Electron surfaces. In the bottom panels of Fig. 4, the squarish electron surface (with k$_{z}$ variation and resulting holes, shown in the lower two panels of Fig. 4) that cuts the aforementioned hole sur- face is pictured, and substantiates the discussion provided just above. In addition, there are simple electron ellipsoids centered along the Γ-M lines. It is curious that in a band structure that is for the most part strongly 2D, all the FSs have a rather definite three-dimensional character. Although the bands show little dispersion along Γ-Z, the bands just above the the Fermi level are quite different de- pending on whether k$_{z}$ = 0 or k$_{z}$ = π . In particular, the lowest band along R-A is rather flat, but the lowest conduction band along X-M has a dispersion of nearly 2 eV. Similar comparisons can be made for the bands along Γ-X and Z-R. The k$_{z}$ dispersion is not nearly as strong near k$_{x}$ = k$_{y}$ = 0, which is clear from both the band structure and the FS. Short discussion. It was noted in the Introduc- tion that the NaAlSi structure is the same as the Fe-pnictide “111” structure. Moreover, in both com- pounds, the relevant bands involve only the (Si-Al-Si or As-Fe-As) trilayer. The top view of the fan-blade surfaces have characteristics in common with those of some of the Fe pnictides, 27,28 all of which have this same trilayer. The similarity is that the top view of the fan blades (if one ignores the diamond-shaped cutout at the intersection, centered at Γ) appears to show intersecting FSs, neither of which has the square symmetry of the lattice. Such occurrence of intersecting FSs, each with lower symmetry than the lattice, has been analyzed for LaFeAsO (a “1111” compound) by Yaresko et al. 29 A symmetry of the Fe$_{2}$As$_{2}$ (also Al$_{2}$Si$_{2}$) sub- structure is a non-primitive translation connecting Fe atoms (respectively, Al atoms) followed by z - reflection. This operation leads to symmetries that allow k$_{z}$ = 0 bands to be unfolded into a larger Brillouin zone (that is, a “smaller unit cell” having only one Fe atom) which un folds the band structure and the intersecting FSs. The NaAlSi FSs appear to have this similar crossing (albeit interrupted by the free-electron bands), and the highly anisotropic dispersion is due to distinct (but symmetry-related) hopping along each of the crystal axes. In this re- spect NaAlSi may clarify the electronic structure of the pnictides: by analogy, there are separate bands that disperse more strongly along the (1 , 1) direc- tion or the (1 , − 1) direction, and give rise to the intersecting, symmetry-related surfaces. In NaAlSi the bands are much more anisotropic in the plane (approaching one-dimensional), making such char- acter much clearer. A difference that complicates the analogy is that in the pnictides the bands near ε$_{F}$ are derived from the Fe atoms, which comprise the center layer of the trilayer, whereas in NaAlSi the bands under discussion are derived from the Si atoms, which comprise the two outer layers. Pictured in Fig. 5 are symmetry-projected Wan- nier functions (WFs) projected onto Si 3 p orbitals. The extension of the WFs shows considerable in- volvement from nearby Al and Si atoms, and in ad- dition have some density extending into the Na lay- ers. The p$_{x}$ WF consists of an atomic p$_{x}$ function, with its density shifted downward by the bonding contribution of Al sp 3 hybrid orbitals. Beyond the p$_{x}$ lobes the nearest Si atoms form a bonding lobe that connects to the “small” side of the Al sp 3 func- tion. The large p$_{x}$ lobes and the extra contribution from nearby Si atoms are responsible for the largest hopping amplitudes shown in Table I, although there is some phase cancellation between the p$_{x}$ lobe and the lobe lying beyond the nodal surface. The p$_{z}$ WF has one lobe extended well into the Na layer; this is responsible for the largest hoppings along b ∗ in Table I, and they create the large disper- sion in the p$_{z}$ bands seen in Fig. 5(c). Again, the Al atoms contribute with an sp hybrid orbital, although it appears to be more sp $^{2}$-like than sp $^{3}$-like. There is also a “ring” structure below the Al layer, where an sp hybrid orbital from the Si atoms forms a bonding combination, but it is antibonding with the p$_{z}$ func- FIG. 4: (Color online.) Views from the xy -plane (left) and top (right) of the FSs of NaAlSi, centered at Γ. The blue (dark) surfaces enclose holes and the yellow (light) surfaces enclose electrons. 5 D. Wannier functions