Temperature-Resolved Anisotropic Displacement Parameters from Theory and Experiment: A Case Study

Abstract

Anisotropic displacement parameters (ADPs) for an organopalladium complex were obtained from synchrotron diffraction data between 100 and 250 K and compared to the results from first-principles calculations at the harmonic approximation. Calculations and experiments agree with respect to the orientation of displacement ellipsoids and hence the directionality of atomic movement, but the harmonic approximation underestimates the amplitudes of motion by about 20%. This systematic but modest underestimation can only be reliably detected with a high-quality experimental benchmark at hand. Our experiments comprised diffraction data at 20 K intervals from 130–250 K on the same crystal. An additional high-resolution data set was collected at 100 K on a second crystal and underlined the robustness of our approach with respect to the individual sample, resolution, and instrumentation. In the temperature range relevant for our study and for many diffraction experiments, the discrepancy between experimentally determined and calculated displacement appears as an almost constant temperature offset. The systematic underestimation of harmonic theory can be accounted for by calculating the ADPs for a temperature 20 K higher than that of the actual diffraction. This entirely empirical “+20 K rule” lacks physical relevance but may pave the way for application in larger systems where a more reliable quasi-harmonic approximation remains computationally demanding or even entirely unaffordable.