In the first step of our calculations, we have carried out the equilibrium lattice constants of Gd₂O₃, and Tb₂O₃ by minimizing the ratio of the total energy of the crystal to its volume using the experimental data [13] [15] . We have compared the present results for lattice parameters of X₂O₃ with previous experimental values [13] - [29] and are given in Table 1. These results are within the accuracy range of calculations based on density functional theory.

The investigation of electronic band structure for understanding the electronic and optical properties of X₂O₃ is very useful. The band structures of the X₂O₃ were calculated using GGA. The electronic band structures were calculated along the special lines connecting the high-symmetry points Г, H, N, and P for X₂O₃ in the k-space. The electronic band structure of Gd₂O₃, and Tb₂O₃ along the high symmetry directions have been calculated by using the equilibrium lattice constants and are given in Figure 1 and Figure 2.

As can be seen in Figure 1, the Gd₂O₃ compound has a direct band gap semiconductor with the value 3.86 eV (in Г-high symmetry point). The band gap with the value 3.82 eV of Tb₂O₃ compound has the same character of that of Gd₂O₃ (Figure 2). The band gap values obtained for X₂O₃ are good agreement with the earlier theoretical resuts, but is less than the estimated experimental results [1] [3] [5] . In these figures (Figure 1 and Figure 2), the lowest valance bands that occur between 0 and −3.5 eV (72 energy states) are dominated by O 2p states while the valence bands that occur between −14 eV and −16.5 eV (24 energy states) are dominated by Gd 6s and Tb 6s states. The lowest occupied valance bands are essentially dominated by O 2s (−19 eV and −21.5 eV and include 48 energy states).

It is well known that the effect of the electric field vector, , of the incoming light is to polarize the material. At the level of a linear response, this polarization can be calculated using the following relation [29] :

where is the linear optical susceptibility tensor and it is given by [30]

where denote energy bands, is the Fermi occupation factor, is the normalization volume. are the frequency differences, is the energy of band n at wave vector. The are the matrix elements of the position operator [30] .

As can be seen in Equation (2), the dielectric function and the imaginary part of, is given by

The real part of, , can be obtained by using the Kramers-Kroning transformation [30] . Be-

cause the Kohn-Sham equations determine the ground state properties, the unoccupied conduction bands as calculated, have no physical significance.

The known sum rules [30] can be used to determine some quantitative parameters, particularly the effective number of the valence electrons per unit cell, as well as the effective optical dielectric constant, which make a contribution to the optical constants of a crystal at the energy. One can obtain an estimate of the distribution of oscillator strengths for both intraband and interband transitions by computing the defined according to

where is the density of atoms in a crystal, e and m are the charge and mass of the electron, respectively, and is the effective number of electrons contributing to optical transitions below an energy of.

Further information on the role of the core and semi-core bands may be obtained by computing the contribution that the various bands make to the static dielectric constant,. According to the Kramers-Kronig relations, one has

One can therefore define an “effective” dielectric constant, that represents a different mean of the interband transitions from that represented by the sum rule, Equation (5), according to the relation

The physical meaning of is quite clear: is the effective optical dielectric constant governed by the

interband transitions in the energy range from zero to, i.e. by the polarization of the electron shells.

We first calculated the real and imaginary parts of the linear dielectric function of the Gd₂O₃, and Tb₂O₃ compounds (Figure 3 and Figure 4). In order to calculate the optical response by using the calculated band structure, we have chosen a photon energy range of 0 - 65 eV and have seen that a 0 - 40 eV photon energy range is sufficient for most optical functions. We first calculated the real and imaginary parts of linear dielectric function of the Gd₂O₃ and Tb₂O₃ compounds (Figure 4 and Figure 5). All the Gd₂O₃ and Tb₂O₃ compounds

studied so far have ε₁ are equal to zero in the energy region between 9 eV and 40 eV for decreasing and increasing of (see, Table 2). Also, values of versus photon energy have main peaks in

the energy region 4 eV and 30 eV. Some of the principal features and singularities of the for both investigated compounds are shown in Table 3.

The peaks of the correspond to the optical transitions from the valence band to the conduction band and are in agreement with the previous results. The maximum peak values of for Gd₂O₃ and Tb₂O₃ are around 9.31 eV and 9.49 eV, respectively.

The corresponding energy-loss functions, , were calculated using Equation (7) and are also presented in Figure 3 and Figure 4. The describes the energy loss of fast electrons traversing the material. The sharp maxima in the energy loss function are associated with the existence of plasma oscillations [30] . The curves of L have a maximum near 35 eV (Gd₂O₃) and 38 eV (Tb₂O₃).

The calculated effective number of valence electrons is given in Figure 5(a). The effective number of valence electron per unit cell, up to 5 eV is zero (below the band gap) then reaches saturation values at

about 30 eV (Gd₂O₃) and 35 eV (Tb₂O₃). This means that deep-lying valence orbitals participate in the interband transitiond as well (see Figure 1 and Figure 2). The effective optical dielectric constant, , is shown in Figure 5(b).

The curves of can be arbitrarily divided into two parts. The first part is characterized by a rapid growth of and extends up to 12 eV. The second part shows a smoother and slower growth of and reaches a saturation values at about 30 eV(Gd₂O₃) and 35 eV (Tb₂O₃). This means that the largest contribution to is made by transitions corresponding to the bands at ~5 eV and ~12 eV.