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Z. Kristallogr. 215 2000) 419±423 419
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The bond character of rutile type SiO 2 , GeO 2 and SnO 2 investigated
by molecular orbital calculation
J. Mimaki, T. Tsuchiya and T. Yamanaka*
Osaka University, Faculty of Science, Department of Earth and Space Science, 1±1 Machikaneyama Toyonaka, Osaka 560±0043, Japan
Received October 18, 1999; accepted February 25, 2000
Abstract. Molecular orbital calculations using descretevariational
Xa method were carried out in order to investigate
the bond nature of SiO 2 , GeO 2 and SnO 2 . The bond
overlap populations and the d-orbital populations of cations
were decreased with increasing the atomic number.
The deformation density maps of electronic charge show
the accumulation of electron in interatomic region which
is most dominant in SiO 2 . The calculated density of states
indicates that the bond character of SiO 2 is more covalent
than that of GeO 2 and SnO 2 . From the excited d-electron
partial density of states, it is inferred that the Si excited-3d
has some connection with the bond nature of SiO 2 covalency.
Introduction
The rutile type structure is the most common MX 2 structure
and the electronic properties of this class of compounds
vary widely from insulating to metallic behavior.
A number of investigations have been conducted by several
theoretical and experimental approaches Baur, Khan,
1971; Baur, 1976; Burdett, 1985; Sorantin, Schwarz,
1992; Camargo, Igualada, BeltraÂn, Llusar, Longo, AndreÂs,
1996). Among them, the bond character of stishovite, a
high-pressure polymorph of SiO 2 , and related substances
having the rutile-type structure are significant subject not
only for the crystal chemistry but also for the earth
science and electronic device.
Since SiO 2 is the most fundamental substance in the
earth crust and plays an important role in modern solid
state technology, many theoretical calculations concerning
SiO 2 polymorphs have been performed with focus on the
crystal structure, phase transition, elastic properties and
electronic states. Electronic and optical properties such as
the band structure, density of stated and valence-charge distribution
of all polymorphs of SiO 2 using self-consistent
orthogonalized linear combinations of atomic orbitals
method have been studied by Xu and Ching 1991). Nada,
Catlow, Dovesi and Pisani 1990) have investigated the
chemical nature of the Si ±O bond of a-quartz and stishovite
* Correspondence author e-mail: mimaki@ess.sci.osaka-u.ac.jp)
by a periodic ab-initio Hartree-Fock method. They concluded
that quartz is more covalent than stishovite and that
the Si d-orbitals play an important role in these materials.
In many SiO 2 polymorphs, Si has a tetrahedral coordination.
Their structural difference is in the way of the linkage
of tetrahedra. Gibbs 1982) has carefully investigated
bond length and angle variations, electron density distributions
and bond characters of silicates by molecular orbital
calculation. On the other hand, the high-pressure polymorph,
stishovite is composed of 6-coordinated Si. X-ray
least-squares refinements of stishovite and molecular orbital
calculation of molecules with 6-coordinated Si has
been performed Hill, Newton, Gibbs, 1983).
Experimental studies on the nature of chemical bonds,
electron emission spectra can provide information about
electronic band structure which can be compared with the
calculated density of states. Despite several theoretical calculations
of stishovite, only little experimental work has
been published Wiech, 1984).
The rutile-type dioxides including the group of IVb
metals is studied for understanding the structural and physical
properties of stishovite. Namely, SnO 2 , GeO 2 and
PbO 2 can be analogue substances for the high-pressure
behavior of stishovite. Haines, LeÂger, Chateau, Bini and
Ulivi 1998) have investigated GeO 2 by Raman spectroscopic
study under pressure and they showed that SiO 2 ,
GeO 2 and SnO 2 transform the CaCl 2 type structure systematically.
The rutile structure consists of straight chains of edgesharing
octahedra which lies parallel to the c axis. There
are two different M-O distances in the each MO 6 octahedron.
The structural difference of rutile-type IVb metal
dioxides have reported by Bolzan, Fong, Kennedy and
Howard 1997) by neutron powder diffraction studies.
Compounds of SiO 2 and GeO 2 have two longer distances
axial M ±O) and four shorter distances equatorial
M ±O), while those of SnO 2 and PbO 2 have two shorter
ones and four longer ones, though the average M ±O distances
display and almost ideal linear relationship with
ionic radii.
Recently, we have investigated the charge density and
bond character of SiO 2 , GeO 2 and SnO 2 by single crystal
X-ray diffraction study using the monopole refinement
technique. The change of valence electron distribution was

420 J. Mimaki, T. Tsuchiya and T. Yamanaka
clarified. The result will be presented in another paper
Yamanaka, Kurashima, Mimaki, submitted).
In the present paper we have studied the bond character
of rutile type SiO 2 , GeO 2 and SnO 2 by molecular orbitals
calculation. The key of this work is the systematic
research of these dioxides. We discuss the differences of
the dioxides in terms of Mulliken population analysis,
density of states and charge density distributions.
DV-Xa molecular orbitals calculation
In order to investigate the bond character of rutile structure
SiO 2 , GeO 2 and SnO 2 , the cluster discrete variational
DV)-Xa molecular orbitals calculations Averill, Ellis,
1973; Rosen, Ellis, Adachi, Averill, 1976) were carried
out. The DV-Xa method is based upon a self-consistentfield
Hartree-Fock-Slater approximation and has been applied
to a wide range of materials Kowada, Adachi, Tatsumisago,
Minami, 1992; Tanaka, Kawai, Adachi, 1995;
Nakatsugawa, Iguchi, 1997). In this method, one-electron
Hamiltonian h is,
h ˆ T i ‡ V n ‡ V c ‡ V xc ;
…1†
where T i ,V n ,V c and V xc is the kinetic energy potential, the
Coulomb interaction with nuclear, the Coulomb interaction
between electrons and exchange-correlation potential, respectively.
In the Hartree-Fock-Slater method, the exchange-correlation
potential is given by
V xa ˆ 3a 3r 1=3
; …2†
4p
Where r…r† is the local charge density
The molecular orbitals of the cluster are represented by
linear combination of atomic orbitals LCAO) expressed
as
w l ˆ P C ij c i ;
…3†
where w l ; c i and C ij is the lth molecular orbital, the ith
atomic orbital and the coefficients of linear combinations
determined by self-consistent procedure, respectively.
Compared with the Hartree-Fock methods, this method
has advantages as follows:
1) It can calculate the electronic structure of the substances
including heavy atoms within same precision as
light ones because the atomic orbitals are numerically generated.
2) By using Xa exchange-correlation potential a
fixed at 0.7), the computation time is greatly reduced and
relatively large-scale calculation is possible. The details of
the computational treatments of the DV-Xa method have
been reported in Adachi, Tsukada, and Satoko 1978).
The atomic basis 1s) 2 ns) 2 np) 2 nd) 0 for each cation
where n is 3, 4 and 5 for Si, Ge, Sn, respectively and
1s) 2 2s) 2 2p) 4 for oxygen were adopted as the initial
state. Moreover, we placed formal charges at each atom
sites around the cluster with cation as ‡4, of oxygen as
2 in order to include the effect of the Madelung potential.
The [MO 6 M 10 O 8 ] 16‡ M ˆ Si, Ge, Sn) and
[MO 6 M 10 O 38 ] 44 M ˆ Si, Ge) clusters in D 2h mmm)
point symmetry were employed. Since there is no large
difference among the results obtained from these clusters,
we concluded the size effects of cluster are small. The
structural data of clusters were taken from experimental
results Yamanaka et al. Submitted).
Results and discussion
Mulliken population analysis
1) [MO 6 M 10 O 8 ] 16‡ M ˆ Si, Ge, Sn) and 2)
[MO 6 M 10 O 38 ] 44 M ˆ Si, Ge) clusters are shown in
Fig. 1. The latter cluster was not calculated in the case of
M ˆ Sn, since we could not have a good self-consistency
in it. Table 1 shows the results of Mulliken population
analysis. The bond overlap population, which is a useful
parameter to discuss the covalency, is decreased on increasing
atomic number. The exited d-orbital population is
also decreased from Si to Sn. The Mulliken population
analysis is widely used in the field of quantum chemistry
in order to analyze the local electronic properties. However,
it is noted that absolute values of Mulliken population
analysis are strongly dependent on the atomic basis
set. On the contrary, they have comparable values when
they are calculated with same conditions.
a
b
Fig. 1. Cluster model of a) [MO 6 M 10 O 8 ] 16‡ and b) [MO 6 M 10 O 38 ] 44 .
The gray and open spheres are Si and O atoms, respectively.
Table 1. The results of Mulliken population analysis.
a) cluster: [MO 6 M 10 O 8 ] 16‡ M ˆ Si, Ge, Sn)
n
ns
np
nd
Si Ge Sn
3
0.50
0.83
0.51
4
0.70
0.88
0.23
5
0.62
0.75
0.18
net charge 2.16 2.18 0.43
ovelap population eq)
overlap population ax)
b) cluster: [MO 6 M 10 O 38 ] 44
n
ns
np
nd
0.44
0.42
M ˆ Si, Ge)
Si
3
0.52
0.87
0.53
0.37
0.33
Ge
4
0.70
0.87
0.21
net charge 2.08 2.20
overlap population eq)
overlap population ax)
0.45
0.43
0.36
0.32
0.26
0.25

The bond character of rutile type SiO 2 , GeO 2 and SnO 2 421
Density of states of SiO 2 , GeO 2 , SnO 2
The calculated total and partial densities of states DOS)
of SiO 2 , GeO 2 and SnO 2 are shown in Fig. 2 and Fig. 3,
respectively. The DOS can provide information on the
electron system as a function of energy. The highest occupied
molecular orbitals HOMO) is set to be zero. These
DOS(arb. units)
SiO 2
GeO 2
SnO 2
-30 -20 -10 0 10
Energy (eV)
Fig. 2. Total density of states TDOS) of SiO 2 , GeO 2 and SnO 2 . The
full, dashed and dotted line corresponds to SiO 2 n ˆ 3), GeO 2
n ˆ 4) and SnO 2 n ˆ 5), respectively.
DOS(arb. units)
0.5
0
0.5
0
0.5
0
4.0
0
4.0
Mns
SiO 2 ( n=3 )
GeO 2 ( n=4 )
SnO 2 ( n=5 )
Mnp
Mnd
O2s
O2p
0
-30 -20 -10 0 10
Energy(eV)
Fig. 3. Partial density of states PDOS) of SiO 2 , GeO 2 and SnO 2 .
The full, dashed and dotted lines correspond to SiO 2 n ˆ 3), GeO 2
n ˆ 4) and SnO 2 n ˆ 5), respectively.
DOS were obtained by convolution of MO levels with a
Gaussian broadening function with a full width at halfmaximum
FWHM) of 0.5 eV. The results of the total density
of states TDOS) for SiO 2 stishovite) is in good
agreement with Xu and Ching 1991). The partial density
of states PDOS) is in very good agreement with Nada
et al. 1990).
Experimental results of X-ray photoelectron spectroscopy
and ultraviolet photoelectron spectroscopy XPS and
UPS) can provide information about density of states of
valence bands. Barr, Mohsenian and Chen 1991) have examined
XPS spectra of sputter-deposited thin films of IVb
oxides. According to their experiment, the covalency ionicity)
affects the bandwidth of these oxides. The more
covalent, the wider the bandwidth. Our results of total
DOS are roughly in agreement with theirs concerning to
bandwidths which gradually shrink from SiO 2 to SnO 2 .
SiO 2 has the most dispersive bandwidth due to the enhanced
covalency and hybridization compared to GeO 2
and SnO 2 , as shown in Fig. 2.
The role played by d-orbitals in Si ±O bond formation
is a controversial problem Gibbs, Downs, Boisen, 1994,
Simunek, Vackar, Wiech, 1993). The electron emission
spectra about a-quartz and stishovite have been analyzed
by Wiech 1984). Si K-emission bands of stishovite are
considerably different from a-quartz, while those of the Si
L 2,3 -emission bands are similar. The Si K- and Si L-emission
bands reflect the Si p-like and Si s/d-like electrons of
the valence band, respectively. Simunek et al. 1993) have
concluded that Si 3d-orbital does not participate in the
Si ±O bond. This conclusion is based upon the fact that Si
L 2,3 -emission bands composed of their s and d constituents
are insensitive to the local crystal structure of a-
quartz and stishovite.
On the other hand, Gibbs et al. 1994) concluded that
the d-orbitals of Si play a small but important role in governing
the geometry, electron density distribution and
spectral properties. As seen from Fig. 3, the population of
d-electron on Ge and Sn is much less than that of Si.
Thus, it is implied that the d-orbitals of cation affects the
difference of local geometry of the crystal structure of
these materials.
Electron density distributions of SiO 2 ,
GeO 2 and SnO 2
The deformation density maps subtracted crystal electron
density from isolated atom electron density) of SiO 2 ,
GeO 2 and SnO 2 are shown in Fig. 4 and the degree of
localization of electron density distribution of Si ±O,
Ge ±O and Sn ±O has been compared.
Many attempts in order to establish a direct link between
the height/position of a build-up of the Drr) map
and the ionic/covalent character of the bond had been made
earlier. However, we can not quantitatively evaluate the
specific covalency of these bonds by deformation density
maps. Dunitz and Sieler 1983) discovered the absence of a
build-up of the Drr) density in interatomic regions for a
number of covalent systems with NN, CN and CO bonds.
Thus, the accumulation of electron density in the bonding
region is not a necessary condition of covalent bonds.

422 J. Mimaki, T. Tsuchiya and T. Yamanaka
(a)
(b)
(c)
O
O
O
O
O
O
Si
Ge
Sn
O
O
O
O
O
O
(d)
O
(e)
O
(f)
O
O
Si
O
O
Ge
O
O
Sn
O
O
O
O
Fig. 4. Deformation density map of SiO 2 , GeO 2 and SnO 2 .a) SiO 2
equatorial bond plane. b) GeO 2 equatorial bond plane. c) SnO 2
equatrial bond plane. d) SiO 2 axial/equatrial bond plane. e) GeO 2
axial/equatrial bond plane. f) SnO 2 axial equatrial bond plane. The
Despite the above comment, the deformation density
maps can provide meaningful information and are worth
while calculation, since the relative charge transfer is at
least shown in these results. There is an increase of electron
density in the interatomic region between Si and O;
such increase of Ge ±O bond is less than that of Si ±O
bond, and that of Sn ±O bond is the last of the three dioxides.
The ratio of the electron density of M ±Oeq)/
M ±Oax) is decreasing with increasing atomic number.
Comparison with monopole refinement results
Together with calculations described above, we have executed
monopole refinements based on X-ray single crystals
diffraction intensities of SiO 2 , GeO 2 and SnO 2 Yamanaka
et al. submitted). Our calculations are in good
agreement with the refined results and other experimental
data Baur, Khan 1971; Hill et al. 1983; Spackman, Hill,
Gibbs, 1987; Bolzan et al. 1997) which indicates that the
electron distributions are more localized with increasing
atomic number.
Conclusion
Molecular orbital calculations using descrete-variational
Xa method have been performed in order to study the
bond nature of SiO 2 , GeO 2 and SnO 2 . Our results indicate
full and dashed lines correspond to positive and negative density, respectively.
Dot-dashed lines indicate the zero level. The contour values
range from 0.04 to 0.04 e a.u.) 3 with an increment of 0.005 e
a.u.) 3 .
that SiO 2 is the most covalent and the ionicity increases
with increase atomic number. Moreover, the population of
d-orbital of cation decreases with increase atomic number.
Thus, we may conclude that there are some correlations
between the d-orbital of cations and the covalency of
M ±O bonds.
Acknowledgments. This work is partly supported by the Grant-in-Aid
A2)-10304044 and B2)-11694076) of Ministry of Education, Sport
and Culture, Japan.
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