In recent years, there has been a growing interest in replacing metallic parts in heat engines by structural ceramics such as zirconium dioxide (ZrO₂). However, the high-temperature form of ZrO₂ with the fluorite structure and high oxide ion mobility is unstable at room temperature. One approach to stabilise the fluorite structure of ZrO₂ is aliovalent substitution of Zr⁴⁺ by Y³⁺ or Ca²⁺. As well as stabilising the fluorite structure, this introduces anion vacancies enhancing the oxide ion conductivity. A second approach is to make an anion substitution of O^2- by N^3- to make a series of zirconium oxynitrides which will have anion vacancies.

The band structures, density of states and optical properties of zirconium dioxide (ZrO₂) in the cubic, the tetragonal and the monoclinic phases and the fuorite-related zirconium oxynitride (Zr₂ON₂) have been calculated using the density functional theory with the generalized gradient approximation (GGA) parameterized with the revised Perdew-Burke-Ernzerh for solids (PBESOL) as an exchange correlation function. Band gaps of 3.30, 4.01, 3.97 and 1.55 eV are obtained respectively for these phases. The underestimation of band gaps by DFT is common, which exists due to the limitation of predicting conduction band properties. As a result, the conduction band minimum of Zr₂ON₂ is reduced compred to ZrO₂, which makes Zr₂ON₂ active under visible light irradiation.

For the ZrO₂ structures, the valence bands and the lower conduction bands are all quite similar but become more fiat in going from the cubic to the monoclinic phase. The valence bands are composed with an higher mixed O 2p, Zr 4d orbitals of about 4 -6 eV width. The conduction band is mainly derived from the Zr 4d orbitals. For the Zr₂ON₂ fluorite structure, the valence band maximum mostly consists of O 2p and N 2p states which are strongly hybridized with Zr 4d orbitals_, while the bottom of the conduction band is composed of a Zr 4d state overlap with O 2p and N 2p. Strong hybridization means potential covalent bonding and a less ionic character. A covalent bond between O or N and the Zr atom can be seen when the O 2p and N 2p states hybridized with Zr 4d between -5.5 eV and 0.0 eV.

The optical properties of ZrO₂ and Zr₂ON₂ are calculated by the dielectric function ε(ω) = ε₁(ω)+Iε₂(ω). The Kramer–Kronig relation is used to obtain the real parts of the dielectric function from imaginary ones. Optical properties such as reflectivity R(ω), energy loss function L(ω), refractive index n(ω) and extinction coefficient k(ω) are calculated from the dielectric constant.

The average value of the refractive index of Zr₂ON₂ is 1.7 at the static limit, n^average(0), and 2.0 at 650 nm. Our calculations show better agreement with the experimental value, 2.2 at 650 nm, than another theoretical value. The refractive indices increased beyond the zero frequency limits and reached their maximum values. Beyond the maximum value they start to decrease and with a few oscillations they go beyond unity

The energy loss function and extinction coefficient were also calculated. The peaks in the energy loss function represent a reduction in the trailing edges of the reflectivity spectrum.