## 1 Introduction  
Currently, Bi-based oxides have received considerable attention due to their particular physical properties and wide applications in different fields like multiferroics [1; 2], superconductivity [3; 4], and photocatalysis [5; 6]. Generally, Bi exists as the trivalent state (Bi\({}^{3+}\)) in most of the Bi-based oxides like Bi\({}_{2}\)O\({}_{3}\)[7], BiVO\({}_{4}\)[8], Bi\({}_{2}\)WO\({}_{6}\)[9], Bi\({}_{2}\)Sn\({}_{2}\)O\({}_{7}\)[10], BiFeO\({}_{3}\)[11], and BiMnO\({}_{3}\)[12]. However, some Bi-containing oxides with the unusual pentavalent state (Bi\({}^{5+}\)) have also attracted research interest. For example, NaBiO\({}_{3}\) has been found to show the absorption of visible light and can be used as a prominent material for photooxidation of organics [13]. A recent work by [PERSON] et al. [14] shows that AgBiO\({}_{3}\) can self-produce significant amounts of reactive oxygen species without light illumination or any other additional oxidant and has an excellent oxidizing reactivity. BaBiO\({}_{3}\), as one kind of Bi-based oxides containing Bi\({}^{3+}\) and Bi\({}^{5+}\) mixed valent states, has been found to show the potential use for the absorber of all-oxide photovoltaics [15] and can be an active photocatalyst under visible-light irradiation [16].  
MgBi\({}_{2}\)O\({}_{6}\) adopting the trirutile-type structure is also a Bi\({}^{5+}\)-containing compound. [PERSON] et al. [17] first successfully prepared MgBi\({}_{2}\)O\({}_{6}\) by the low-temperature hydrothermal method and characterized its crystal structure in detail. In 2003, [PERSON] et al. [18] investigated the optical and electrical properties of MgBi\({}_{2}\)O\({}_{6}\) and found that MgBi\({}_{2}\)O\({}_{6}\) is a degenerate n-type semiconductor with the band gap of about 1.8 eV, and it is possible to produce an optical band gap that extends into the visible region of the spectrum by tuning the Bi\({}^{5+}\) 6s-O 2p interaction in order to produce a disperse conduction band. The band structure makes MgBi\({}_{2}\)O\({}_{6}\) a good candidate of visible light-sensitive photocatalysts for decomposition of organic species. Comparing with other pentavalent bismuthates such as LiBiO\({}_{3}\), NaBiO\({}_{3}\), KBiO\({}_{3}\), ZnBi\({}_{2}\)O\({}_{6}\), SrBi\({}_{2}\)O6, AgBiO\({}_{3}\), BaBi\({}_{2}\)O\({}_{6}\), and PbBi\({}_{2}\)O\({}_{6}\), MgBi\({}_{2}\)O\({}_{6}\) does not possess the highest photocatalytic activity, but there is no adsorption observed in the decomposition of methylene blue [19]. It may be a good idea to adjust the wide band gap of some traditional photocatalysts such TiO\({}_{2}\) by building complex compounds with MgBi2O\({}_{6}\). In our recent work, the photocatalytic activity of MgBi\({}_{2}\)O\({}_{6}\) has been found to be significantly enhanced via constructing AgBr/MgBi\({}_{2}\)O\({}_{6}\) heterostructured composites [20]. Theoretically, the band gap of MgBi\({}_{2}\)O\({}_{6}\) was calculated to be about 1.10 eV using the Heyd-Scuseria-Ernzerhof (HSE) functional method within the framework of the density functional theory (DFT), and was found to be widely tuned by applying external strain [21]. On the basis of theoretical calculations, however, the deeper understanding of the physical properties of MgBi\({}_{2}\)O\({}_{6}\) is still in lacking. In this work, we investigated the electronic structure, optical, mechanical, and lattice dynamical properties MgBi\({}_{2}\)O\({}_{6}\) using first-principles calculations.