View source for Basis set convergence of RPA-ACFDT calculations

To reach technical convergence, a number of flags are available to control the evaluation of the ACFDT-RPA correlation energy in the fourth step. The expression for the ACFDT-RPA correlation energy reads:
<math>E_{\rm c}^{\rm RPA}=\int_{0}^{\infty} \frac{\mathrm{d}\omega}{2\pi} \sum_{{\mathbf{q}}\in \mathbf{BZ} }\sum_{{\mathbf{G}}} \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\} </math>.

The sum over reciprocal lattice vectors has to be truncated at some <math>\mathbf{G}_{\mathrm{max}}</math>, determined by <math>\frac{\hbar^2|{\mathbf{G}}+{\mathbf{q}}|^2}{2\mathrm{m}_e}</math> < {{TAG|ENCUTGW}}, which can be set in the {{TAG|INCAR}} file. The default value is <math>\frac{2}{3}\times</math> {{TAG|ENCUT}}, which experience has taught us not to change. For systematic convergence tests, instead increase {{TAG|ENCUT}} and repeat steps 1 to 4, but be aware that the "maximum number of plane-waves" changes when {{TAG|ENCUT}} is increased. Note that it is virtually impossible, to converge absolute correlation energies. Rather concentrate on relative energies (e.g. energy differences between two solids, or between a solid and the constituent atoms).

Since correlation energies  converge very slowly with respect to <math>\mathbf{G}_{\mathrm max }</math>, VASP automatically extrapolates to the infinite basis set limit using a linear regression to the equation: {{cite|harl:2008}}{{cite|harl:2010}}{{cite|klimes:2014}}

<math>E_{\mathrm{c}}({\mathbf{G}})=E_{\mathrm{c}}(\infty)+\frac{A}{{\mathbf{G}}^3}</math>.

Furthermore, the Coulomb kernel is smoothly truncated between {{TAG|ENCUTGWSOFT}} and {{TAG|ENCUTGW}} using a simple cosine like window function (Hann window function). The default for {{TAG|ENCUTGWSOFT}} is 0.8<math>\times</math>{{TAG|ENCUTGW}} (again we do not recommend to change this default).

The integral over <math>\omega</math> is evaluated by means of a highly accurate minimax integration.{{cite|kaltak:2014}} The number of <math>\omega</math> points is determined by the flag {{TAG|NOMEGA}}, whereas the energy range of transitions is determined by the band gap and the energy difference between the lowest occupied and highest unoccupied one-electron orbital. VASP determines these values automatically (from vasp.5.4.1 on), and the user should only carefully converge with respect to the number of frequency points {{TAG|NOMEGA}}. A good choice is usually {{TAG|NOMEGA}}=12, however, for large gap systems one might obtain <math>\mu</math>eV convergence per atom already using 8 points, whereas for metals up to {{TAG|NOMEGA}}=24 frequency points are sometimes necessary, in particular, for large unit cells.

Strictly adhere to the steps outlines above. Specifically, be aware that steps two and three require the {{TAG|WAVECAR}} file generated in step one, whereas step four requires the {{TAG|WAVECAR}} and {{TAG|WAVEDER}} file generated in step three (generated by setting {{TAG|LOPTICS}}=''.TRUE.'').
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