The systems discussed inPubl. 4and Sec. 4.7, i.e., organic molecules and hydrogen chains, were calculated using the Bayreuth version [MKHM06] of the program packagePARSEC[KMT+06], a real-space code based on finite-difference methods. InPARSEC, core electrons are treated via the pseudopotential approach described in Appendix A.4. Here, norm-conserving pseudopotentials of Troullier-Martins type are used.
While pseudopotentials for semilocal functionals can be created in a straightforward manner (cf. Appendix A.4), their construction for orbital-dependent functionals is more involved [KK08].
Therefore, it seems inadvisable to construct pseudopotentials for each orbital-dependent functional anew, and instead a different approach is employed. This approach consists of using orbital-dependent functionals on top of pseudopotentials that were constructed from different functional approximations. Such a strategy has already proven to be justified for GSIC using LSDA pseu-dopotentials [HKKK12].
For the global hybrid PBEh one can show that good agreement in the KS eigenvalues with all-electron calculations can be obtained if either PBE or EXX [EHS+01] pseudopotentials are employed. In Appendix A.6 a direct comparison is provided for the CO and N2 molecule, with deviations typically ranging between 0.05−0.1 eV for the valence states. For the local hybrids ISO and ISOII, however, such an approach does not yield eigenvalues that agree with the all-electron results within an acceptable accuracy (see supplemental material ofPubl. 4). Instead, rather big deviations for the KS eigenvalues and the asymptotic slopesγσ occur. In fact, the asymptotic slope of a local hybrid calculated on top of, e.g, an EXX pseudopotential is systematically smaller than the correct all-electron result. Based on Eq. (4.10) and the assumption that the shape of the ho KS orbital does not differ greatly between the all-electron and the pseudopotential calculation, it is conclusive that these differences must be caused by an incorrect representation of the LMF.
In Fig. A.7 the LMF fxISOII(r)of ISOII(c∗=0) computed in different schemes is plotted along the interatomic axis for the CO molecule. It becomes evident that the LMF obtained on top of an EXX pseudopotential substantially differs from the all-electron result in the vicinity of the atomic core regions. Here, besides strong oscillations due to numerical instabilities, the LMF is overall too large, while the long-range behavior of fxISOII(r)is described rather well.
0.0 0.2 0.4 0.6 0.8 1.0
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
fISOII x
(r)
z(a.u.) ISOII(c∗=0)
C O
all-electron PP EXX PP EXX + cd
Figure A.7:LMF of ISOII with c∗=0 for the CO molecule along the interatomic z-axis obtained inDARSEC(black line) and inPARSECusing an EXX pseudopotential with (blue line) and without (red line) explicit consideration of the core density.
A.5 Compatibility with Pseudopotentials
The incorrect behavior of the LMF around the atomic positions can be attributed to the fact that, when computed in a pseudopotential context, only the valence densitynv(r) is used for its construction. In said regions, however, the core density nc(r) is dominating and exhibits great influence on, e.g, the detection functionτW(r)/τ(r)[PKZB99]. To obtain agreement between the LMF of an all-electron and a pseudopotential calculation, it is therefore necessary to explicitly includenc(r)via a core-density correction (denoted with the superscript "cd") according to
τWcd(r) = |∇(nv(r) +nc(r))|2
Here,ϕiσv(r)denotes the KS orbitals of the valence states.
In Fig. A.7 the corrected LMF is given in blue for fxISOII(r)of ISOII(c∗=0), indicating great improvement in the description of the core regions. Besides small remaining instabilities directly at the atomic positions, the core-density-corrected LMF describes the all-electron function much more accurately. A similar result is illustrated in Fig. A.8 for the LMF of ISOII(c∗=0.5). Here, due to the finite value ofc∗, the core-density-corrected reduced density gradient
tcd(r)2 that also for a LMF using the reduced density gradient, explicit inclusion of the core density yields great improvement in the vicinity of the nuclei in contrast to the uncorrected LMF.
0.0
Figure A.8:LMF of ISOII withc∗=0.5 for the CO molecule along the interatomic z-axis obtained inDARSEC(black line) and inPARSECusing an EXX pseudopotential with (blue line) and without (red line) explicit consideration of the core density.
Importantly, the core-density-corrected LMFs are inserted at the level of the xc energy for the corresponding local hybrid, providing a corrected basis for the functional derivativesuiσ(r).
Fur-ther, numerical instabilities can be limited by decreasing the expansion order of the finite-difference calculation. In the supplemental material ofPubl. 4 detailed numerical results are provided for the molecules CO, N2 and NH. It is demonstrated that core-density-corrected LMFs in the spirit of Eqs. (A.3), (A.4) and (A.5) systematically enhance the agreement of the asymptotic slopeγσ with all-electron results. Governed by this improvement, direct comparisons of KS eigenvalues systematically yield improved agreement between all-electron and pseudopotential results.
Note that the core-density correction of the KS kinetic energy density in Eq. (A.4) explicitly assumes that
∑
kν core states|∇ϕkνc (r)|2≈ |∇(nc(r))12|2, (A.6) withϕkνc (r)denoting the KS orbitals of the core states obtained by all-electron calculations. While Eq. (A.6) is in fact exact for atoms with only one doubly occupied core orbital ofs-character (as, for instance, the C, N, and O atom), this is not the case for other atoms, since here
|∇(nc(r))12|2=|∇
∑
core stateskν
|ϕkνc (r)|2
12
|26=
∑
core stateskν
|∇ϕkνc (r)|2. (A.7)
Hence, τWcd(r) and tcd(r)2
are accurately represented for such atoms while τcd(r) is not. For a description of the latter quantity the individual core orbitals are required, andτcd(r) cannot be reproduced if only the core density is available. This issue is illustrated in Fig. A.9, which shows
fxISOII(r)of ISOII(c∗=0) for the SiO molecule calculated on top of an LDA pseudopotential.
0.0 0.2 0.4 0.6 0.8 1.0
-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 fISOII x
(r)
z(a.u.)
ISOII(c∗=0) Si O all-electron
PP LDA + cd
Figure A.9:LMF of ISOII withc∗=0 for the SiO molecule along the interatomic z-axis obtained in DARSEC (black line) and in PARSEC (blue line) using an LDA pseudopotential with cd.
It becomes evident from Fig. A.9 that the LMF around the oxygen atom is described acceptably well besides a small instability at the atomic position. In the vicinity of the silicon atom, which contains five doubly occupied core orbitals in the pseudopotential representation (two withs- and three withp-character),PARSECerroneously obtains fxISOII(r)≈0 and does not resolve the shell structure given by the all-electron result. This behavior is rooted in the fact that around the atomic positions the core density is dominating and, ifτcd(r) is obtained via (A.4), one yieldsτW≈τ in
A.5 Compatibility with Pseudopotentials
this region3. Hence, for systems with several core orbitals the approximation of Eq. (A.6) is not resulting in a reliable representation of the LMF.
As a summary, Table A.3 provides a comparison of the asymptotic slope according to Eq. (4.10) obtained inDARSECandPARSECby using ISO and ISOII for different values ofcandc∗ for the CO and SiO molecule. Since both the carbon and the oxygen atom contain only one core orbital, good agreement ofγ is obtained for CO with both ISO and ISOII for all parameters investigated.
For SiO, however, ISOII produces notably larger deviations inγthan ISO. This behavior is plausible considering that SiO is a spin-unpolarized system. Thus, the LMF of ISO only contains the reduced density gradient, which is described accurately even for systems with several core orbitals. ISOII, on the other hand, uses the incorrectly represented detection functionτW(r)/τ(r) also for spin-unpolarized systems, which leads to larger deviations inγ. More precisely, the deviations observed in ISOII decrease with increasing parameter values, since larger values ofc∗ increase the effect of the reduced density gradient and suppress the error introduced byτW(r)/τ(r)at the position of the silicon atom.
Table A.3:The asymptotic slopes of ISO and ISOII for CO and SiO.
CO SiO
γ ∆γ γ ∆γ
all- PP EXX all- PP LDA
Functional electron + cd electron + cd
cISO=0.5 0.634 0.634 0.000 0.625 0.626 −0.001
c=1.0 0.700 0.700 0.000 0.688 0.690 −0.002
c=2.5 0.796 0.797 −0.001 0.786 0.789 −0.003 ISOII
c∗=0.0 0.827 0.827 0.000 0.806 0.814 −0.008
c∗=0.5 0.871 0.871 0.000 0.857 0.862 −0.005
c∗=1.0 0.894 0.893 −0.001 0.883 0.887 −0.004
c∗=2.5 0.928 0.927 −0.001 0.922 0.925 −0.003
The discussion of the differences between the CO and the SiO molecule indicates a fundamental difficulty to describe systems with several core orbitals via the approach of correcting LMFs using the core density. The inability to accurately reproduce the functionτW(r)/τ(r) by using only the core density restricts the use of this approach to period 1 or period 2 elements. While this range of elements is sufficient to calculate the organic molecules discussed in Sec. 4.7, many interesting elements such as, for instance, transition metals are out of reach. Especially for larger atoms, the effect of an incorrect representation of the detection function is expected to be more severe. One possibility to overcome this obstacle is to provide the individual core orbitalsϕkνc (r)of all-electron calculations as input in analogy to how it is currently handled in PARSEC for the core density.
Alternatively, an orbital-free detection function could be employed, which further potentially limits the influence of orbital nodal planes on this quantity [dSC15].
3Note that the von Weizsäcker kinetic energy density can be expressed asτW(r) =|∇n(8n(rr))|2 =12|∇(n(r))12|2.