Single pivoted motions

Atomic level mechanics of the single pivot

For the PCS, the lanthanide ion to nuclear vector is

r = pN - pLn3+, (12.53)

where pN is the Cartesian coordinates of the nucleus of interest and pLn3+ is the position of the aligning lanthanide ion. r is defined in the alignment frame, and pLn3+ is constant in this frame. After a forward rotation to the discrete state i, the new atomic position in the reference frame is

pN' = Ri$\displaystyle \left(\vphantom{ p_{\textrm{N}} - p_{\textrm{P}} }\right.$pN - pP$\displaystyle \left.\vphantom{ p_{\textrm{N}} - p_{\textrm{P}} }\right)$ + pP. (12.54)

where pP is the pivot point of the rotation. Hence the transformed vector is

\begin{subequations}\begin{align}r_i &= p_{\textrm{N}}^i - p_{\textrm{Ln}^{3+}} ...
...} \right) + p_{\textrm{P}} - p_{\textrm{Ln}^{3+}} .\end{align}\end{subequations}

The set of three vectors are defining this pivoted system are

\begin{subequations}\begin{align}& r_{\textrm{LN}} = p_{\textrm{N}} - p_{\textrm...
...trm{LP}} = p_{\textrm{P}} - p_{\textrm{Ln}^{3+}} . \end{align}\end{subequations}

Let the pre-rotation vectors be

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

The post-rotation vectors are

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

The vector rPN is independent of alignment so can be calculated once per atom, and rLP is independent of alignment and atom position so can be calculate once.

PCS and single pivoted motions

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

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

Expanding for the single motion of the lanthanide-atom vector rLN(1), this becomes

\begin{subequations}\begin{align}\delta &= \frac{c}{\left\vert r_{\textrm{LN}}^{...
...0)^T} \cdot A \cdot r_{\textrm{LP}}^{(0)} \Bigg] . \end{align}\end{subequations}

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 2016-10-28.