In this section a direct comparison of KS eigenvalues between all-electron and pseudopotential calculations is provided for the CO and N2molecule computed with the PBEh global hybrid. The all-electron results were obtained inDARSEC, while the pseudopotential eigenvalues were attained withPARSECusing both PBE and EXX pseudopotentials. The PBE pseudopotential cutoff radii were chosen asrc(s) =1.49 a.u.,rc(p) =1.53 a.u. for C,rc(s) =1.50 a.u.,rc(p) =1.50 a.u. for N, andrc(s) =1.45 a.u.,rc(p) =1.45 a.u. for O, while the specifications for the EXX pseudopotential arerc(s) =1.20 a.u.,rc(p) =1.20 a.u. for C,rc(s) =1.19 a.u.,rc(p) =1.19 a.u. for N, andrc(s) = 0.99 a.u.,rc(p) =0.94 a.u. for O.
The corresponding valence-state eigenvalues of PBEh for various values of the parameteraare listed in Table A.4 for a comparison based on a PBE pseudopotential and in Table A.5 for an EXX pseudopotential. Additionally, the latter table provides the results of a full EXX calculation.
In summary, both tables show eigenvalue differences for the valence states that are typically in the range of≈0.05−0.1 eV. Thus, the global hybrid PBEh can be evaluated on top of either a PBE or an EXX pseudopotential with satisfying accuracy in the KS eigenvalues.
Table A.4:Eigenvalue comparison of all-electron and PBE pseudopotential results (denoted PP PBE) for the CO and N2molecule using PBEh(a). The quantity∆εidenotes the eigenvalue difference for the corresponding state.
CO N2
val- εi (eV) ∆εi (eV) εi(eV) ∆εi (eV)
state all- PP all- PP
a i electron PBE electron PBE
0.25 2 −16.022 −15.968 −0.054 −15.439 −15.523 0.084 3 −13.737 −13.706 −0.030 −13.619 −13.689 0.070
4 −13.737 −13.706 −0.030 −13.619 −13.689 0.070
5 −10.757 −10.742 −0.015 −12.208 −12.229 0.021 6 −3.661 −3.629 −0.032 −3.793 −3.786 −0.007 0.50 2 −17.923 −17.850 −0.073 −17.395 −17.455 0.060 3 −15.623 −15.584 −0.039 −15.610 −15.672 0.062
4 −15.623 −15.584 −0.039 −15.610 −15.672 0.062
5 −12.481 −12.449 −0.032 −14.160 −14.152 −0.007 6 −5.326 −5.305 −0.021 −5.645 −5.637 −0.008 0.75 2 −19.833 −19.740 −0.092 −19.359 −19.393 0.034
3 −17.516 −17.469 −0.047 −17.609 −17.660 0.051
4 −17.516 −17.469 −0.047 −17.609 −17.660 0.051 5 −14.210 −14.162 −0.048 −16.120 −16.082 −0.038
6 −7.001 −6.992 −0.008 −7.511 −7.500 −0.011
A.6 Comparison of Eigenvalues for the Global Hybrid Functional Table A.5:Eigenvalue comparison of all-electron and EXX pseudopotential results (denoted
PP EXX) for the CO and N2molecule using PBEh(a) and pure EXX. The quantity
∆εi denotes the eigenvalue difference for the corresponding state.
CO N2
val- εi (eV) ∆εi (eV) εi(eV) ∆εi (eV)
state all- PP all- PP
a i electron EXX electron EXX
0.25 2 −16.022 −16.089 0.067 −15.438 −15.522 0.084 3 −13.737 −13.738 0.001 −13.619 −13.689 0.070
4 −13.737 −13.738 0.001 −13.619 −13.689 0.070
5 −10.757 −10.806 0.049 −12.208 −12.229 0.021 6 −3.661 −3.619 −0.042 −3.793 −3.786 −0.007 0.50 2 −17.924 −17.972 0.048 −17.395 −17.455 0.060
3 −15.623 −15.618 −0.005 −15.610 −15.672 0.062
4 −15.623 −15.618 −0.005 −15.610 −15.672 0.062 5 −12.481 −12.506 0.025 −14.160 −14.151 −0.007
6 −5.326 −5.294 −0.032 −5.645 −5.637 −0.008
0.75 2 −19.833 −19.860 0.027 −19.359 −19.393 0.034
3 −17.516 −17.503 −0.013 −17.609 −17.660 0.051
4 −17.516 −17.503 −0.013 −17.609 −17.660 0.051 5 −14.210 −14.210 −0.001 −16.120 −16.082 −0.038
6 −7.001 −6.979 −0.022 −7.511 −7.500 −0.011
EXX 2 −20.679 −20.723 0.044 −20.291 −20.346 0.055
3 −18.336 −18.315 −0.021 −18.529 −18.568 0.039 4 −18.336 −18.315 −0.021 −18.529 −18.568 0.039 5 −15.038 −15.085 0.047 −17.150 −17.158 0.007 6 −7.763 −7.726 −0.037 −8.436 −8.410 −0.026
Acknowledgments
I am greatly indebted to numerous persons for their support in the process of creating this thesis. In particular, I wish to thank ...
Stephan Kümmel for providing infinite amounts of both guidance and partnership. This thesis would not have been possible without his advice and ability to inspire, and I am grateful to have met a person like him along my way.
Leeor Kronikfor his hospitality during my research stay in Israel as well as his scientific support as my second supervisor.
Eli Kraisler for a fruitful and pleasant scientific collaboration that turned into a cordial friendship.
the current and former members of the Kronik groupfor their openness, in particularAdi Makmalfor her technical advice regardingDARSEC.
Rodrigo Q. Albuquerque for the discussions here in Bayreuth as well as his advice and hospitality in Brazil.
Matthias Dauthfor providing the right balance of support and distraction as my office mate during the past five years and for his friendship for many more years to come.
all current and former members of the TP IV groupfor creating a pleasant and inspiring working environment. In particular, I want to thankThilo,Ingo,Philipp, andMatthiasfor carefully proofreading the manuscript of this dissertation.
Monika Birkelbach,Markus Hilt, andBernhard Winklerfor their technical and adminis-trative assistance.
all my friendsfor contributing to the precious moments and memories that are inseparably linked to my time in Bayreuth.
my parentsDagmarandVolkeras well as my brotherMirkoand his family for supporting me in countless ways during the past years.
Annette for her appreciation and care. Walking this path would not have been the same without her.
I further acknowledge financial support by the SFB 840 of the DFG, the German-Israeli Foundation, and theElite Network of Bavaria("Macromolecular Science" program).
B3LYP global hybrid functional with 20% exact exchange, LSDA exchange and corre-lation, B88 exchange, and LYP correlation . . . 22 B88 exchange functional of Becke from 1988 . . . 20 BHLYP global hybrid functional with 50% EXX, 50% B88 exchange, and LYP correla-tion . . . 35 BLYP xc energy functional using B88 exchange with LYP correlation . . . 19 DFA density-functional approximation . . . 15 DFT density-functional theory. . . 3 DOS density of states . . . 39 EA electron affinity . . . 12 EXX exact exchange. . . 10 GGA generalized gradient approximation . . . 19 GKLI generalized approximation of Krieger, Li, and Iafrate . . . 21 GKS generalized Kohn-Sham . . . 17 GOEP generalized optimized effective potential . . . 21 GSIC generalized self-interaction correction . . . 21 HF Hartree-Fock . . . 54 ho highest occupied . . . 12 IP ionization potential . . . 12 ISO local hybrid functional of Schmidt, Kraisler, Makmal, Kronik, and Kümmel . . . 31 ISOII modification of the ISO local hybrid functional. . . 34 KLI approximation of Krieger, L,i and Iafrate . . . 17 KS Kohn-Sham. . . 6 L(S)DA local (spin-)density approximation . . . 15 LMF local mixing function . . . 25
lu lowest unoccupied . . . 13 LYP correlation functional of Lee, Yang, and Parr . . . 20 many-error many-electron self-interaction error . . . 16 meta-GGA meta-generalized gradient approximation . . . 20 NTCDA 1,4,5,8-naphthalene tetracarboxylic dianhydride . . . 40 OEP optimized effective potential . . . 16 one-error one-electron self-interaction error . . . 15 PBE xc energy functional of Perdew, Burke, and Ernzerhof . . . 19 PBE0 global hybrid functional with 25% exact exchange and PBE components for semilocal exchange and correlation . . . 22 PBEh global hybrid functional with a certain amount of exact exchange and PBE com-ponents for semilocal exchange and correlation . . . 22 PES photoemission spectroscopy . . . 39 PKZB meta-generalized gradient approximation of Perdew, Kurth, Zupan, and Blaha . . . 20 RSH range-separated hybrid functional . . . 23 SIC self-interaction correction . . . 20 TPSS meta-generalized gradient approximation of Tao, Perdew, Staroverov, and Scuse-ria . . . 20 TPSSh global hybrid functional with 10% EXX, 90% TPSS exchange, and TPSS corre-lation . . . 35 xc exchange-correlation . . . 3
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