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dielectric constants with good convergence within 0.001 for E || x ( y ) and 0.09 for E || z . The value of ε LR mic can be compared with experimental data. Obviously, ε ∞ is 5.785 and 11.028 in E || x ( y ) and E || z directions, respec- tively. The obtained values can be as important reference for further experimental measurements of Na-hp4. 

Our calculated optical spectra cover the photon energy range of 30 eV. In order to show the details clearly, we use logarithm longitudinal scale for the imaginary part of dielectric constant Im$_{(}$$_{ε}$$_{)}$ and absorption coefficient I ( ω $_{)}$. Im$_{(}$$_{ε}$$_{)}$ shows mainly five peaks both in E || z direc- tion and in E || x ( y ) directions. In E || z direction, these peaks correspond to the photon energy of 2.54, 4.31, 13.37, 17.7 and 24.64 eV, while in E || x ( y ) directions, 4.31, 10.54, 16.34, 20.8 and 28.8 eV, respectively. The peak at 2.54 eV originates from the transition from 3 p to 3 d corresponding to the topmost valence band to bot- tommost conduction band, while the peak at 4.31 eV is attributed to the transition from 3 s to 3 $_{p}$. At 4.31 eV and 13.37 eV in E || z direction and 4.31 eV and 16.34 eV in E || x ( y ) directions, the sign of Re$_{(}$$_{ε}$$_{)}$ changes, which predict the anisotropy of the polarized Re$_{(}$$_{ε}$$_{)}$ spectra. In the range around 13.37 eV and 16.34 eV, Im$_{(}$$_{ε}$$_{)}$is very low and thus the conditions for a plasma resonance are satisfied 21 . Hence, there are strong resonance maxima at these energies in the calculated energy-loss spectra L ( ω ) for these polarizations, as can be seen from Fig. 6(c). The other seven high-energy peaks can all be at- tributed to transitions from the top two valence bands to the upper conduction bands, without deep-lying valence orbitals participating in the interband transitions. These also explain the origin of the peak structure in the ab- sorption coefficient I ( ω ) and reflectivity R ( ω ) spectra. The absorption spectra remains nearly zero in the visible photon energy, which implies that Na-hp4 is transparent to visible light. As seen from Fig. 5(c), Na-hp4 remains transparent to light with higher frequency in E || x ( y ) di- 

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rection than in E || z direction. In Fig. 5(d), we present I ( ω ) in the photon energy range from 1.4 eV to 2.4 eV at 200 GPa along with the previously reported experimental measurement 8 . Here we assume the sample thickness is 8 $_{µ}$m. As all know, the band gap can be deduced by ex- trapolating the absorption to zero. It is concluded that the results from theoretical calculation and experiment agree well except that the theoretically predicted band gap is smaller than the experiment, the origin of which is the local-density approximation. In addition, it should be noted that the present calculation only pertains to the electronic response, and does not include the effects of lattice vibrations that dominate the experimental ab- sorption spectrum in the low frequency region. Thus we can not derive the complete agreement with experimental results. 

In Fig. 6, we show the main optical spectra includ- ing optical conductivity, reflective spectrum, energy-loss spectrum, extinction and refractive indices in a photon energy width of 30 eV. The peak of optical conductivity in z direction is almost twice as large as that in x and y directions. The reflectivity spectrum is distinct from the drude-type spectrum, which means Na-hp4 is no longer metallic. The reflectivity is small at visible optical en- ergy, which ensures the transparency of Na-hp4. During the range about 5 eV to 10 eV, the reflective index re- mains almost constant as 60% in E || z direction and 20% in E || x ( y ) direction. Refractive indices of a crystal are closely related to electronic polarizability of ions and the local field inside the crystal. Another interesting point connected with the refractive indices is the electro-optic effect. We present the refractive indices n ( ω ) in Fig. 6(d). The low-frequency refractive index along E || x ( y ) and E || z direction are calculated as n$_{x}$$_{(}$$_{y}$$_{)}$ = 2 . 38 and n$_{z}$ = 3 . 26 respectively. It is clear that the light polarized parallel to the z axis is more refracted than that with polarization along the x and y axes. This indicates the 

