Double pivoted motions

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.

Figure 12.1: Frame order in the double pivot system. The lanthanide position is denoted by Ln3+ or simply L, the position of the first pivot by Piv1, the position of the second pivot by Piv2, and the position of the nucleus of interest by 15N. In the vector notation these are L, P1, P2 and N. The original position is denoted by (0), the position after the first rotation by (1), and the position after the second rotation by (2).
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Atomic level mechanics of the double pivot

The six vectors at the original position are

\begin{subequations}\begin{align}& r_{\textrm{LN}}^{(0)} = r_{\textrm{LP}_2}^{(0...
...textrm{P}_1}^{(0)}, \\ & r_{\textrm{LP}_2}^{(0)} . \end{align}\end{subequations}

The six vectors after the first rotation for state i, Ri(1), are

\begin{subequations}\begin{align}& r_{\textrm{LN}}^{(1)} = r_{\textrm{LP}_2}^{(0...
..._{\textrm{LP}_2}^{(1)} = r_{\textrm{LP}_2}^{(0)} . \end{align}\end{subequations}

The six vectors after the second rotation for state i, Ri(2), are

\begin{subequations}\begin{align}& r_{\textrm{LN}}^{(2)} = r_{\textrm{LP}_2}^{(1...
..._{\textrm{LP}_2}^{(2)} = r_{\textrm{LP}_2}^{(1)} . \end{align}\end{subequations}

PCS and double pivoted motions

As defined in equation 12.59 on page [*], the PCS for state i is

δ = $\displaystyle {\frac{{c}}{{\left\vert r_i \right\vert^5}}}$  riTAri. (12.64)

For the double motion of the lanthanide-atom vector rLN(2), this becomes

\begin{subequations}\begin{align}\delta &= \frac{c}{\left\vert r_{\textrm{LN}}^{...
... \cdot r_{\textrm{P}_1 \textrm{N}}^{(0)} \right) . \end{align}\end{subequations}

The relax user manual (PDF), created 2016-10-28.