  The structural, mechanical, electronic, optical and thermodynamic properties
of the t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ (X $=$ Si, Ge and Sn) with
tetragonal structure have been investigated by first principles calculations.
Our calculated results show that these compounds are mechanically and
dynamically stable. By the study of elastic anisotropy, it is found that the
anisotropic of the t-Sn$_{\mathrm{3}}$As$_{\mathrm{4}}$ is stronger than that
of t-Si$_{\mathrm{3}}$As$_{\mathrm{4}}$ and
t-Ge$_{\mathrm{3}}$As$_{\mathrm{4}}$. The band structures and density of states
show that the t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ (Si, Ge and Sn) are
semiconductors with narrow band gaps. Based on the analyses of electron density
difference, in t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ As atoms get electrons, X
atoms lose electrons. The calculated static dielectric constants,
$\varepsilon_{1} (0)$, are 15.5, 20.0 and 15.1 eV for
t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ (X $=$ Si, Ge and Sn), respectively. The
Dulong-Petit limit of t-X$_{\mathrm{3}}$As$_{\mathrm{4}}$ is about 10 J
mol$^{\mathrm{-1}}$K$^{\mathrm{-1}}$. The thermodynamic stability successively
decreases from t-Si$_{\mathrm{3}}$As$_{\mathrm{4}}$ to
t-Ge$_{\mathrm{3}}$As$_{\mathrm{4}}$ to t-Sn$_{\mathrm{3}}$As$_{\mathrm{4}}$.
