After sufficient justification of the subsystem approach, a direct investigation of the obtained DFT DOS provides the next step on the path to a characterization of the supported Pd nanoparticle in terms of its catalytic activity. In Fig. 6.4(a) the DOS obtained with PBE is plotted for the subsystem R=8.0 Å, which, as argued above, can be regarded as converged with respect to the integrated DOS. The DOS of the combined Pd@SiCN in the direct vicinity of the Fermi level is evidently
1The program packageTURBOMOLE[tur] was used to perform self-consistent DFT calculations. Semilocal functionals were utilized using a TZVPP def2-basis set, while hybrid functionals were evaluated with an SVP def2-basis set in the GKS scheme. The accuracy of both the basis sets and the resolution-of-the-identity approximation (see Ref. [Ahl04]
and references therein) were checked and both were found to have negligible influence onΣ∆(R).
6.3 Analyzing the Density of States
dominated by the DOS of the Pd atoms. In this region, both graphs exhibit a similar height and structure, whereas the DOS of the SiCN support is found to have little effect on the overall curve.
-13.0 -12.0 -11.0 -10.0 -9.0 -8.0 -7.0 -6.0 -5.0 -4.0
DOS(arb.units)
energy (eV) Pd@SiCN
SiCNPd
(a)R=8.0 Å with PBE
-13.0 -12.0 -11.0 -10.0 -9.0 -8.0 -7.0 -6.0 -5.0 -4.0
DOS(arb.units)
energy (eV) Pd@SiCN
SiCNPd
(b)R=5.5 Å with PBE Figure 6.4:DFT DOS obtained with PBE for the subsystem withR=8.0 Å (a) andR=5.5 Å
(b) for Pd only, SiCN only, and both combined (Pd@SiCN). The positions of the corresponding ho states, i.e. the Fermi levels, are given by the colored ticks at the top.
-13.0 -12.0 -11.0 -10.0 -9.0 -8.0 -7.0 -6.0 -5.0 -4.0
DOS(arb.units)
energy (eV) Pd@SiCN
SiCNPd
(a)R=5.5 Å with PBE0
-13.0 -12.0 -11.0 -10.0 -9.0 -8.0 -7.0 -6.0 -5.0 -4.0
DOS(arb.units)
energy (eV) Pd@SiCN
SiCNPd
(b)R=5.5 Å with BHLYP Figure 6.5:DFT DOS obtained with PBE0 (a) and BHLYP (b) for the subsystem withR=
5.5 Å for Pd only, SiCN only, and both combined (Pd@SiCN).
A similar observation, even though less distinct, can be made in Fig. 6.4(b) for the smaller subsystem withR=5.5 Å calculated with PBE. The same subsystem is recalculated with the PBE0 and BHLYP global hybrid functionals in order to demonstrate that the conclusion drawn above is not merely a feature of the PBE functional. The resulting DOS are shown in Fig. 6.5(a) for PBE0 and Fig. 6.5(b) for BHLYP. In principle, both results underline the argumentation that the overall DOS is predominantly influenced by Pd in the region close to the Fermi level. Importantly, these functionals counteract electronic self-interaction that affects the energetic positioning of states emerging from localized orbitals (cf. Secs. 4.7 and 4.8). This effect becomes apparent from the corresponding figures since the DOS is significantly shifted towards lower energies with an increasing amount of EXX involved in the calculation. It is also shown in Figs. 6.3(a) and 6.3(b) that the integrated DOS decreases with larger amounts of nonlocal EXX. Note that investigations regarding the electronic structure of Pd@SiCN based on local hybrid functionals are not presented due to inaccuracies
arising for local hybrids in the context calculating transition metals with pseudopotential methods (see Appendix A.5 for details). Yet, the results of Sec. 4.8 strongly indicate a similar sensitivity to localizedd-states for both global and local hybrid functionals. Based on this reasoning, it can be expected that local hybrids provide a characterization of the DOS and the catalytic properties of Pd@SiCN that is comparable to the presented results obtained with global hybrid functionals.
In summary, it is demonstrated that the electronic structure of a Pd nanoparticle within SiCN, as obtained with molecular-dynamics simulations, can be accessed reliably via the subsystem ap-proach. DFT calculations of subsystems with sufficient size yield the observation that electronic-structure properties of the supported Pd@SiCN are essentially influenced by Pd instead of SiCN.
This conclusion is supported by calculations with different DFAs, especially addressing the question of self-interaction and localized states. In this sense, the results presented in this chapter support the observation that Pd preserves its excellent catalytic properties within a matrix of SiCN and provide a theoretical reasoning for the use of this particular catalyst in the synthesis ofPubl. 5.
A Appendix
A.1 Modifications in DARSEC
The numerical results which set the basis forPubl. 1, Publ. 2, and Publ. 3were almost entirely obtained with the program package DARSEC. I here give a concise outline of the fundamental principle of DARSEC and highlight relevant modifications of the code that I implemented in the context of the work presented in this thesis. More detailed information about DARSEC can be found in Refs. [MKK09, Mak10].
DARSECis a parallelized electronic-structure code based on a finite-difference approach with a representation of space-dependent functions on a real-space numerical grid. It allows for self-consistent KS DFT calculations of systems with up to two atomic centers, i.e., atoms and diatomic molecules, with a wide range of DFAs and an explicit consideration of all electrons present. The rotational symmetry inherent to such systems enables an analytical treatment of the dependence on the azimuthal angle in all occurring functions. Thus, the problem effectively reduces to two dimensions, resulting in a drastically diminished numerical effort. The numerical grid is expressed in prolate spheroidal coordinates, which naturally provide a dense sampling in the vicinity of the atomic center and a more coarse representation of the asymptotic regions. In DARSEC, a local multiplicative potential of orbital-dependent functionals is obtained either via the full OEP based on theS-iteration method [KP03b, KP03a] or via the KLI approximation (cf. Sec. 2.6). DARSEC further facilitates DFT calculations with user-specified electronic configurations of the molecule or atom of interest.
In the following, I provide a brief overview of important modifications that I applied toDARSEC, listed by the name of the modified subroutine with a short description of the applied changes:
– programdarsec: general cleaning and reduction of output;
– get_xc: rearrangement of the computation of the xc energy and local potential for both density- and orbital-dependent functionals, enabling a straightforward combination of the corresponding functions for DFAs with several parts such as, for instance, hybrid functionals;
– restart: enabling restarted calculations on charge-density or KS orbital input files ob-tained from previousDARSECcalculations;
– b88_x: implementation of the B88 exchange functional;
– lyp_c: implementation of the LYP correlation functional;
– pbe_x,pbe_c andpbe_xc: implementation of the PBE exchange and correlation func-tionals, which are combined with EXX to the PBEh hybrid functional;
– get_uiwf_eiso and get_uiwf_coco: numerically stable implementation of the xc energy and potential for the ISO and ISOII local hybrid functionals based on their functional derivativeuiσ(r), including an evaluation of the asymptotic slopeγσ;
– get_pot_bar,get_vxc_kli,get_uiwfandoep_S_iter: rearrangement and mod-ification of subroutines related to the OEP/KLI procedure, enabling an evaluation of function-als consisting of several orbital-dependent components and different convergence schemes for theS-iteration;
– get_v0_DD_ddep,get_v0_orb,create_red_electronic_structand print_out_shift: implementation of the ensemble potential shiftv(0)for density- and orbital-dependent functionals in a spin-polarized formalism;
– ionpot: inclusion of the electrostatic potential of a positively charged sphere ("jellium potential") as external potential for debugging;
– usrinput_90: introduction of several flags to the input file darsec.in related to the changes described above.