The problem of numerical accuracy in the calculation of vibrational frequencies of crystalline compounds from the hessian matrix is discussed with reference to alpha-quartz (SiO(2)) as a case study and to the specific implementation in the CRYSTAL code. The Hessian matrix is obtained by numerical differentiation of the analytical gradient of the energy with respect to the atomic positions. The process of calculating vibrational frequencies involves two steps: the determination of the equilibrium geometry, and the calculation of the frequencies themselves. The parameters controlling the truncation of the Coulomb and exchange series in Hartree-Fock, the quality of the grid used for the numerical integration of the Exchange-correlation potential in Density Functional Theory, the SCF convergence criteria, the parameters controlling the convergence of the optimization process as well as those controlling the accuracy of the numerical calculation of the Hessian matrix can influence the obtained vibrational frequencies to some extent. The effect of all these parameters is discussed and documented. It is concluded that with relatively economical computational conditions the uncertainty related to these parameters is smaller than 2-4 cm(-1). In the case of the Local Density Approximation scheme, comparison is possible with recent calculations performed with a Density Functional Perturbation Theory method and a plane-wave basis set.

We have performed plane wave density functional theory calculations of atomic and molecular interstitial defects and oxygen vacancies in monoclinic hafnia (HfO 2 ). The atomic structures of singly and doubly positively charged oxygen vacancies, and singly and doubly negatively charged interstitial oxygen atoms and molecules are investigated. We also consider hafnium vacancies, substitutional zirconium, and an oxygen vacancy paired with substitutional zirconium in hafnia. Our results predict that atomic oxygen incorporation is energetically favored over molecular incorporation, and that charged defect species are more stable than neutral species when electrons are available from the hafnia conduction band. The calculated positions of defect levels with respect to the bottom of the silicon conduction band demonstrate that interstitial oxygen atoms and molecules and positively charged oxygen vacancies can trap electrons from silicon.

We performed plane wave density functional theory ͑DFT͒ calculations of formation energies, relaxed structures, and electrical levels of oxygen vacancies and interstitial oxygen atoms in monoclinic zirconia. The atomic structures of positively and negatively charged vacancies and interstitial oxygen atoms are also investigated. The ionization energies and electron affinities of interstitial oxygen atoms and oxygen vacancies in different charge states are calculated with respect to the bottom of the zirconia conduction band. Using the experimental band offset values at the interface of ZrO 2 films grown on silicon, we have found the positions of defect levels with respect to the bottom of silicon conduction band. The results demonstrate that interstitial oxygen atoms and positively charged oxygen vacancies can trap electrons from the bottom of the zirconia conduction band and from silicon. Neutral oxygen vacancy serves as a shallow hole trap for electrons injected from the silicon valence band. The calculations predict negative U for the O Ϫ center and stability of V ϩ centers with respect to disproportionation into V 2ϩ and V 0 in monoclinic zirconia.

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