We calculated the phase transition, elastic constants, full phonon dispersion curves, and thermal properties of molybdenum (Mo) for a wide range of pressures using density functional theory. Mo is stable in the bodycentered-cubic (bcc) structure up to 703 (19 GPa and then transforms to the face-centered close-packed (fcc) structure at zero temperature. Under high temperature and pressure, the fcc phase of Mo is more stable than the bcc phase. The calculated phonon dispersion curves accord excellently with experiments. Under pressure, we captured a large softening along H-P in the TA branches. When the volume is compressed to 7.69 A3, the frequencies along H-P in the TA branches soften to imaginary frequencies, indicating a structural instability. When the pressure increases, the phonon calculations on the fcc Mo predict the stability by promoting the frequencies along Γ to X and Γ to L symmetry lines from imaginary to real. The thermal equation of state was also investigated. From the thermal expansion coefficient and the heat capacity, we found that the quasiharmonic approximation was valid only up to about melting point at zero pressure. However, under pressure, the validity can be extended to a much higher temperature.

We apply a finite-difference pseudopotential density functional theory in real space and the Langevin molecular dynamics annealing technique as well as the adiabatic time-dependent density functional theory within the local density approximation (TDLDA) to the descriptions of structures and optical absorption spectra of carbon clusters C8. It is shown that the odd-numbered clusters have the linear structures and most of the even-numbered clusters prefer cyclic structures. The calculated spectra exhibit a variety of features that can be used for comparison against future experimental investigations.

We report a detailed first-principles calculation to investigate the structures, elastic constants, and phase transition of Ti. The axial ratios of both a-Ti and w-Ti are nearly constant under hydrostatic compression, which confirms the latest experimental results. From the high pressure elastic constants, we find that the a-Ti is unstable when the applied pressures are larger than 24.2 GPa, but the w-Ti is mechanically stable at all range of calculated pressure. The calculated phonon dispersion curves agree well with experiments. Under compression, we captured a large softening around point of w-Ti. When the pressure is raised to 35.9 GPa, the frequencies around the point along-M-K and -A in transverse acoustical branches become imaginary, indicating a structural instability. Within quasiharmonic approximation, we obtained the full phase diagram and accurate thermal equations of state of Ti. The phase transition w-Ti→a-Ti→b-Ti at zero pressure occurs at 146K and 1143K, respectively. The predicted triple point is at 9.78 GPa, 931 K, which is close to the experimental data. Our thermal equations of state confirm the available experimental results and are extended to a wider pressure and temperature range.