First-principles calculations as a tool for structure validation in electron crystallography

The crystal structures of Ti₁₁Se₄ [Weirich, Ramlau, Simon, Hovmöller & Zou (1996). Nature (London), 382, 144–146] and Ti₄₅Se₁₆ [Weirich (2001). Acta Cryst. A57, 183–191] determined previously from selected-area electron diffraction (SAED) data have been checked for their correctness by means of total energy calculations within the non-local density functional theory. The reliability of the used method was verified by test calculations carried out for the structurally related compound Ti₈Se₃, which is well known from single-crystal X-ray diffraction [Weirich, Pöttgen & Simon (1996). Z. Kristallogr. 212, 929–930]. For Ti₈Se₃, structural models from both experiment and calculation show a perfect match (average agreement 0.01 Å). This proves that the geometrical optimized models from first-principles calculation can be used as a reliable alternative when good-quality X-ray results cannot be obtained. Calculations carried out for the two structures determined from electron crystallography yielded average improvement of the atomic coordinates of 0.04 and 0.09 Å for Ti₁₁Se₄ and Ti₄₅Se₁₆, respectively. The optimized cell parameters of the monoclinic structures (both space group C2m, No. 12) are a = 25.51, b = 3.43, c = 19.19 Å, β = 117.9° for Ti₁₁Se₄ and a = 36.31, b = 3.45, c = 16.59 Å, β = 92.1° for Ti₄₅Se₁₆. These results prove that crystals that are too small for single-crystal X-ray diffraction and are difficult to solve by powder diffraction may nevertheless be amenable to accurate structure determination by electron diffraction structure analysis using data from standard SAED and the assumption of quasi-kinematical scattering. Moreover, this study shows that geometry optimization by first-principles calculations is the perfect tool for validation and improvement of complex structural models, which are suspected to have errors owing to the poor quality of experimental data.