For the PCS, the lanthanide ion to nuclear vector is

r = p_{N} - p_{Ln3+}, |
(12.53) |

where
*p*_{N} is the Cartesian coordinates of the nucleus of interest and
*p*_{Ln3+} is the position of the aligning lanthanide ion.
*r* is defined in the alignment frame, and
*p*_{Ln3+} is constant in this frame.
After a forward rotation to the discrete state *i*, the new atomic position in the reference frame is

p_{N}' = R_{i}⋅p_{N} - p_{P} + p_{P}. |
(12.54) |

where
*p*_{P} is the pivot point of the rotation.
Hence the transformed vector is

The set of three vectors are defining this pivoted system are

Let the pre-rotation vectors be

The post-rotation vectors are

The vector
*r*_{PN} is independent of alignment so can be calculated once per atom, and
*r*_{LP} is independent of alignment and atom position so can be calculate once.

For a single state *i*, the PCS value when substituting 12.55b into 12.35 is

Expanding for the single motion of the lanthanide-atom vector
*r*_{LN}^{(1)}, this becomes

Due to the distance normalisation factor in these equations, the symbolic integration for the modelling of specific motional modes is currently intractable.

The relax user manual (PDF), created 2019-06-14.