Structure and dynamics of the symmetric hydrogen ... - ResearchGate

396( )F. Fillaux et al.rChemical Physics 244 1999 387–403Ž . Ž . Ž Ž ..Fig. 8. Potential function left and wave functions right for the n OHO mode see Table 6, Eq. 1 .aŽ Table 6 ., in agreement with previous observationsin the infrared w19,21 x. This is an example wherefrequency ratios close to unity are not due to adouble-well potential.Intuitively, the naOHO spectrum can be picturedas a single transition at 665 cm y1 , due to the centralminimum, and a doublet at 500–530 cm y1 , due tothe double minimum. This view is in qualitativeagreement with the shapes of the wave functionsŽ Fig. 8 .: C1 and C2are similar to wave functionsfor a double-well potential with minima located at;"0.8 A, ˚ except that the antisymmetric state isnow below the symmetric state. C3is largely delocalizedover all three wells. However, in the rangespanned by C0it is similar to the ns1 state in aharmonic potential.In the ground state, the proton is fairly welllocalised at the center of the hydrogen bond, inagreement with diffraction data. The shape of thewave function resembles that of a harmonic oscillatorat frequency ;670 cm y1 , close to the 03transition. It is almost impossible to determine theanharmonicity of the potential function from measurementsof the probability distribution of protonsat low temperature Žfor example, neutron diffraction,neutron Compton scattering, etc. . The calculatedmean-square amplitude Ž 0.025 A˚2. compares wellwith the temperature factor determined by neutrondiffraction at low temperature Ž 0.016 A˚2in Table 4 ..A major consequence of the potential shape is theŽy1large zero-point energy ;600 cm .. In the harmonicapproximation, the mean square amplitudeshould be ;0.016 A ˚2. However, the very largeanharmonicity spreads the wave function across theregion normally forbidden within classical mechanics.6. Maleate-ring dynamicsThe out-of-plane bending of the maleate ring canbe represented as an anti-rotation of the CHCOOentities around an axis parallel to the C5C bond andgoing through the center of gravity of each residueŽ see Fig. 1a .. This axis is very close to the OŽ . 2˚2atoms and the moment of inertia is about 44 amu Afor each entity. The most intense bands at 153, 303,605 cm y1 and some weaker bands above 800 cm y1Ž Table 5.are not amenable to either harmonic orvery flat Ž square-like.potentials. A symmetric doubleminimum potential is also unlikely and should beincompatible with the crystal structure. Followingthe same line of reasoning as for the naOHO bands,the energy level scheme can be fitted with a tripleminimum potential function Ž Fig. 9 and Table 7 ..INS relative intensities calculated assuming the Ž C.Hprotons ride the HCOO entities are in agreementwith the observations Ž see Table 7 ..