R.A. Evarestov, A.V. Leko, and V.A. Veryazov
(Submitted July 15, 1997; accepted August 8, 1997)
Titanium oxides form a wide row of compounds with stoichiometric and nonstoichiometric chemical composition with different valence states of the cation atom. The aim of this work is study of local electronic properties of titanium oxides in oxidation states of Ti (IV, III, II).
In quantum chemistry of molecules a scheme of chemical bonding analysis has been developed [1]-[5]. Using the density matrix elements calculated on an orthogonal atomic basis one can obtain atomic charges QA [1], covalent bond orders WAB [2], atomic covalence CA [3] and valence VA [4]. These definitions are easily generalized to a crystal in band structure calculation by transformation of density matrix elements from Bloch basis to atomic one. For metallic compounds it was offered in [5] to neglect in the calculation of CA the homoatomic bond orders between atoms of the same element. Corrected in such a way the heteroatomic valence VAhetero correlates well with classical oxidation states for both insulating and metallic compounds.
For non-orthogonal basis Mulliken's population analysis is commonly used to calculate atomic charges and bond overlap populations. Although chemical bonding can be analyzed on an arbitrary basis [6], we used an alternative way – symmetric (Löwdin) orthogonalization of the atomic basis [1] with subsequent use of formulas for an orthogonal basis.
Band structure calculations of TiO2 (rutile, anatase, brookite), Ti2)3, TiO (hexagonal modification) were made in Hartree-Fock approximation with the Durand and Barthelat effective core pseudopotentials using CRYSTAL 95 code [7] and by the CNDO band theory technique [8].
For HF calculations we used two basis sets. Set I was taken from [9] and includes PS-31G for oxygen and PS-411G for titanium atom. To investigate the influence of polarizing atomic functions on the local properties we used set II with truncated basis for the titanium atom (PS-41G). Semiempirical parameters in CNDO calculations were calibrated to reproduce the experimental geometry of rutile.
For all titanium oxides we obtained the insulating state in the restricted closed shell Hartree-Fock band structure calculation. The results describing the chemical bonding in titanium oxides are presented. We did not find significant differences between the rutile, anatase, brookite modifications of TiO2, thus only rutile data are presented in the tables.
In the row of Ti(IV)-Ti(III)-Ti(II) the ionicity degree is increased, whereas the covalency of the Ti-O bond is decreased. The change in d-shell occupancy of titanium 3dn obtained for basis set I contradicts the qualitative notions. Using the truncated basis (set II) in HF calculation gives results close to those of CNDO and the obtained atomic valencies are close to the expected values.
We conclude that standard methods of population analysis cannot be applied correctly when extensive basis sets with broad polarizing functions are used. This effect is known in molecular calculations [10].
The most interesting result is the high bond order for the short Ti-Ti bond (R=2.58 Å) in Ti2O3 crystal. A significant difference between total valence and heteroatomic valence, as obtained for this crystal, usually indicates metallic bonding in a compound. Although the more stable form of Ti2O3 is insulating, it can undergo an insulator-to-metal transition.
Our results show the possibility to use the CNDO technique to study chemical bonding in nonstoichiometric compounds TiO2–x.
| crystal | HF set I | HF set II | CNDO | |||
| Mulliken | Löwdin | Mulliken | Löwdin | Löwdin | ||
| TiO2 | QTi nd |
2.66 1.46 | 1.73 2.18 | 3.32 0.81 | 3.09 0.82 | 3.26 0.62 |
| Ti2O3 | QTi nd |
1.74 2.38 | 1.43 2.50 | 2.80 1.29 | 2.62 1.31 | 2.47 1.43 |
| TiO(hex) | QTi nd |
1.62 2.38 | 1.27 2.70 | 1.96 2.04 | 1.92 2.05 | 1.70 2.22 |
| crystal | method | WTi–O | WTi–Ti | CTi | VTi | VTi hetero | VO |
| TiO2 | HF set I HF set II CNDO |
0.557 0.270 0.224 | 0.030 0.021 0.006 | 3.46 1.62 1.37 |
4.18 4.00 4.02 | 4.04 3.74 3.81 | 2.36 2.09 2.02 |
| Ti2O3 | HF set I HF set II CNDO |
0.414 0.128 0.163 | 0.855 0.937 0.922 | 3.46 1.71 1.94 |
3.98 3.61 3.62 | 3.12 3.03 3.02 | 2.30 2.07 2.02 |
| TiO(hex) | HF set I HF set II CNDO |
0.211 0.031 0.092 | 0.041 0.000 0.003 | 1.60 0.19 0.60 |
2.30 2.02 2.03 | 2.07 2.02 1.98 | 2.15 2.04 2.03 |
We thank Prof. C. Pisani and Prof. R. Dovesi for CRYSTAL 95 code. This work was supported by Russian Foundation of Basic Research (grants 96-03-33796-a and 96-03-33991).