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# Basis set convergence of RPA-ACFDT calculations

The expression for the ACFDT-RPA correlation energy written in terms of reciprocal lattice vectors reads:

${\displaystyle E_{\rm {c}}^{\rm {RPA}}=\int _{0}^{\infty }{\frac {\mathrm {d} \omega }{2\pi }}\sum _{{\mathbf {q} }\in \mathbf {BZ} }\sum _{\mathbf {G} }\left\{(\mathrm {ln} [1-{\tilde {\chi }}^{0}({\mathbf {q} },\mathrm {i} \omega )V({\mathbf {q} })])_{\mathbf {G,G} }+V_{\mathbf {G,G} }({\mathbf {q} }){\tilde {\chi }}^{0}({\mathbf {q} },{\mathrm {i} }\omega )\right\}}$.

The sum over reciprocal lattice vectors has to be truncated at some ${\displaystyle \mathbf {G} _{\mathrm {max} }}$, determined by ${\displaystyle {\frac {\hbar ^{2}|{\mathbf {G} }+{\mathbf {q} }|^{2}}{2\mathrm {m} _{e}}}}$ < ENCUTGW, which can be set in the INCAR file. The default value is ${\displaystyle {\frac {2}{3}}\times }$ ENCUT, which experience has taught us not to change. For systematic convergence tests, instead increase ENCUT and repeat steps 1 to 4, but be aware that the "maximum number of plane-waves" changes when 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 ${\displaystyle \mathbf {G} _{\rm {max}}}$, VASP automatically extrapolates to the infinite basis set limit using a linear regression to the equation: [1][2][3]

${\displaystyle E_{\mathrm {c} }({\mathbf {G} })=E_{\mathrm {c} }(\infty )+{\frac {A}{{\mathbf {G} }^{3}}}}$.

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

The integral over ${\displaystyle \omega }$ is evaluated by means of a highly accurate minimax integration.[4] The number of ${\displaystyle \omega }$ points is determined by the flag 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 NOMEGA. A good choice is usually NOMEGA=12, however, for large gap systems one might obtain ${\displaystyle \mu }$eV convergence per atom already using 8 points, whereas for metals up to 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 WAVECAR file generated in step one, whereas step four requires the WAVECAR and WAVEDER file generated in step three (generated by setting LOPTICS=.TRUE.).