Theory#

Below, we outline the key models and equations implemented in simpnmr.

Pseudocontact shift#

Pseudocontact shift (PCS) depends on the anisotropic part of the magnetic susceptibility tensor and the dipolar part of the hyperfine coupling tensor (HFC). Depending on the source of the HFC (output from a density functional theory (DFT) calculation or simply a file containing coordinates), three approximations are implemented in simpnmr.

1. PCS with point-dipole approximation#

Assuming that a paramagnetic metal centre is at the origin and a nucleus of interest has coordinates (x, y, z), the isotropic part of the pseudocontact shift (PCS) tensor \(\delta_{\mathrm{PCS}}\) can be calculated as a third of the trace of the magnetic susceptibility tensor \(\chi\) multiplied by the reduced dipolar hyperfine tensor, which in the point-dipole approximation is a matrix that depends only on the nuclear coordinates.

(1)#\[\begin{split} \delta_{\mathrm{PCS}}=\frac{1}{12 \pi r^5} \operatorname{tr}\left[\left(\begin{array}{ccc} \chi_{x x} & \chi_{x y} & \chi_{x z} \\ \chi_{y x} & \chi_{y y} & \chi_{y z} \\ \chi_{z x} & \chi_{z y} & \chi_{z z} \end{array}\right) \cdot\left(\begin{array}{ccc} 3 x^2-r^2 & 3 x y & 3 x z \\ 3 x y & 3 y^2-r^2 & 3 y z \\ 3 x z & 3 y z & 3 z^2-r^2 \end{array}\right)\right]\end{split}\]

If the coordinates are specified in Å and \(\chi\) is in Å³, then the equation above, multiplied by 10⁶, gives the PCS in ppm.

2. PCS with non-relativistic hyperfine from DFT#

To account for the non-point nature of the paramagnetic center, the normalised reduced dipolar hyperfine can be obtained from a simple single-point non-relativistic DFT calculation \(\mathbf{A}^{\mathrm{dip}}\).

(2)#\[\delta_{\mathrm{PCS}}=\frac{1}{3} \operatorname{tr}\left(\Delta \boldsymbol{\chi} \cdot \mathbf{A}^{\mathrm{dip}}\right)\]

Note that, since the dipolar hyperfine tensor is traceless, the isotropic part of the PCS does not depend on the isotropic part of the magnetic susceptibility tensor. Hence the equation features the traceless \(\Delta \boldsymbol{\chi}\).

3. PCS with relativistic hyperfine from DFT#

If a relativistic contribution to the HFC \(\mathbf{A}^{\text {orb }}\) and the g-tensor are calculated in the same DFT output, then an additional contribution to the PCS coming from \(\mathbf{A}^{\text {orb }}\) can be evaluated.

(3)#\[\delta_{\mathrm{PCS}}^{\text {orb }}=\frac{1}{3} \operatorname{tr}\left(\Delta \boldsymbol{\chi} \cdot\left(\frac{\mathrm{~g}_{\mathrm{e}}}{\mathbf{g}^{\mathrm{T}}} \cdot\left(\mathbf{A}^{\operatorname{dip}}+\mathbf{A}^{\text {orb }}\right)^{\mathrm{T}}-\mathbf{A}^{\operatorname{dip}}\right)\right)\]

This contribution should decay to zero with increasing distance from the paramagnetic centre for light nuclei.

Fermi-contact shift#

The Fermi-contact shift is often described as the contribution arising from the spin density at the nucleus.

1. FC with spin-only magnetic susceptibility#

In the simplest model, the Fermi-contact shift is proportional to the reduced Fermi-contact hyperfine interaction at the nucleus of interest and the spin-only magnetic susceptibility.

(4)#\[ \delta_{\mathrm{FC}}=\chi_{iso}^S A^{FC}\]

where the spin-only magnetic susceptibility in SI units is

(5)#\[ \chi_{iso}^S=\frac{\mu_0 \mu_B^2 \mathrm{g}_{\mathrm{e}}^2 S(S+1)}{3 k T}\]

where \(\mu_0\) is the vacuum permeability, \(\mu_B\) is the Bohr magneton, \(\mathrm{g}_{\mathrm{e}}\) is the free-electron g-factor, \(S\) is the total spin, \(k\) is the Boltzmann constant and \(T\) is the temperature.

2. Accounting for g-tensor anisotropy#

In order to account for the effect of g-tensor anisotropy on the FC shift, both the magnetic susceptibility tensor and the g-tensor must be calculated at the same level of theory (e.g. SOC-NEVPT2).

(6)#\[\delta^{F C}=\frac{\mathrm{g}_{\mathrm{e}}}{3} \operatorname{tr}\left(\frac{\boldsymbol{\chi}}{\mathbf{g}^{\mathrm{T}}}\right) A^{FC}\]

3. Relativistic hyperfine from DFT#

If the relativistic hyperfine contribution \(\mathbf{A}^{\text {orb }}\) and the g-tensor are calculated using a DFT method in ORCA, an additional contribution to the shift, which does not depend on the spin-only HFC \(A^{FC}\) but on the isotropic magnetic susceptibility tensor can be evaluated.

(7)#\[\delta^{\text {orb }}=\chi_{\text {iso }} \frac{1}{3} \operatorname{tr}\left(\frac{\mathrm{~g}_{\mathrm{e}}}{\mathbf{g}^{\mathrm{T}}} \cdot\left(\mathbf{A}^{\text {dip }}+\mathbf{A}^{\text {orb }}\right)^{\mathrm{T}}\right)\]

As with other terms arising from relativistic contributions to the hyperfine tensor, the term above vanishes at large distances from the paramagnetic centre.