Rubies multiplet structures

Separation of the effects of charge transfer, covalency and electron correlations on the multiplet structure of ruby based on first-principles cluster calculations... 

In our previous report, however, the calculated multiplet energies tend to be overestimated especially for the doublets. This is due to the underestimation of the effect of electron correlations. Recently, we have developed a simple method to take into account the remaining effect of electron correlations. In this method, the electron-electron repulsion integrals are multiplied by a certain reduction factor (correlation correction factor), c, and the value of c is determined by the consistency between the spin-unrestricted one-electron calculations and the multiplet calculations. The details of this method will be described in another paper (5). In the present paper, the effect of electron correlations on the multiplet structure of ruby is investigated by the comparison between the results with and without the correlation corrections. 

Although the ligand field theory is based on the several critical approximations, a first-principles calculation based on the ligand field theory can also provide a useful information when the results are compared to those of the DV-ME calculations. For example, a comparison between the calculations based on the LFT using the pme atomic orbitals (AOs) and the DV-ME calculations using the molecular orbitals (MOs) provide a clear separation of the effect of covalency. Therefore, in the present work, we also carried out the calculation of the multiplet structure of ruby based on the LFT. In this approach, the parameters representing the electron-electron repulsion are calculated using the pure 3d atomic orbitals of the... 

However, as we have already pointed out, the value of Sgg — t2g is quite insensitive to the electronic configuration (2). Therefore, in the present paper, we approximated all of these effective crystal-field splittings by the value of eg — t2g Calculated in the ground state. On the other hand, the Racah parameters represent the electron-electron repulsion interaction and can be calculated by the radial part of the pure 3d atomic orbitals of the impurity chromium ion (1). After evaluating the value of these parameters, the multiplet structure of ruby can be obtained by diagonalizing the Tanabe-Sugemo matrices. 

Fig 1. The cluster model adopted for the first-principles calculation of the multiplet structure of ruby. The small black sphere at the center of the cluster represents the impurity chromium ion. Small gray spheres and lEtrge gray spheres represent aluminum ions emd oxygen ions, respectively. 

The multiplet structures of ruV calculated by the DV-LFT method using the point charge model and the cluster model are shown in Fig. 3. For comparison, the peak positions of the experimental absorption spectra of ruby obtained by Fairbank et al. (13) are shown together. Here each state is labeled according to the notation in the octahedral symmetry for simplicity. 

The multiplet structures of ruby calculated by the DV-ME method with and without the correlation correction are shown in Fig. 4. The experimental values are also shown together. In this case, the multiplet structures are calculated directly using the molecular orbitals of the impurity states obtained by the cluster calculation. In the calculated results, each level is broadened by the presence of the trigonal crystal field. Although the split of each peak seems to be somewhat overestimated due to the computational errors, it can be improved by increasing the number of sampling points (2). 

Fig 4. The multiplet structures of ruby calculated by the DV-ME method using the (CrAli4048) cluster, together with the peak positions in the absorption spectra of ruby reported by Fairbank et al. (13). In each result, the doublets are shown at left and quartets are at right. 

As we know, a few first-principles calculations for multiplet structure have been tried by several researchers. Ohnishi and Sugano calculated the energy positions of the (R line) and Ti (U band) states in ruby, under one-electron approximation (12). Xia et al. carried out similar calculations using more realistic model cluster (13). They could, however, only consider the energies of lower-lying two states in multiplet structure. Watanabe and Kamimura combined one-electron calculations with ligand field theory, and carried out first-principles calculation for the "full" multiplet structure of several transition metal impurities... 

We recently developed a general method, to directly calculate the electronic stracture in many-electron system DV-ME (Discrete Variational MultiElectron) method. The first apphcation of this method has been reported by Ogasawara et al. in ruby crystal (17). They clarified the effects of covalency and trigonal distortion of impurity-state wave functions on the multiplet structure. 

The effect of intrinsic trigonal distortion on the multiplet structures of ruby and emerald... 

In solid-state laser materials, such as ruby (chromium doped alumina, AljOjiCr " ) (1) and emerald (chromium doped beryl, Be,Al,(Si03)5 Cr ) (2), transitions between multiplets of impurity states are utilized. These states mainly consist of 3d orbitals of the impurity chromium ions. For the analysis of these multiplet structures, the semi-empirical ligand-field theory (LFT) has been frequently used (3). However, this theory can be applied only to the high symmetry systems such as O, (or T ). Therefore, the effect of low symmetry is always ignored in the analysis based on the LFT, although most of the practical solid-state laser materials actually possess more or less distorted local structures. For example, in ruby and emerald, the impurity chromium ions are substituted for the aluminum ions in the host crystals and the site symmetry of the aluminum ions are C, in alumina and D, in beryl. Therefore, it is important to clarify the effect of low symmetry on the multiplet structure, in order to understand the electronic structure of ruby and emerald. 

Effect of Intrinsic Trigonal Distortion on the Multiplet Structures of Ruby and Emerald 99... 

Chromium doped alumina, or ruby, is needless to say, a beautiful gemstone and known as the first solid-state laser in history. It is also a material of central importance to high pressure science since the ruby pressure gauge using its fluorescence lines is particularly popular in the diamond anvil cell (DAC) experiments. The electronic structure of ruby has been studied extensively based on the ligand-field theory with some additional parameters such as the trigonal-field parameter or the spin-orbit interaction parameter . However, the reports on the first-principles calculation of the multiplet structure of ruby are rather limited . The electronic structure of a-A]2 03 V + has also been studied in details based on a similar semiem-pirical approach . ... 

Recently, a first-principles calculation of the entire multiplet structure of ruby has been carried out by Duan et al and the pressure dependence of the multiplet structure of ruby has been well reproduced. They predicted an anomalous local relaxation which could explain the observed frequency shifts. However, their calculation was based on the analytic multiplet approach using the atomic Racah parameters and the matrix elements were calculated in the octahedral approximation. Although the effect of the covalency was taken into account by multiplying the orbital deformation parameters on the electron-electron repulsion integrals, these parameters were adjusted to the optical spectra of ruby under zero pressure for the quantitative analysis of the pressure dependence of the multiplet structure. Moreover, it would be difficult for their approach to predict the intensity of the optical spectra, since the optical spectra of ruby are dominated by the electric-dipole transitions arising... 

Ishii T, Ogasava K, Tanaka A, Adachi H (1999) Theoretical calculation for multiplet structure of chromium irai parr in ruby. Mafia Trans JIM 40 416-419... 

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