OEP calculation for a Be atom

The optimized effective potential (OEP) framework allows one to use orbital-dependent functionals while remaining in the Kohn-Sham picture. In Octopus, the default OEP approximation is the Krieger-Li-Iafrate (KLI) scheme¹, which is much faster than the full OEP solution². In this tutorial we therefore start from a KLI calculation for an isolated Be atom, and then restart from that result to run the full OEP calculation. While we do here exact exchange (EXX) calculations, this can also be used for mGGA functionals with dependence on the kinetic energy density for instance.

KLI approximation

We begin with the KLI input:

CalculationMode = gs
FromScratch = yes

PseudopotentialSet = hgh_lda_sc
ExperimentalFeatures = yes

%Coordinates
 'Be' |  0 | 0 | 0
%

Spacing = 0.2
Radius = 10

XCFunctional = oep_x
OEPLevel = oep_kli

%Output
  potential
  density
%
OutputFormat = axis_x

The OEP-specific ingredients are:

• XCFunctional = oep_x: use exchange-only OEP.
• OEPLevel = oep_kli: use the KLI approximation to the OEP equation.

We use here a moderate grid spacing for a fast but physical example (Spacing = 0.2) and we define a compact simulation box for this atom (Radius = 10).

This KLI calculation converges in 23 SCF iterations and provides the restart data for the following full-OEP step. The corresponding eigenvalues are:

SCF converged in   23 iterations

Some of the states are not fully converged!
Eigenvalues [H]
 #st  Spin   Eigenvalue      Occupation
   1   --    -4.174637       2.000000
   2   --    -0.309001       2.000000
   3   --    -0.056433       0.000000

Full OEP restarted from KLI

To perform the full OEP calculation, we now rerun Octopus in the same directory with:

CalculationMode = gs
FromScratch = no

PseudopotentialSet = hgh_lda_sc
ExperimentalFeatures = yes

%Coordinates
 'Be' |  0 | 0 | 0
%

Spacing = 0.2
Radius = 10

XCFunctional = oep_x
OEPLevel = oep_full

%Output
  potential
  density
%
OutputFormat = axis_x

OptimizeChebyshevFilterDegree = no
LRMaximumIter = 1
LRTolFinalTol = 1.0e-9

The important change is:

• OEPLevel = oep_full: switch from KLI to the full OEP equation.
• FromScratch = no: reuse the converged KLI ground-state from restart/gs. While we could do the calculation from scratch, we restarted here from the KLI result, as this results in less iterations, and hence a shorter wall time for the calculation.

The final full-OEP eigenvalues are:

Some of the states are not fully converged!
Eigenvalues [H]
 #st  Spin   Eigenvalue      Occupation
   1   --    -4.172733       2.000000
   2   --    -0.309035       2.000000
   3   --    -0.062219       0.000000

The total energies of the two runs are very close because we considered a simple Be atom. More complicated molecules or solids show more pronounced differences between the approximate and exact solutions for the OEP equation.

References