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Exact exchange plane-wave-pseudopotential calculations for slabs
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10.1063/1.4863219
/content/aip/journal/jcp/140/18/10.1063/1.4863219
http://aip.metastore.ingenta.com/content/aip/journal/jcp/140/18/10.1063/1.4863219

Figures

Image of FIG. 1.
FIG. 1.

--integrated density of highest occupied band of graphene in supercell approach: (a) complete width of vacuum for = 6 ( = 2.461 Å); (b) half of vacuum for three different (for technical details see Sec. III ).

Image of FIG. 2.
FIG. 2.

-dependence of highest occupied and lowest unoccupied states of graphene in supercell approach: bands resulting from = 7 (solid line) versus = 5 (dashed line) for fixed (both the Γ and the Dirac-point are shown; for technical details see Sec. III ).

Image of FIG. 3.
FIG. 3.

--averaged asymptotic exchange potential of graphene: supercell result (36) with -point sampling function (38) for = 21 versus single slab potential (15) — (a) (); (b) (for technical details see Sec. III ).

Image of FIG. 4.
FIG. 4.

Si(111) slab with 3 bilayers.

Image of FIG. 5.
FIG. 5.

Asymptotic decay of radial 2 orbital of atomic Carbon: exact numerical EXX pseudo-orbital versus Fourier representations truncated at different cut-off energies.

Image of FIG. 6.
FIG. 6.

EXX band structure of graphene with sheet separation of 3 : KLI approximation versus full OPM.

Image of FIG. 7.
FIG. 7.

--averaged total KS potential of graphene with vacuum of 3 : KLI approximation versus full OPM. = 6.97 bohrs corresponds to the middle of the vacuum.

Image of FIG. 8.
FIG. 8.

EXX/KLI band structure of graphene for different sheet separations: 7 versus of 3 .

Image of FIG. 9.
FIG. 9.

Non-normalized EXX/KLI potential (in Ha) of graphene in planes spanned by -axis and (a) -axis, (b) diagonal of --plane, with the graphene sheet being located at = 0 and the two atoms in the unit cell sitting at , . The -range covers exactly half of the supercell, for (a) , for (b) , ∈ [0, 2 ].

Image of FIG. 10.
FIG. 10.

Derivative of --averaged potential (52) for graphene: total EXX/KLI KS potential for different width of the vacuum (7 , 6 , and 5 ) versus EXX/KLI exchange potential for vacuum width of 7 as well as asymptotic exchange potential (15) (evaluated with the --averaged density of the highest occupied band) and −1/.

Image of FIG. 11.
FIG. 11.

Asymptotic form of total KS potential of graphene: --averaged EXX/KLI potential (normalized at the point indicated, vacuum width of 7 ) versus total LDA potential (including VWN correlation) and −1/.

Image of FIG. 12.
FIG. 12.

--averaged total KS potential of graphene: EXX/KLI potential versus LDA potential (including VWN correlation) and −1/.

Image of FIG. 13.
FIG. 13.

Derivative of --averaged potential (52) for Si(111) slab with 3 bilayers: total EXX/KLI KS potential for different width of the vacuum ( and ) and two different -point samplings (6 × 6 × 1 and 6 × 6 × 2) versus EXX/KLI exchange potential for vacuum width of as well as asymptotic exchange potential (15) (evaluated with the --averaged density of the highest occupied band) and −1/.

Image of FIG. 14.
FIG. 14.

Band structure of graphene with sheet separation of 7 : EXX/KLI versus LDA (including VWN correlation 27 ). The right-hand scale shows the energies relative to the EXX/KLI vacuum limit.

Image of FIG. 15.
FIG. 15.

EXX/KLI exchange potential of Si(111) slab with 3 bilayers in plane spanned by -axis and primitive vector . The middle of the slab corresponds to = 0, the outermost atoms sit at = = 6.66 bohrs with one atom placed at (0, 0, ), the middle of the outermost bilayer is at 5.92 bohrs (compare Fig. 4 ).

Image of FIG. 16.
FIG. 16.

LDA exchange potential of Si(111) slab with 3 bilayers (all details as in Fig. 15 ).

Image of FIG. 17.
FIG. 17.

--averaged exchange potential of Si(111) slab with 3 bilayers: EXX/KLI result versus LDA and two asymptotic forms (all details as in Fig. 15 ).

Tables

Generic image for table
Table I.

Work function of graphene as well as Si(111) slabs (without relaxation): EXX-only (in KLI approximation) versus LDA and PBEsol-GGA 36 results with and without correlation. The uncertainties given for the EXX values have been estimated by variation of the normalization point, the width of the vacuum and the -point sampling. The experimental value for graphene 37 is also listed (all energies in eV).

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/content/aip/journal/jcp/140/18/10.1063/1.4863219
2014-01-31
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Exact exchange plane-wave-pseudopotential calculations for slabs
http://aip.metastore.ingenta.com/content/aip/journal/jcp/140/18/10.1063/1.4863219
10.1063/1.4863219
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