_{2}

^{1}

^{1}

^{1}

First principles computations of second-order elastic constants (SOECs) and bulk moduli (

Titanium dioxide (

The

There has been no systematic effort to explore the elastic properties of rutile

The HF, local and non-local DFT potentials have consistently resulted in inadequate results of the lattice parameters. First principles computations using PW codes [

Therefore, we employed hybrid potentials in our computations of elastic properties as suggested in [

In addition to using hybrid potentials, the proper choice of basis sets, SCF tolerances, ELASTCON, and EOS parameters can achieve the optimum efficiency as well as accuracy (see Sections

We selected the O-8411d1 [

The sensitivity of the elastic properties with respect to deviations in the lattice parameters [

Further, we have also compared our computational results for each potential and basis set combination with [

We employed two different basis sets and a variety of potentials to compute the optimized lattice constants and elastic properties (see Section

The two basis sets employed are named as basis set 1 and 2 (see Section

Optimized lattice parameters, elastic constants, and bulk moduli, obtained with basis sets 1 and 2, are also compared with the experimental values where possible. A possible contribution of this research work is to assist a general reader in understanding the complex dependance of the elastic properties on the quality of basis sets, potentials, SCF process, ELASTCON, and EOS parameters. Experimental values of lattice parameters, elastic constants, and bulk moduli of rutile

A significant number of computations and experiments have been conducted on rutile

A vast majority of

The computation of elastic constants and bulk moduli is an automated procedure within the ELASTCON algorithm. The computational process begins with determining the crystalline symmetry of rutile

The elastic constants can be extracted from the second derivative of the total energy as

The appropriate number of strains is applied in a systematic manner, the elastic constants are calculated, and the compliance coefficients are computed from (

The terms

The EOS algorithm [

The EOS algorithm in CRYSTAL09 contains a diversity of equations of state such as Birch Murnaghan, third-order Birch Murnaghan, logarithmic, Vinet, and polynomial. The third-order Birch Murnaghan equation of state algorithm utilized for computing the bulk moduli from the energy versus volume computations for rutile

The bulk moduli results are obtained with Levenberg-Marquardt curve fitting of the

Due to its technological importance, a significant number of experiments have been conducted on rutile

The experimental and computational values of elastic constants and bulk moduli [

However, there are deviations in the experimental values of elastic constants and bulk moduli due to their dependence on the experimental details, pressure, and temperature conditions [

An increase in pressure has shown an increase in the elastic constants and bulk moduli values [

Another important area where experiments were performed is the volume charge density and chemical bonding of rutile

The charge density influenced by highly localized

Unlike the precision in the experimental lattice parameters, the computational values of lattice parameters vary in the second significant figures. The variation in the lattice parameter values is partially due to the complex nature of the chemical bonding of rutile

In general, the

For the computation of lattice parameters and elastic properties, the SCF tolerances and other computational parameters were carefully chosen. The ELASTCON, EOS, and SCF tolerances were adjusted due to the highly localized nature of transition metal

Table

The values of relaxed lattice constants (in Å), ambient volume (in

HF | 4.568 (−.54) | 2.980 (.74) | 62.11 (0) |

4.561 (−.70) | 2.991 (1.10) | 62.24 (0) | |

LDA | 4.559 (.74) | 2.932 (.87) | 60.98 |

4.539 (−1.18) | 2.904 (−1.85) | 59.84 | |

PWGGA | 4.640 | 2.976 (.33) | 64.08 |

4.619 (.56) | 2.946 (−.40) | 62.82 | |

PBE | 4.647 (1.17) | 2.978 (.67) | 64.32 |

4.625 (.69) | 2.949 (−.30) | 63.05 (1.03) | |

BLYP | 4.66 (1.45) | 3.01 (1.75) | 65.67 (5) |

4.657 (1.37) | 2.971 (.43) | 64.41 (3.12) | |

B3LYP | 4.629 (.78) | 2.976 (.60) | 63.78 |

4.607 (.21) | 2.957 (−.03) | 62.75 (.55) | |

B3PW | 4.599 (.13) | 2.961 (.10) | 62.63 (0) |

4.583 (−.21) | 2.942 (-.54) | 61.81 (.95) | |

PBE0 | 4.627 (.73) | 2.973 (.50) | 63.69 |

4.571 (−.48) | 2.940 (−.61) | 61.46 (1.52) | |

Exps. [ | 4.593 | 2.958 | 62.40 |

The rutile

Elastic constants and bulk moduli computed with ELASTCON are presented in Tables

The elastic constants and bulk modulus computational results using the Hartree Fock and DFT LDA, PWGGA, BLYP, B3LYP, and B3PW potentials with basis set 1. All values are in GPa. The numbers within parentheses are percent differences from experiment. (See Sections

HF | 364. (36) | 216. (23) | 184. (25) | 626. (29) | 164. (32) | 277. (456) | 270. (27,17) |

LDA | 311. (16) | 211. (21) | 175. | 504. (28) | 159. (33) | 253. (15) | 243. (15,6) |

PWGGA | 266. (−1) | 177. | 149. | 462. (−5) | 138. (11) | 223. (17) | 208. (−2,−10) |

PBE | 262. (−2) | 176. | 146. (−1) | 458. (−6) | 135. (9) | 221. (16) | 205. (−3,−11) |

BLYP | 256. (−5) | 133. (−24) | 143. | 478. (−1) | 154. (−19) | 206. (8) | 188. (−11,−18) |

B3LYP | 281. (5) | 186. (6) | 157. (7) | 506. (5) | 140. (13) | 236. (24) | 220. (4,−4) |

B3PW | 294. (10) | 194. (11) | 164. (12) | 517. (7) | 147. (19) | 246. (29) | 229. (8,−1) |

PBE0 | 277. | 185. (6) | 155. (5) | 494. | 139. (12) | 234. (23) | 217. (2,−6) |

Exps. [ | 268.00 | 175.00 | 147.00 | 484.00 | 124.00 | 190.00 | 212.00, 230.00 |

The elastic constants and bulk modulus computational results using the Hartree Fock and DFT LDA, PWGGA, BLYP, B3LYP, and B3PW potentials with basis set 2. All values are in GPA. The numbers within parentheses are percent differences from experiment. (See Sections

HF | 393. (47) | 237. (35) | 210. (43) | 662. (37) | 171. (38) | 303. (60) | 295. (39,28) |

LDA | 171. (−36) | 385. (120) | 205. (39) | 577. (19) | 130. (5) | 279. (47) | 266. (23,16) |

PWGGA | 223. (−17) | 273. (56) | 182. (24) | 522. (8) | 123. (−1) | 246. (29) | 237. (12,3) |

PBE | 220. (−18) | 270. (54) | 181. (23) | 517. (7) | 122. (−2) | 244. (28) | 235. (11,2) |

BLYP | 260. (−3) | 226. (29) | 175. (19) | 509. (5) | 123. (−1) | 236. (24) | 231. (9,1) |

B3LYP | 295. (10) | 240. (37) | 193. (31) | 562. (16) | 137. (10) | 261. (37) | 255. (20,11) |

B3PW | 269. (0) | 269. (54) | 197. (34) | 569. (18) | 137. (10) | 267. (41) | 257. (21,12) |

PBE0 | 267. (−1) | 282. (61) | 203. (38) | 579. (20) | 139. (12) | 273. (44) | 263. (24,14) |

Exps. [ | 268.00 | 175.00 | 147.00 | 484.00 | 124.00 | 190.00 | 212.00, 230.00 |

The computational and experimental values of elastic constants and bulk moduli are shown. All values are in GPa.

The computational and experimental values of bulk moduli are shown. All values are in GPa.

Table

Equation of state results for rutile TiO_{2} with the Birch Murnaghan third-order equation. The energy-volume curve was fitted with eleven points, and the range of volume around equilibrium was chosen as ±10% using basis sets 1 and 2 (computational results with basis 2 shown in even-numbered rows). (See Sections

HF | 267. | 62.29 | 270. |

293. | 62.31 | 295. | |

LDA | 241. | 60.98 | 243. |

279. | 59.84 | 283. | |

PWGGA | 206. | 64.09 | 208. |

239. | 62.87 | 237. | |

PBE | 202. | 64.33 | 205. |

236. | 63.10 | 235. | |

BLYP | 191. | 65.93 | 188. |

229. | 64.45 | 231. | |

B3LYP | 217. | 63.81 | 220. |

252. | 62.81 | 255. | |

B3PW | 226. | 62.66 | 229. |

256. | 61.83 | 257. | |

PBE0 | 232.36 | 62.24 | 217.12 |

263. | 61.49 | 263. | |

Exps.[ | — | — | 212.00, 230.00 |

However, there is a considerable disagreement between the computational and experimental values of the elastic constants and

The hybrid potentials have shown better agreement due to the adequate percentage of exchange and correlation contributions to total energy of the crystal specifically important for the highly correlated physics of the

The HF, local DFT, and nonlocal DFT potentials cannot predict results as effectively as hybrid potentials. It can be easily seen in Tables

It must be mentioned that DFT-PWGGA and DFT-PBE exhibit better agreement with experimental results due to the nonlocalized nature of the rutile

The technological applications of titanium dioxide (

We have separated the factors that determine the quality of computational results of the lattice parameters and elastic properties. The nonlocal DFT and hybrid potentials present better agreement with the experimental values of lattice parameters, elastic constants, and bulk moduli. However, the disagreement between the computational and experimental values of the elastic constants and bulk moduli for HF and DFT-LDA potentials [

The results presented for rutile

One of the authors (W. F. Perger) gratefully acknowledges the support of the Office of Naval Research Grant N00014-01-1-0802 through the MURI program.

_{2}rutile and anatase polytypes: performances of different exchange-correlation functionals

_{2}thin films by magnetron sputtering: a review

_{3}

_{2}phase explored by the lattice constant and density method

_{2}

_{3}, and lead azide, Pb(N

_{3})

_{2}

_{2}polymorphs

_{ 2}

_{2}

_{2}: a phonon approach

_{2}

_{2}and VO

_{2}

_{3}

_{2})

_{2}

_{2}polymorphs under pressure:

_{2}under pressure

_{2}