4.5 Converging SCF Calculations 4.5.5 Level-Shifting 4.5.7 Geometric Direct Minimization (GDM)

4.5.6 Pseudo-Fractional Occupation Number Method (pFON)

(September 1, 2024)

An alternative to level-shifting for cases exhibiting small (or zero) HOMO/LUMO gaps is the pseudo-fraction occupation number (pFON) approach, ¹⁰⁶⁵ Rabuck A. D., Scuseria G. E.
J. Chem. Phys.
(1999), 110, pp. 695. Link which corresponds to a “smearing out” of the occupation numbers at the HOMO level. Often, this improves the stability and accelerates the convergence by eliminating the discontinuous occupancy changes (from one SCF iteration to the next) that can arise in small-gap systems. Essentially, more than one electron configuration is allowed during the same orbital optimization, with fractional occupancies. This is formally equivalent to a finite-temperature formalism.

The pFON method introduces a density matrix

P_{\mu\nu}=\sum_{p=1}^{N}\;n_{p}C_{\mu p}C_{\nu p}

(4.37)

with occupancies $0\leq n_{p}\leq 1$ that can be fractional, whereas for a conventional SCF calculation either $n_{p}=1$ (occupied) or $n_{p}=0$ (virtual). In pFON, the occupation numbers follow a Fermi-Dirac distribution,

n_{p}=\bigl{(}1+e^{(\epsilon_{p}-\epsilon_{\mathrm{F}})/kT}\bigr{)}^{-1}\;,

(4.38)

where $\epsilon_{p}$ is an SCF eigenvalue (orbital energy) and $T$ is a temperature. In Q-Chem’s implementation, the Fermi energy is set to $\epsilon_{\mathrm{F}}=(\epsilon_{\mathrm{HOMO}}+\epsilon_{\mathrm{LUMO}})/2$ . To ensure conservation of the total number of electrons, the pFON approach re-scales the occupation numbers so that $\sum_{p}\;n_{p}=N_{\mathrm{el}}$ .

There are several parameters to control the electronic temperature $T$ throughout a pFON SCF run. The temperature can either be held constant at finite temperature ( $T_{\mathrm{init}}$ = $T_{\mathrm{final}}$ ), or the system can be cooled from a higher temperature down to the final temperature. So far, no zero-temperature extrapolation has been implemented.

OCCUPATIONS

OCCUPATIONS
       Activates pFON calculation.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Integer occupation numbers 1 Not yet implemented 2 Pseudo-fractional occupation numbers (pFON)
RECOMMENDATION:
       Use pFON to improve convergence for small-gap systems.

FON_T_START

FON_T_START
       Initial electronic temperature (in K) for FON calculation.
TYPE:
       INTEGER
DEFAULT:
       1000
OPTIONS:
       Any desired initial temperature.
RECOMMENDATION:
       Pick the temperature to either reproduce experimental conditions (e.g. room temperature) or as low as possible to approach zero-temperature.

FON_T_END

FON_T_END
       Final electronic temperature for FON calculation.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       Any desired final temperature.
RECOMMENDATION:
       Pick the temperature to either reproduce experimental conditions (e.g. room temperature) or as low as possible to approach zero-temperature.

FON_NORB

FON_NORB
       Number of orbitals above and below the Fermi level that are allowed to have fractional occupancies.
TYPE:
       INTEGER
DEFAULT:
       4
OPTIONS:
       $n$ number of active orbitals
RECOMMENDATION:
       The number of valence orbitals is a reasonable choice.

FON_T_SCALE

FON_T_SCALE
       Determines the step size for the cooling.
TYPE:
       INTEGER
DEFAULT:
       90
OPTIONS:
       $n$ temperature is scaled by $0.01\cdot n$ in each cycle (cooling method 1) $n$ temperature is decreased by n K in each cycle (cooling method 2)
RECOMMENDATION:
       The cooling rate should be neither too slow nor too fast. Too slow may lead to final energies that are at undesirably high temperatures. Too fast may lead to convergence issues. Reasonable choices for methods 1 and 2 are 98 and 50, respectively. When in doubt, use constant temperature.

FON_E_THRESH

FON_E_THRESH
       DIIS error below which occupations will be kept constant.
TYPE:
       INTEGER
DEFAULT:
       4
OPTIONS:
       $n$ freeze occupations below DIIS error of $10^{-n}$
RECOMMENDATION:
       This should be one or two numbers bigger than the desired SCF convergence threshold.

FON_T_METHOD

FON_T_METHOD
       Selects cooling algorithm.
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       1 temperature is scaled by a factor in each cycle 2 temperature is decreased by a constant number in each cycle
RECOMMENDATION:
       We have made slightly better experience with a constant cooling rate. However, choose constant temperature when in doubt.

Example 4.11 pFON calculation of a metal cluster.

$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
   MAX_SCF_CYCLES  200
   ECP             fit-lanl2dz
   BASIS           lanl2dz
   OCCUPATIONS     2    ! pseudo-fractional occupation numbers
   FON_NORB        10   ! 10 fractionally occupied orbitals above and below the Fermi level
   FON_T_START     1000 ! starting electronic temperature: 1000 K
   FON_T_END       0    ! final electronic temperature: 0 K
   FON_T_METHOD    2    ! constant cooling scheme
   FON_T_SCALE     25   ! reduce the temperature by 25 K per cooling step
   FON_E_THRESH    5    ! freeze occupation numbers once DIIS error is 10-5
   GEN_SCFMAN        false
   INTEGRAL_SYMMETRY false
$end

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4.5.6 Pseudo-Fractional Occupation Number Method (pFON)