Least-squares estimates of equivalent isotropic temperature factors: an application to H atoms in crystalline sucrose

An anisotropic harmonic oscillator has mean-square amplitudes of vibration U₁, U₂ and U₃. The geometric mean, U_gm, is (U₁ U₂ U₃)^1/3 and the arithmetic mean U_am, is (U₁ + U₂ + U₃)/3. It is shown that estimates of U_iso which minimize the least-square error of structure factors is bounded by the inequalities U_gm ≤ U_N ≤ U_x ≤ U_am where U_N is the equivalent isotropic U for the neutron diffraction experiment and U_x is the corresponding estimate of U_iso for the X-ray diffraction experiment. The results have been applied to H atoms in sucrose where the U₁, U₂ and U₃ are taken from the neutron diffraction analysis of Brown & Levy [Acta Cryst. (1973). B29, 790-797]. It is found that U_x ≃ 0.75U_am + 0.25U_gm. When U_x is compared to U_iso (X-ray) from Hanson, Sieker & Jensen [Acta Cryst. (1973). B29, 797-808], U_x's for the fourteen H atoms bonded to C atoms exceed U_iso (X-ray) by ~ 0.022 Ų (or equivalently ~ 1.7 Ų in B units, where B = 8π²U). A systematic trend for the eight hydroxyl hydrogens is not found; the average difference U_x - U_iso (X-ray) is - 0.006 Ų. The discrepancy for H bonded to C is consistent with quantum-chemical calculations of small molecules; the apparent agreement of U_x and U_iso (X-ray) for H bonded to O may reflect the influence of hydrogen bonding in crystalline sucrose.