Results

The simultaneous optimization of the volume and the c/a ratio for the paraelectric phase of LiNbO₃ resulted in the values of lattice parameters (in the hexagonal setting) a_H=5.1378 Å and c=13.4987 Å. As compared to the experimental data (a_H=5.1483 Å and c=13.8631 Å, see Ref. (1)), this corresponds to the volume underestimated by $\sim3\%$ and the c/a ratio deviating by $\sim$2% from experiment, i.e. quite good agreement by the standards of first-principles calculations based on the density functional theory. In the subsequent optimization of atomic positions, we kept the lattice parameters fixed. The fully optimized paraelectric structure was found energetically instable with respect to the symmetry-lowering atom displacements. The paraelectric phase has one internal coordinate (other atomic positions are fixed by symmetry) whereas the ferroelectric phase has four. The experimental and calculated atomic positions (in the hexagonal coordinates, following (1)) for both phases are given in Table I.

Table I. Atomic positions in hexagonal coordinates (e: experiment (1); t: theory)

phase	Nb			Li			O
para	0	0	0	0	0	1/4	$\begin{array}{c}0.049^e\\ *[-0.25cm]0.041^t\end{array}$	1/3	1/12
ferro	0	0	0	0	0	$\begin{array}{c}0.283^e\\ *[-0.25cm]0.279^c\end{array}$	$\begin{array}{c}0.049^e\\ *[-0.25cm]0.041^t\end{array}$	$\begin{array}{c}0.346^e\\ *[-0.25cm]0.344^t\end{array}$	$\begin{array}{c}0.067^e\\ *[-0.25cm]0.066^t\end{array}$

The $\Gamma$-TO frequencies in the ferroelectric structure are splitted by symmetry into four A₁, five A₂ and nine E modes. Calculated (in the harmonic approximation) and measured (Ref. (11)) frequencies of A₁ modes are listed in Table II. The deviation from the experimental frequencies is large for the softest and the hardest modes. The softest mode correspond primarily to the displacement of Li (and also - to a smaller extend - Nb) along the polar axis, against the relatively rigid oxygen sublattice. This kind of displacement is also typical for perovskite-type ferroelectrics. The hardest mode is essentially related to the torsion deformation of rigid oxygen octahedra. The substantial anharmonicity of these vibrations may be the reason of the difference between the calculated and experimental frequencies.

TABLE II
Experimental and calculated frequencies (in cm^-1) for the A₁ zone-center TO modes.

exp.	250	275	332	632
calc.	163	273	347	588