20.1 Contact Density and Isomer Shift
Mössbauer spectroscopy measures the energy shift of the resonance for the \(\gamma\)-transition of an absorbing nucleus and the source nucleus. This isomer shift can be related to the effective contact density \(\rho^{\text{e}}\), which is the derivative of the energy with respect to the radius of the finite nucleus model [348, 349, 350, 351, 352]. Then, the isomer shift reads \[\begin{equation} \delta^{\text{IS}} = \alpha \left(\rho_\text{A}^{\text{e}} - \rho_{\text{S}}^{\text{e}} \right) \end{equation}\](20.1) and \(\alpha\) is a nuclear calibration function, which is typically obtained by fitting theoretical predictions to experimental measurements. See Ref. [352] for further details and fitting options. Note that we apply a Gaussian charge distribution for the finite nucleus model, see Sec. 6.5. Thus, the finite nucleus model should be set and the derivative of the energy with respect to the root-mean-square radius \(\sqrt{ <R>^2}\) of Ref. [353] is evaluated. Thus, the effective contact density follows as \[\begin{equation} \rho_\text{A}^{\text{e}} = \left[ \frac{3}{4 \pi Z_\text{A} \sqrt{ <R>^2}} \frac{\partial E}{\partial \sqrt{ <R>^2}} \right]. \end{equation}\](20.2) Alternatively, the energy derivative with respect to the exponent of the finite nucleus model can be formed. This only changes the prefactor [349, 350]. The respective energy derivative and the effective contact density can be obtained within the non-relativistic framework or exact two-component (X2C) theory. For the latter, the derivatives of the decoupling matrix are calculated by solving the X2C linear response equations [170, 42].
The effective contact density is calculated in the proper section of the respective module, e.g. dscf or ridft. For post-HF or post-KS modules such as mpgrad, ricc2, or rirpa, a gradient calculation has to be performed. The following Hamiltonians are available [108].
Non-relativistic, only
$mossbaueris required.X2C or local X2C, please also use the picture-change correction and the X2C response equations. To include spin-orbit effects, set
$soghf. Here, the screened nuclear spin–orbit approach is also available. See Sec. 6.5 for details. The spin–orbit treatment is only available with the two-component Hartree–Fock, density functional theory, and current density functional theory level [43]. This is only available withridft. By default, the diagonal local approximation to the unitary decoupling transformation (DLU) is chosen. The DLH or linear-scaling DLU ansatz is available as described in Sec. 6.5.
Please make sure to use the $nucsel keyword followed by the number of the atom in your coordinate file to only calculate the effective contact density at the nucleus of interest. For comparison with other programs, the option without diagonalization of the overlap matrix to remove linear dependencies is recommended with Dyall basis sets, see Sec. 6.5 for details.
The contact density \(\rho^{\text{c}}\) can be approximated with the density at the respective nucleus. That is, the expectation value of the density operator is calculated. For X2C, the picture-change correction of the density operator is of great importance and should always be used. Note that the absolute value of the density is returned. Therefore, the isomer shift follows as \[\begin{equation} \delta^{\text{IS}} = \alpha \left(\rho_\text{A}^{\text{c}} - \rho_{\text{S}}^{\text{c}} \right). \end{equation}\](20.3)
Application of effective core potentials for the nucleus of interest results in non-physical calculations. Thus, only relativistic all-electron calculations are meaningful for heavy elements.
Note that the m-grids such as gridsize m3 should never be used for these calculations. Please make sure to always use the full grid for the SCF iterations, i.e. gridsize 3 or 3a.