Frequency calculations, hydrogen-atom

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

Jaquet and Miller [1985] have studied the transfer of hydrogen atom between neighbouring equilibrium positions on the (100) face of W by using a model two-dimensional chemosorption PES [McGreery and Wolken 1975]. In that calculation, performed for fairly high temperatures (T> rj the flux-flux formalism along with the vibrationally adiabatic approximation (section 3.6) were used. It has been noted that the increase of the coupling to the lattice vibrations and decrease of the frequency of the latter increase the transition probability. 

The mirror plane is defined by the dummy atoms. The migrating hydrogen H le is not allowed to move out of the plane of symmetry, and must consequently have the same distance to C4 and C5. A minimization will locate the lowest energy structure within the given Cs symmetry, and a subsequent frequency calculation will reveal that the optimized structure is a TS, with the imaginary frequency belonging to the a" representation (breaking the symmetry). 

Using Bohr s model, one could calculate the energy difference between orbits of an electron in a hydrogen atom with Planck s equation. In the example of a system with only two possible orbits, the equation of the emitted radiation as the electron went from a higher energy state 2 to a lower one j would be - E = hf, where h is Planck s constant and/is the frequency of the emitted radiation. 

Calculate the frequency, wavelength, and wave number for the series limit of the Balmer series of the hydrogen-atom spectral lines. 

Hydrogen atoms on Cu( 111) can bind in two distinct threefold sites, the fee sites and hep sites. Use DFT calculations to calculate the classical energy difference between these two sites. Then calculate the vibrational frequencies of H in each site by assuming that the normal modes of the adsorbed H atom. How does the energy difference between the sites change once zero-point energies are included ... 

The second theoretical development is due to Sokolov (p. 385). This writer attempts to calculate the additional energy of the H-bond in terms which appear to correspond to our effects (A), (B) and (C). The effective parameter is the positive formal charge Z on the hydrogen atom. Plausible variations of the various terms with Z and with the bond distances enable the change of frequency v in the 0—H valence vibration to be estimated, in good agreement with experiment. 

The frequencies of the spectral lines emitted by a hydrogen atom when it undergoes transition from one stationary state to a lower stationary state can be calculated by the Bohr frequency rule, with use of this expression for the energy values of the stationary state. It is seen, for example, that the frequencies for the lines corresponding to the transitions indicated by arrows in Figure 2-2, corresponding to transitions from states with n 3, 4, 5, to the state with n = 2,... 

The F and G matrices may now be combined to give two two-dimensional equations, one for the A, modes and one for the E modes. To illustrate what these modes actually look like in a real case, they are depicted for ND3 in Figure 10.10. These drawings are based on calculations (cf. Wilson, Decius, and Cross for details) from experimentally observed frequencies. The molecule ND3 is used rather than NH3 because the inverse mass dependence of the amplitudes would make the vectors on the nitrogen atom of NH3 im-practically small compared to those on the hydrogen atoms. 

The connection of the 36 hydrogen atoms to the C60 cage lowers the molecular symmetry and activates Raman scattering from a variety of initially forbidden phonon modes (Bini et al. 1998). In addition, the appearance of the C-H stretching and bending modes and those related to various isomers of C6)0n%, results in a very rich Raman spectrum. The comparison of the phonon frequencies for live principal isomers of C60I f 6, obtained by molecular dynamics calculations, with experimentally observed phonon frequencies has led to the conclusion that the material prepared by the transfer hydrogenation method contains mainly two isomers, those with symmetries DM and S6 (Bini et al. 1998). 

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