TURBOMOLE manual
v8.0

18.5 ZFS tensor

The ZFS tensor can be calculated for compounds with a spin \(S > 1/2\). It is only implemented within a non- and scalar-relativistic one-component treatment using spin–orbit perturbation theory in the mpshift module [338]. It is commonly divided into the spin–dipolar (SD) and spin–orbit (SO) contributions (see Ref. [344] for details on the derivations with GGAs and their hybrids, and [338] for the extension to mGGAs, local hybrids, and X2C). The SD contribution consists of four-center-two-electron integrals, which are evaluated in a seminumerical scheme on a grid [144] and are contracted with the spin-excess density matrix. There are two approaches for constructing the spin-excess density matrix. On the one hand, we can use the canonical HF or KS MOs (direct method). On the other hand, unrestricted natural orbitals (UNOs) can be employed to reduce the spin contamination (UNO method) [345]. We generally recommend the UNO method and it is chosen by default both for calculations with the keywords $epr and $pnmr. The SO contribution consists of the same-spin and spin-flip contributions and requires the calculation of first-order orbital-rotation matrices with respect to a spin–orbit perturbation. This is evaluated with the coupled-perturbed Hartree–Fock (CPHF) or Kohn–Sham (CPKS) equations. Here, the following spin–orbit perturbation operators are available: (Effective) Pauli operator, (modified) screened nuclear spin–orbit (mSNSO, SNSO), and spin–orbit mean-field (SOMF) ansatz. We recommend the SNSO Hamiltonian in non-relativistic calculations and the mSNSO Hamiltonian in X2C calculations. Note that X2C or DLU-X2C always include the relativistic picture-change correction.

After the calculation, the ZFS tensor is transformed to its principal axis system. The axial ZFS parameter D, the rhombic parameter E, and the rhombicity parameter \(\text{E}/\text{D}\) are calculated via the elements of the diagonalized tensor by \[\begin{align} \text{D} & = D_{zz} - \frac{1}{2} \left( D_{xx} + D_{yy} \right), \\ \text{E} & = \frac{1}{2} \left( D_{xx} - D_{yy} \right), \\ 0 & \leq {\text{E}} / {\text{D}} \leq 1/3. \end{align}\](18.2–18.4) Additionally, a traceless version of the tensor is calculated as well.

The general keywords for the mpshift module are as follows.

  • followed by zfs; ZFS tensor is calculated additional to the other properties calculated when doing a pNMR calculation. Note, that the calculaton of the paramagnetic shift does NOT use the ZFS tensor, but the simplified equation for doublets, even for \(S > 1/2\)! The UNO approach is chosen as a default for the SD contribution.

  • followed by zfs-only; Only the ZFS tensor is calculated.

  • followed by zfs-dip; Only the SD contribution to the ZFS tensor is calculated.

  • followed by zfs-soc; Only the SO contribution to the ZFS tensor is calculated.

  • followed by zfs direct; Direct approach for the SD term is used. Works analogously with zfs-only and zfs-dip.

  • followed by spin = real; Uses a user-defined spin of as a prefactor instead of the expectation value of \(\hat{S}_z\) calculated from the UHF/UKS occupation.

  • can be used to compute all EPR parameters in one run. Then, the $pnmr settings are ignored. The UNO approach is used for the SD contribution to the ZFS tensor.

  • is used to set a gridsize for the seminumerical SD contribution by writing gridsize a below this keyword (similar to $dft). Recommended is the default -2. Other usable grid sizes are tiny and -1 ... 9, where for 9 more specifications are necessary (see Section 25.2.10).

  • followed by an integer int. It is used to set a threshold of \(10^{-int}\). SD integrals over shell pairs are negelected if the corresponding elements of the spin-density matrix fall below this threshold. The default for int is 14.

  • The effective Pauli or SNSO/mSNSO terms are used for the SO contribution. Available options for the additional keyword $snsopara are

    snsopara 0

      SNSO Hamiltonian [167]

    snsopara 1

      mSNSO Hamiltonian [168, 169], only with X2C

    snsopara 2

      Effective nuclear charges of [340], not available with X2C.

    snsopara 3

      Effective nuclear charges of [341], not available with X2C

    snsopara 4

      mSNSO Hamiltonian with universal DC parameters [190], only with X2C

    snsopara 5

      mSNSO Hamiltonian with universal DCB parameters, [190], only with X2C

    snsopara 6

      mSNSO Hamiltonian with row-dependent DCB parameters, [190], only with X2C

    snsopara 7

      mSNSO Hamiltonian with the later parameters of the Cremer group [191], only with X2C

    With snsopara 2 effective nuclear charges for the elements B-F and Al-Cl are available. With snsopara 3 effective nuclear charges for the elements Li-Ar are available. For H and He, the respective actual nuclear charges are used as the effective charges. If a compound contains elements for which no effective nuclear charges are available and which are not H or He, all elements in the compound are described with their actual nuclear charges. We recommend snsopara 0 for non-relativistic calculations and snsopara 1 for X2C calculations on heavier compounds.

  • followed by nuclear real-a coulomb real-b exchange real-c

    The three parameters real a–c are used to scale the nuclear, two-electron Coulomb, and exchange contribution. Using 1.0 for all three parameters gives the standard expression for the SOMF ansatz (see e.g. [342]).

Note that using both SNSO and SOMF is not possible. The SO contribution supports the current-dependent generalization of \(\tau\) [145, 321] for mGGAs and local hybrid functionals ($curswitchengage, $hfc_curswitchengage) and the ZFS tensor is available for all functionals up to the class of local hybrids. The non-generalized kinetic energy density can be used with $curswitchdisengage, $hfc_curswitchdisengage. We recommend GGA and current-dependent mGGA based hybrid functionals such as \(\omega\)B97X-D with a substantial admixture of HF exchange for the simultaneous calculation of all EPR properties, see [338].

When looking at the output and extracting the ZFS parameters D and E, it is important to check, whether \(\text{E}/\text{D}\) is close to 1/3. If this is the case, the sign of D becomes unreliable, as small changes in the diagonal elements of the tensor may lead to a flip of the sign.

Finally, note again that the ZFS tensor is not used for the calculation of the pNMR shift, even when using the keyword $pnmr zfs. The pNMR shielding tensor including the ZFS contribution can be computed as outlined in [346] based on the scripts of the Autschbach group at https://github.com/jautschbach/pnmr-shift.