NMR in Hydrate Crystals: Correction for Vibrational Motion

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Abstract
A theoretical analysis of the effect of vibrational motion of the water molecule on the line splitting observed in protonmagnetic resonancespectra of hydrates is presented. For an assemblage of magnetically equivalent water molecules, the observed line splitting is given by the Pake formula properly averaged over the motion, 〈(3μ/R 3) (3 cos2θ—1)〉. The intramolecular stretching and deformation vibrations are assumed to be uncoupled from the rigid body librational vibrations. The intramolecular vibrations affect the (3μ/R 3) term only and are given as 0.980 times the value at equilibrium from an anharmonic model of the vibrations. The angular factor is averaged over the librational oscillations assumed to be of small amplitude and harmonic. The resulting expression is a function of the mean‐square amplitude of the rocking, 〈θ x 2〉, and the twisting, 〈θ z 2〉, oscillations of the H2O molecule. From a simple model of the librational motion the relative magnitude of 〈θ x 2〉 and 〈θ z 2〉 is calculated, and an expression is developed for 〈θ z 2〉 which only contains the potential barrier for rotation of the H2O molecule and the temperature. From the developed formulas it is estimated that, on the average, the uncorrected intramolecular proton—proton distance is about 5% larger than the equilibrium value. This is mainly due to the zero point motion of the H2O molecule; the line splitting is expected to decrease about 5% going from 0°K to room temperature. The theory presented has been used in interpreting splitting data obtained from Ba(ClO3)2·H2O rotated about the baxis. The set of line splittings are shown not to follow the simple Pake formula, but the data can be satisfactorily interpreted on the basis of the developed formulas with the numerical values of 〈θ x 2〉 and 〈θ z 2〉 available in the literature. A reinterpretation of the experimental results by McGrath and Silvidi is given at the end of the paper. The equilibrium intramolecular proton—proton distances in the lattice‐bound H2O molecules are derived from literature data in three hydrates and ice. The values obtained are close to the equilibrium distance found in the isolated H2O molecule, 1.51 Å.