Pseudocontact shift analysis
This tutorial shows how to use Spinach for analysis of paramagnetic NMR data. The examples are designed in a way to familiarize users with Spinach functions as well as the theory of paramagnetic shift.
Paramagnetic chemical shift consists of two contributions: Fermi-contact (FC) and pseudocontact shift (PCS). It can be expressed via trace of the product of hyperfine (A) and susceptibility tensors (χ). There are four different approaches for PCS analysis currently presented in Spinach. Difference in the presented methods appears only at the evaluation of the hyperfine tensor while susceptibility is either defined by user or optimized. Note that PCS depends only on the traceless part of χ, whereas Fermi contact part of the paramagnetic shift depends on the Tr(χ).
Point approximation
The simplest approach is based on the point-dipole approximation where dipolar hyperfine tensor depends on the position of a nucleus relative to the paramagnetic center \(\left(\vec r\right)\) as \(\frac{1}{{4\pi {r^3}}}\left( {\frac{{\vec r \cdot {{\vec r}^{\rm{T}}}}}{{{r^2}}} - \frac{1}{3}} \right)\). Therefore PCS is defined as \(\sigma = \frac{1}{{4\pi {r^3}}}{\rm{Tr}}\left( {\left( {\frac{{\vec r \cdot {\vec r^{\rm{T}}}}}{{{r^2}}} - \frac{1}{3}} \right) \cdot {\bf{\chi }}} \right)\). This approach is valid at the sufficient distance from paramagnetic center where it can be viewed as a point. To solve a direct problem use the function ppcs.m. This function takes structure of a molecule and susceptibility tensor as an input and gives PCS at the nuclei as an output. To solve an inverse problem use ippcs.m. The function evaluates susceptibility tensor and position of the paramagnetic center for provided PCS data with corresponding nuclear coordinates.
Example 1a. Direct problem for PCS using the point-dipole approximation: Metal porphyrin proton PCS.
In this example we will compute PCS on protons of the Co(II)/Cu(II) porphyrin and see how magnetic anisotropy affects PCS.
Load the structural parameters. Specify susceptibility tensor. How to construct it from g-tensors. Easy axis/ easy plane examples. Non-axial example.
Example 1b. Inverse problem for PCS using point-dipole approximation. Calbidin PCS.
Corrections to PCS from non-point source
The next level of theory that corrects the point approximation is based on the multipole expansion of spin distribution. It works for nuclei outside the bounding sphere of the paramagnetic density and accounts for its anisotropy that affect PCS (see paper). It should be noted that any isotropic distribution of the spin gives the same PCS as the point source. Direct problem is solved using lpcs.m and inverse problem is solved using ilpcs.m. This method requires from user list of the ranks of spherical harmonics (L). If L=0 the density is isotropic and result is identical to point approximation. Normally L=[0 1 2] accounts for the major effects of the anisotropy in the spin density distribution. We do not recommend to go beyond L=[0 1 2 3 4].
Example 2a. Correction to point PCS from the prolate/oblate Gaussian spin density.
