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large optical anisotropy in Na-hp4. 

Since the optical properties of Na-hp4 are anisotropic, it is particularly interesting to calculate the effective number of valence electrons per Na atom n$_{eff}$ contribut- ing to the optical properties in each direction. This can be accomplished by means of the sum rule 22 : 

where E$_{m}$ denotes the upper limit of integration, m is the electron mass, and N stands for the electron density. The effective number of valence electrons per Na atom participating in the interband transitions along the crys- tallographic directions at 320 GPa is shown in Fig. 7 as a function of$_{E}$$_{m}$. One can see that$_{n}$$_{eff}$ remains lager be- fore 13 eV and smaller after 13 eV in E || z direction than in E || x ( y ) direction. This is connected with the larger reflectivity feature observed in E || z direction at low en- ergy range than in E || x ( y ) direction. In addition, the n$_{eff}$ doesn’t reach a saturation value up to 30 eV. This shows that deep-lying valence orbitals do participate in the interband transition. 

## IV. CONCLUSIONS 

In summary, we have performed systematic study of the electronic and optical properties of the transpar- 

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ent insulating phase Na-hp4 using the first-principles method. The insulating mechanism that exclusionary ef- fect induces the valence electrons being excluded from the core region is clearly demonstrated from the calculated 

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band structure, angular moment decomposed density of states and valence charge density. The dielectric func- tions and all main optical spectra indicate that Na-hp4 is optically anisotropic. The macroscopic dielectric con- stants ε ∞ have been obtained and are expected to be useful for future reference. Especially, our calculated ab- sorption spectrum in the energy range from 1.4 to 2.4 eV agrees well with the experimental observation. Fi- nally, we have shown that during the interband transi- tion, deep-lying valence electrons of Na-hp4 also partici- pate. 

## Acknowledgments 

We thank Yanming Ma for dicussions. This work was supported by NSFC under Grant No. 60776063, No. 10734140 and No. 10674021, the National Basic Re- search Program of China (973 Program) under Grant No. 2009CB929103 and No. 2005CB724500. 

