A.3 Details of the Potential Asymptotic Behavior
In this section, the important findings ofPubl. 2as well as Secs. 4.5 and 4.6 are investigated in more detail. Based on the illustrative example of the F2molecule I discuss the influence of orbital nodal planes and nodal axes1 in the ho KS orbital on the asymptotics of the local xc potential for local hybrid functionals. Further, a summary of the asymptotic behavior of relevant DFAs, including local and global hybrids as wells as GSIC, is provided in tabular form.
As a starting point, Fig. A.1 shows relevant quantities the for F2molecule along the(xz)-plane as facilitated by the program packageDARSEC(see Appendix A.1 for details). The F2 molecule is evaluated on its experimental bond length ofRexpF−F =2.6695 a.u.(see Appendix A.2). The two separate fluorine atoms are located atrF= (x,y,z) = (0,0,±RexpF−F/2). The spin indexσis neglected in the following since F2is fully spin unpolarized. I here focus on the ISOII local hybrid functional for the reasons explained in Sec. 4.3.
-4 -2 0 2 4
Figure A.1:(a), (b): the highest lying KS orbitals of F2 plotted between their respective maximum and minimum values (in atomic units); (c), (d): the detection function τW/τ(r)obtained by ISOII withc∗=0; (e), (f): the ISOII LMF forc∗=0 and c∗=0.5.
Both the ho and the ho−1 KS orbital are twofold degenerate (labeled "2x deg.") and exhibit a nodal axis along thez-axis. The ho KS orbital contains an additional nodal plane along the (xy)-plane. The corresponding orbitals are plotted in Figs. A.1(a) and A.1(b). The resulting detection function τW(r)/τ(r) is depicted in Fig. A.1(c) for a large region of the numerical grid, a more
1Nodal planes and nodal axes of KS orbitals have the same effect onτW(r)/τ(r)and only differ in the dimension of their influence (2D vs. 1D). Therefore, I mainly refer to nodal planes, but all arguments hold for nodal axes as well.
detailed view of the core area is provided in Fig. A.1(d). Note that these quantities are obtained with the ISOII local hybrid usingc∗ =0, but it can readily be assumed that fundamental orbital features such as nodal planes are not influenced by the DFA put to task. Therefore, the KS orbitals and consequentlyτW(r)/τ(r)appear similar when calculated with other functionals.
The detection function reaches its intended asymptotic limit along all spatial directions where ϕho(r)andϕho−1(r)do not exhibit nodal features. Along the(xy)-plane and thez-axis,τW(r)/τ(r) behaves as predicted by Eq. (4.11). More precisely, the detection function deviates more strongly along thez-axis since heren(r)∼ |ϕho(r)|2+|ϕho−1(r)|2+|ϕho−2(r)|2, yielding
τW
τ −→n.a. |∇ϕho−2|2
|∇ϕho|2+|∇ϕho−1|2+|∇ϕho−2|2 <1. (A.1)
vxc(hartree)
-20-16-12-8-4048121620
z(a.u.)
-20 -16 -12 -8 x-4(a.u.)0 4 8 12 16 20 -0.5
-0.4 -0.3 -0.2 -0.1 0.0
F F
Figure A.2:The local xc potential of F2on the (xz)-plane obtained using ISOII(c∗=0.5). The black grid marks−γ/rwithγ=0.852. The grey line flags grid points located at R=20 a.u..
Based on these properties of the detection function, two different scenarios arise for the long-range behavior of the LMF. For a LMF that employsτW(r)/τ(r)without any additional function, as given by, e.g, Eq. (3.12) and ISOII(c∗=0), the asymptotic behavior follows directly from the properties of the detection function. This case is illustrated in Fig. A.1(e). The other scenario is given by a LMF that uses the detection function, but its long-range behavior is dominated by a different functional ingredient. Examples are the ISO and ISOII local hybrids with finite values for their respective parameter. Here, the reduced density gradient in Eq. (4.2) and Eq. (4.6) suppresses τW(r)/τ(r)in the asymptotic limit, and the LMF approaches 0 as intended. This is demonstrated in Fig. A.1(f) on the example of ISOII withc∗=0.5. In the following, I discuss the influence of orbital nodal planes on the asymptotics ofvxc(r)based on these two scenarios.
Fig. A.2 shows the xc potential for F2 calculated via the KLI approximation using ISOII with c∗=0.5. In general,vxc(r)decays withγ=0.852 as determined according to Eq. (4.10), but along
A.3 Details of the Potential Asymptotic Behavior
thex-andz-axis special potential features occur. These bumps and valleys are manifestations of the nonvanishing asymptotic constants of Eq. (2.60).
Due to the orbital structure of F2, two different constants can be observed in Fig. A.2. Along the x-axis, the nonvanishing asymptotic constant is determined by the ho−1 state, which gives Cx=0.019 hartree. Along thez-axis, both the ho and ho−1 KS orbital vanish and the asymptotic constant is determined by the ho−2 state, yieldingCz=−0.090 hartree for ISOII withc∗=0.5. In Fig. A.3 thevxc(r)of Fig. A.2 is replotted at a fixed radiusR=20 a.u.as function of the polar angle φ together with the relevant asymptotic quantities. It becomes evident thatvxc(r)generally decays with−γ/r. In the vicinity of nodal planes, the potential bends towards the asymptotic constantsCx
atφ=0,πandCzatφ=π2,3π2 2.
−R1 -0.12
-0.09 -0.06 -0.03 0.00
0 π/2 π 3π/2 2π
vxc(hartree)
φ
−Rγ
Cz−Rγ Cx−γR
Figure A.3:The xc potential of F2computed with ISOII(c∗=0.5)plotted along the polar angleφat fixed radiusR=20 a.u.as marked by the grey line in Fig. A.2.
A similar representation is given in Fig. A.4 for the xc potential of ISOII withc∗=0. Here, the potential generally decays withγ =0.776 as determined via Eq. (4.10). Yet, the asymptotic constantsCx=0.016 hartree andCz=−0.071 hartree are not correctly attained byvxc(r). Instead, the potential takes up larger values along the x- andz-axis and their vicinities, as indicate by the bend ofvxc(r) close toφ= π2,3π2. The reason for this deviation from the predicted behavior are the incorrect asymptotic properties of the LMF of ISOII(c∗=0) as depicted in Fig. A.1(e). As mentioned earlier, such a LMF does not fulfill Eq. (4.7) and violates the requisite for Eq. (4.10).
This leads to a different asymptotical behavior for which no general expression is known at this stage. Importantly, the xc potential does not achieve the correct−1/rbehavior in this case either.
Table A.2 provides an overview of the asymptotic behavior of the local xc potential of rele-vant DFAs discussed in this thesis, including the local hybrids ISO and ISOII, the global hybrid PBEh, and the GSIC. The asymptotics of local hybrids are listed for the various in terms of spin polarization and orbital nodal planes. In Table A.2, the first column denotes the functional with a specification of the corresponding parameter, while the second column distinguishes between the cases of spin-unpolarized (spin = 1) and spin-polarized (2) systems. In the third column, it is specified if a nodal plane/axis in the ho KS orbital is assumed. In the last column, the explicit analytical form of the asymptotic decay is given if available. In case no general expression is known, it is denoted as "not generally defined" (n.g.d.). For spin-polarized cases, the index σho
denotes the spin channel that contains the global ho KS state, while the other spin channel is given by ¯σho(cf. Appendix B ofPubl. 2).
2Atφ= π2,3π2, the non-vanishing asymptoticCzis not fully attained byvxc(r)since grid points directly on thez-axis were excluded fromDARSEC-calculations [MKK09].
-0.12 -0.09 -0.06 -0.03 0.00
0 π/2 π 3π/2 2π
vxc(hartree)
φ
−γR
Cz−Rγ Cx−γR
−1R
Figure A.4:The xc potential of F2computed with ISOII(c∗=0)plotted along the polar angle φat fixed radiusR=20 a.u..
Table A.2:Overview of the general potential asymptotics of PBEh, GSIC and the local hybrid functionals ISO and ISOII.
nodal plane/axis asymptotic behavior functional spin in ho KS orbital? vxc(σ)(r) −→
|r|→∞...
PBEh(a>0) 1,2 no −|r|a
1,2 yes C(σ)−|r|a
(G)SIC 1,2 no −|r|1
1,2 yes C(σ)−|r|1
ISO(c=0) 1 no, yes exp(−const· |r|)
2 no σho:−γσ|r|ho; ¯σho: n.g.d.
2 yes σho: n.g.d.; ¯σho: n.g.d.
ISO(c>0) 1,2 no −γ|r|(σ)
yes C(σ)−γ|r|(σ)
ISOII(c=0) 1 no −|r|γ
1 yes n.g.d.
2 no σho:−γσ|r|ho; ¯σho: n.g.d.
2 yes σho: n.g.d.; ¯σho: n.g.d.
ISOII(c>0) 1,2 no −γ|r|(σ)
yes C(σ)−γ|r|(σ)