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Hyperfine Parameters

To have VASP (as of version 5.3.2) compute the hyperfine tensors at the atomic sites, set

The hyperfine tensor $ A^I$ describes the interaction between a nuclear spin $ S^I$ (located at site $ {\mathbf{R}}_I$) and the electronic spin distribution $ S^e$ (in most cases associated with a paramagnetic defect state):

$\displaystyle E=\sum_{ij} S^e_i A^I_{ij} S^I_j

In general it is written as the sum of an isotropic part, the socalled Fermi contact term, and an anisotropic (dipolar) part.

The Fermi contact term is given by

$\displaystyle (A^I_{\mathrm{iso}})_{ij}= \frac{2}{3}\frac{\mu_0\gamma_e\gamma_I...
...ij}\int \delta_T({\mathbf{r}})\rho_s({\mathbf{r}}+{\mathbf{R}}_I)d{\mathbf{r}}

where $ \rho_s$ is the spin density, $ \mu_0$ is the magnetic susceptibility of free space, $ \gamma_e$ the electron gyromagnetic ratio, $ \gamma_I$ the nuclear gyromagnetic ratio of the nucleus at $ {\mathbf{R}}_I$, and $ \left< S_z \right>$ the expectation value of the $ z$-component of the total electronic spin. $ \delta_T({\mathbf{r}})$ is a smeared out $ \delta$ function, as described in the Appendix of Ref. [152].

The dipolar contributions to the hyperfine tensor are given by

$\displaystyle (A^I_{\mathrm{ani}})_{ij}=\frac{\mu_0}{4\pi}\frac{\gamma_e\gamma_...
...thbf{r}}+{\mathbf{R}}_I)}{r^3}\frac{3r_ir_j-\delta_{ij}r^2}{r^2} d{\mathbf{r}}

In the equations above $ r=\vert{\mathbf{r}}\vert$, $ r_i$ the i-th component of $ {\mathbf{r}}$, and $ {\mathbf{r}}$ is taken relative to the position of the nucleus $ {\mathbf{R}}_I$.

The nuclear gyromagnetic ratios should be specified (in MHz, for $ H_0=1$ T) by means of the NGYROMAG-tag:

NGYROMAG = gamma_1  gamma_2 ... gamma_N
where one should specify one number for each of the species on the POSCAR file. If one does not set NGYROMAG in the INCAR file, VASP assumes a factor of 1 for each species.

As usual, all output is written to the OUTCAR file. VASP writes three blocks of data, that look something like:

 Fermi contact (isotropic) hyperfine coupling parameter (MHz)
  ion      A_pw      A_1PS     A_1AE     A_1c      A_tot
   1       ...       ...       ...       ...       ...
  ..       ...       ...       ...       ...       ...

with an entry for each ion on the POSCAR file. $ A_{\mathrm{pw}}$, $ A_{\mathrm{1PS}}$, $ A_{\mathrm{1AE}}$, and $ A_{\mathrm{1c}}$ are the plane wave, pseudo one-center, all-electron one-center, and one-center core contributions to the Fermi contact term, respectively. The total Fermi contact term is given by $ A_{\mathrm{tot}}$. Beware: for the moment we have chosen NOT to include the core contributions $ A_{\mathrm{1c}}$ in $ A_{\mathrm{tot}}$. If you so want, you should add them by hand to $ A_{\mathrm{tot}}$. Core electronic contributions to the Fermi contact term are calculated in the manner proposed in Ref. [153].

The dipolar constributions are listed next:

 Dipolar hyperfine coupling parameters (MHz)
  ion      A_xx      A_yy      A_zz      A_xy      A_xz      A_yz
   1       ...       ...       ...       ...       ...       ...
  ..       ...       ...       ...       ...       ...       ...

Again one line per ion in the POSCAR file.

The total hyperfine tensors are written as:

 Total hyperfine coupling parameters after diagonalization (MHz)
 (convention: |A_zz| > |A_xx| > |A_yy|)
  ion      A_xx      A_yy      A_zz     asymmetry (A_yy - A_xx)/ A_zz
   1       ...       ...       ...         ...
  ..       ...       ...       ...         ...

i.e., the tensors have been diagonalized and rearranged.

N.B.: The Fermi contact term is strongly dominated by the all-electron one-center contribution $ A_{\mathrm{1AE}}$. Unfortunately, this particular term is quite sensitive to the number and eigenenergy of the all-electron partial waves that make up the one-center basis set, i.e., to the particulars of the PAW dataset you are using. As a result the Fermi contact term may strongly depend on the choice of PAW dataset.

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