^{1}

### Abstract

The exact exchange of density functional theory is applied to both free-standing graphene and a Si(111) slab, using the plane-wave pseudopotential (PWPP) approach and a periodic repetition of the supercell containing the slab. It is shown that (i) PWPP calculations with exact exchange for slabs in supercell geometry are basically feasible, (ii) the width of the vacuum required for a decoupling of the slabs is only moderately larger than in the case of the local-density approximation, and (iii) the resulting exchange potential v x shows an extended region, both far outside the surface of the slab and far from the middle of the vacuum region between the slabs, in which v x behaves as −e ^{2}/z, provided the width of the vacuum is chosen sufficiently large. This last result is corroborated by an analytical analysis of periodically repeated jellium slabs. The intermediate −e ^{2}/z behavior of v x can be used for an absolute normalization of v x and the total Kohn-Sham potential, which, in turn, allows the determination of the work function.

Very helpful discussions with J. Braun, D. Ködderitzsch, J. Minar, and T. Stroucken are gratefully acknowledged. The calculations for this work have been performed on the computer cluster of the LOEWE Center for Scientific Computing of J. W. Goethe University Frankfurt am Main.

I. INTRODUCTION

II. THEORY

A. States and density of single jellium slab

B. Exchange energy and potential of single jellium slab

C. States and density of jellium slab in supercell geometry

D. Exchange energy and potential of jellium slab in supercell geometry

E. Exchange energy density of arbitrary slab

III. COMPUTATIONAL DETAILS

IV. RESULTS

A. Comparison of full OPM solution with KLI approximation

B. Decoupling of slabs

C. Normalization of total and exchange potential

D. Comparison of EXX and LDA results

V. SUMMARY

### Key Topics

- Laser Doppler velocimetry
- 34.0
- Graphene
- 28.0
- Work functions
- 17.0
- Brillouin scattering
- 6.0
- Eigenvalues
- 6.0

## Figures

x-y-integrated density of highest occupied band of graphene in supercell approach: (a) complete width of vacuum for a = 6a 0 (a 0 = 2.461 Å); (b) half of vacuum for three different a (for technical details see Sec. III ).

x-y-integrated density of highest occupied band of graphene in supercell approach: (a) complete width of vacuum for a = 6a 0 (a 0 = 2.461 Å); (b) half of vacuum for three different a (for technical details see Sec. III ).

q-dependence of highest occupied and lowest unoccupied states of graphene in supercell approach: bands resulting from a = 7a 0 (solid line) versus a = 5a 0 (dashed line) for fixed k (both the Γ and the Dirac-point are shown; for technical details see Sec. III ).

q-dependence of highest occupied and lowest unoccupied states of graphene in supercell approach: bands resulting from a = 7a 0 (solid line) versus a = 5a 0 (dashed line) for fixed k (both the Γ and the Dirac-point are shown; for technical details see Sec. III ).

x-y-averaged asymptotic exchange potential of graphene: supercell result (36) with k-point sampling function (38) for a = 21a 0 versus single slab potential (15) — (a) v x, av(z); (b) (for technical details see Sec. III ).

Si(111) slab with 3 bilayers.

Si(111) slab with 3 bilayers.

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

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

EXX band structure of graphene with sheet separation of 3a 0: KLI approximation versus full OPM.

EXX band structure of graphene with sheet separation of 3a 0: KLI approximation versus full OPM.

x-y-averaged total KS potential of graphene with vacuum of 3a 0: KLI approximation versus full OPM. z = 6.97 bohrs corresponds to the middle of the vacuum.

x-y-averaged total KS potential of graphene with vacuum of 3a 0: KLI approximation versus full OPM. z = 6.97 bohrs corresponds to the middle of the vacuum.

EXX/KLI band structure of graphene for different sheet separations: 7a 0 versus of 3a 0.

EXX/KLI band structure of graphene for different sheet separations: 7a 0 versus of 3a 0.

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

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

Derivative of x-y-averaged potential (52) for graphene: total EXX/KLI KS potential for different width of the vacuum (7a 0, 6a 0, and 5a 0) versus EXX/KLI exchange potential for vacuum width of 7a 0 as well as asymptotic exchange potential (15) (evaluated with the x-y-averaged density of the highest occupied band) and −1/z.

Derivative of x-y-averaged potential (52) for graphene: total EXX/KLI KS potential for different width of the vacuum (7a 0, 6a 0, and 5a 0) versus EXX/KLI exchange potential for vacuum width of 7a 0 as well as asymptotic exchange potential (15) (evaluated with the x-y-averaged density of the highest occupied band) and −1/z.

Asymptotic form of total KS potential of graphene: x-y-averaged EXX/KLI potential (normalized at the point indicated, vacuum width of 7a 0) versus total LDA potential (including VWN correlation) and −1/z.

Asymptotic form of total KS potential of graphene: x-y-averaged EXX/KLI potential (normalized at the point indicated, vacuum width of 7a 0) versus total LDA potential (including VWN correlation) and −1/z.

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

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

Derivative of x-y-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 k-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 x-y-averaged density of the highest occupied band) and −1/z.

Derivative of x-y-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 k-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 x-y-averaged density of the highest occupied band) and −1/z.

Band structure of graphene with sheet separation of 7a 0: EXX/KLI versus LDA (including VWN correlation ^{27} ). The right-hand scale shows the energies relative to the EXX/KLI vacuum limit.

Band structure of graphene with sheet separation of 7a 0: EXX/KLI versus LDA (including VWN correlation ^{27} ). The right-hand scale shows the energies relative to the EXX/KLI vacuum limit.

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

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

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

x-y-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 ).

x-y-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

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 k-point sampling. The experimental value for graphene ^{37} is also listed (all energies in eV).

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 k-point sampling. The experimental value for graphene ^{37} is also listed (all energies in eV).

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