19.1 Theory and Formalism

The program mpshift can also calculate paramagnetic NMR (pNMR) shielding constants and shifts with gauge-including atomic orbitals or a common gauge origin at the HF and DFT levels [337, 347]. For a doublet system, the pNMR shielding tensor based on the EPR properties, outlined in the previous chapter, reads \[\begin{equation} \bm{\sigma}_N^{\text{tot}} = \bm{\sigma}_N^{\text{orb}} + \bm{\sigma}_N^{\text{temp}} = \bm{\sigma}_N^{\text{orb}} - \frac{10^6 h \mu_e}{g_N \mu_N} \frac{S (S + 1)}{3 k_B T} \bm{g} \cdot \bm{A}_N^{\text{T}} \end{equation}\](19.1) \(\bm{\sigma}^{\text{orb}}\) is the orbital contribution which is the straightforward UHF/UKS generalization of the closed-shell NMR shifts in RHF/RKS [337, 321]. \(\bm{g}\) and \(\bm{A}\) are the g-tensor and HFC tensor. The prefactor includes the nuclear g-factor \(g_N\), the electron magneton \(\mu_e\), the nuclear magneton \(\mu_N\), Planck’s constant \(h\), Boltzmann’s constant \(k_B\), the temperature \(T\), and the (effective) spin \(S\). Therefore, mpshift only requires the user to define the spin and temperature in addition to calculating \(\bm{\sigma}^{\text{orb}}\), \(\bm{g}\), and \(\bm{A}\) as described in the previous sections. Note that the g and HFC tensor may be calculated at the one-component level within a non-relativistic or scalar X2C framework or in the two-component formalism with X2C. The one-component ansatz uses spin–orbit perturbation theory for the g and HFC tensors, where X2C includes the proper picture-change correction [85]. Therefore, the one-component non-relativistic or X2C approach is the most consistent option.

Matters are more complicated for triplet systems and beyond, as this necessitates the zero-field splitting, \[\begin{equation} \bm{\sigma}_N^{\text{tot}} = \bm{\sigma}_N^{\text{orb}} + \bm{\sigma}_N^{\text{temp}} = \bm{\sigma}_N^{\text{orb}} - \frac{10^6 h \mu_e}{g_N \mu_N k_B T} \bm{g} \cdot \bm{Z} \cdot \bm{A}_N^{\text{T}} \end{equation}\](19.2) with the \((3 \times 3)\) matrix [346] \[\begin{align} Z_{kl} = & \frac{1}{Q} \sum_{\lambda} e^{-\frac{E_\lambda}{k_B T}} \left[\sum_{a,a'} \ensuremath{\langle S\lambda a | \hat{S}_{k} | S\lambda a' \rangle} \ensuremath{\langle S \lambda a' | \hat{S}_{l} | S\lambda a \rangle}\right. \\ &\left. + 2 k_B T \; \Re \sum_{\lambda' \neq \lambda} \sum_{a,a'} \frac{\ensuremath{\langle S\lambda a | \hat{S}_{k} | S\lambda'a' \rangle} \ensuremath{\langle S\lambda'a' | \hat{S}_{l} | S\lambda a \rangle}}{E_{\lambda'}-E_{\lambda}} \right] \end{align}\](19.3) including the eigenvalues \(E_{\lambda}\) of the ZFS Hamiltonian \(\hat{\vec{S}} \cdot \bm{D} \cdot \hat{\vec{S}}\). \(\lambda\) and \(a\) refer to the \((2S + 1)\) eigenfunctions, i.e. \(a\) counts the components of degenerate states. \(Q\) is the partition function, i.e. \(Q = \sum_{\lambda, a} \exp \left[ - E_{\lambda}/ (k_B T) \right]\). \(Z_{kl}\) is consequently obtained from the ZFS tensor, those calculation [338] was discussed in the previous chapter. To evaluate Eq. 19.2,, mpshift calculates all four individual tensors, i.e. the orbital contribution, the g-tensor, the HFC tensor, as well as the ZFS tensor. However, the paramagnetic NMR shielding tensor has to be computed with the the PNMRShift program of the Autschbach group at https://github.com/jautschbach/pnmr-shift.