6.6 Finite magnetic fields

General framework of finite-magnetic-field calculations

Calculations of magnetic properties in finite magnetic fields can be carried out in the 2c framework (see also section 6.5 for details on 2c). The non-relativistic Hamiltonian in a magnetic field ($\mathcal{B}$) can be written as:

\[\begin{equation*} \hat{H} = \hat{H}_0 + \frac{1}{2} \sum_i \mathcal{B} \cdot \mathbf{l}_{iO} + \sum_i \mathcal{B} \cdot \mathbf{s}_i + \frac{1}{8} \sum_i \left( \mathcal{B}^2 \mathbf{r}_{iO}^2 - (\mathcal{B} \cdot \mathbf{r}_{iO})^2 \right) \end{equation*}\]

Here, $\hat{H}_0$ is the zero-field Hamiltonian. The terms dependent on the angular momentum operator $\mathbf{l}_{iO}$ and spin operator $\mathbf{s}_i$ are called orbital-Zeeman term and spin-Zeeman term, respectively. Both are referred to as paramagnetic terms since they are linearly dependent on the magnetic field. The terms which are quadratically dependent on the magnetic field are called diamagnetic terms. In order to avoid a dependence on the gauge origin $O$ of the system, London orbitals are used. An extension to relativistic exact two-component and its local variants is also available and uses the same keywords as field-free calculations. However, the finite nucleus model and picture-change correction schemes for expectation values are not yet available.
Using this framework, calculations in finite magnetic fields can be performed at various levels of theory. The starting point of any such calculation is always a 2c calculation using the ridft module which has to include the $soghf and $magnetic field keywords. The x, y and z coordinates of the magnetic field as well as its absolute value have to be specified. An exemplary input into the control file might look like this:

$soghf
$magnetic field
  1 1 2 0.1
  z-field
  nomag

You may also add the option ’z-field’ to this data group in the next line to tell the program that only the z-component will have a non-vanishing field component. This will set the x and y magnetizabilities to zero, which is valid for non-symmetry broken references. Furthermore, the keyword ’nomag’ will assume that all magnetization densities are 0, allowing for TD-DFT treatment.

This input assumes atomic units and thus this would correspond to a magnetic field of $|\mathcal{B}| = 0.1$ being applied in the $(1 1 2)^T$ direction. Within the $magnetic field keyword, a number of additional useful things may be specified. Writing ’tesla’ will assume that the magnetic field is not given in atomic units but in units of tesla instead. Writing ’spin xxx’ will scale the Spin-Zeeman term by a constant of xxx. This might help if one wants to converge towards different spin states. Adding a larger, positive number will make it easier to converge to spin states with larger multiplicity. In the last iteration, the Spin-Zeeman term is not scaled anymore and thus the correctly non-scaled result is displayed at the end of any calculation.

Available quantum-chemical methods

Calculations in finite magnetic fields are possible employing a number of different quantum-chemical methods. The RI approximation may be used throughout. If an exact calculation of the Coulomb-part is desired, the $coulex keyword needs to be added.

Generalized Hartree-Fock (GHF) and Density functional theory (DFT) using the ridft module. Molecular gradients of a converged GHF or DFT calculation can be obtained using the rdgrad module. Consequently, it is possible to optimize the structure of a molecule on the GHF level using the jobex script if the ’-ri’ option is specified. It should be noted that most symmetries are broken using this ansatz and the resulting wave function needn’t be an eigenfunction to either $\hat{S}^2$, $\hat{S}_z$ or the time-reversal operator $\hat{T} = -\text{i} \sigma_y$. Therefore, the results are often spin-contaminated (check $<\hat{S}^2>$!) and not Kramers-symmetric.
Post-Hartree-Fock methods MP2 and CC2 for ground states using the Ricc2 module. As always, a converged GHF solution needs to be present.
Calculation of excited states and related properties using TDHF, TDDFT or GW/BSE employing the escf module. Excitation energies (but not properties!) can be calculated on the CC2 and ADC(2) level using the Ricc2 module. Not that TDDFT does not support non-negligible spin densities at the moment.