ESF_SPLINES

ESF_SPLINES = .FALSE. | .TRUE.
Default: ESF_SPLINES = .FALSE.

Description: Enable k-point interpolation of the electronic structure factor using tricubic splines in ACFDT/RPA calculations.

With ESF_SPLINES =T, the electronic structure factor (ESF) is interpolated using tricubic splines to accelerate k-point convergence of the RPA-correlation energy in ACFDT/RPA calculations. The default settings of the maximum number of iteration steps (ESF_NINTER) and convergence threshold (ESF_CONV) typically yield similar k-point convergence compared to the k-p perturbation theory approach.

Tip: By means of ESF interpolation, one can obtain the RPA-correlation energy for metals and insulators, in contrast to the k-p method that fails for metals.

Algorithm

This feature follows the same idea as in coupled cluster calculations.^[1] To compute the RPA-correlation energy, the electronic structure factor in the RPA

${\displaystyle S({\bf q}+{\bf G}) =\int {\rm d}\omega \left\{(\mathrm{ln}[1-\tilde\chi^0({\mathbf{q}},\mathrm{i}\omega)V({\mathbf{q}})])_{{\mathbf{G,G}}} +V_{{\mathbf{G,G}}}({\mathbf{q}})\tilde\chi^0({\mathbf{q}},{\mathrm{i}}\omega) \right\} }$

is evaluated on the k-point grid defined in KPOINTS and the correlation energy (as its trace) is stored.^[2] To obtain the correlation energy on a finer k-point grid, more q-points are added using tricubic spline interpolation. The resulting energy is compared to the previous correlation energy. This procedure is repeated ESF_NINTER times or until the difference in energy between the interpolation steps is less than ESF_CONV.

ESF-interpolation method vs k-p perturbation theory

Warning: Remove WAVEDER and avoid setting LOPTICS=T when running a job with ESF_SPLINES=T.

Note that the ESF-interpolation method is incompatible with k-p perturbation theory, where the largest q-point integration error

{\displaystyle \lim_{\bf q\to 0} \tilde\chi^0_{{\bf G G}'}({\bf q},{\rm i}\omega) \cdot {\bf V}_{\bf G G'}({\bf q}) }

is added explicitly to the RPA integral. The long-wave limit is ill-defined for metallic systems; hence, the k-p method fails for metals. For the k-p method, the long-wave contribution is stored in the WAVEDER file, and VASP assumes you want to add this term if the file is present in the working directory.

Output

The result of the ESF interpolation is reported to the OUTCAR file in the following format

     cutoff energy     smooth cutoff   RPA   correlation   Hartree contr. to MP2  RPA spline-interp.
-----------------------------------------------------------------------------------------------------
           166.667           133.333      -12.9738715106      -19.7255874374      -13.4968000908
           158.730           126.984      -12.8840657072      -19.6294580403      -13.4017404001
           151.172           120.937      -12.7775593388      -19.5151822998      -13.3005326847
           143.973           115.178      -12.6604147404      -19.3892142669      -13.1868498210
           137.117           109.694      -12.5530911576      -19.2733151174      -13.0861120393
           130.588           104.470      -12.4659186304      -19.1786165194      -12.9778587892
           124.369            99.495      -12.3690601643      -19.0725742983      -12.8709666989
           118.447            94.758      -12.2461267475      -18.9372318755      -12.7590723870
 linear regression    
 converged value                          -14.0340307585      -20.8751715586      -14.5828037654

The last column contains the result from the spline interpolation for the selected energy cutoffs reported in the first column.

Mind: Available as of VASP.6.5.0

References

[liao:jcp:2016-1] K. Liao and A. Grueneis, J. Chem. Phys. 145, 141102 (2016).

[gelbenegger:thesis2018-2] K. Gelbenegger, Thesis: Finite size corrections in the RPA (2018).

[1]

[2]

Algorithm

ESF-interpolation method vs k-p perturbation theory

Output

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References