SCF calculations for systems with zero or small HOMO-LUMO gap (such as metals) can exhibit very slow convergence or may even fail to converge. This problem arises because the energetic ordering of orbitals and states can switch during the SCF optimization leading to discontinuities in the optimization. Using fractional MO occupation numbers can improve the convergence for small-gap systems. In this approach, the occupation numbers of MOs around the Fermi level are allowed to assume non-integer values. This “occupation smearing” allows one to include multiple electron configurations in the same optimization, which improves the stability of the optimization.
We follow the pseudo-Fractional Occupation Number (pFON) method of Rabuck and Scuseria763 that scales the MO occupation used to construct the AO density:
For a conventional (integer occupation number) SCF run, the occupation number is either one (occupied) or zero (virtual). In pFON, the occupation numbers are following a Fermi-Dirac distribution,
where is the respective orbital energy and the Boltzmann constant and temperature, respectively. The Fermi energy is set to in our implementation. To ensure conservation of the total number of electrons, the pFON approach re-scales the occupation numbers so that .
There are several parameters to control the electronic temperature throughout a pFON SCF run. The temperature can either be held constant at finite temperature ( = ), or the system can be cooled from a higher temperature down to the final temperature. So far, no zero-temperature extrapolation has been implemented.
$molecule 0 1 Pt -0.20408 1.19210 0.54029 Pt 2.61132 1.04687 0.66196 Pt 0.83227 0.03296 -1.49084 Pt 0.95832 -1.05360 0.92253 Pt -1.66760 -1.07875 -1.02416 $end $rem METHOD pbe ECP fit-lanl2dz SYMMETRY false OCCUPATIONS 2 ! pseudo-fractional occupation numbers FON_NORB 10 FON_T_START 300 ! electronic temperature: 300 K FON_T_END 300 FON_E_THRESH 5 ! freeze occupation numbers once DIIS error is 10^-5 GEN_SCFMAN false $end