BRESCANSIN, LM

FERREIRA, LG

Repositorio da Producao Cientifica e Intelectual da Unicamp

PHYSICAL REVIEW B VOLUME 20, NUMBER 8
Crystalline cluster model for ionic solids: NaC1*
15 OCTOBER 1979
Luiz M. Brescansin
Instituto de Fisica Gleb Wataghin, Universidade Estadual de Campinas, Caixa Postal 1170, Campinas, Brazil
)
Luiz G. Ferreira
Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 20.516, Sao Paulo, Brazil
(Received 14 March 1979)
A crystalline cluster model for ionic crystals is presented, in the large context of the self-consistent-field
multiple-scattering method, by imposing that the cluster potential has the same features as the crystal
potential. The model is particularly investigated with respect to NaC1, and the results are compared with
those of a conventional cluster calculation. It is seen that the main features of the electronic structure
emerge from the occupied cluster states and the obtained results of our model provide a good interpretation
of experimentally observed properties, e.g., the experimental one-electron spectrum.
I. INTRODUCTION
Self-consistent band-structure calculations
usually require a large amount of computational
work. For this reason, it is becoming increasingly
popular to calculate crystalline properties by
using small clusters of atoms arranged as in the
crystal. ' ' The cluster method can also be used to
describe localized states of impurities in crys-
tals" and in some cases seems to be much better
than the description of such states in terms of
Bloch-wave expansions.
The ability of a cluster to describe a crystal
depends on how well the occupied crystal orbitals
can be localized in small volumes. The localized
crystal orbitals are Wannier states' whose spread
in space is inversely proportional to the energy
band gaps. ' Thus, one may expect to obtain a
good cluster description in insulators and large-
gap semiconductors.
The present work began from a simple hypothe-
sis: large clusters should yield a better descrip-
tion than small clusters for a good insulator as
NaC1. A self-consistent-field (SCF) cluster cal-
culation for this material with one central Cl atom
and six neighboring Na atoms gave a band gap of
8.65 eV, a quite remarkable result. Thus, we had
a strong reason to suspect that clusters with more
atoms, such as the 1C1 6Na+ 12Cl 8Na', should
yield even better results. Our expectations turned
out to be wrong, as reported in Sec. II. On the
other hand, a restudy of the cluster method showed
us what was the origin of the errors, and how to
correct them. This is reported in Sec. III and the
results are shown therein.
II. CONVENTIONAL CLUSTER MODEL
A. Method
The conventional self -consistent multiple-scat-
tering Xjy method of Slater and Johnson was ap-
plied to NaCl clusters. This method has been suc-
cessfully applied to many molecular systems and
is described in detail elsewhere. ' " The method
assumes the muffin-tin approximation for the
cluster potential, iri which the potential is spher-
ically averaged inside the spheres surrounding
the cluster constituent atoms, volume averaged in
the region between the atomic spheres, and is
again spherically averaged in the extramolecular
region. The exchange parameters n for Na and
Cl were taken from Schwarz's compilation":
n N, = 0.731, e ci = 0.723,
and, for the extramolecular region, a weighted
average was used, resulting in
(y,„,= 0.727.
The cluster model used by us consists of a cen-
tral atom, six nearest neighbors, 12 next-nearest
neighbors, and eight third-order neighbors, orien-
ted in an octahedrally coordinated configuration.
We used tao clusters to represent the NaCl perfect
crystal: one with the ion Na' as the central atom
(1Na' 6C1 12Na' 8C1 ) and one with the ion Cl
as the central atom (1C1 6Na' 12C1 8Na'). Due
to the charge unbalance of minus or plus one elec-
tron in each cluster, they were neutralized by dis-
tributing a charge of plus or minus one electron in
the Watson sphere. "
The radii of the atomic and outer spheres are
shown in. Table I. The atomic radii were chosen
so that the non-self-consistent molecular poten-
tial of the Na' ion, along the Na'-Cl direction,
was equal to the potential of Cl . As the ion clus-
ter potential depends on its neighborhood, it fol-
lows that the Na+ and Cl radii, calculated by this
criterion, are different in the two Na'-eentered
and Cl -centered clusters. The radius of the
surrounding outer sphere was chosen so that it
touched the cluster atomic spheres. The secular
20 1979 The American Physical Society
34l6 LUIZ M. BRESCANSIÃ AND LUIZ G. FKRREIRA
TABLE I. Radii of atomic and outer spheres of the two three-shell conventional clusters of
NaC1 crystal (atomic units).
Sphere
Radius (a.u.)
1Na' 6Cl 12Na' 8Cl 1C1 6Na+ 12Cl 8Na'
Central sphere
Near est-neighbor sphere
Next-nearest-neighbor
sphere
Next-next-nearest-neighbor
sphere
Outer sphere
2.5398
2.7890
2.5398
2.7890
12.0188
2.8508
2.4780
2.8508
2.4780
11.7078
determinant was formed using partial waves of
angular momenta
l=0, 1, 2, 3
for the outer sphere, and
1=0, 1
for the atomic spheres, as the Na+ and Cl ions
have only s- and p-type orbitals.
During the self-consistent procedure the core
electrons of the first and second shells were fro-
zen, and only the third-shell electrons were per-
mitted to relax. The Na'-centered cluster has
112 valence electrons while the Cl -centered
cluster had 104. Both clusters have 2VO core
electrons.
B. Results
The self-consistent energy spectra of the 1Na'
6Cl 12Na' 8Cl and 1Cl 6Na' 12Cl. 8Na' clus-
ters representing a perfect NaCl crystal are
shown in Fig. 1. The energy levels are labeled
according to the irreducible representations of the
point group 0„.The electron distributions for
these cluster are shown in Table II. They in-
clude atomic core electrons.
Due to the strong ionic character of the crystal,
the cluster orbitals are well localized inside the
atomic spheres, thus permitting an easy identifi-
cation of the occupied orbitals. Then the band gap
is the energy difference between the first unoccu-
pied state and the last occupied valence state. The
band gap turns out to be 4.88 eV for the Na'-cen-
tered cluster and 5.51 eV for the Cl -centered
cluster, values which are much smaller than the
experimentally observed gap of 8.97+ 0.07 eV."
The reason for this failure may be understood as
follows. Due to different environments, chemi-
cally identical atoms have different self-consis-
tent potentials in the cluster. Figure 2 is a sche-
matic plot of the Cl -centered cluster potential
along the direction of the touching spheres. The
figure shows that the potentials of the central Cl
ion and first Na. ' shell are, respectively, deeper
than those of the second (Cl ) and third (Na')
shells. Following these potential differences,
there is an electronic charge migration from the
outermost Na' and Cl spheres to the innermost
spheres. Also due to the potential differences,
TABLE II. Charge distributions of the conventional clusters 1Na+ 6CI 12Na' 8C1 and 1Cl
6Na' 12C1 8Na' expressed in number of electrons per sphere. For the interatomic and ex-
tramolecular regions the total number of electrons is given. The self-consistent potential
(Ry) is also shown
Atom
Central atom
Nearest-neighbor atom
Next-nearest-neighbor atom
NeN-next-nearest-neighbor
atom
Extr amolecular region
Interatomic region
Interatomic constant
potential
12Na' 8C1
10.1984
16.9173
10.1176
16.7002
0.6198
14.3477
-0.4850
1Cl- 6Na' 12C1- 8Na+
17.0870
10.1092
16.9000
10.0680
0.2586
12.2444
-0.4850
20 CRYSTALLINE CLUSTER MODEL FOR IONIC SOLIDS: NaC1
-0.3
1NII6Q 12NO~Bgl
' E (Ry)
ICI 6Na~ I2CI 8Na+
E (Ry)
I1-0.3
-0.6
l2 I25
~25 p c))~)
)r 'r 'r' 'r)5
0 y 1 12 12
-0.r
-0.8
C12~5,
C25 25 C&
15
12
I C15 25
r'
f2
0.8 .
-0.9-
r1'5
«C12«l
15
r» sc))&)~15
-0.9-
-I.5-
C15 PCI(1&
FIG. 1. Energy levels
(rydbergs) of 3s and 3p Cl
valence states and first
conduction state for the con-
ventional clusters 1Na'
6C1 12 Na'SC1 and 1CI"
6Na+ 12Cl SNa+. Cl &g)s
C1(2), Cl (3) and Cl &4& mean
the Cl orbitals of the cen-
tral ion, first, second, and
third shells of the cluster,
respectively.
~12 ~ 25~fr
Ic)),)
-l.5-
—I 15 SCI(~)
C25
-l.6
-l.6- -I.7
~l )5 scl)p)
the energy levels corresponding to the Cl orbi-
tals of the second shell lie between the valence
levels of the Cl central ion and the orbital cor-
responding to the Na' ion (the conduction energy
state). Thus, whenever molecular clusters of
ionic materials made up by more than one shell
of atoms are investigated, the identification of the
crystal band edges from the cluster spectrum
turns out to be misleading and one gets an energy
gap much smaller than the experimenta. l one. On
the other hand, the valence-band width of the
Na, Cl crystal, which is an interesting feature of
its energy-level structure, is a.iso not properly
described by this method: One sees from Fig. 1
that the states are spread too much in energy.
The failure reported above moti'vated the work
described in Sec. III. As will be seen shortly, the
main idea is to make the cluster potential as per-
iodic a,s possible, thus avoiding the effect sketched
in Fig. 2.
3418 LUIZ M. BRESCAA SIN AND LUIZ G. FERREIRA
CI
No&
dtrection of touching Spheres
from which one determines the values of B~. The
second charge distribution has a uniform electron-
ic density n~ and nuclei with B~ protons. The po-
tential due to this latter distribution is determined
by the Ewald method. " The sum of the potentia. ls
due to the two distributions ha. s the following ex-'
pressions.
(i) Inside the sphere p,
FIG. 2. Schematic profile of the potential in a conven-
tional cluster of an ionic crystal. v and c designate,
respectively, a valence and a conduction state centered
at the ion. The figure explains the shrinkage of the gap
in the energy spectrum.
III. CRYSTALLINE CLUSTER MODEL
A. Method
We show in this section the modifications to be
introduced in the conventional SCF-Xn multiple-
scattering method with which the one-electron clus-
ter potential acquires crystalline properties.
Among the many Na' and Cl atoms of the clus-
ter, we select two atoms as representatives of the
families. Practically, the selected atom are the
innermost of the cluster. I et
n, (r)
be the electronic density of the representative of
the family of atoms p. Let b~ be its sphere radius.
A perfect crystal made up of atoms identical to
the representatives would have an average elec-
tronic density n~ in the intersphere region given
by
bp
Q~n~=g Z~ —g 4w r'n~(r) dr,
where Z~ are the proton numbers and Qz is the
intersphere volume in the unit cell of the crystal.
The determination of the crystal Coulomb po-
tential due to densities
n (r),
inside the spheres, and nn in the intersphere re-
gion, is a solved classical problem. One considers
two charge distributions. The first ha, s neutral
spheres with electronic densities
n, (r) —n~
and proton numbers equal to
2Z, 8g
v~(r) = — '+— r'n~(r) dr +8m rn~(r) dr
0 r
—4mbqnq —2 Q B,M(R~,) .
(ii) In the intersphere region,
8~ B b', 2~
Q ~ 2 1508 p
+~ Q B,M(R~,)),
c
(4)
where R~, is the radius vector from nucleus p to
nucleus q, both in the unit cell, and
2 2
e~ ~ e-«4' ~ 2t:l
Q~oo Q ~ g
2 2
4& V'~(R) =lim ~ cosg R, — 2 —
~
(5b)
a „g„o g. GQ)
are Madelung-like constants expressed as sums in
the reciprocal-lattice vectors g, and where 0 is
the volume of the unit cell.
Once the crystal potentia, l is calculated, we se-
lect a piece of the crystal, the cluster, and en-
velop it with a sphere outside which we set the po-
tential equal to zero. As a matter of fact, it is
convenient to set the outside potential equal to a
positive constant sufficiently higher than the inter-
sphere potentia, l V& so as to avoid the cluster elec-
trons escaping outside the enveloping sphere. The
value of the positive constant is irrelevant and a
few Rydbergs will do.
B. Results
We have selected to investigate two Cl -centered
clusters whose geometric parameters are pre-
sented in Table III. The two clusters are of the
type 1Cl 6Na' 12Cl 8Na' and differ by the sphere
radii. Cluster Cl 1 has radii proportional to ionic
radii, while in cluster Cl 2 the radii a.re chosen
to minimize the integrated
Since the spheres are neutral, one has [v(r) —v(r)]' d r
(2)
bp
Zp —B~ = 4n r'n~(r) dr bpn~, —
0
squared deviation of the potential v(r) from its
muffin-tin average v(r). The o. parameters for
20 CRYSTALLINE CLUSTER MODEL FOR IONIC SOLIDS: NaCl 3419
Sphere
Radius
Cluster Cl 1 Cluster Cl 2
C I central sphere
Na' internal sphere
Cl external sphere
Na' external sphere
Outer sphere
3.4580
1.8708
3.4580
1.8708
11.1006
2.9308
2.3980
2.9308
2.3980
11.6278
TABLE IG. Radii of atomic and outer spheres of the
two three-shell crystalline clusters of NaC1 crystal
(atomic units) .
Na+, Cl, and the outer-sphere region are the
same as in the conventional cluster calculation.
The self-consistent energy spectra of the clus-
ters Cl 1 and Cl 2 are shown in Fig. 3. The en-
ergy levels are labeled according to the irreduc-
ible representations of the octahedral point group
0„.The 104 valence electrons fill the valence
levels up to the last F» level and there is no dif-
ficulty in identifying the crystal energy gap as the
difference between the first unoccupied state and
the last I'» valence state. The energy gap found
in this way is 9.29 eV in the Cl 1 cluster and
LC1- &)
' E(Ry)
-1.2-
(Cl- 2)
E(Ry)
-1.5-
-1.) - 1( 1.4-
-19
-20-
l I I ~1(a la "(a -1 (5
ls
I j I fps—"a's
Iis
Ias
(pCI&(~ 4pCI~~&)
r( ~(z
rl f(s
I (5
-2.0-
-21
C~ 1(s I(s
I ] I f(p
~as „,
t (5
as / is ( p C I(, I +pCIIsI )
C(s
FIG. 3. Energy levels
(rydbergs) of the valence
states and first conduction
state for the crystalline
clusters Cl 1 and Cl 2.
Both clusters are of the
type 1C1 6Na'12C1 8Na'
but have spheres with dif-
ferent radii. The energy
zero was arbitrarily cho-
sen.
-2.1 2t2
-2.8 -2.8
-2.9
I J /~(z r25
teel( i ) 4 sCl&&& )
y ~as
f(s
-2.9-
fi ~iia ias
les tsCI~,
&
4sCI~~&)
8420 LUIZ M. BRESCANSIN AND LUIZ G. FERREIBA 20
TABLE IV. Charge distribution of the crystalline clusters Cl 1 and Cl 2 expressed in
number of electrons per sphere. For the interatomic and extramolecular regions the total
number of electrons is given.
Atom Cluster Cl 1 Cluster Cl 2
Central atom
Nearest-neighbor atom
Next-nearest-neighbor atom
Next-next-nearest-neighbor atom
Extramolecular region
Inter atomic region
17.5323
9.8636
17.5323
9.8636
0.1379
5.4755
17.0661
10.0806
17,0661.
10.0806
0.0878
11.6247
8.73 eV in the Cl 2 cluster. Both values are in
very good agreement with the experimental find-
ings. ""The upper valence band arises from the
3p states of the Cl atoms. According to Page and
Hygh,
"the width of this band is 0.5'7 eV. From
Figs. 1 and 3 it is immediately seen that the va-
lence-band width of NaCl calculated by the cry-
stalline-cluster model compares much better with
the mentioned value, although the bandwidth is
still larger than the experimental one.
The conduction-band states are supposed to be
rather delocalized and could not be well described
by a small cluster. Therefore, no attempt was
made by us to investigate and identify the higher
unoccupied levels with optical spectroscopy data
from band-to-band transitions, even for the cry-
stalline-cluster calculations.
The charge distributions of the clusters Cl 1
and Cl 2 are shown in Table IV. They are ex-
pressed in numbers of electrons per region, in-
cluding the atomic core electrons. One readily
sees there no'longer is the inconvenient effectof
charge migration from one sphere to another
chemically equivalent.
IV. CONCLUSIONS
We have shown in this paper how a cluster-mod-
el technique may be successfully utilized to de-
scribe electronic structure and related properties
of ionic compounds. The case of Na'Cl clus-
ters, containing up to 27 atoms (three shells), was
particularly studied, involving two kinds of cluster
calculations. Our presented results show that the
conventional cluster method applied to the three-
shell cluster of Na'Cl leads to a wrong energy
spectrum for the bulk material; however, by propo-
sing a new way of calculating the cluster poten-
tial, we have shown that a localized description of
the electronic structure is satisfactory. . It ap-
pears as a good alternative to the conventional
band theory description, for the main aspects of
the electronic energy structure emerge from the
cluster occupied states. We believe that the cry-
stalline-cluster technique developed for NaCl
should be useful whenever one deals with insula-
tors, for which the occupied orbitals are well
localized, and especially for ionic materials, when
the self-consistent calculation in conventional
clusters may result in very dissimilar potentials
for identical atoms. Of course, our model is not
suitable for describing crystal-surface proper-
ties, in which the differing potentials for the same
chemical species as a function of position is clear-
ly meaningful.
ACKNOWLEDGMENTS
Helpful discussions with Professor Huberto
Closs and Professor P. S. GuimarKes during the
calculations are gratefully acknowledged. Thanks
are also due to Professor N. J. Parada and
C. M. C. Castilho for their encouragement and
constant help during the implementation of the pro-
gram at the Universidade de Campinas.
*Paper based on a doctoral thesis submitted to the Uni-
versidade Estadual de Campinas by one of the authors
(L.M.B.).
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20 CRYSTALLINE CLUSTER MODEL FOR IONIC SOLIDS: 3421
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CRYSTALLINE CLUSTER MODEL FOR IONIC SOLIDS - NACL

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