the mean-field potential from the harmonic oscillator, and a similar treatment as the present one is possible.


ACKNOWLEDGMENTS

We would like to acknowledge discussions with Nguyen Van Giai, N. Vinh Mau, P. Schuck, and M. Grasso. K.H. thanks the IPN Orsay for their warm hospitality and financial support. G.F.B. also thanks the IPN Or-say as well as CEA Ile de France for their hospitality and financial support. Additional financial support from the Guggen- heim Foundation and the U.S. Department of Energy (G.F.B.) and from the the Kyoto University Foundation (K.H.) is acknowledged.

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