Variational RPA for the dipole surface plasmon in metal clusters

ELSEVIER

Nuclear

Physics

A731

(2004)

347-354 www.elsevier.comilocatelnpe

Variational K.

Hagino”,

“Yukawa

RPA for the dipole surface plasmon G.F. Bertschb

Institute

clusters

and C. Guet ’

for Theoretical

“Institute for Nuclear Seattle, WA 98195

in metal

Theory

Physics,

Kyoto

and Department

“DCpartement de Physique ThCorique 91680 Bruyhres le Chgtel, France

University,

of Physics,

et AppliquCe,

CEA-Ile

Kyoto University

606-8502,

Japan

of Washington,

de France,

Boite

Postal

12,

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, independently 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 approximated 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 doi:i0.1016/j.nuc1physa.2003.11.047

0 2004 Elsevier

B.V.

All rights

reserved.

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

348

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It is well established that the photoabsorption spectra for clusters with a “magic” number of valence electrons, which behave as close shell jellium spheres, are properly described within the linear response theory using either the time-dependent local-density approximation (TDLDA)[6-81 or the random phase approximation with exact exchange (RPAE)[S,lO]. In order to illustrate 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 includes 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,

6

Aw,, =

WMie(l

-

dm)~

(2)

where AN/N is the fraction of electrons in the ground state that is outside the jelFigure 1. Strength function of Nazo in lium sphere radius, is shown as w,,. One the jellium model. Upper panel shows sees that the strength function is fragmented the dipole strength function, broadened into two large components that are considerby a artifical width. Lower panel shows ably red-shifted from the Mie frequency, and the integerated strength function. Dashed smaller contributions at higher frequencies. line is the results of the computation in The corresponding spectrum with the jelwhich the jellium background potential is lium background potential replaced by a pure replaced by a harmonic oscillator. 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

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 RPA theory, which we present in the Hamiltonian [14]. 0 ur approach will be a variational in the next section. The rest of the paper is organized as follows. In Section 3 we apply The model Hamiltonian describes the formalism to a system of interacting electrons. 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)]“. from the ground state, (hi --t 4; + qzpt

The RPA equations

are obtained

by considering

+ yp).

(4

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

(G%- 4% + 6p* -(f4Pol

where

-

%)Yi

-

the transition

E

* (bi

=

WCC,,

6p * $

* (fJi

=

WY,

density

Sp is defined

= w/z).

number,...)

is used

by

and the symbol * denotes an operator or matrix multiplication. resent linear eigenvalue problem for a nonhermitean operator (~1, y1,22, yz, . ..). We will write the equations compactly as Rlz)

small deviations

Eqs. (5) and (6) repR and the vector 1~) =

(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

350

K. Hagino

et al. /Nuclear

Physics

the complete spectrum of Ro, the perturbation down in the usual way,

A731

(2004)

347-354

series for R = Ro + AR could be written

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

w

XwlRlzo

3- Xw)

(zo+xwlzo+xw)

(10)

’

where ]w) is a vector to be specified later and X is to be varied Carrying out the variation and assuming that the perturbation the minimum is given by

to minimize the expression. is small, the value of X at

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

x-

and the energy

shift is

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

WI =

(zolRjzo)

= wo +

3. COLLECTIVE

LIMIT

(z~IARIz~).

OF THE SURFACE

We apply the RPA variational perturbation theory the surface plasmon of small metal clusters. We write h

ho

=

ho +

=

h2 -2mv2

where u*po=

(12)

PLASMON derived in the previous section to the single particle Hamiltonian as

AV(r),

(13)

+ ;mw;,P

+ u * PO,

(14

u * pa is the mean field potential, .I

U(T, 7-‘)po(4) d37-‘.

(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)= respectively,

[+-

(-4-G) R

?$]

being the sharp-cutoff

@(r-R); radius

for the ion distribution.

(16)

K. Hagino et al. /Nuclear The RPA equations The solution is bo)-

(;)

Physics A731 (2004) 347-354

can be solved exactly

for the dipole

resonance

351

if AV is neglected.

(17)

=-~(:$)+&g$$

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 Hamiltonian Hamiltonian use the same

wave function zo must be taken with the collective ansatz applied to the h. This is different from the ~0 defined in Eq. (17), which was based on the ho. In the following, we have no further use for the original zo and we will name here. Applying the RPA operator R to ~0, we find (19)

Rlzo) = wobo) + 14, where

u is given by

(20) The expectation

value eq. (18) then reduces to

AN Aw = (z&L) = -wo -, 2N with AN=

s

,” 47rr2dr /q)(T).

Eq. (21) is just the well-known

4. SECOND

ORDER

spill-out

ENERGY

formula,

Eq.(2),

to the first order in AN/N.

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)

352

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

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

after

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)

with K2/2m also expand dAV dr

2

RI,

(28)

= E, where E is the ionization energy. AV(r) and take the first term,

N -3mwi(r

The result

(r

In order

to simplify

the algebra,

we

situations,

E is

- R).

is [14],

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

AN . wo-. N

Note that the perturbation theory breaks down at E = w0/2. In realistic 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. Notice 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. Otherwise, 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 sh 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” ; -RPA @. 0 Spill out L- -A Perturbation

0.8

I

10

I

I

20

I

I

30

,

I

40

I

I

50

I

I

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.

K. Hagino et al. /Nuclear

354

Physics A731 (2004) 347-354

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 Guggenheim Foundation and the U.S. Department of Energy (G.F.B.) and from the the Kyoto University Foundation (K.H.) is acknowledged.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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