When the motion of a multiple rigid body system can be described as two rotations about two different pivots, the modulation of the PCS becomes more complicated. Figure 12.1 shows this motion for a single lanthanide to atom vector.

The six vectors at the original position are
The six vectors after the first rotation for state i, R_{i}^{(1)}, are
The six vectors after the second rotation for state i, R_{i}^{(2)}, are
As defined in equation 12.59 on page , the PCS for state i is
δ = r_{i}^{T}⋅A⋅r_{i}.  (12.64) 
For the double motion of the lanthanideatom vector r_{LN}^{(2)}, this becomes
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