Prediction of emission energy of Cr3+ in oxides based on first-principles calculations and machine learning

The first-principles calculations of the energies of the emission level (2Eg) of Cr3+ in 10 oxide crystals were performed using the first-principles configuration-interaction calculations. In order to improve the accuracy of the prediction, a machine learning model was created by using the results of the first-principles calculations and the experimental 2Eg energies as the training data. The predicted values using this model showed good correlation with the experimental values, where the correlation coefficient is 0.92. The obtained predicted model indicated that the Cr 3d component of t2g molecular orbital is the most important quantity for prediction of the 2Eg energy.


Introduction
Transition-metal ions with d 3 electronic configuration such as Cr 3+ and Mn 4+ are utilized as emission centers in various luminescent materials such as ruby (Cr 3+ -doped alumina) [1] or alexandrite (Cr 3+doped chrysoberyl) [2]. Since it is difficult to investigate the multiplet energy levels of Cr 3+ in a wide variety of crystals experimentally, the theoretical prediction of the emission energy is indispensable for the efficient development of novel Cr 3+ -doped luminescent materials. In the high field environment, the emission of octahedrally coordinated Cr 3+ occurs from 2 Eg state. Since both this state and the ground state belong to the same (t2g) 3 electronic configuration, the Stokes shift due to the structural relaxation in the excited state is negligibly small. Therefore the 2 Eg energy calculated in the ground state can be regarded as the emission energy. In this study, we predicted the multiplet energy of the emission level ( 2 Eg) of Cr 3+ in oxides by first-principles calculations based on the discrete variational multi-electron (DVME) method [3] using relatively small clusters consisting of seven atoms. In order to improve the accuracy of the prediction, we also performed a machine learning modeling using the Waikato environment for knowledge analysis (WEKA) [4].

First-principles calculation
We performed the first-principles calculations of the multiplet energies of Cr 3+ in oxides using the discrete variational multi-electron (DVME) method [3], which is a configuration-interaction (CI) calculation program based on the discrete-variational X (DV-X) molecular orbital (MO) method. We considered 10 oxide crystals listed in Table 1. In these crystals the impurity Cr 3+ ions occupy the Al sites coordinated with six O 2ions. Therefore, we constructed 7-atom clusters centred at Al 3+ ion based on the crystal structures of the host crystals [6][7][8][9][10][11][12][13][14][15]. Then the model clusters for Cr 3+ in oxides were constructed by substituting Cr 3+ for Al 3+ . The space group, lattice constants, the symmetry of the model clusters are summarized in Table 1. Table 1. Crystal structure data considered in this work and the symmetry of the model cluster.

Machine learning
The machine learning was performed using the software developed in the University of Waikato known as WEKA (Waikato Environment for Knowledge Analysis) [31]. The predictive model of 2 Eg energy was created based on the linear regression algorithm. As the attributes, the theoretical 2 Eg energy obtained by the first-principles calculations, the crystal field splitting 10Dq, the Cr 3d compositions of the t2g or eg MOs were considered.

Results
The correlation between the theoretical 2 Eg energy obtained by the first-principles calculation and the experimental ones is shown in Figure 1. The correlation coefficient is 0.52 and the theoretical values are significantly overestimated. The main cause of the poor correlation is the small size of the model clusters while the main cause of the overestimation is underestimation of the electron correlation due to the finite number of the Slater determinants in the CI calculation. The accuracy of the firstprinciples calculation can be improved by using larger clusters and introducing some corrections such as configuration-dependent correction (CDC) and correlation correction (CC) [3]. However, in this work, we tried to improve the accuracy of the prediction by creating a machine learning model.
where a is the theoretical 2 Eg energy obtained by the first-principles calculation, b and c are the Cr 3d orbital component of t2g and eg MOs, respectively. In this model, 10Dq is not included since consideration of 10Dq did not improve the accuracy of the prediction. The correlation between the predicted 2 Eg energy using Eq (1) and the experimental ones [16] is shown in Figure 2. In this case, the correlation coefficient is 0.92. Therefore, the correlation is significantly improved compared to the simple CI calculations using small clusters.

Figure 2.
Correlation between the predicted 2 Eg energies of Cr 3+ in oxides using the machine learning model and the experimental ones [16].

Discussion
In order to investigate the physical meaning of Eq. (1), we have investigated the correlation between the theoretical 2 Eg energy and the Cr 3d orbital component of the MOs mainly consisting of Cr 3d orbitals. The relationship between the theoretical 2 Eg energy and the Cr 3d orbital component of eg MO is shown in Figure 3. Although the correlation is not good, it should be noted that the data could be classified into three groups, namely, oxides, borates, and molybdates. They clearly appear in different regions in this map. This fact indicates that the Cr 3d orbital component of eg MO is a good quantity to correct the computational errors depending on the group. The relationship between the theoretical 2 Eg energy and the Cr 3d orbital component of t2g MO is also shown in Figure 4. There is a relatively high correlation between these quantities. This is due to the fact that the 2 Eg energy is mainly determined by  3 configuration. In Eq. (1), the coefficient of the Cr 3d orbital component of t2g MO (b) is much larger than that of the theoretical 2 E energy (a), indicating that it is the better quantity to predict the actual 2 Eg energy.

Conclusion
First-principles calculations of 2 Eg energy of Cr 3+ in 10 oxides were calculated based on simple CI calculations using small clusters. The correlation coefficient between the calculated 2 Eg energies and the experimental ones was 0.52, indicating poor correlation. In order to improve the accuracy of the prediction, we created a machine learning model of the 2 Eg energy, using the results of the firstprinciples calculations and the experimental 2 Eg energies as the training data. The correlation coefficient between the predicted 2 Eg energies using this model and the experimental ones is 0.92, indicating significantly improved predictive capability. The obtained predictive model indicated that the Cr 3d component of t2g MO is the most important quantity to predict the actual 2 Eg energy while the Cr 3d component of eg MO is also effective for corrections of computational errors depending on the groups of the host crystals.