Simulated annealing study of neutral and charged clusters - General

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Simulated annealing study of neutral and charged clusters: Aln and Gan
R. O. Jones

Citation: J. Chem. Phys. 99, 1194 (1993); doi: 10.1063/1.465363
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Simulated annealing study of neutral and charged clusters: Aln and Ga
n
R. O. Jones
Institut fUr Festkorperjorschung, Forschungszentrum lillich, D-52425 lillich, Germany
(Received 15 December 1992; accepted 29 March 1993)
Density functional calculations with simulated annealing have been performed for clusters of
aluminum Aln and gallium Gan up to n = 10. There are many local minima in the energy
surfaces, with a rich variety of structures and spin multiplicities. With increasing cluster size we
find transitions from planar to nonplanar structures at n = 5, and to states with minimum spin
degeneracy at n = 6. Isomers (n >5 ) with buckled planar structures reminiscent of the layers in
crystalline a-gallium are generally less stable than "three-dimensional" isomers. All structures
show regular patterns of bond and dihedral angles. Systematic differences between Al and Ga
clusters-bonds in the latter are shorter and bond angles closer to 90°--can be understood in
terms of atomic properties. Trends in binding and ionization energies are compared with ex
periment and with the predictions of other calculations.
I. INTRODUCTION (Ref. 14) revealed pronounced discontinuities in cohesive
and ionization properties that were discussed in terms of
There has been a dramatic growth in interest in atomic the level structure of the S1M.
clusters in recent years. This is true for both metallic and There are, however, many experimental data that can
nonmetallic clusters, and the developments in both theory not be explained by this model. The static polarizabilities
and experiment have led to a better understanding of many in Aln show a transition from "non-jellium" to "jellium"
properties of atomic aggregates, including those that reflect behavior near n=4O,3 and ionization potentials in Aln and
the bulk behavior. Inn show an initial linear increase with increasing n, an
The group III a elements aluminum, gallium, and in abrupt leveling near n = 5, and only then a gradual ap
15
dium have been amongst the favorite metallic elements for proach to the results of a jellium calculation. The shell
cluster studies. Work on Aln has included magnetic prop structure observed in larger clusters of Al (n < 1400) (Ref.
2
erties, I ionization thresholds and reactivities, and the 16) is inconsistent with the predictions of the S1M, and the
3
static polarizabilities. There have also been measurements recent extension of the cluster size for which shell structure
of collision induced dissociation of AI;; (Ref. 4), and the can be observed (10 000 atoms) (Ref. 17) showed a cor
photoelectron spectroscopy of AI; (Refs. 5 and 6) and relation with the number of atoms needed to cover succes
Ga; (Ref. 6), where transitions between states of the an sively larger faces on close-packed octahedra. The impor
ions and states of the neutral clusters can be observed. tance of the electronic state of the cluster is shown by
Gallium clusters with up to more than ten atoms have been magnetic deflection measurements I that indicate a transi
detected following laser vaporization of gallium arsenide. 7 tion to states with low spin multiplicity as n increases. All
Gallium clusters are of particular interest in intermetallic these results show that detailed calculations of geometries
compounds with alkali metals, where Gas-dodecahedra, s and electronic properties are essential to understand much
IO
GaI2-icosahedra,9,IO and Gal5 clusters have all been of the experimental data.
found. The structure of bulk (a)-gallium, which we dis The stable isomers of a cluster can be found by locating
cuss further later, has been interpreted by von Schnering the low-lying minima in the energy surface. In the alumi
and Nesperl I as icosahedra that have been dissected and num dimer there is excellent agreement between the most
condensed via edge-sharing. detailed experimentslS and calculations, 19 but the results in
Particular attention has been paid to the existence of larger clusters are less extensive and often contradictory.2o
prominent or unusually stable clusters with "magic num The most stable structures in Alr A15, for example, are
y
bers" of atoms or electrons. The electronic structures of predicted by Pacchioni and Kouteck 21 to be deformed
the bulk metals are characterized by small departures from fragments of the bulk (face-centered cubic) lattice, with
22
free-electron behavior, and several theoretical studies of high spin multiplicities, while Upton predicted three
small metal clusters have adopted the "spherical jellium" dimensional structures with C symmetry and minimum
12 2v
model (S1M), where both the electronic charge and pos spin degeneracy for the same molecules (A16 had a struc
itive background distributions are uniform within a sphere ture with D2k symmetry). Other calculations for Al4 led to
of appropriate size. Several predictions of this model are a planar rhombic (D2h) ground state,21,19,23 and Pettersson
supported by measurements. Reactions of aluminum clus et af. 23 also found a planar ground state in A15, with Al6
ters with oxygen (AI;;, n=I-33 and AI;; for n=5- 37),13 having Ok symmetry. Semiempirical molecular orbital cal
for example, are not observed for Ali, Alj3, and Ali3, culations using the SINDOI method for Al3 to AlIO (Ref.
corresponding to predicted shell closings with 20, 40, and 24) predicted that three-dimensional structures are favored
70 electrons, respectively. Photoionization mass spectrom for clusters with more than four atoms.
etry measurements on Aln (n < 430) and Inn (n < 120) Density functional calculations focusing on larger Al
1194 J. Chern. Phys. 99 (2), 15 July 1993 0021-9606/93/99(2)/1194/13/$6.00 @) 1993 American Institute of Physics
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R. O. Jones: Neutral and charged clusters: Aln and Ga 1195
n
clusters have been described by Cheng et al. (AI13, A143, by performing calculations for the atoms and the positive
A1ss).25 In AI13 the icosahedral (h) structure is favored ions both with and without periodic boundary conditions.
over the cuboctahedral (Oh), whereas for Al55 the fcc The correction to the energy differences (2.02 eV in AI,
structure is more stable than the icosahedral. This sug 2.15 eV in Ga for the unit cell used here) leads to ioniza
gested to the authors that there is a transition in the 13 to tion energies (AI, 6.00 eV; Ga, 6.15 eV) in satisfactory
55 atom range from polyhedral to lattice-based structures. agreement with experimental values (AI, 5.984 eV; Ga,
Density functional calculations on AIl3 and Al55 by Yi 6.00 eV),40 and with the results of all-electron local spin
et al.26 showed that there is a substantial Jahn-Teller dis density (LSD) calculations (AI, 6.00 eV; Ga, 6.06 eV). It
tortion in the icosahedral and cuboctahedral forms of each. has been added to the ionization energies in all other clus
These calculations did not include the effects of spin po ters.
larization. In the case of gallium, Balasubramanian and The local density approximation for the exchange
co-workers have performed extensive calculations on states correlation energy does not lead to bound solutions of the
of Ga2 (Ref. 27) and Ga3 (Ref. 28) and their ions, and (DF) equations for the isolated ions Al- and Ga -, i.e., the
Meier et al. carried out calculations on neutral and outermost electron has a positive energy eigenvalue and is
charged clusters up to Ga4 (Ref. 19). Density functional not localized to the region near the nucleus. The periodic
calculations with simulated annealing have been reported boundary conditions used in the present calculations lead
29
recently by Gong and Tosatti for single isomers of Ga2 to to localized, bound solutions. The focus in the present
Gag. We discuss the results later. work is on the structures of the anions. However, if we
The number of isomers (the number of minima in the adjust the energies of the negative ions by the amounts
energy surface) of a cluster increases exponentially with (1.73 eV in AI; 1.85 eV in Ga) needed to reproduce the
increasing size, so that it is essential to perform reliable measured electron affinities of the atoms,41 we obtain a
calculations of the energy surfaces and to avoid energeti good description of the electron affinities of clusters of Al
cally unfavorable minima. We study here the structures and Ga with up to 13 atoms. Details will be given in an
42
and energies of neutral and charged clusters of aluminum analysis of the electron photodetachment measurements
and gallium clusters with up to 10 atoms, using a method of Cha et al.6
30
that does both: ,31 a combination of density functional The calculations reported previously37 were performed
32
(DF) calculations with finite-temperature molecular dy with an fcc unit cell with lattice constant 30 a.u. This large
namics (MD). This approach has been applied with suc unit cell guarantees that the interaction between clusters in
cess to crystalline phases of Ga (Ref. 33), and predicts adjacent cells is almost always negligible. In some near
stable, previously unknown structures in Sn (Ref. 34) and planar structures in Al7 to AllO, however, the distances
P n clusters (Refs. 35 and 36), and in the present work. The between these clusters can become small and overestimates
method we use has been described in detail elsewhere,35,34 in the relative stabilities can result. To avoid this, we have
and we discuss the aspects particular to the present calcu repeated all calculations for clusters with n>6 using a unit
lations in Sec. II. We present our results for aluminum and cell with lattice constant 36 a. u.
gallium clusters in Secs. III and IV, respectively. We study The calculations are performed for starting geometries
the structures of the neutral clusters Xn and some of their found previously by ourselves and others in clusters of Al
anions X;, and the vertical and adiabatic ionization ener and other elements, and have no restrictions on cluster
gies of Xn• We compare with experimental information symmetry. We have also used ionic structures as starting
where available. The similarities and differences between geometries for neutral clusters (and vice versa) and Aln
aluminum and gallium, as well as the general bonding geometries for Gan clusters (and vice versa). For the start
trends, are discussed in Sec. V, and the results are summa ing geometries we use steepest descents techniques to bring
rized in Sec. VI. Preliminary results have been reported the electrons into their ground state, locate the nearest
previously.37 minimum in the energy surface, and alternate simulated
annealing (typically at 300--500 K) and ionic steepest de
II. METHOD OF CALCULATION scents to locate minima in the potential energy surface.
Numerous new structures were found by allowing clusters
The calculations have been performed using the
to move in a heat bath for long periods.
method described in earlier applications to clusters of sul
34 35 Bulk aluminum and gallium are metallic. The gap be
phur and phosphorus. ,36 The present calculations use
tween the highest occupied and lowest unoccupied molec
the same basis functions (plane waves with energy cutoff
ular orbitals is small in most of the clusters we have cal
5.3 a.u.) and nonlocal pseudopotential form as in the lat
culated, leading to numerous structures with similar
ter. In aluminum, we use the pseudopotential parameters
energies but different symmetries. To describe the symme
of Bachelet et al., 38 and in gallium we use those of Stumpf
try [in particular, the (spin) multiplicity] of a state, we fix
et al.,39 both with sp nonlocality. The use of periodic
the ordering of the occupation (and spin occupation) num
boundary conditions means that care is required in calcu
bers, e.g., 22221 (00001) or 22212 (00010), and allow the
lating ionization energies and electron affinities, since cal
structure to evolve as described earlier. In the present
culations for systems with a net charge include a contribu
example, we obtain two doublet states with different struc
tion to the energy that is absent in calculations for isolated
tures and energies. A quartet state is described by occupa
ions. The contribution depends on the size of the unit cell
tion numbers 222111 (000 111). "Level crossings" are of
and the localization of the charge, and has been estimated
ten observed, and there are cases, particularly in ionic
J. Chern. Phys., Vol. 99, No.2, 15 July 1993
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1196 R. O. Jones: Neutral and charged clusters: Aln and Ga
n
dimers, where electronic steepest descents calculations lead TABLE I. Molecular parameters re (atomic units) and we (em-I) for
to apparently stable solutions of the density functional low-lying states of neutral and charged aluminum dimers, together with
equations that do not correspond to the lowest energy with energies relative to the ground state (!!..E). Vertical ionization energies
Up) and electron affinities (A) are given in eV.
this set of occupation numbers. While the present calcula
tions have been very extensive, no scheme can guarantee to re We !!..E lpi'A
find all the important minima of energy surfaces of the
complexity encountered here. A12: 3nu(ag1Tu)
(a) 5.135 284.97 6.19
(b) 5.150 277
III. ALUMINUM CLUSTERS AND IONS (c) 5.19 284
(d) 5.095 290 6.45
In the present section we present results for neutral Expt. (e) 5.10 284.2
and charged aluminum clusters. We discuss the most stable
A12: 31:i(~)
structures that we have found, together with selected ones (a) 4.687 355.15 +0.06 6.46
that lie higher in energy. In all clusters, fl.E denotes the (b) 4.711 343 +0.02
energy of a state relative to that of the most stable isomer. (c) 4.78 340 +0.02
"Bonds" in the figures represent all interatomic separations (d) 4.672 340 -0.08 6.77
Expt. (e,f) 4.660 350.01 >0
less than 6 a.u. For comparison purposes, we note that the
nearest-neighbor separation in bulk (face-centered-cubic) A12: 51:; (~PU~ag)
aluminum is 5.411 a.u.43 (d) 4.444 435 + 1.59 5.24
A. AI2 Ali: 21:: (a;,ag)
(a) 6.062 169
The dimer is the best studied of the aluminum clusters. (c) 6.00
It has been the subject of considerable interest recently, (d) 5.961 161
and the nature of the ground state has only recently been Al2+ ·• 2nu(a;1Tu)
determined. The two candidates for the ground state are (a) 5.276 216 +0.54
the 3I1u (ag1ru) and 3~g (~), and the ease of transfer (d) 5.143 +0.31
between a-and 1r-e1ectrons is reflected in the fact that each Ali: 41:i(ag~)
has been favored at different times. Recent experimental (a) 4.834 335 1.44
workl8 supports theoretical predictions44,45 that the 3I1u (c) 4.97
state is slightly (less than 0.025 eV) more stable. (d) 4.775 325 1.58
Experimental and theoretical spectroscopic parameters Ali: 2nu(~)
for some low-lying states of Al2 and its ions are shown in (a) 4.651 355 +0.58
Table I, and energy curves for Al2 and Ali in Fig. 1. The (d) 4.505 378 +0.58 0.87
present calculations agree well with available data for
Ali: 2nu(ai1T.)
A12,18,46 although the 3I1u and 3~g states are reversed in (a) 5.140 287 +0.64 0.85
stability, with the latter being slightly (0.08 eV) more sta (d) 5.100 284 +0.75 0.80
ble. The equilibrium separations re and vibration frequen
"Reference 45. Coupled-cluster doubles+ST(CCD).
cies We are in excellent agreement with experiment for both bReference 44. Complete active space SCF/second order CI CASSCF/
states. The well depth (2.03 eV compared with the exper SOCI.
imental value 1.5 eV) (Ref. 46) agrees with other DF <Reference 19. Multireference configuration interaction (MRD-CI).
dThis work.
estimates (Ref. 47) and shows an overestimate similar to
"Reference 18.
those found in other sp-bonded systems (Ref. 32). The
fReference 46.
5~;;- state has not been studied before in this molecule, but
it was an early candidate for the ground state of B2, and
lies within 1300 cm -, of the X 3~ g state of that mole
48 bond angles a-60°. Configuration interaction (CI) calcu
cule.
Table I also shows vertical excitation energies from lations lead either to (in C2v notation) a 2AI ground
states of Al2 -+ Ali and from Ali -+ A12. The latter are of state,22,49,50 or a near degeneracy between 2A I and 4A2
states. 19,23 Electron spin resonance measurements of matrix
particular interest, since they can be observed in photoelec
isolated Al3 indicate a quartet ground state,51 while mag
tron detachment spectroscopy of negative ions. A detailed
comparison with these data will be given elsewhere.42 The netic deflection measurements' point to a doublet.
overall agreement of the present results with available data The present calculations indicate that the most stable
on Al2 and its ions is encouraging for the application to form of Al3 is an equilateral triangle eA1, re=4.65 a.u.)
larger clusters, for which there is much less spectroscopic The 2B I state (4.83 a.u., 60°) is 0.32 eV above the ground
information. state, with the 4A2 (4.82 a.u., 68°) and 4B, (5.04 a.u., 56°)
states -0.1 eV higher. The most stable linear state of A13,
a quartet (au~) related to the 4A2 state, has bonds 4.86
B. AI3
a.u. in length and lies 0.9 eV above the most stable trian
There have been several calculations of low-lying states gular form.
of the aluminum trimer. The most stable isomers have The picture of bonding in Al3 apparent from the
J. Chern. Phys., Vol. 99, No.2, 15 July 1993
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R. O. Jones: Neutral and charged clusters: Aln and Gan 1197
4
(Q)5L-~
3 u
~ 2 A12
<I (a) 3
Ali
o ~
(b)
4 5.35
5L~ (c) 3 (d) 5
3 ,
~2 31Ittu -----
<I 3L~~
9
2rru --t-
Gcii
o
(el
40 4.2 4.4 4.6 48 5.0 5.2
r (uu)
FIG. 2. Structures and spin multiplicities of isomers of A14• The inter
FIG. I. (a) Energy curves for low-lying states of AI; and A12; (b) Ga; anruec l5e.a5r7 saenpda r4a.t6io6n as. ua.r e given in a.u. The unmarked separations in (e)
and Ga2'
present results is similar to that described previously by some of them. The most stable structure [Fig. 2(a)] is a
other authors.19,22,49 The low-lying states have the same planar rhombus (D2h, bond angle a = 56.5"), with a related
orbital structure derived from the s electrons (laI2aT1b~), singlet oflower symmetry [C2h, Fig. 2(b)] 0.12 eV higher
but differ in the occupancies of molecular orbitals derived (am = 63.7°). A quintet "roof" structure [Fig. 2 (d)]
from p-atomic orbitals. These are Ibi3al eA\), (with aE=0.45 eV and dihedral angle 122.7°) may be
1b13aie B1), Ib13a12bi(4A2), and 1b13al4al(4B,). The 1bl viewed as a distorted rhombus. There are local minima for
and 3a, orbitals are 1T- and a-bonding orbitals that are square (D4h) structures with triplet (re=4.929 a.u., energy
symmetric with respect to rotations through 120°, so that above ground state aE=0.23 eV) and singlet (re=4.812
the two most stable states have bond angles of 60°. The a.u., aE=0.48 eV) multiplicities. The undistorted tetrahe
relative energies of the two states shows that 1T bonding is dral structure has a highest occupied orbital that is three
favored in this molecule. fold degenerate, and we find a minimum only for a quintet
Removal or addition of an electron results in changes state (aE=0.64 eV, re=5.071 a.u.). Structures that may
of the lb and 3a, occupancies, so that the most stable be viewed as distorted tetrahedra include (1) a triplet C
l 3v
forms of both Alt and AI; also have bond angles of 60°. structure with aE=0.39 eV [Fig. 2(c)], the structure is
The bond lengths are 4.70 a.u. and 4.63 a.u., respectively. somewhat flattened from T d symmetry; (2) a singlet C2v
The bond length in the anion is only 0.02 a.u. different structure [Fig. 2(e)] with aE=0.47 eV. This structure
from that in the neutral molecule. This is consistent with represents a tetrahedron with the opposite distortion, with
the very sharp transition found in the photoelectron de the apex atom moved away from C symmetry.
3v
tachment of AI;.6
D. A1s
c. AI4
There have been several studies of the aluminum pen
There have been several calculations of the aluminum tamer. Upton22 found the Jahn-Teller distorted pyramidal
tetramer, and they have led to different predictions. A pla structure (C2v) to be the lowest in energy, Jug et al.24
nar rhombus (D2h) structure is favored by Pettersson found that the pyramidal form (C4v) was the most stable,
et al.,23 while Upton predicted a three-dimensional de and Pettersson et al. found that a planar (C2u) form-with
formed rhombus with C symmetry, and Jug et al. a trig bond lengths constrained to be equal-is 0.2 eV more sta
2v
onal pyramid (C3v) that is a Jahn-Teller distortion of a ble than the pyramid. As in the case of A14, we find nu
tetrahedron. Pacchioni and Koutecky,2' and Meier et al. 19 merous structures corresponding to local minima in the
also predict a rhombic ground state, but the square (D4h) energy surface.
is very close in energy (0.005 eV and 0.04 eV higher, re The pentamer is the first cluster for which a three
spectively) higher in energy. dimensional (3D)-structure has comparable energy to the
The present calculations have found numerous local most stable planar form. In the present calculations, the
minima in the energy surface of A14• and Fig. 2 shows two most stable structures-a Cs form [Fig. 3(a)] and the
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1198 R. O. Jones: Neutral and charged clusters: Aln and Ga
n
(oj 2 (b) 2
~ (oj
(e) 2 (d) 4
(e) 1 (d) 1
(e) 4
FIG. 3. Structures and spin multiplicities of isomers of Als.
FIG. 4. Structures and spin multiplicities of isomers of A16•
planar structure [C Fig. 3(b)]-have almost identical
2V'
energies. The former may be viewed as a (substantially)
ometry is very similar to that in Fig. 4(a)], so that the
deformed pyramid, but it is better described as a deforma
hexamer is the first molecule where we have (i) a singlet
tion of Fig. 3 (b), as the bond lengths differ by at most 0.2
a.u. Doublet states found by reversing the occupation ground state and (ii) a "three-dimensional" structure
clearly more stable than planar structures.
numbers of the uppermost two occupied orbitals have en
The prism structure [D3h' Fig. 4(b), triplet, bond
ergies within 0.3 eV of the first two, but the structural
lengths 4.89 a.u. (triangle) and 4.84 a.u.] and the related
relaxation can be significant. In Fig. 3(a), for example, the
singlet [C Fig. 4(d)] have aE of 0.22 and 0.35 eV, re
central in-plane bond increases in length by 0.30 a.u. The 2V'
spectively. The structure in Fig. 4(c) [aE=0.23 eV] may
ideal pyramid [C Fig. 3(d)] can be realized for a quartet
4V'
state with aE=0.36 eV. The length of the in-plane bonds be derived from the most stable form of Al5 by addition of
a single, doubly coordinated atom. We have found several
is 5.14 a.u., and the edges of the pyramid are 4.85 a.u.long.
planar structures that correspond to local minima in the
A further quartet state, a buckled planar structure [Fig.
energy surface. The most stable of these is shown in Fig.
3(e)], and the triangular biprism [D3h' Fig. 3(f)] lie 0.52
4(e).
eVand 1.13 eV, respectively, above the ground state. Sev
eral planar structures not shown in Fig. 3 have higher
energies and transform readily into more stable forms on
F. AI7
annealing.
The most stable form of Al7 was predicted by Jug
E. A1s et af. 24 and by Raghavachari20 to be a C3v structure found
by capping the trigonal antiprism form of A16• A doublet
The aluminum hexamer has been studied by a number structure of this form [Fig. 5(a)] is found to be the most
of groups. The most stable form found by Upton22 was a stable isomer in the present work.
distorted octahedron, and Pettersson et af. 23 found that the While the aluminum heptamer is similar to other clus
octahedron was the most stable of the (symmetric) struc ters in that its structure can be found by capping that of a
tures they studied. Jug et af. 24 found that the octahedron smaller cluster, it shows a pronounced difference in the
underwent a Jahn-Teller distortion to a trigonal antiprism relative energies of the different stable isomers. The doublet
( D3d) structure, where the bond lengths were 4.724 and structure [CJ obtained by reversing the occupation num
5.212 a.u. bers of the two highest-lying orbitals [Fig. 5 (b)], for ex
The present study led to a large number of local min ample, is 0.9 eV less stable than the C structure, and the
3v
ima in the energy surface, some of which are shown in Fig. related quartet structure is a further 0.2 eV higher. This is
4. We also find that the octahedral (Oh) structure distorts a consequence of the large separation (1.15 eV) between
to the D3d trigonal antiprism [Fig. 4(a)], with bond the energy eigenvalues of electrons 20 and 21, and we re
lengths (4.74 and 5.40 a.u.) similar to those found by Jug turn to this point in Sec. III H.
et al.24 The lowest lying state (a singlet) is less than 0.1 Local minima also exist for a C structure [Fig. 5 (c),
2v
eV more stable than the lowest lying triplet [D2d' the ge- aE= 1.35 eV] and the planar form, Fig. 5(d). The exten-
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R. O. Jones: Neutral and charged clusters: Aln and Ga 1199
n
(a) 1
(b) 2
(a) 2
(b) 1
(c) 1
FIG. 6. Structures and spin multiplicities of isomers of Als.
FIG. 5. Structures and spin multiplicities of isomers of A17•
7(d), where the more symmetric Fig. 7(c), is 0.19 eV
sion of the planar form of Al6 [Fig. 4(e)] is, however,
more stable than Fig. 7 (d). Annealing of "open" arrays of
unstable against buckling. One local minimum we found is
triangles in the buckled planes distort towards "closed"
shown in Fig. See).
bulklike structures. We also found a structure comprising
Al3 and Al7 units, an example of linked clusters of group
G. Al -Al
a 1o III a elements that have been discussed by Burdett and
These clusters have been studied previously in the Canadell.52 The large number of local minima in the en
SINDOI calculations of Jug et 01.,24 who found that the ergy surface for AIIO indicates how difficult it would be to
stable structures are related to those of smaller clusters by find the structures of all the stable isomers of larger clus
bonding additional atoms to three atoms in a face or two ters.
atoms at an edge. In contrast to A17, we find that the
energy surfaces have many low-lying minima with similar
energies. Structures of Aig obtained by capping the ground
state of the heptamer are shown in Figs 6(a) and 6(b), the
most stable being a singlet [C2h, Fig. 6(a)] and the related
triplet [C 6oE=0.2S eV]. These two structures are related
31!'
to the primitive cell of the face-centered-cubic lattice), but
the distortion is much less in the latter. The singlet struc
ture derived by symmetrically capping the heptamer form
in Fig. S(c) is shown in Fig. 6(b) (6oE=0.30 eV) and (a) (bl
there is a related triplet with 6oE=0.33 eV.
In addition to these "three-dimensional" structures, we
found buckled planar arrays that conform to the pattern
found in AIs: almost equilateral triangles connected so
r
that they are either coplanar (dihedral angle near 0°) or
have r-50°. The buckled structures, an example of which
(el
is shown in Fig. 6(c) (AE=2.1 eV), are significantly less
stable than the 3D or "bulklike" forms. Annealing at 300-
500 K leads to interconversion between the buckled forms
and, eventually, to one of the 3D structures.
In Al9 and AIIO as well, "bulklike" structures are more
stable than buckled planes. Two stable isomers of Al9 are
(d)
shown in Figs 7(a) and 7(b). The former is more stable
(by 0.25 eV) and both are capped forms of Aig structures.
Further cappings lead to the Alto structures in Figs. 7(c)- FIG. 7. Isomers of (a) Al9 (doublets) and (b) AIIO (singlets).
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1200 R. O. Jones: Neutral and charged clusters: Aln and Ga
n
isomer of P 4S3' 54 Viewed from the center of mass of A17,
the apex atom moves much closer on ionization (4.11 a. u.
compared with 4.85 a.u.), while the atoms of the next layer
relax outward (3.80 a.u. compared with 3.26 a.u.). Six of
the atoms in the ion are essentially equidistant from the
center, with the apical atom only 9% farther removed. The
perturbation from radial symmetry in Ali is then rela
tively small.
The stability of the Ali ion has been related to the
"magic number" of 20 valence electrons in the spherical
jellium model. In the present calculations, the dissociation
energy of Al7 is larger than those of Al6 and AIs, and we
have noted earlier the unusually large separation (1.15 e V)
~ between the highest (singly) occupied orbital and the next
2 highest (doubly degenerate) orbital. Since the sum of the
W- energy eigenvalues is one of the contributions to the total
I
W 00 energy, the large gap between the eigenvalues of valence
-Ie
electrons 20 and 21 will lead to a low ionization energy in
Al7 and the pronounced stability of Ali. We have noted
3 earlier the similarity of the Ali structure to that of the
most stable isomer of P 4S3' It is interesting to note that
there is also an unusually large gap (-1.6 eV) between the
36
eigenvalues of electrons 20 and 21 in the latter.
FIG. 8. Energy differences in aluminum clusters: (a) Ionization poten
tial Ip for both vertical (solid curve) and adiabatic (chain curve) transi The energies of the most stable isomers of neutral alu
tions. The two values for the dimer are for the two almost degenerate minum clusters are shown in Fig. 8 (b) as functions of the
states of A12• Experimental values after Cox et al. (Ref. 1) (bars) and number of atoms n. The binding energy approaches the
Jarrold et al. (Ref. 4) (dotted-dashed curve). The experimental (left
scale) and theoretical curves (right scale) are displaced by 1 eV. (b) bulk cohesive energy with increasing n,55 but the variation
Binding energies for most stable isomers of Aln clusters. with n is irregular. In the case of alkali metal clusters, '2 it
has been useful to discuss such irregularities in terms of
.:l2(n) =E(n+ 1) +E(n-l) -2E(n), (1)
H. Ionization energies
which is related to the second derivative of the energy with
The energy required to remove an electron from a clus
respect to n. A peak in .:l2(n) indicates that a cluster with
ter depends on the amount of structural relaxation al
n atoms is particularly stable, and the measured abundance
lowed. We show in Fig. 8 the results of two calculations of
of alkali metal clusters can be related directly to gaps in the
the ionization energy I p : (a) for the structure of the most
eigenvalue spectrum calculated using the spherical jellium
stable isomer found for Aln; (b) for the relaxed structure
model. The present calculations for Aln clusters give a
of AI;;. Also shown are experimental bounds on the ion
maximum of.:l2 for n=7, but there are few other parallels
ization thresholds' and Ip values derived from the product
with the results of jellium calculations. The structures of
branching ratios in collision induced dissociation of AI;;
4 aluminum clusters are characterized by transitions from
clusters. Although the uncertainties in the measured ion
ization energies',4 are substantial, the overall agreement planar to nonplanar at A15, and to ground states with min
imum spin degeneracy for n > 6. The structural changes, in
between theory and experiment supports our estimate of
particular, are difficult to relate to a model based on spher
the effect of the periodic boundary conditions on the total
ical symmetry. The slower increase in the binding energy
energies of the ions: Ip increases initially to a maximum
for n > 7 is reflected in longer bonds in AI8-AIIO than those
at A16,4 has a sharp minimum at AI7,' and the dissociation
energies of the ions decrease in the order in the most stable form of A17.
Ali> Alt > Alia> Alt > Alt > (Alt ,Ali) > Alt. The
IV. GALLIUM CLUSTERS AND IONS
high stability of Ali and the low stability of Alt have been
4
noted elsewhere. The odd-even variation in Ip with the Gallium clusters were calculated using the same
number of (trivalent) atoms reflects the relative ease of method as described earlier. The structures found were in
removal of an electron from a singly occupied orbital, and most cases very similar to the aluminum analogs, although
is also found in (monovalent) alkali metal clusters. 53 the bond lengths and bond angles show characteristic dif
The Ali ion is one of the most prominent in beam ferences. In this section we show structures only if they
experiments, and the energy change on relaxing the struc differ qualitatively from those given in Figs. 2-7.
ture from that of the neutral hexamer is much larger
A. Ga
(-0.5 eV) than in the other ions ( <0.2 eV). Atoms in the 2
central layer of the final structure (C3v) have bonds of There is relatively little experimental information on
almost equal length (4.78 a.u.) to basal and apical atoms, gallium clusters, but Raman spectra of matrix-isolated Ga2
w;
and the compact structure is reminiscent of the most stable leads to a ground state vibrational frequency = 180
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R. O. Jones: Neutral and charged clusters: Aln and Gan 1201
TABLE II. Molecular parameters re (atomic units), We (em-I) for low the aluminum and gallium dimers is that the bond lengths
lying states of neutral and charged gallium dimers, together with energies in Ga2 are 3-7 % shorter than those in the lighter A12, a
relative to the ground state (aE). Vertical ionization energies Up) and feature particularly evident in Fig. 1 and not found in the
electron affinities (A) are given in eV.
previous calculations for Ga2' It is unusual to find bonds
r. we that are shorter than those between lighter atoms in the
same main group, but it is a general feature of these clus
Ga2: 3nu(ug1Tu)
ters and we discuss the reasons for it later.
(a) 5.220 158
(b) 5.24 158
(c) 4.864 184 6.87
Expt. (d) 180
B. Ga
Ga2: 3~i(~) 3
(a) 4.736 197 +0.05 The gallium trimer and its ions have been investigated
(b) 4.78 197 +0.09 by Balasubramanian and Fu28 and by Meier et al.19 There
(c) 4.396 217 -0.06 7.09
are several low-lying states in each calculation, but there
Ga2: 5~; (a2pu~Ug) are some notable differences. Balasubramanian and Feng
(c) 4.146 308 +2.42 4.89 predict a 2Al ground state (r=4.88 a.u., a=61.2°) sepa-
Gat: 2~: (if"ug) rated by 0.24-0.34 eV from the 4A2, 2BI, and 4BI states,
(a) 6.12 108 while Meier et al. favor quartet states amongst the four
(b) 6.20 97 that occur within 0.1 eV. The bond lengths found by these
(c) 5.810 101 authors are significantly (0.3-0.4 a.u.) longer than those of
(cr.
Gat: 2nu 1Tu) Balasubramanian and Fu.
(a) 5.40 119 The present calculations lead to similar results for
(b) 5.51 III +0.06 those in A13, with the 2Al state Cre=4.39 a.u., a=600)
(c) 4.959 +0.46 lying ~0.38 eV below the 2B I (4.61 a.u., 60°) and 4A2
Gai": 4~i(Ug~) (4.56 a.u., 74°) states. The most stable linear form (~1Tu'
(a) 4.876 191 re=4.56 a.u.) lies 0.9 eV above the ground state. The
(b) 5.03 171 1.25 bonds are shorter than those found in previous calcula-
(c) 4.502 215 1.44 28
tions ,l9 and, as in the case of the dimers, shorter than
n-t)
Gai: 2n.( those in the corresponding aluminum cluster.
(a) 4.736 179 In contrast to the previous results on Gat, we find
(b) 4.97 134 0.68 that the most stable state is triangular (4.46 a.u., 60°) and
(c) 4.222 255 +0.58 1.43
not linear. The most stable linear form in this ion is ~ 1 eV
Gai: 2nu(~1Tu) higher in the present calculations.
(a) 5.221 167
(b) 5.20 184 0.46
(c) 4.854 182 +0.79 1.59
"Reference 27. Complete active space SCF calculations followed by first C. Ga4
order CI.
Meier et al. 19 have studied selected structures of the
bReference 19. Multireference configuration interaction (MRD-CI).
7his work. gallium tetramer. Almost equally stable are the square
dReference 56. (D4h, r=5.30 a.u.) and rhombus (D2h, r=5.30 a.u.) struc
tures. The undistorted tetrahedron (Td, R=5.56 a.u.) and
T shape (C2", r=5.34 a.u.) lie 0.48 eV higher, and the
cm - 1.56 Thermodynamic studies yield an upper bound to linear form (with bond constrained to be equal, r= 5.10
the dissociation energy of 1.4 eV.46 Calculations have been a.u.) is a further 0.40 eV higher.
published on the neutral dimer as well as the anion and The results found in the present work reflect those in
27
cation by Balasubramanian and by Meier et al. 19 A14. The most stable state is the rhombus (D2h, triplet,
Calculated spectroscopic constants for states of Ga2 r=4.49 a.u., a=71.6°), with the related singlet 0.12 eV
and its ions are given in Table II, and the energy curves for higher (C2h, r=4.28, 4.71 a.u., a=67.7°). The triplet
Ga2 and Gal" are shown in Fig. 1. There are several sim square structure (r=4.63 a.u.) is only 0.03 eV less stable,
ilarities between the results of the calculations of alumi and the singlet square structure reverts to the rhombic
num and gallium dimers. The calculated well depth of the form on annealing at 300 K. The bond lengths are all much
3"};; state (2.08 eV) in Ga2 is similar to that found in A12, shorter than those of Meier et al. 19 We note that the flat
and shows a comparable overestimate. The 3"};; and 3IIu tened tetrahedral (triplet, sides 4.49 a.u. and 5.63 a.u.,
states are nearly degenerate in both molecules, with the ilE=0.60 eV) and tetrahedral structure (quintet, re=4.94
present calculations favoring the former in both cases. The a.u.) are relatively less stable than in A14. There are two
agreement between the calculated vibration frequency of roof structures (triplet, ilE=0.42 eV; singlet, ilE=0.53
the 3IIu state and the only measured value is an indication eV). A comparison with the results for Al4 shows that the
that this is the ground state. bonds are 3-7 % shorter, and the bond angles in the planar
The most significant difference between the results for structures are all closer to 90°.
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1202 R. O. Jones: Neutral and charged clusters: Aln and Gan
(Q)
(Q)
( b)
FIG. 9. Two structures of Gas.
( b)
D. Gas
FIG. 10. Isomers of Ga7.
The neutral gallium pentamer shows structures similar
to those found in A15, although there is no barrier between
the "planar" and "buckled" structures of Figs. 3 (a) and
eigenvalues is greater (1.75 eV) than in A17, and the en
3(b). There is a shallow energy surface with a minimum
ergy separation between ground and excited states is cor
at the planar structure. The doublet state found by revers
respondingly greater. The doublet found by reversing the
ing the occupancies of the uppermost two orbitals lies 0.53
occupation numbers of the uppermost two levels (I:l.E
eV higher. Two quartet states, a rectangular pyramid and a
= 1.22 eV ) is related to the prism structure of Ga6 and
twisted plane analogous to Figs. 3(d) and 3(e), respec
may be viewed as a (distorted) rhombus overlaid by a
tively, are almost degenerate with I:l.E=O.72 eV.
triangle. The C structure shown in Fig. lOeb) (I:l.E= 1.35
The structure of the negative ions in similar to that of 2v
eV) is more open than that in A17, and the lowest-lying
the neutral cluster in most cases we have studied. In the
quartet structure (1:l.E= 1.41 eV) is very similar to that in
case of Gas, however, the most stable form is the (slightly
Fig. 5(c).
asymmetric) envelope shown in Fig. 9(a). The corre
The differences between aluminum and gallium clus
sponding triplet [Fig. 9(b)] has C symmetry and lies 0.5
s ters are evident in the structural parameters, and can be
eV higher. The preference for gallium bonds to favor bond
made clearer by starting a calculation for an aluminum
angles closer to 90° is also evident here.
cluster with the geometry of the corresponding gallium
cluster (or vice versa). The final geometries of the most
E. Ga
6 stable isomer of (a) Al7 and (b) Ga7 are shown from the
The low-lying structures in the gallium hexamer are same perspective in Fig. 11. The bond from the apical atom
the same as found in A16, although the energy differences
are smaller. The most stable isomer is the (singlet) trigonal
bipyramid, with bond lengths 4.49 and 5.28 a.u. 2-5 %
shorter than those in the same state of A16• The prism
structures [Fig. 4(b) singlet and Fig. 4(d) triplet] are al
most degenerate with I:l.E=O.03 eV, and have bonds that
are -5% shorter than in A16. The bond angle in the C2v
prism Fig. 4(d) (75.5") is _6° larger than in A16. The
singlet structure Fig. 4(c) is only a further 0.01 eV less
stable. The triplet structure related to Fig. 4(a) has I:l.E
=0.29 eV, and there are planar structures at higher ener
gies.
F. Ga7
(Q) (b)
The situation in Ga7 is related to that in A17, but the
structures show some differences. The most stable isomer,
a distortion of the compact C structure, is shown in Fig.
3v FIG. 11. The differences between the most stable heptamer structures in
lO(a). The gap between the two highest occupied energy (a) A17; (b) Ga7.
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