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 \(\mathbf{A}^{\mathrm{SD}}+\mathbf{A}^{\mathrm{ORB}}\).

Paramagnetic relaxation

simpnmr evaluates nucleus-resolved longitudinal (\(R_1\)) and transverse (\(R_2\)) paramagnetic relaxation rates from the Solomon–Bloembergen–Morgan (SBM) dipolar and Fermi-contact terms and the Guéron Curie-spin term. All terms share the Lorentzian spectral density

\[J(\omega,\tau) = \frac{\tau}{1+\omega^2\tau^2}\]

where \(\omega_I\) and \(\omega_S\) are the nuclear and electron Larmor angular frequencies, \(r\) is the electron–nucleus distance, \(\gamma_I\) is the nuclear gyromagnetic ratio, \(A_{\mathrm{iso}}\) is the isotropic Fermi-contact coupling (in angular-frequency units), and \(T\) is the temperature. The correlation times are the dipolar correlation times \(\tau_{c1}, \tau_{c2}\), the electronic correlation times \(\tau_{e1}, \tau_{e2}\), and the rotational correlation time \(\tau_R\).

Note

\(g_{\mathrm{eff}}\) and \(S_{\mathrm{eff}}\) are the effective electron g-factor and angular-momentum quantum number. For spin-only systems they are the free-electron value \(g_e\) and the spin \(S\); for systems with a well-defined total angular momentum they are the Landé \(g_J\) and \(J\).

SBM dipolar relaxation

\[R_1^{\mathrm{DD}} = \frac{2}{15}\left(\frac{\mu_0}{4\pi}\right)^2 \frac{\gamma_I^2\, g_{\mathrm{eff}}^2\, \mu_B^2\, S_{\mathrm{eff}}(S_{\mathrm{eff}}+1)}{r^6} \Big[\,3\,J(\omega_I,\tau_{c1}) + 6\,J(\omega_I+\omega_S,\tau_{c2}) + J(\omega_I-\omega_S,\tau_{c2})\,\Big]\]

\[R_2^{\mathrm{DD}} = \frac{1}{15}\left(\frac{\mu_0}{4\pi}\right)^2 \frac{\gamma_I^2\, g_{\mathrm{eff}}^2\, \mu_B^2\, S_{\mathrm{eff}}(S_{\mathrm{eff}}+1)}{r^6} \Big[\,4\,J(0,\tau_{c1}) + 3\,J(\omega_I,\tau_{c1}) + 6\,J(\omega_S,\tau_{c2}) + 6\,J(\omega_I+\omega_S,\tau_{c2}) + J(\omega_I-\omega_S,\tau_{c2})\,\Big]\]

SBM Fermi-contact relaxation

\[R_1^{\mathrm{SC}} = \frac{2}{3}\, A_{\mathrm{iso}}^2\, S_{\mathrm{eff}}(S_{\mathrm{eff}}+1)\, J(\omega_I-\omega_S,\tau_{e2})\]

\[R_2^{\mathrm{SC}} = \frac{1}{3}\, A_{\mathrm{iso}}^2\, S_{\mathrm{eff}}(S_{\mathrm{eff}}+1) \Big[\,J(0,\tau_{e1}) + J(\omega_I-\omega_S,\tau_{e2})\,\Big]\]

Guéron Curie relaxation

The Curie-spin term uses the point-dipole approximation and the thermally averaged (static) electron moment:

\[R_1^{\mathrm{Curie}} = \frac{2}{5}\left(\frac{\mu_0}{4\pi}\right)^2 \frac{\omega_I^2\, g_{\mathrm{eff}}^4\, \mu_B^4\, \big[S_{\mathrm{eff}}(S_{\mathrm{eff}}+1)\big]^2}{(3 k_B T)^2\, r^6}\, 3\,J(\omega_I,\tau_R)\]

\[R_2^{\mathrm{Curie}} = \frac{1}{5}\left(\frac{\mu_0}{4\pi}\right)^2 \frac{\omega_I^2\, g_{\mathrm{eff}}^4\, \mu_B^4\, \big[S_{\mathrm{eff}}(S_{\mathrm{eff}}+1)\big]^2}{(3 k_B T)^2\, r^6} \Big[\,4\,J(0,\tau_R) + 3\,J(\omega_I,\tau_R)\,\Big]\]

Note

For both the SBM dipolar and the Curie terms the \(R_1\) prefactor is twice the \(R_2\) prefactor (\(\tfrac{2}{15}\) vs \(\tfrac{1}{15}\), and \(\tfrac{2}{5}\) vs \(\tfrac{1}{5}\)). In the high-field, fast-motion limit (\(\omega_S\tau \gg 1\), \(\omega_I\tau \ll 1\)) the dipolar spectral densities reduce to \(3\tau\) for \(R_1\) and \(4\tau+3\tau=7\tau\) for \(R_2\), so \(R_2/R_1 \to 7/6\).

Susceptibility tensor and Euler angle convention

The magnetic susceptibility tensor \(\boldsymbol{\chi}\) is decomposed into an isotropic part and a traceless anisotropic (deviatoric) part:

\[\boldsymbol{\chi} = \chi_{\mathrm{iso}} \mathbf{I} + \Delta\boldsymbol{\chi}\]

where \(\chi_{\mathrm{iso}} = \tfrac{1}{3}\operatorname{tr}(\boldsymbol{\chi})\).

The anisotropic part is characterised by two invariants:

• Axiality \(\Delta\chi_{\mathrm{ax}}\) — the largest principal deviation from isotropy.

• Rhombicity \(\Delta\chi_{\mathrm{rh}}\) — the in-plane asymmetry.

Euler angles describe the passive ZYZ rotation that maps the input (molecular) frame to the eigenframe of \(\boldsymbol{\chi}\). The three angles \((\alpha, \beta, \gamma)\) are defined as:

\[\begin{split}\alpha &= \operatorname{arctan2}(R_{31},\,-R_{11}) \\ \beta &= \arccos(R_{21}) \\ \gamma &= \operatorname{arctan2}(-R_{23},\, R_{21})\end{split}\]

where \(R_{ij}\) are elements of the rotation matrix whose columns are the eigenvectors of \(\boldsymbol{\chi}\), sorted in ascending order of deviation from \(\chi_{\mathrm{iso}}\). All angles are reported in degrees in the range \([0°,\,180°]\) for \(\beta\) and \((-180°,\,180°]\) for \(\alpha\) and \(\gamma\).

Note

The eigenvector ordering places the eigenvector with the smallest deviation from \(\chi_{\mathrm{iso}}\) first (index 0) and the one with the largest deviation last (index 2). The axial direction therefore corresponds to the last eigenvector, and the angles describe how to align the molecular-frame Y-axis with the axial eigenvector.

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