Translation-Libration-Screw (TLS) representation was initially presented by Verner Schomaker and Ken Trueblood in a seminal 1968 Acta Cryst. paper (DOI: 10.1107/S0567740868001718) as a mechanism for understanding the motion of rigid bodies from their ADPs. They expanded the representation developed by D. W. J. Cruickshank who used two tensors to describe group motion, a six-term tensor for anisotropic translational motion, that is labeled “T” (for translations) and another a six-term tensor for anisotropic rotational motion, labeled “L” for libration. Schomaker and Trueblood added a third eight-term “S” tensor (named for screw motion, which combines translations and librations.) This is now known as the TLS model. Note that for a body that has an origin placed at the center of the rotary motion, the S terms are all zero, greatly simplfying the full TLS representation from 20 terms to 12. For an isolated body, the center of rotation will be the center of mass, but in a crystal, bonds and Van der Waals interactions make the center of rotation less clear.
Initially, crystallographers fit individual anisotropic ADP values for each atom in the group and then fit TLS terms to the ADP values to extract information about body’s motions. Now when TLS fitting is used, the model is invoked directly and ADP values are generated from the T, L and S terms.
Note that the effect of the T matrix terms, if asymmetrical, will be to have different motion in different crystallographic directions, but motion of each atom due to the contribution of T will be the same. In contrast, the L matrix terms will have greater impact the farther an atom is from the origin of the body. S terms have both effects. In GSAS-II, when atoms in a rigid body are set as isotropic, the ADP will not show the anisotropy generated by the T, L and S terms, as the \(\rm U_{iso}\) value will show an average for all directions, but if the rigid body atoms are set to be anisotropic, the anisotropy will be clear from the \(\rm U_{\it ij}\) values. In powder diffraction, one seldom has sufficient data to refine as many as six anisotropic parameters for an atom. However, with rigid bodies, since these values are being generated from the TLS terms, there is no disadvantage to using anisotropic parameters for atoms in a rigid body.
Fitting TLS terms. With powder data, I never try to fit all 20 TLS terms. I will assume that my origin choice is close enough to the center of rotation to leave the S terms as all zero. Even the remaining 12 terms may be too many to refine reliably. I may refine only the diagonal T and L terms (for a total of 6 terms) and have been known to even constrain the diagonal terms to all be the same, so that I have only 2 terms to refine for the entire group, but that still allows me to have a feel if rotation or translation is the primary form of group motion.