Determination of the electronic band structure of the rutile
polymorph of TiO2: a quantum chemical approach.
Piotr Józef Bardziński1
1
Imperial College London, Thomas Young Centre, Computational Materials Science
Group; South Kensington Campus, Exhibition Road, London SW7 2AZ, Great Britain
1
Wroclaw University of Technology, Institute of Materials Science and Technical
Mechanics, B1 building - room 110, ul. Smoluchowskiego 25, 50-370 Wroclaw,
Rzeczpospolita Polska
Abstract
The aim of this work is the investigation of the relationship between the electronic band
structure of the TiO2 rutile and the dimensionality of the system. For three dimensional
system the bulk form of rutile was considered, while a slab model was chosen in order to
represent the titanium (IV) dioxide (110) surface. The influence of changing the number of
atomic layers on the bandgap value for the (110) surface was also examined. Density of states
was covering the bands from the first valence band up to the bottom of the conduction band
and was projected on the whole set of atomic orbitals as well as on the significant shells of the
Titanium and Oxygen atoms. Ab initio calculations with a B3LYP functional were carried
out. Basis sets used were modified Ti 86-411(d31)G darco unpub and O 8-411 muscat 1999.
The results are compared with experimental and computational data already present in the
literature. Surface termination problem was discussed and the application of the obtained
results as a starting point to obtain the first model of the rutile titania nanotube was proposed. The surface formation energies for rutile planes with a different surface termination
were compared and the modification to the equation needed for surface energy calculation
was introduced.
Keywords: rutile; band structure; B3LYP; surface termination; surface formation energy
1
Corresponding author
Tel: (004871)320-34-96
Mob: (0048)605-611-284
E-mail address: piotr.bardzinski@pwr.wroc.pl
1
The following document contains the Appendix to the article Determination of the electronic
band structure of the rutile polymorph of TiO2 : a quantum chemical approach, where the basis
set chosen for titanium atoms was shown.
References
[1] BREDOW T., HEITJANS P., and WILKENING M. Electric field gradient calculations for
LixTiS2 and comparison with Li-7 NMR results. Phys. Rev. B, 70:115111, 2004.
[2] CORA F. The performance of hybrid density functionals in solid state chemistry: the case
of BaTiO3. Mol. Phys., 103:2483 – 2496, 2005.
[3] MUSCAT J. PhD Thesis. University of Manchester, 1999.
[4] SCARANTO J. and GIORGIANNI S. A quantum-mechanical study of CO adsorbed on
TiO2: A comparison of the Lewis acidity of the rutile (1 1 0) and the anatase (1 0 1)
surfaces. J. Mol. Struct. THEOCHEM, 858:72 – 76, 2008.
A
Basis sets
The basis set used in this work for titanium (IV) dioxide is as following:
22 7
0 0 8 2. 1.
225338.0 0.000228
32315.0 0.001929
6883.61 0.011100
1802.14 0.05
543.063 0.17010
187.549 0.369
73.2133 0.4033
30.3718 0.1445
0 1 6 8. 1.
554.042 -0.0059 0.0085
132.525 -0.0683 0.0603
43.6801 -0.1245 0.2124
17.2243 0.2532 0.3902
7.2248 0.6261 0.4097
2.4117 0.282 0.2181
0 1 4 8. 1.
24.4975 0.0175 -0.0207
11.4772 -0.2277 -0.0653
4.4653 -0.7946 0.1919
1.8904 1.0107 1.3778
0 1 1 0. 1.
0.8126 1.0 1.0
0 1 1 0. 1.
0.3297 1.0 1.0
0 3 4 2. 1.
16.2685 0.0675
4.3719 0.2934
1.4640 0.5658
2
0.5485 0.5450
0 3 1 0. 1.
0.26 1.0
8 5
0 0 8 2. 1.
8020.0 0.00108
1338.0 0.00804
255.4 0.05324
69.22 0.1681
23.90 0.3581
9.264 0.3855
3.851 0.1468
1.212 0.0728
0 1 4 7. 1.
49.43 -0.00883 0.00958
10.47 -0.0915 0.0696
3.235 -0.0402 0.2065
1.217 0.379 0.347
0 1 1 0. 1.
0.4567 1.0 1.0
0 1 1 0. 1.
0.1843 1.0 1.0
0 3 1 0. 1.
0.6 1.0
99 0
The core part of the basis set chosen for Ti atom is the same as the Ti 86-411(d31)G darco unpub
[1, 2]. Exponents in the functions describing two additional sp shells were changed from 0.8099
and 0.3242 to 0.8126 and 0.3297, respectively. For the first titanium d shell, number of primitives
GTF n g was changed from 3 to 4. All of the exponents as well as their corresponding contraction
coefficients were changed from 7.6781 0.1127, 1.8117 0.3927, 0.4630 0.5206, to: 16.2685 0.0675,
4.3719 0.2934, 1.4640 0.5658 and one extra function was added to desribe this d shell, with an
exponent (which is the most diffuse) set to 0.5485 and contraction coefficient of 0.5450. Note
that the formal charge of this d shell was set to 2 instead of 0, as it was in the previous version
of the basis set. In the last d shell from the original basis set, the exponent was changed from
0.23 to 0.26.
The basis set for oxygen was the O 8-411 muscat 1999 [3, 4] with a minor modification.
Namely, the extra d shell was added, described with the function with an exponent of 0.6 and
the value of 1.0 of its corresponding coefficient.
3