PHYSICAL REVIEW B VOLUME 47, NUMBER 18 1 MAY 1993-II Theoretical analysis of the structures of titanium dioxide crystals Adil Fahmi and Christian Minot* Laboratoire de Chimie Organique Theorique, Uniuersite Pierre et Marie Curie, Bohte 53, B&timent F 642, 4 Place Jussieu, 75252 Paris CEDEX 05, France Bernard Silvi Laboratoire de Dynamique des Interactions Moleculaires, Universite Pierre et Marie Curie, Tour 22 Place Jussieu, 75252 Paris CEDEX 05, France Mauro Causa. Dipartamento di Chimica Inorganica, Chimica Fisica e Chimica dei Materiali, Universita di Torino, Uia Pietro Giuria 5, I-10125 Torino, Italy (Received 4 May 1992; revised manuscript received 24 July 1992) This paper presents results of pseudopotential Hartree-Fock calculations on the titanium dioxide crys- tals. The optimized structures are in good agreement with experimental data. Both anatase and rutile structures have a large ionic character. The distortions from the undistorted crystals made of regular octahedra are discussed and qualitative trends for the distortions are justified by semiempirical calcula- tions (extended Huckel theory). When correlation is taken into account, a large part of the binding ener- gy is recovered and the rutile structure is found to be more stable than the anatase one. I. INTRODUCTION Titanium dioxide has been the subject of considerable interest. Many experimental' ' and theoretical' studies have been performed. Using extended Hiickel tight-binding calculations (EHT), Burdett studied the electronic structure of anatase and rutile. ' The most stable crystal structures are rutile (P4z/mmm space group) and anatase (I4, /amd space group). Rutile struc- ture is more stable than anatase structure by 1.2 — 2. 8 kcal mol ' ~ ' This paper presents a qualitative analysis of both structures and explains the importance of the distortions relative to regular crystal lattices, made of undistorted octahedra. It shows that the main difference between the stability of the two phases is the angular dis- tortion. Anatase is less dense and also found to be less stable. Next, this paper presents results of pseudopoten- tial periodic Hartree-Fock calculations on the anatase structure and compares them with results on the rutile structure already calculated by one of us (Table I). II. THE TiO2 STRUCTURES Formally, the Ti02 oxides are made of d titanium ions (+IV) at the center of octahedra of six 0 ions. Each oxygen atom has three titanium neighbors and therefore belongs to three different octahedra. The two structures differ by the distortion inside each octahedron and by their assemblage. The rutile structure is defined by three crystallographic parameters, the two lattice parameters a, c, and the oxy- gen fractional coordinate u. Alternatively, three parame- ters can be chosen, as shown in Fig. 1: the Ti-O bond lengths, dip and deq and one angle. The local symmetry is D2&. There are two apical and four equatorial oxygen TABLE I. Enthalpies and entropies of TiO2 (Ref. 24). 298. 15 K (25 C) AHf (kcal mol ') AGf (kcalmol ') S' (cal/deg mol) Rutile 225. 8 212. 6 12. 03 Anatase 224. 6 211. 4 11. 93 atoms. The OTi3 pattern has a planar Y' shape. Let us call 28 the smallest Ti-0-Ti angle (the smallest 0-Ti-0 angle which concerns two equatorial Ti-0 bonds is a=sr 28) T— he ne. ighboring octahedra share opposite edges in the equatorial plane. Extending this pattern plane leads to a linear chain. Conversion from our pa- rameters to the crystallographic ones are the following: a =&2(d, +d, cos8)=4. 593 A c 2d qsin8 = 2. 959 A d 0— =0. 305 . 2(d, +d, cos8) The cell volume (2 Ti02 units) is V=a c =4(d, +d, cos8) d, sin8 . In the undistorted structure, the 0 angle is 45 and dip deq so that locally the building block around the ti- tanium atom is a regular octahedron. Similarly, for the anatase structure, we can define three parameters (Fig. 2): the two Ti-0 bond lengths and the angle 28 (the largest Ti-0-Ti angle; the smallest 0-Ti-0 angle which concerns one apical and one equatorial Ti-0 bond is a=8). The local symmetry is D2d. There are two apical and four equatorial oxygen atoms. The OTi3 11 717 1993 The American Physical Society