Calculated difference of the electron densities between the ground state and the state with in the equilibrium structure in the ground state of benzene molecule, which can be compared with Fig. 4(d) in Ref. 27. The dark and light spheres indicate carbon and hydrogen atoms, respectively. The contours are on the plane including two opposite C–H bonds of benzene and perpendicular to the molecular plane. The solid and dotted lines correspond to an increase and decrease, respectively, in the electron density with the contour values of , , and electrons. This is consistent with the result by the SAC-CI calculations.
(a) Structure of benzene molecule with labeled carbon atoms. (b) Time evolutions of the energies and the selected bond lengths after the Franck-Condon excitation with the excitation density of 4.8%. The broken line denotes the total energy of the system. The constant feature indicates the numerical stability of the RTP-TDDFT simulation. The potential energy for the ions (the total energy minus the kinetic energy of the ions) is shown by the solid line. The three C–C bond lengths are shown by cross, open and filled circles, respectively. Those oscillations suggest a harmonic oscillation of a single mode.
Fourier spectra of the potential energy trajectory for the ions in the 2.4% (broken line) and 4.8% (solid line) excitations. The horizontal axis is rescaled by the wave number of ionic vibration. (Period of the potential energy oscillation is half of the vibration period.) The height is also rescaled by the maximum value. Note that the difference in the peak width between the two spectra arises from that in the duration used in the Fourier transformation. The widths thus have no physical meaning.
(a) Schematic view of the phase III (monoclinic, space , ) of the crystalline benzene. The dark and light gray spheres indicate C and H ions, respectively. (b) Pressure dependence of the unit cell parameters calculated in our constant-pressure first-principles calculations with STATE (Ref. 32). The open circle, triangle, and square denote the parameters , , and . The lines are their interpolations. The corresponding experimental values are shown by the closed circle, triangle, and square.
Time evolutions of the total and potential energies in the 4.8% average excitation of the crystalline benzene (phase III), calculated with the unit cells at 2.4 and . The plus and cross marks denote the total energy of the system. Their constant features indicate the numerical stability of the RTP-TDDFT simulations. The potential energy trajectory at is shown by open (closed) circles. Compare to the molecule case, the oscillations involve a variety of the vibration as well as phonon modes.
Fourier spectra of the potential energy trajectories shown in Fig. 5. The broken and solid lines denote the results at 2.4 and , respectively. The horizontal axis is rescaled by the wave number of ionic vibration. The height is also rescaled so that the peak heights around match each other. Owing to the sufficient durations of the trajectories, the peak width can be also examined. The peak around appearing at , however, is likely to be an artifact.
Snapshots of the RTP-TDDFT simulation of the excited state of the crystalline benzene in the unit cell at . The dark and light gray spheres indicate C and H ions, respectively. (a) The initial structure without any deformation. The atoms are labeled for later discussion. (b) C–H swing is clearly seen. (c) ring twist appears. (d) The slight rotation of the molecules occurs in association with a lattice phonon. (e) The large deformation with tetrahedral coordination is observed at .
Time evolutions of the dihedral angles of (; cross, ; open circle, ; filled circle) and (; plus, ; open square, ; filled square). See Fig. 7 for the labels of the ions. The results using the unit cell at 2.4 and are shown in (a) and (b), respectively. In the bonding, the C–C–C–C and C–C–C–H should be 0° and 180°, respectively. In the tetrahedral configuration, and 180° are the ideal angles for both.
Time evolutions of the Kohn-Sham orbital energies. The energy origin is set to the midpoint of the gap at the initial state. The results using the unit cell at 2.4 and are shown in (a) and (b), respectively. The white up arrows show the orbitals from which electrons are equally removed, while the black downward arrows denote the orbitals to which electrons are equally added, both at the initial state.
Calculated vibration frequencies (in ) of the representative modes of the benzene molecule with experimental results (Refs. 28 and 29). The symmetry of each mode is shown in the parentheses. The calculated value in the excited state is obtained by the Fourier transformation of the potential energy trajectory of the 2.4% average excitation, corresponding to the one-electron excitation. The values in the ground state are calculated by the harmonic approximation along the normal coordinates.
Article metrics loading...
Full text loading...