First-Principles Calculation on the Emission Energy Level of Ruby Based on DV-Xα Molecular Orbital Method and Ligand Field Theory
Authors: Novita Mega , Ito Akane , Ogasawara Kazuyoshi 
Published: February 2020

The transition metal 3d3 ions activated in various host crystals have been studied in a great number of experiments and calculations. Their energy levels in strong crystal field namely 4A2, 2E, 2T1, 2T2, 4T2, and 4T1a play important roles in the luminescence process. Especially, the transition energy from 2E state to 4A2 state or R-line has been utilized for the emission process. Since the development of ligand field theory (LFT), the first-principles calculations have been helpful to understand the behavior of R-line. In 1982, Ohnishi and Sugano [1] carried out the semi-empirical first-principles calculation based on the discrete variational Xα (DV-Xα) method [2] to estimate the emission energy of ruby (Al2O3 doped with Cr3+) at various bond lengths under Oh
symmetry. In their estimation, the barycenter of t
2g

3 configuration was an important parameter to estimate the R-line energy. Although the results are reasonable, the barycenter was defined as the average of only 3 multiplet states i.e. 2E, 2T1, and 2T2 states. However, according to LFT, the t
2g

3 configuration consists of 4 multiplet states not only 2E, 2T1, and 2T2 states but also the ground state, 4A2. Therefore, here we proposed a new calculation formula to improve the accuracy of R-line energy by considering 4 multiplet states such as 4A2, 2E, 2T1, and 2T2 states.
Two types of model clusters, non-optimized and optimized model clusters consisting of 63 atoms were used for the calculations. The impurity Cr3+ ion was located at the center of the model clusters. The non-optimized model cluster was constructed based on the experimental crystal structures of α-Al2O3 under pressures with hexagonal A2X3 structure. On the other hand, since the local structure in the actual materials is different from the original ones, the optimized model cluster was constructed to consider the effect of lattice relaxation. It was estimated by performing geometry optimization using the CAmbridge Serial Total Energy package (CASTEP) code [3]. Firstly, the structural optimizations of pure α-Al2O3 crystal under various pressures were carried out and then followed by the geometry optimization of α-Al2O3 crystal doped with Cr3+ ion. The molecular orbitals energies were then estimated by performing one-electron calculations using DV-Xα method for both model clusters. During the calculations, the local C
3 symmetry was maintained to minimize the computational error. Next, the R-line energies were then estimated by two different approaches i.e., (1) based on the approach proposed by Ohnishi’s group and (2) based on LFT.
Figure 1 shows the theoretical R-line energies of ruby under pressures calculated using the optimized model clusters. E

R1 and E

R2 indicate the estimated R-line energies based on the approach proposed by Ohnishi’s group and those based on LFT, respectively. For comparison, the R-line energies of ruby at various Cr-O bond length calculated under Oh
symmetry [1] and the experimental R-line energies of ruby under pressures observed within C3 symmetry [4] are shown together in Fig. 1. Our results show that by both approaches, the R-line energies decrease as the applied pressure increases. Accordingly, simple estimation based on one-electron calculations has successfully reproduced the tendency of ruby’s emission energies. Compare to the experimental values, the R-line energies estimated based on Ohnishi’s approach were significantly underestimated. On the other hand, the calculated R-line energies based on LFT improved the accuracy towards the experiment excellently.
References
[1]. S.Ohnishi and S.Sugano. Jpn. J. Appl. Phys. 21 (1982) L309-L331.
[2]. H. Adachi, M. Tsukada, and C. Satoko. J. Phys. Soc. Jpn. 45 (1978) 875-883.
[3]. M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos. Rev. Mod. Phys. 64 (1992) 1045-1097.
[4]. J. H. Eggert, K. A. Goettel, and I. F. Silvera. Phys. Rev. B 40 (1989) 5724-5732.

Figure 1