Theory

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

Total shift

In simpnmr, the diamagnetic, spin-dipolar, Fermi-contact, Fermi-contact g-correction, and orbital contributions are treated as separate components of the total shift. The total predicted chemical shift is therefore written as

(1)\[ \delta^{\mathrm{TOTAL}}=\delta^{\mathrm{DIA}}+\delta^{\mathrm{SD}}+\delta^{\mathrm{FC}}+\delta^{\mathrm{FC}}_{\mathrm{g-corr}}+\delta^{\mathrm{ORB}}_{\mathrm{iso}}+\delta^{\mathrm{ORB}}_{\mathrm{aniso}}\]

Note

The calculation without \(\delta^{\mathrm{ORB}}\) remains physically reasonable because the orbital contribution is expected to become negligible at sufficiently large distances from the paramagnetic centre. [Lang2020]

Diamagnetic shift

When the diamagnetic contribution is obtained from DFT shielding data, a reference value is required in order to convert shielding into a chemical shift. In simpnmr, the diamagnetic shift contribution is written as

(2)\[ \delta^{\mathrm{DIA}}=\sigma_{\mathrm{ref}}-\sigma\]

where \(\sigma\) is the calculated shielding for the nucleus of interest and \(\sigma_{\mathrm{ref}}\) is the corresponding reference shielding.

This diamagnetic term is then added directly to the other shift contributions when forming \(\delta^{\mathrm{TOTAL}}\).

Note

For a consistent diamagnetic shift, the calculated shielding \(\sigma\) and the reference shielding \(\sigma_{\mathrm{ref}}\) must be obtained at the same level of theory.

Spin-dipolar contribution

In simpnmr, the contribution arising from the anisotropic magnetic susceptibility and the traceless spin-dipolar hyperfine interaction is treated as a separate spin-dipolar shift channel, denoted \(\delta^{\mathrm{SD}}\). This term depends on the anisotropic part of the magnetic susceptibility tensor and the spin-dipolar part of the hyperfine coupling tensor (HFC).

1. Spin-dipolar contribution with hyperfine from DFT

To account for the non-point nature of the paramagnetic centre, the normalised traceless spin-dipolar hyperfine contribution can be obtained from a simple single-point DFT calculation of \(\mathbf{A}^{\mathrm{SD}}\):

(3)\[\delta^{\mathrm{SD}}=\frac{1}{3} \operatorname{tr}\left(\Delta \boldsymbol{\chi} \cdot \mathbf{A}^{\mathrm{SD}}\right)\]

In simpnmr, this defines the spin-dipolar contribution independently of the FC, FC g-correction, and orbital shift terms.

Note

The spin-dipolar hyperfine tensor \(\mathbf{A}^{\mathrm{SD}}\) is always traceless.

2. Spin-dipolar contribution 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 spin-dipolar contribution \(\delta^{\mathrm{SD}}\) can be calculated as a third of the trace of the magnetic susceptibility tensor \(\chi\) multiplied by the traceless spin-dipolar hyperfine tensor, which in the point-dipole approximation is a matrix that depends only on the nuclear coordinates:

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

If the coordinates are specified in Å and \(\chi\) is in ų, then the equation above, multiplied by 10⁶, gives the spin-dipolar contribution in ppm.

Fermi-contact shift

The Fermi-contact contribution is split into a spin-only term \(\delta^{\mathrm{FC}}\) and an additional g-correction term \(\delta^{\mathrm{FC}}_{\mathrm{g-corr}}\). Both depend on the isotropic Fermi-contact hyperfine interaction at the nucleus.

Note

By construction, the Fermi-contact hyperfine tensor \(\mathbf{A}^{\mathrm{FC}}\) is isotropic. In matrix form, it is diagonal with equal diagonal elements.

1. FC with spin-only magnetic susceptibility

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

(5)\[ \delta^{\mathrm{FC}}=\chi_{iso}^S A^{FC}\]

In simpnmr, \(\delta^{\mathrm{FC}}\) is evaluated from the isotropic part of the Fermi-contact hyperfine tensor, i.e. from \(\frac{1}{3}\operatorname{tr}(\mathbf{A}^{\mathrm{FC}})\), where the spin-only magnetic susceptibility in SI units is:

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

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. FC g-correction from g-corrected magnetic susceptibility

In order to account for the effect of \(\mathbf{g}_{\mathrm{ab-initio}}\) anisotropy on the FC term, simpnmr treats the corresponding correction as a separate contribution.

(7)\[\delta^{\mathrm{FC}}_{\mathrm{g-corr}}=\left[\chi^{\mathrm{g-corr}}_{\mathrm{iso}}-\chi^S_{\mathrm{iso}}\right]\,\frac{1}{3}\operatorname{tr}\left(\mathbf{A}^{\mathrm{FC}}\right)\]

where the g-corrected isotropic susceptibility is

(8)\[ \chi^{\mathrm{g-corr}}_{\mathrm{iso}}=\frac{g_{\mathrm{e}}}{3}\left(\frac{\chi_x}{g_x}+\frac{\chi_y}{g_y}+\frac{\chi_z}{g_z}\right)\]

This correction remains proportional to the spin-only Fermi-contact hyperfine term and isolates the additional contribution arising from g-tensor anisotropy.

Note

Here, \(\mathbf{g}_{\mathrm{ab-initio}}\) should be taken from the same level of theory as the susceptibility tensor used to compute \(\chi^{\mathrm{g-corr}}_{\mathrm{iso}}\).

Orbital shift contribution

In simpnmr, the orbital contribution is treated as two additional shift channels, \(\delta^{\mathrm{ORB}}_{\mathrm{iso}}\) and \(\delta^{\mathrm{ORB}}_{\mathrm{aniso}}\). These do not modify the definitions of \(\delta^{\mathrm{SD}}\), \(\delta^{\mathrm{FC}}\), or \(\delta^{\mathrm{FC}}_{\mathrm{g-corr}}\).

Note

The orbital contribution is evaluated only when both of the following are available from the same QC source:

• the orbital hyperfine contribution \(\mathbf{A}^{\mathrm{ORB}}\)

• the associated \(\mathbf{g}_{\mathrm{DFT}}\) tensor

If the orbital hyperfine contribution \(\mathbf{A}^{\mathrm{ORB}}\) and the associated \(\mathbf{g}_{\mathrm{DFT}}\) tensor are available from the same QC source, the isotropic orbital contribution is evaluated as

(9)\[ \delta^{\mathrm{ORB}}_{\mathrm{iso}}= \chi_{\mathrm{iso}}\frac{1}{3}\operatorname{tr}\left[\frac{g_{\mathrm{e}}}{\mathbf{g}^{\mathrm{T}}_{\mathrm{DFT}}}\left(\mathbf{A}^{\mathrm{SD}}+\mathbf{A}^{\mathrm{ORB}}\right)^{\mathrm{T}}\right]\]

and the anisotropic orbital contribution is evaluated as

(10)\[ \delta^{\mathrm{ORB}}_{\mathrm{aniso}}=\frac{1}{3}\operatorname{tr}\left[\Delta\boldsymbol{\chi}\frac{g_{\mathrm{e}}}{\mathbf{g}^{\mathrm{T}}_{\mathrm{DFT}}}\left(\mathbf{A}^{\mathrm{SD}}+\mathbf{A}^{\mathrm{ORB}}\right)^{\mathrm{T}}-\Delta\boldsymbol{\chi}\mathbf{A}^{\mathrm{SD}}\right]\]

For reporting purposes, the total orbital shift contribution is the sum

(11)\[ \delta^{\mathrm{ORB}}=\delta^{\mathrm{ORB}}_{\mathrm{iso}}+\delta^{\mathrm{ORB}}_{\mathrm{aniso}}\]

Therefore, evaluating orbital shift contributions requires both the orbital hyperfine contribution and the associated \(\mathbf{g}_{\mathrm{DFT}}\) tensor from the same QC source.

Note

Another common source of confusion is the role of \(\mathbf{A}^{\mathrm{SD}}\) in the orbital expressions above. In simpnmr, \(\mathbf{A}^{\mathrm{SD}}\) still defines the spin-dipolar term on its own, while the orbital term is evaluated separately from the transformed combination \(\mathbf{A}^{\mathrm{SD}}+\mathbf{A}^{\mathrm{ORB}}\).

References

[Lang2020]

Lang, L.; Ravera, E.; Parigi, G.; Luchinat, C.; Neese, F. J. Phys. Chem. Lett. 2020, 11 (20), 8735-8744. DOI: 10.1021/acs.jpclett.0c02462