Quantum-mechanical analysis of the equation of state of anatase TiO 2 M. Calatayud, 1 P. Mori-Sa ´ nchez, 2 A. Beltra ´ n, 1 A. Martı ´ n Penda ´ s, 2 E. Francisco, 2 J. Andre ´ s, 1 and J. M. Recio 1,2 1 Departament de Cie`ncies Experimentals, Universitat Jaume I, E-12080 Castello´, Spain 2 Departamento de Quı ´mica Fı ´sica y Analı ´tica, Universidad de Oviedo, E-33006 Oviedo, Spain ~Received 30 May 2001; published 23 October 2001! Quantum-mechanical simulations have been performed to investigate pressure effects on the crystal geom- etry, chemical bonding, and the electronic structure of anatase TiO 2 . Total energy calculations are carried out using the density functional formalism under the nonlocal B3LYP approximation. The optimized unit cell equilibrium parameters and the bulk and linear compressibilities are determined to be in good agreement with recent experimental data. The topology of the electron density is examined by means of the atoms in molecules ~AIM! theory. Computed AIM charges and topological properties of the bond critical points reveal a partially ionic behavior of the crystal that complements the description obtained from the band structure and the projected density of states analysis. A microscopic interpretation of the crystal response to hydrostatic pressure is given in terms of the elementary polyhedra and the AIM atomic volumes that fill the unit cell space. DOI: 10.1103/PhysRevB.64.184113 PACS number~s!: 64.30.1t, 61.50.Ah, 71.15.Nc, 71.20.2b I. INTRODUCTION The anatase phase of titanium dioxide raises a great inter- est in a wide range of scientific and industrial fields such as the photochemistry of solar cells, ceramics, catalysis, pig- ments, and optoelectronics. 1–4 Many of these applications rely on the good performance of this material under variable conditions of temperature and hydrostatic pressure, exposure to electromagnetic radiation, and surface reactivity with a variety of chemical agents. The investigation of the funda- mental properties of this polymorph appears to be crucial in order to understand such interesting behavior. Although com- paratively to the rutile structure anatase was claimed to have received less attention, new experimental and theoretical re- sults are now stimulating the study of the structural, elastic, electronic, and optical properties of this polymorph. 5–8 Recently, high-pressure measurements have characterized the rich polymorphism of TiO 2 using single-crystal and poly- crystalline anatase as the starting material. 6 The volume- pressure room-temperature data yield values for anatase bulk modulus ( B 0 517962 GPa in single crystals and 190 610 GPa in polycrystalline samples! in sharp contrast with the other two known experimental data ( B 0 55965 GPa, Ref. 9, and B 0 5360 GPa, Ref. 10!. Previous theoretical val- ues of B 0 ranged from 194 GPa ~Ref. 11! to 272 GPa ~Ref. 12!. The most recent quantum-mechanical calculations 6,7 and atomistic simulations 8 predict B 0 to be around 190 GPa, giv- ing thus support to the experimental equation of state ~EOS! of Arlt et al. The scattering in the reported data demands a comprehensive interpretation of the response of the crystal- line structure to hydrostatic pressure beyond the evaluation of EOS parameters. A detailed decomposition of the crystal compressibility in terms of linear ~lattice parameters and bond lengths! and volumetric ~polyhedral and atomic! con- tributions becomes appropriate to evaluate the proposed EOS. Such an analysis has proved to be useful to explain the pressure behavior of alkali halides and several nitrogen- and oxygen-based spinels. 13–15 Given its predictive capabilities, it is worthwhile to extend this type of analysis to the anatase structure. The electronic structure of anatase is also a subject of debate, as shown by the discrepancies among the theoretical studies reported so far. First-principles pseudopotential 7 ~PS!, all-electron 5 ~AE!, and orthogonalized linear combina- tion of atomic orbitals 12 ~OLCAO! calculations within the local density approximation ~LDA! are qualitatively consis- tent with each other, but disagreement with the PS Hartree- Fock results of Fahmi et al. 16 has been pointed out 5 ‘‘even in the valence bands.’’ Moreover, the band gap is predicted to be indirect according to the PS- and OLCAO-LDA analyses, but direct when computed at the AE-LDA optimized equilib- rium geometry. Asahi et al. 5 concluded that the nature of the band gap is ‘‘quite sensitive to the crystal configuration’’ since they obtained the same kind of indirect absorption edge as Mo and Ching 12 ( M →G ) when the experimental lattice parameters were chosen. Mikami et al. 7 found the minimum band gap between the conduction band at G and the valence band near X ~and not M ) because the high-symmetry direc- tions of the Brillouin zone ~BZ! should correspond to a body-centered tetragonal lattice, and not to a simple tetrago- nal one as in rutile. All these calculations estimated the band gap around 1.4 eV below the experimental value of 3.2 eV, 17 due to the inherent shortcomings associated with the LDA. Analyses of the electronic density of states ~DOS! have been presented by Fahmi et al., 16 Mo and Ching, 12 and Ashami et al. 5 Core and valence DOS have also been investigated from the photoemission spectromicroscopy experiments of Sanjine ´ s et al. 18 and implications related to chemical bond- ing in anatase deduced. It is then desirable to use the correct BZ geometry to- gether with nonlocal exchange-correlation functionals in or- der to obtain reasonable anatase band structures at changing unit cell geometries. The outcome of this exploration should contribute to clarify some of the questions posed in previous works. It is also interesting to complement these results with the less common analysis of the topology of the electron density in the light of the atoms in molecules ~AIM! formalism. 19 Within the AIM framework, a quantum system is rigorously partitioned into subsystems that obey the gen- eral quantum-mechanical rules. In this way it is possible, for PHYSICAL REVIEW B, VOLUME 64, 184113 0163-1829/2001/64~18!/184113~9!/$20.00 ©2001 The American Physical Society 64 184113-1