Charge Density and Hydrogen-Bond Energies

The correction for stretching anharmonicity can be made in a similar way using the differences in the mean square amplitudes parallel to the direction of the X-H bond using the leading term in the correction for a Morse-curve oscillation [205-207] 

More complex mathematical treatment is necessary when the thermal motion is very large, as for hydrogen atoms in a room-temperature neutron structure analysis, or when it is curvilinear as in a hindered-rotor. The Uy second-rank tensor does not adequately describe the nuclear or electron-scattering density when the motion is far from harmonic or when it deviates from the familiar ellipsoidal probability form. Tb deal with such examples, more complex mathematic expressions using Gram Charlier or Edgeworth expansions are available [210]. 

However, these refinements are made at a heavy cost in the observation-to-parameter ratio. The third-order terms, Cyk, of the Gram Charlier expansion add ten more parameters per atom to the nine Uy terms. These expressions are therefore only used when the experimental data are of exceptionally high quality, as in the neutron diffraction analysis of ice, Ih, discussed in Part IV, Chapter 21. They may also be necessary in experimental deformation density analysis, where a very precise description of the atomic thermal motion is required. 

Deformation density and electrostatic potential measurements are used to study the electronic structure associated with hydrogen bonding. These methods seek to 

Electrostatic Properties of Molecules Derived from Fourier Summations of Crystal Structure Factors, Fh [211] 

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