Fractional occupation density

Fractional occupation number weighted density (FOD), introduced by Grimme and Hansen Grimme2015, provides a qualitative indication of static electron correlation and its spatial distribution. It is not a definitive test for multireference character and does not replace a multireference calculation.

For each spin channel CP2K uses the converged Fermi-Dirac occupations and chemical potential:

\[\begin{split} \rho_{\mathrm{FOD}}(\mathbf r)=\sum_{i\sigma}w_{i\sigma}|\psi_{i\sigma}(\mathbf r)|^2, \qquad w_{i\sigma}=\begin{cases} f_{\max}-f_{i\sigma}, & \epsilon_{i\sigma}\leq\mu_\sigma,\\ f_{i\sigma}, & \epsilon_{i\sigma}>\mu_\sigma. \end{cases} \end{split}\]

Here f_max is two for a restricted calculation and one for each unrestricted spin channel. At the chemical potential both expressions give half the maximum occupation. The reported N_FOD (orbital sum) is the sum of these weights, with no real-space quadrature error. The SCF wavefunction and density are not changed by the analysis.

Enable FORCE_EVAL/DFT/PRINT/FOD explicitly. This initial implementation supports GPW at the Gamma point, with positive-temperature SCF/SMEAR/METHOD FERMI_DIRAC. Other smearing schemes, GAPW and explicit k-point calculations are rejected rather than producing a different quantity.

&SCF
  ADDED_MOS 20
  &DIAGONALIZATION
  &END
  &SMEAR
    METHOD FERMI_DIRAC
    ELECTRONIC_TEMPERATURE [K] 5000
  &END
&END
&PRINT
  &FOD
    CUBE T
    STRIDE 1
  &END
&END

This is an input fragment, not a complete calculation. Choose ADDED_MOS for the system and check convergence of N_FOD with the number of available orbitals. A still-occupied last orbital produces a warning. The reference molecular protocol uses TPSS/def2-TZVP and 5000 K. Changing the functional, temperature, basis or pseudopotential changes the diagnostic; a PBE/MOLOPT calculation is not a numerical reproduction of that protocol. Metallic thermal occupations must not be interpreted directly as molecular multireference character.

CUBE defaults to false: scalar-only analysis avoids copying orbitals or allocating density grids. With CUBE T, CP2K additionally prints the full-grid integral and writes the nonnegative FOD density in electrons/bohr cubed. Converge the density grid when comparing that integral to the orbital sum. STRIDE subsamples only the output cube; use STRIDE 1 to integrate the cube without additional sampling error. The cube writer is collective and works in serial and MPI.