An Introduction to the Theory of Crystalline Elemental Solids and ...

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TABLE 2: Bulk properties of C, Si, and Ge, as computed with the LDA, PBE, TPSS, and
PBE0 exchange-correlation functionals. All data is taken from Refs. [63] and [56]. Ref. [63] is
an all-electron study with a Gaussian basis set. Data from ref. [56] has been obtained with the
plane-wave plus PAW method. Note that the PBE0 functional has not yet been employed to
treat Ge.
a 0 (Å) B 0 (Mbar) E coh (eV)
Ref. [63] Ref. [56] Ref. [63] Ref. [56] Ref. [63] Ref. [56]
C LDA 3.54 4.54 8.83
PBE 3.58 3.57 4.22 4.31 7.62 7.71
TPSS 3.58 4.17 7.12
PBE0 3.55 4.67 7.59
expt. 3.57 4.43 7.37
Si LDA 5.43 0.95 5.26
PBE 5.49 5.47 0.89 0.88 4.50 4.56
TPSS 5.48 0.92 4.36
PBE0 5.43 0.99 4.56
expt. 5.43 0.99 4.62
Ge LDA 5.63 0.76 4.72
PBE 5.77 0.63 3.82
TPSS 5.73 0.66 3.78
PBE0
expt. 5.65 0.76 3.87
with the PBE0 functional) PBE0 does, indeed, offer slightly improved performance
compared to the other functionals. PBE0 gets the lattice constants of both materials
correct to within 1 %, the bulk moduli to within 5 % and the cohesive energies to within
3 % of the corresponding experimental values.
In conclusion, we again see that all functionals discussed perform reasonably well in
capturing the variation in lattice constants and bulk moduli for the small collection of
covalently bonded solids considered. As we observed for metals, the situation with regard
to the quantitative determination of E coh is far from satisfactory with the choice of
exchange-correlation functional crucial to the value obtained. Finally, before closing this
section we briefly discuss one further important aspect of DFT calculations of covalent
elemental solids, namely how the computed band gaps compare to the experimental ones.
We have indicated already that the computed Kohn-Sham band gap depends sensitively
on the exchange-correlation functional used and generally it is found that the LDA
and GGA functionals predict Kohn-Sham band gaps that are considerably smaller than
the experimentally observed optical band gaps. The LDA and PBE band gaps of C in
the diamond lattice, for example, are ∼4.2 and ∼4.8 eV, respectively, compared to the
corresponding experimental value of ∼7.3 eV [56, 64]. Moving to the hybrid functionals
such as PBE0 leads to increased band gaps through the inclusion of a fraction of Hartree-
Fock exchange and generally (but not always) this improves the band gap compared
to the pure DFT functionals. The PBE0 band gap for C, for example, is at ∼6.7 eV