Physics LettersA 160 (1991) 179-183 PHYSICS LETTERS A North-Holland
Triaxially deformed sodium clusters
in a self-consistent microscopic description
G. Lauritsch, P.-G. Reinhard
Institut J~r Theoretische Physik, Universitlit Erlangen, Erlangen, Germany
J. Meyer
Institut de Physique Nucleaire, Universitd Lyon-I, Lyon, France
and M. Brack
Institut J~r Theoretische Physik, Universitiit Regensburg, Regensburg, Germany
Received 17 June 1991; revised manuscript received 11 September 1991; accepted for publication 18 September 1991 Communicated by B. Fricke
We have performed self-consistent Kohn-Sham calculations for the triaxially deformed sodium clusters Nal2 and Nat4. We use the jellium approach and compute the properties of the valence electrons with an exchange-correlation energy functional in the local density approximation. In both clusters a pronounced shape isomerism is obtained. We find a well-developed triaxial ground- state deformation for Na~2 which produces a triple-peaked dipole surface plasmon, as experimentally observed, whereas Na~4 maintains axial symmetry but with two almost degenerate oblate and prolate minima.
Recent p h o t o a b s o r p t i o n measurements on free so- d i u m clusters [ 1 ] have revealed a multipeak struc- ture o f the absorption cross section which can be at- tributed to surface plasmon resonances in deformed systems. In fact, both the phenomenological ellip- soidal shell model o f Clemenger [2 ] and self-con- sistent K o h n - S h a m calculations [ 3 ] with imposed axial s y m m e t r y for the valence electron system in the jellium approach have revealed that sodium clusters Na~ with atomic numbers 8 < N < 20 and 20 < N < 40 m a y be deformed in their ground states. Depending on whether the shape is axial or triaxial, one there- fore can expect a splitting o f the dipole plasmon res- onance into two or three peaks, respectively, corre- sponding to collective vibrations o f the valence electrons in the directions o f the principle axes. In- deed, the splittings f o u n d in ref. [ 1 ] for the clusters with N = 9, 10 and 12 are in fair qualitative agree- ment with the axis ratios o f the corresponding ellip- soidal deformations predicted by the Clemenger
model [ 2 ]. In particular, the predicted triaxial shape o f Na12 has been confirmed by a triple-peaked struc- ture o f the cross section.
In this note we present fully self-consistent micro- scopic calculations for triaxially deformed sodium clusters. We have solved the K o h n - S h a m equations for the valence electrons o f the clusters Na~2 and Nal4, treating the ionic b~tckground in the jellium approximation [ 3 - 5 ]. We use the same energy den- sity functional as Ekardt and Penzar [ 3,5 ] which in- cludes exchange and correlation energies in the local density approximation. In contrast to these authors, we use a jellium density with a diffuse surface pro- file. This modified jellium model has been shown [6,7] to lead to a better fit o f experimental dipole polarisabilities and plasmon surface resonances in spherical clusters than is the case for the standard model with a sharp-edged jeUium density [ 8 - 1 0 ] . We describe the spherical jellium density pj (r) by a Fermi function with a surface width o f 1.0 a.u. and
Volume 160, number 2 PHYSICS LETTERS A 11 November 1991 a half-density radius adjusted such that pj ( 0 ) has the
value corresponding to a Wigner-Seitz radius rs=
3.96 a.u. [ 7 ]. For deformed clusters, the Fermi func- tion is scaled by the transformation flj(X, y, Z)---~
p j (t/:cX , rlyY ,
tlzZ),
where ~lx=~ly=tl; -2 produces an ax- ially symmetric quadrupole deformation and rb,# r/y generates triaxial shapes. Volume conservation is maintained by t/d/d/z= 1.In ref. [ 3 ] the total energy o f the cluster was min- imized with respect to axial quadrupole deforma- tions. The idea behind this approach is to assume that the ionic background is very "ductile" and adapts itself to the shape preferred by the valence electrons; this assumption has lead to the successful prediction [ 5 ] o f the magic numbers N = 8, 20, 40, 58, 92, ... o f spherical clusters. In the present work we do not only want to find the (eventually triaxial) deformed ground-state deformation o f a given clus- ter, but to exhibit the shape isomerism by presenting explicitly the total energy as a function o f the de- formations given to the jellium background. We therefore perform the K o h n - S h a m calculations on a series o f fixed points in deformation space and ex- hibit the results in the form o f potential energy sur- faces (PES). Since triaxial deformations require rather elaborate three-dimensional calculations, we have also developed a code with imposed axial sym- metry; we shall present and compare results o f both codes in this paper.
O u r three-dimensional ( 3 D ) code makes use o f a mixed coordinate- and m o m e n t u m - s p a c e represen- tation connected by the fast Fourier transformation technique. The kinetic energy is evaluated in mo- m e n t u m space whereas the potential acts in coor- dinate space. That has been found to be the opti- m u m strategy for saturating systems o f this size [ 11 ].
We employ spin-symmetry, occupying each spatial single-particle state twice, and assume reflection s y m m e t r y in x- and y-direction. The C o u l o m b prob- lem is solved using the Fourier representation o f the Laplacian operator [12]. T h e efficiency o f the Fourier representation allows to use a grid Nx X ivy × N. = 32 × 32 × 64 points with a spatial grid size o f A x = A y = A z = 0 . 9 a.u. The K o h n - S h a m equations are solved with the d a m p e d gradient it- eration method [ 1 l, 13 ]. The average total C P U time on a CRAY Y / M P for one deformation point in Na~4 is 30 s.
The reflection symmetry in x- and y-direction de- fines the parities/-/x and lly. We therefore have to consider various competing single-particle configu- rations for each cluster. For both Na12 and Na14, we fill the four lowest states (giving N = 8 ) with the se- quence (/-/x,/Ty) = ( + , + ) , ( - , + ) , ( + , - ) , ( + , + ) , each state having two spin directions. The further filling differs. We list in table 1 the energetically most favorable configurations. Each o f them produces its own PES with a stable m i n i m u m . The lowest min- i m u m determines the stable ground-state configu- ration, whereas the other m i n i m a correspond to iso- meric states.
The triaxial shapes o f the deformed jellium den- sity are most conveniently classified in terms o f the Hill-Wheeler coordinates (fl, y) for quadrupole de- formations [ 14 ]. fl describes the overall quadrupole deformation. 7 = 0 °, 120 °, 240 ° describe prolate and y = 60 °, 180 °, 300 ° oblate deformations; all other y give truly triaxial shapes. There is the usual sym- metry in the (fl, y) plane: the PES is fully determined if it is known in the first sextant 0~<y~<60 °. This holds only if all possible electronic configurations (i.e. parities in the occupied states) are considered.
The PES for one fixed configuration is possibly dif- ferent in the first three sextants 0~< y~< 180 ° (see fig.
1 ). For 7 = 0 °, the often used ratio q o f the major axes is related to the deformation fl by
For first explorations and systematic investiga- tions, it is more economical to perform a K o h n - S h a m Table 1
Selection of parity quantum numbers for the electron states above N= 8 for different configurations in Na~2 and Na~4.
Name (Hx,//y) of state no.
5 6 7
Nal2 triaxial ( + , + ) ( --,-- ) Na~2 prolate ( + , - ) ( - , + ) Nalz oblate ( + , + ) ( + , + ) Nat4 prolate ( +,+ ) ( + , - ) Na~4 oblate ( +, + ) ( +, + ) Nal4 spher. I ( - , - ) ( - , + ) Nat4 spher. II ( +,+ ) ( + , + )
( - , + ) ( - , - ) ( + , - ) ( + , + )
Volume 160, n u m b e r 2 PHYSICS LETTERS A 11 November 1991
"~=0 ~ ¥=-60 °
" - " ' C ' " , ~ , N
¢ • • • , , • , , x
I !
, , , , ,, , , , . . . .
1 1 1 1 ! I I ~ ~ •
i i | i " " I l
i I I
l O , , \ _ ' ~ / , .' ~ ' \ , , V " . " . - x ~ \
.~=60 o , ~ ~ , , , ~ = - 1 2 0
I I I I i f i ,~
0.7 0.6 0.5 0.~ 0.3 0.2 0.1 0.0 0.1 0.2 0.3 OA- 0.5 0.6 0.7
Fig. 1. Potential energy surface in steps o f 0.01 Ry (full lines every 0.05 Ry) in the (,8, 7) plane for N a t : in the triaxial configuration (see table 1 ). The triaxial m i n i m u m with fl=0.54, y = 15 ° has an energy o f - 1.768 Ry.
calculation with imposed axial symmetry and then to use the triaxial code to check whether the axial minima are stable against triaxial deformations. We have therefore developed a two-dimensional ( 2 D ) axial code, adapting a method used in nuclear phys- ics [15]. Hereby the K o h n - S h a m equations are solved by diagonalizing the total potential in an ax- ially deformed harmonic oscillator basis using cylin- drical coordinates; the electrostatic (direct) Cou- lomb potential is obtained in each iteration by straightforward integration. For the clusters inves- tigated here, we truncate the oscillator basis at a con- stant energy corresponding to the inclusion of 12 major shells at sphericity (see ref. [ 15 ] for the tech- nical details). We have ensured that the results are stationary with respect to changes of the bases pa- rameters. The integrations are done by Gauss-Her- mite and Gauss-Laguerre quadrature in the two re- spective spatial coordinates, using 32X32 mesh points.
Fig. 1 exhibits the PES for Na12 resulting from our 3D calculations for the energetically most favorable configuration, listed as "triaxial" in table 1. The ground state at E = - 1.768 Ry is clearly triaxial with the deformation fl= 0.54, 7= 15 ° and is energetically well separated from the competing "prolate" and
"oblate" configurations (see fig. 3 ). This triaxial de- formation compares favourably with the result ,8= 0.54, y = 19 ° of the phenomenological shell model [ 2 ]. For Na14, the 3D calculations yield axially sym- metric minima in all the configurations listed in ta- ble 1. We show in fig. 2 the PES in the (fl, y) plane
•/•=60°
%. " d d .
, / , / / .. . . . 2 - . . /.. ', ', ,' I : , ," -2.0.90 - , "
/ ' , ~ ~ ; I ' ' , ..i-. "~
/ ~ ~ ' ! ' ': ' ',' ~ ' ~ : ' ! ' : "~=0
0 0.1 0.2 0.3 O.t, 0 5 0.6 0 2 O.B
Fig. 2. Same as fig. 1, but for Na~4 (prolate and oblate configu- rations as listed in table l ). The thick dash-dotted line indicates the line where the two surfaces cut each other.
for the two lowest configurations ("prolate" and
"oblate"). Note that the two minima are energeti- cally almost degenerate. The oblate m i n i m u m is rather soft in the y-direction whereas the prolate m i n i m u m seems to predict stiffer y-vibrations.
In fig. 3 we show cuts of the PES along the axes 7 = 0 ° (fl>~0) and y = 6 0 ° ( f l ~ 0 ) versus the quad- rupole deformation p. The agreement in the energies between the 2D and the 3D results, within a few parts per thousand, lies within the limits of the estimated numerical accuracy of either code and is therefore very rewarding. We see that the prolate and oblate
Volume 160, n u m b e r 2 P H Y S I C S L E T T E R S A I l N o v e m b e r 1991
- 1 . 6 4 r ~ - ~ . . . , . . . . - ~ - ~ - . . ~ y ~ ~ - - ~
! ',,
,N a - 1 2
,,'-1.~i \ ; \ \, / / /
.1.68 1 I ~ /
. . ,
-,.~o~ :~"
- 1 . 7 2 P ",_ . /
- 1 . 7 4 1
-1.76 F
M i n i m u m a t ~ : 0 . 5 ~ . ~ : 0 . 1 5 "
I l I l I%1 l I I l . l I I l [ [ '. : : : I ~ tr : : : :/; : : : : : ~ [1: ! !
F ' ' N a - 1 4 / I
~ i I /
F 1 '
-1.98 ~ ~ / /
x 11 I
ii ~ , 30sphet l / /
\ ~ ' x i I / / / /
-2. ! \ . - . / "
- 2 . 0 ~ ~ / "
i ob~
-2.08 ~ ~
-2.10~ . . . , . . . . , . . . , . . .
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
P jellium
Fig. 3. Deformation energy (in Ry) for axially symmetric defor- mations of Na~2 (upper part) and Na~4 (lower part), shown ver- sus the quadrupole deformation p. (2D results: dashed lines. Cuts of 3D results: full lines (lowest configuration) and dotted lines (higher configurations ); single-particle configurations named as in table 1. ) The energy of the
triaxial
Na~2 ground state is indi- cated by a full horizontal line.minima of Na~2, too, are nearly degenerate. The axis ratios q at the axially symmetric m i n i m a agree fairly well with the published values of both the phenom- enological shell model [2] and the sharp-edged spheroidal jellium model [ 3 ]. For instance, for the oblate m i n i m u m in Nal4 we find q=0.57 (2D) or q=0.60 ( 3 D ) , compared with q = 5 7 in ref. [3] and q=0.53 in ref. [2].
The cusps in the PES at spherical symmetry are unphysical. The residual interaction between the dif- ferent electronic configurations will smooth out these
cusps and thereby lower the barriers separating the isomeric minima (cf. the Jahn-Teller effect [ 16] ).
Moreover, the free variation of higher multipole de- formations of the jellium may further modify some- what the relative positions of the stationary points in the PES.
The pronounced shape isomerism found here for both clusters bears a certain resemblance to that found in the totally different approach of quantum chemistry, where the clusters are described fully mic- roscopically in ab initio calculations using the Born- Oppenheimer approximation for the relative posi- tions of the atoms [ 17 ]. Clearly, the latter approach is a priori more adequate for the description of mi- croclusters. But our results seem to indicate that a full exploitation of the three-dimensional deforma- tion degrees of freedom in the jellium model will lead to a partial reconciliation of these two otherwise quite orthogonal approaches. However, a quantitative comparison exceeds the limits of this Letter and will the object of future studies.
We have finally investigated the resonance ener- gies of collective dipole oscillations of the valence electrons against the ionic background. As a first es- timate, we have approximated them as pure surface plasmons (i.e., purely translational dipole modes) by the random phase approximation (RPA) sum rule expression [ 10,18 ] hoJ = [ m3 ( D ) / m i ([~) ] 1/2, where D is the dipole operator and
mk
is the kth energy- weighted moment of the RPA dipole strength func- tion. For deformed clusters, the expression found in ref. [ 10 ] generalises trivially toh2 f 32
(h°Ji)2= ~mm
d3rpel(r) ~
Vj(r) , (1)where
pea(r)
is the electron density, Vj(r) the elec- trostatic jellium potential, and i runs over the spatial directions, i.e.,rie {x, z,
y} for the triaxial case andri~{r, z}
for the axial case. We have evaluated the resonance energies hcoi for the most prominant min- ima in both clusters. The results are shown in table 2. We find three different energies for Na12, related to its triaxial shape in the ground state. It is most in- teresting to compare them with the experimental data of ref. [ 1 ] quoting peaks athcoi=
0.156, 0.182, and 0.203 Ry. Our results are too high in energy by about 10-15% due to the simple dipole sum-rule approx-Volume 160, number 2 PHYSICS LETTERS A 11 November 1991 Table 2
Dipole surface plasmon energies (in Ry ) in the RPA dipole sum- rule approximation ( 1 ). The numbers in the upper row are from the 3D code, those in the lower row from the axial 2D code. (Note the perfect agreement. )
Na12 Nal4
triaxial
oblate prolate
hO.) x ~OJy ~O) z h(.O r hO.)g hO) r hO) z
0.211 0.239 0.171 0.187 0.251 0.229 0.171
0.188 0.252 0.229 0.170
i m a t i o n ( 1 ), b u t we r e p r o d u c e r e m a r k a b l y well t h e relative t r i p l e s p l i t t i n g o f t h e d i p o l e p l a s m o n peak.
T h e t w o c o m p e t i n g axial states i n N a l 4 p r o d u c e each a d o u b l e - p e a k e d s t r u c t u r e w h e r e
hO~r
has d o u b l e weight c o m p a r e d to ho~=. T h e a c t u a l s t r e n g t h distri- b u t i o n is to b e s u p e r p o s e d i n c o h e r e n t l y f r o m t h e two i s o m e r i c m i n i m a . O n e t h e r e f o r e expects f o u r peaks o r a w a s h e d - o u t s t r u c t u r e i f b o t h m i n i m a h a v e c o m - p a r a b l e o c c u p a t i o n weight. B u t s u c h q u a n t i t a t i v e de- tails d e p e n d s e n s i t i v e l y o n t h e f o r m a t i o n h i s t o r y o f t h e clusters, a n d this is b e y o n d o u r p r e s e n t approach.I n a l o c a l - c u r r e n t R P A [ 10,18 ] it was s h o w n t h a t t h e c o u p l i n g to v o l u m e p l a s m o n s (i.e., d i p o l e m o d e s i n v o l v i n g local c o m p r e s s i o n ) lowers t h e f r e q u e n c i e s o f t h e surface p l a s m o n s b y a b o u t 1 5 - 2 5 % for light s p h e r i c a l clusters. T h i s will persist for d e f o r m e d clusters a n d lower t h e r e s o n a n c e energies. I n de- f o r m e d clusters, we also h a v e to a c c o u n t for t h e c o u - p l i n g o f d i p o l e m o d e s w i t h m o d e s o f o t h e r m u l t i - polarities. Both k i n d s o f c o u p l i n g s are p r e s e n t l y b e i n g i n v e s t i g a t e d a n d t h e i r effects i n d e f o r m e d clusters will he d i s c u s s e d elsewhere [ 7 ] .
We t h a n k O. G e n z k e n for m a n y i n s p i r i n g discus- s i o n s c o n c e r n i n g this w o r k a n d T. H i r s c h m a n n for t h e help i n i m p r o v i n g t h e a c c u r a c y o f t h e 2 D code.
We a c k n o w l e d g e t h e H 6 c h s t l e i s t u n g s r e c h e n z e n t r u m Jtilich a n d t h e L e i b n i z R e c h e n z e n t r u m M t i n c h e n for t h e p r o v i d e d c o m p u t i n g time.
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