TURBOMOLE manual
v8.0

18.2 EPR g-Tensor

One-component treatment: The g-tensor is calculated with the (effective) Pauli operator or the SNSO/mSNSO approach. Here, X2C applies the picture-change correction to the spin–orbit perturbation. By default, gauge-including atomic orbitals (GIAOs) are used throughout and the g-tensor is implemented in the mpshift module. All functionals including local hybrids are supported [85, 321]. The generalization using the current density is applied by default or with the keyword $curswitchengage [320, 321]. This is strongly recommended. Alternatively, the kinetic energy density is generalized with the external magnetic field to ensure gauge-origin invariance [45] with $curswitchdisengage.

The general keywords for the mpshift module are as follows.

  • followed by g-only; g-tensor is calculated. The g-tensor is transformed into the principal axis system and the \(\Delta\)g-shift with respect to the g-factor of the free electron is given in ppt (parts per thousand).

  • 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 effective Pauli or SNSO/mSNSO terms are used for PSOSO. 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

      Universal DC mSNSO Hamiltonian [190], only with X2C.

    snsopara 5

      Universal DCB mSNSO Hamiltonian [190], only with X2C.

    snsopara 6

      Row-dependent DCB mSNSO Hamiltonian [190], only with X2C.

    snsopara 7

      mSNSO Hamiltonian [168, 169], only with X2C (fixed typo in original paper by Cremer).

    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. The SNSO parameters defined in the $relham group overwrite the other SNSO options.

In non-relativistic calculations (no X2C), the g-tensor may be partitioned into the relativistic mass correction (RMC), the diamagnetic correction (DC), and the orbital Zeeman (OZ) term. The derivation of the RMC term is based on the virial theorem for molecules ($rmc) and is only rigorous when considering the complete potential, i.e. one and two-electron contributions. Hence, the Pauli kinetic energy (PKE) term is formally more correct, better suited for heavy elements and is used with the keyword $pke. By default, the PKE term is considered. X2C calculations use the PKE term only. For the g-tensor and g-shift of EPR experiments, the two-component treatment is suggested for heavy elements.

The common gauge origin (CGO) approach is applied with $cgo_epr. By default, the heaviest atom is chosen as common gauge origin. To manually set the CGO, use $cgo integer, where the atom number of the coordinate file selects the respective nucleus. $cgo 0 selects the center of mass as common gauge origin. \(-1\) selects the Cartesian origin. However, the GIAO-based ansatz is strongly recommended.

Two-component treatment: The g-tensor can be obtained with a common gauge origin (CGO) in the ridft and mpshift modules or gauge-including atomic orbitals in the mpshift module only. The latter is recommended [188]. To use a CGO with mpshift set $cgo_epr, ridft always uses a CGO. The work flow is similar to the HFC tensor. This requires three independent two-component (2c) calculations with the spin magnetization vector aligned along the Cartesian axes. Again, this should be done with converged one-component MOs to accelerate the 2c SCF convergence. The spin magnetization is aligned with the keywords $sxeig, $syeig, and $szeig. Then, each spin direction yields 3 tensor elements. The g-tensor contributions are calculated with converged spinors in ridft or mpshift. ridft requires the keyword $x2c_gtensor or $x2c_gtensor rkb and mpshift needs $pnmr g-only. The restricted magnetic balance (RMB) condition is employed by default. Alternatively, the flag $epr can be used to compute all EPR parameters in one mpshift run per spin orientation. We strongly recommend the DLU-X2C approach in combination with the mSNSO correction, see also Sec. 6.5.

The complete two-component work flow is as follows:

  1. Delete all previously inserted two-component keywords and the spinor files. Alternatively, set up a new directory from scratch. We need to start from UHF/UKS orbitals to align the spin.

  2. Run a UHF/UKS calculation with the X2C or DLU-X2C Hamiltonian. The finite nucleus model is strongly recommended. Please make sure to use the proper grids for X2C and DFT (gridsize 3a etc.). Do not use gridsize m3 or m4.

  3. Use gtensprep.sh to prepare your 2c input (see gtensrep.sh –h for more information; gtensprep.sh -msnso is recommended) The HFC and g-tensor can be computed simultaneously with gtensprep.sh -msnso -hfc.

  4. Run the 2c SCF calculations in the directories x, y, and z. ridft computes the g-tensor components using a common gauge origin and stores the results in the control file. Alternatively, run an mpshift calculation with $pnmr g-only based on the converged spinors. Alternatively, you may use $epr and mpshift.

  5. Go to parent directory and call calcgtens.py, which computes the g-tensor in its principal axis system based on the rank-2 tensor. Additionally, the \(\Delta\)g-shift is computed in ppt.

By default, the heaviest atom is chosen as common gauge origin. To manually set the CGO, use $cgo integer, where the atom number of the coordinate file selects the respective nucleus. $cgo 0 selects the center of mass as common gauge origin. Note that 2c calculations with local hybrid functionals are currently restricted to the CGO approach [107]. The same holds for 2c CDFT approaches.

For systems with multiple metal centers, we strongly recommend the GIAO ansatz in mpshift. This only leads to a minor computational overhead.