The considered by locating ~6400 point charges at external

The structure optimization was performed by sampling the Brillouin zone of the primitive cell on 2 x 2 x 1 Monkhorst-pack grids and the cutoff energy of the plane wave was 600 eV. A supercell consisting of 96 atoms was then constructed by expanding the conventional cell by 2 x 1 x1. One d3 ion, Mn4+ or Cr3+, was substituted into one Ti1 site or one Ti2 site. Next, the Brillouin zone of the supercell was calculated on 2 x 2 x 2 Monkhorst-pack grids and the cutoff energy of the plane wave was 600 eV. The numerical error was estimated to be less than 1 meV/atom by cutoff and k-point convergence tests for all calculations. The geometry optimization was conducted until the residual forces and stresses dropped below 0.01 eV/Å and 0.02 GPa, respectively. The local symmetry of Ti1 and Ti2 sites was maintained in an effort to minimize the computational error. During the calculations, the total charges of Li2TiO3: Mn4+ or Li2TiO3: Cr3+ were set to 0 or -1, respectively. In this step, the effect of spin polarization was also taken into account. Model cluster Two different size of Li2TiO3: d3 ion model clusters; TMO6 (7 atoms) and MLi15Ti3O44 (63 atoms) were prepared, where M represents Mn or Cr. Figure 1b and 1c show those model clusters which are drawn by VESTA 38. Each model cluster was constructed by substituting one Mn4+ or Cr3+ ion into one of the Ti1 or Ti2 site. For example, in the case of Li2TiO3: Mn4+ we used 8 model clusters used in the calculations. For 7-atom model cluster we adopted non-optimized MnO68- (site1), non-optimized MnO68- (site2), optimized MnO68- (site1), and optimized MnO68- (site2) while for 63-atom model cluster we adopted non-optimized Li15MnTi3O4457- (site1), non-optimized Li15MnTi3O4457- (site2), optimized Li15MnTi3O4457- (site1), and optimized Li15MnTi3O4457- (site2). One-electron calculations The one-electron molecular orbital calculations were carried out using the discrete variational X? (DV-X?) method which was developed by Adachi and co-workers 39. Here, the electronic structures of the model clusters were calculated using the numerical atomic orbitals 1s-4p for TM ions and titanium while 1s-2p for oxygen and lithium, as the basis functions. The Madelung potential was considered by locating ~6400 point charges at external atomic sites, i.e., +4 for Ti site, +1 for Li site and -2 for O site. Sample point of 160,000 and 315,000 were used for 7-atom and 63-atom model clusters, respectively. Many-electron calculations After one electron approximation calculations, the configuration interaction calculations were carried out using the DVME method which was developed by Ogasawara and co-workers 40. It is an electronic state calculation which combines the density-functional theory (DFT) 41 and the configuration interaction (CI) calculations. This method has been shown to be powerful in various studies such as optical absorption spectra of 3d ions 42-46 and calculations of the X-ray Absorption Near Edge Structure (XANES) or Electron energy-Loss Near-Edge Structure (ELNES) spectra of TM ions 47-50. In this work, 10C3 = 120 Slater determinants were constructed with the 10 molecular orbitals (MOs) mainly consisting of TM 3d orbitals. The many-electron wave functions, the multiplet energies and the theoretical absorption spectra were obtained by diagonalizing the many-electron Hamiltonian. Since it has been known that the multiplet energies obtained by CI calculations are generally overestimated, several corrections such as configuration-dependent correction (CDC) and correlation correction (CC) were used. The detailed computational methods employed in this study were originally described in Ref. 40. Using the eigenvector obtained by the diagonalization of the many-electron Hamiltonian, the many-electron wave functions corresponding to each multiplet state can be obtained explicitly as a linear combination of the Slater determinants. Therefore, the anisotropic of the oscillator strength for the electric-dipole transition (transition probability) between multiplets can be calculated directly using 5 Iif=2Ef-Ei?fk=1nrk.e?i2(1) where ?i and ?f are the many-electron wave functions of the initial and final states, while Ei and Ef are the energy eigenvalues of these states. r denotes the position of the electron, and e denotes the unit vector parallel to the direction of the electric field of the incident light. Results and Discussion Geometry Optimization We first discuss the results of the optimized structure for Li2TiO3 host crystal performed by CASTEP code. In order to confirm the accuracy, the lattice parameters of the unit cell for pure Li2TiO3 were optimized and compared with the experimental values. The theoretical and experimental lattice constants are listed in Table 1. The lattice parameters of the primitive cell are estimated to be a = 5.1105 Å and c = 9.8503 Å while the experimental values are a = 5.0707 Å and c = 9.7533 Å. On the other hand, the lattice parameters a, b and c of the conventional cell are estimated to be 5.0997, 8.8578, and 9.8503 Å, respectively, whereas the experimental values are 5.0623, 8.7876, and 9.7533 Å, respectively. In both cases of primitive and conventional cells, the difference between theoretical and experimental lattice parameters is about 1%, therefore the optimized structure is considered to be in a good agreement. Next, we estimated the optimized structure around doped Mn4+ and doped Cr3+ ions in Li2TiO3 using CASTEP code. In order to estimate the lattice relaxation rate, we estimated the three different bond lengths thoroughly. The theoretical and experimental bond lengths indicated by d1, d2, and d3 are listed in Table 2. The bond length of the optimized structure around doped Mn4+ and doped Cr3+ in both Ti1 and Ti2 sites are analyzed. Our theoretical estimation shows that almost all TM-O bonds of Li2TiO3: Mn4+ and Li2TiO3: Cr3+ were longer than the original Ti-O bonds. In the case of Li2TiO3: Mn4+, the relaxation rate was found to be ca. 100%, which indicates that the Mn-O and Ti-O bonds are almost the same. On the other hand, the relaxation rate of Li2TiO3: Cr3+ was found to be ca. 102% which indicates that the Cr-O bonds are slightly longer than Ti-O bonds. Furthermore, based on our estimation, the Mn-O bonds have longer bond length than the Cr-O bonds. MO Energy After the optimized structures of Li2TiO3: Mn4+ and Li2TiO3: Cr3+ were obtained, we performed first-principles calculations based on DVME method. The one-electron MO energy levels of the 7-atom and 63-atom model clusters are shown in Fig. 2. The dashed and solid black lines represent the conduction band and the valence band, respectively. The conduction bands mainly consist of Ti 3s, 3p, 3d, and 4p orbitals. On the other hand, the valence bands mainly consist of O 2p and Li 2p orbitals. The uppermost levels of valence bands are set to zero. The red lines represent the impurity levels mainly consisting of TM 3d orbitals. For all cases shown in Fig. 2, the impurity levels decreased from Mn4+ to Cr3+. In this study, the notations of Oh symmetry were used; the five-fold degenerate impurity orbitals split further into the doubly degenerate eg level and the triply degenerate t2g level. The difference between the average energy of the eg level and that of the t2g level is described as 10Dq. When one TM d3 ion was substituted in Ti1 or Ti2 site, the values of 10Dq are almost the same. However, a small but noticeable effect in the crystal field splitting of Li2TiO3: Cr3+ was found due to the lattice relaxation. The 10Dq was estimated to decrease by about 0.13 eV and 0.05 eV for 7-atom and 63-atom model clusters, respectively. Detailed investigation also shows that in the case of 7-atom model clusters, 10Dq decreases from Mn4+ to Cr3+. Unexpectedly, the tendency is reversed when the 63-atom model clusters are used. Since the TM-O bond length decrease from Mn4+ to Cr3+, the crystal field splitting was supposed to decrease from Mn4+ to Cr3+. This inconsistency might be due to the orbital mixing between the impurity Cr 3d level and the conduction band. Absorption spectra The theoretical d–d absorption spectra of Li2TiO3: Mn4+ and Li2TiO3: Cr3+ obtained by the CI calculations using 7-atom model clusters are shown in Fig. 3 and 4, respectively. The ? spectra are shown in blue lines while the ? spectra are shown in red lines. Several computational conditions such as dopant sites at Ti1 or Ti2 sites, consideration of lattice relaxation effect, and consideration of CDC-CC correction are investigated. The right panels show the summation of theoretical spectra of Li2TiO3: d3 doped into Ti1 and Ti2 sites. In the case of Li2TiO3: Mn4+, the experimental spectra reported by Seki’s group 25 is shown on the bottom part. As it might be seen in the theoretical spectra, there are 3 peaks which indicate the transition energies from 4A2 to 4T2, 4T1a, and 4T1b, respectively. On the other hand, in the case of experimental Li2TiO3: Mn4+ spectra, the sharp peak indicates the 2E ? 4A2 transition energy derives from the emission process, while the two broad peaks indicate the 4A2 ? 4T2 and 4A2 ? 4T1a transition energies derive from the excitation process. The peak positions of 4A2 ? 4T2 and 4A2 ? 4T1a are shown in Table 3. However, since neither of the absorption spectra nor the multiplet energies have been reported to our knowledge, the results on the absorption spectra and the multiplet energies of Li2TiO3: Cr3+ shown in this work are our prediction. In the case of theoretical absorption spectra of Li2TiO3: Mn4+, the lattice relaxation increases the peak positions by 0.01-0.07 eV, depending on the computational conditions. But in the case of theoretical absorption spectra of Li2TiO3: Cr3+ does not, the lattice relaxation decreases the peak positions by 0.05-0.16 eV. On the other hand, the CDC-CC correction decreases the peak positions of both Li2TiO3: Mn4+ and Cr3+ which are found to be 0.60-0.85 eV and 0.24-0.49 eV, respectively. In all cases, the results estimated using different computational conditions show that the peak position of Li2TiO3: Mn4+ are higher than those of Li2TiO3: Cr3+. The effect of CDC-CC correction in both cases is stronger than the lattice relaxation effect. The theoretical spectra obtained from Li2TiO3: d3 doped in either Ti1 or Ti2 sites shows the similar behavior. Despite of the different intensity, the peaks owing to ? and ? spectrum appear almost in the same region. The experimental excitation spectra of Li2TiO3: Mn4+ shows that 4A2 ? 4T2 and 4A2 ? 4T1a are located at about 2.48 and 3.54 eV, respectively. Our calculations using CDC-CC correction with or without considering lattice relaxation give good agreement. However, since the consideration of lattice relaxation in the case of Cr3+ is especially important to reproduce the tendency of d3 ions, we suggest a calculation considering both lattice-relaxation effect and CDC-CC correction would be effective to predict both the theoretical absorption spectra and the multiplet energies. Multiplet energies of Li2TiO3: d3 Table 4 shows the calculated multiplet energies of Li2TiO3: Mn4+ and Li2TiO3: Cr3+ using the 7-atom model clusters under different computational conditions together with the CC factor c 40. In the case of Li2TiO3: Mn4+, the lattice relaxation increased the multiplet energies especially the quartet states by 0.04 – 0.05 eV. The corrections so called CDC-CC decreased the energies of all the multiplet energies. The c factor was estimated to be 0.93 under all computational conditions. It means that the separation of the barycenter of (t2g)3, (t2g)2(eg)1, (t2g)1(eg)2, and (eg)3 configurations was reduced by approximately 7%. The results show that the multiplet energies estimated by using the optimized model cluster with considering CDC-CC correction are in good agreement with the experiment. On the calculations of Li2TiO3: Cr3+, a charged supercell was used to deal with the charge difference between Ti4+ and Cr3+. The validity of the geometry optimization was then investigated based on the multiplet energies. The results show that the geometry optimizations decreased the estimated multiplet energies by 0.05 – 0.11 eV. Moreover, the consideration of CDC-CC corrections also decreased the multiplet energies significantly. The c factors were estimated to be 0.86 and 0.88 for calculations using the non-optimized and the optimized model clusters respectively, which implies that the multiplet splitting of Li2TiO3: Cr3+ was reduced more than those of Li2TiO3: Mn4+. The detailed comparison shows that the quartet energies decreased from Mn4+ to Cr3+. This tendency is mainly due to the decreasing crystal field splitting from Mn4+ to Cr3+, as shown in Table 3. This result has been expected based on our previous studies of 3d3-ion-doped compounds 15-17. On the other hand, we found that the doublet energies increased from Mn4+ to Cr3+. However, when the CDC-CC correction was considered, the doublet energies of Mn4+ were slightly higher than those of Cr3+. Similar tendency has also been shown in our previous study of 3d3 ions in MgO 29 which was suggested due the inverse nephelauxetic effect 51. Therefore, our prediction for the multiplet energies of Li2TiO3: Cr3+ is consistent. Nevertheless, as has been shown in the theoretical absorption spectra, the estimated multiplet energies of Li2TiO3: d3 doped in either Ti1 or Ti2 sites are almost the same. Figure 3 shows the multiplet energies of Li2TiO3: Mn4+ and Cr3+ calculated using the optimized 7-atom model cluster including the effect of CDC-CC correction. The experimental data of Li2TiO3: Mn4+ are shown on the left. Coulomb integrals In this work we also calculated Coulomb integrals for the pure AOs (JAO) and for the MOs (JMO), the orbital deformation parameters (?=JMOJAO) 52,53, and the effective Coulomb integral (Jeff=c?JAO), as presented in Table 5. JAO is expressed in terms of Racah parameters 5 as JAO=A+4B+3C. On the other hand, JMO is directly calculated using the MO’s of the impurity states in both t2g and eg states which are defined in atomic units by JMOt2g,t2g=?t2gr121r12?t2gr22dr1dr2,(3) JMOeg,eg=?egr121r12?egr22dr1dr2,(4) ri is the position of the ith electron while rij is the distance between the ith electron and the jth electron. ?t2g and ?eg are the wave function of the t2g and eg states, respectively. The results explain that: (1) unlike JAO, JMO in both t2g and eg levels increased from Mn4+ to Cr3+; (2) the orbital deformation parameters with or without considering the CC factor c (? or c?) increased from Mn4+ to Cr3+; and (3) Jeff in the t2g level decreased while those in the eg level increased. Since the doublet states belong to the (t2g)3 configuration, their energy levels are mainly determined by the electron-electron repulsion in this configuration. In the calculations with considering CDC-CC corrections, the doublet states were found to decrease from Mn4+ to Cr3+ which was consistent with the decreasing tendency of Jeff in the t2g level. Conclusion We have non-empirically calculated the absorption spectra and the multiplet structures of Li2TiO3: d3 ions using DVME method. The detailed investigations on (1) ion dependence i.e., Mn4+ and Cr3+; (2) cluster size dependence i.e., 7- and 63-atom model clusters; (3) dopant site dependence i.e., Ti1 and Ti2 sites; (4) effect of lattice relaxation using CASTEP code; and (5) effect of CDC-CC correction have been performed. The results of the MO calculations show that due to the strong mixing between the impurity levels and the conduction band of Li2TiO3: Cr3+ even after the consideration of lattice-relaxation effect, the many-electron calculations using the 63-atom model clusters were not easy. Therefore, the theoretical absorption spectra and the multiplet structures of Li2TiO3: d3 ions were estimated based on the 7-atom model clusters. On the other hand, the results of geometry optimization using CASTEP code show that the relaxation rates were estimated to be 99.8 – 100.5% for Li2TiO3: Mn4+ and 101.8 – 102.7% for Li2TiO3: Cr3+. The results on the absorption spectra and the multiplet structures show that the calculation considering this lattice-relaxation effect is very important to reproduce reasonable tendency of d3 ions. The behavior of quartet states such as 4T2 and 4T1a can be ascribed to the decrease of crystal field splitting which originates from the increase of TM-O bond length. However, in the case of Mn4+, since the effect of lattice relaxation is negligibly small, the CDC-CC correction plays an important role to improve the agreement with the experiment.