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The optical conductivity then follows immediately from the imaginary part of the dielectric constants, 

## III. RESULTS AND DISCUSSION 

## A. Atomic and electronic structures of Na-hp4 

The crystal structure of sodium in hp4 phase have been studied experimentally using x-ray diffraction and reflection 8 . The insulating phase considered here is a highly distorted double-hexagonal close-packed structure (D_2h space symmetry). It belongs to the space group of P 6$_{3}$ /mmc and contains four atoms per cell. There are two inequivalent atomic positions in unit cell, Na1 located at the 2 a site (0 . 0 , 0 . 0 , 0 . 0) and Na2 at 2 d site ( 2 / 3 , 1 / 3 , 1 / 4 $_{)}$. All the crystal parameters are optimized. Atoms in this dense structure are six-coordinated. Note that the stacking of close-packed layers of Na atoms is CACBCACB. . . as in any d.h.c.p structure. Top view along the c-axis within 2 a × 2 b plane of Na-hp4 is show in the Fig. 1(b), in which A, B, and C denote the atoms occupying the A, B, and C layers in the Fig. 1(a). 

Electronic structure is first calculated since the op- tical spectra are calculated from interband transition. It should be noted that theoretical predicted transition pressure of Na-hp4 is about 260 GPa, which is higher than experimental reported value 200 GPa 8 . The dis- crepancy may result from the well-known overstabiliza- tion of metallic states by DFT calculations. For this rea- son, we calculate the electronic and optical properties of Na-hp4 at 320 GPa, at which Na-hp4 exists stably. The calculated band structure of hp4 phase at 320 GPa in the high-symmetry directions in the Brillouin zone are shown in Fig. 2. The energy scale is in eV, and the valence band maximum is set to be the Fermi energy, in- dicating that the bands below E$_{f}$ are all occupied. From 

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this figure it is clear that a direct band gap appears be- tween the top-most valence band and the bottom-most conduction band at $_{Γ}$. Our calculated value of the direct band gap is about 1.75 eV, which should be smaller than the true value due to the discontinuity of the exchange- correlation potential, while consistent with other theo- retical results 8 . The bands have large dispersion. There are only two valence bands in the valence region shown in Fig. 2, the bottom one mainly arising from 3 s orbitals and the above one from the hybridization between 3 p and 3 d orbitals. These two valence bands are degenerate along A − L − H − A directions. 

Our calculated total density of states (DOS) along with site-projected and angular momentum decomposed DOS of hp4 phase are shown in Fig. 3. The valence maximum is set to be zero on the energy scale. The most intriguing feature of the DOS is the emergence of 3 d state. Un- der the compression of 320 GPa, the overall density is increased to about six times as large as the atmospheric density of Na. The nearest Na-Na distance is 1.878 Å, while the 3 s and 2 p orbital radii are 1.71 and 0.28 Å, ionic radius of Na + is 1.02 Å. That means strong valence- core and core-core overlap exist between the neighboring atoms. Due to the exclusionary effects, the Bloch states with the angular momentum components of 3 s and 3 p are largely excluded from the core region by the core states with the same angular momentum components. However, the more asymmetric states which have lower kinetic energy is apt to sampling the full nuclear poten- tial. Thus 3 d states drops below the Fermi energy and hybridize with 3 p states, which induce strong electron localization as seen in valence charge density shown in Fig. 4. Figure 3(b) indicates that the DOS for p$_{x}$ + p$_{y}$ is different from that for $_{p$_{z}$}$, which predicts the anisotropy of Na-hp4. As shown in Fig. 3(c), the DOS peaks for Na1 and Na2 are largely different, which implies evident charge transfer from Na1 to Na2 and thus weak ionicity 

