Single pivoted motions

Atomic level mechanics of the single pivot

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⋅ $\displaystyle \left(\vphantom{ p_{\textrm{N}} - p_{\textrm{P}} }\right.$ p_N - p_P $\displaystyle \left.\vphantom{ p_{\textrm{N}} - p_{\textrm{P}} }\right)$ + p_P.

(12.54)

where p_P 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

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

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 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.

PCS and single pivoted motions

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

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

(12.59)

Expanding for the single motion of the lanthanide-atom vector r_LN⁽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 2024-06-08.