A.4 Pseudopotential Generation
The Pseudopotential Principle
The core electrons of atoms are chemically inert, i.e., their density does not vary greatly if the atom is subject to different chemical environments. Based on this reasonable assumption, one can construct a general, effective potential felt by the valence electrons that reproduces the potential of the nuclei screened by the core electrons [KK08]. Thus, only the valence electrons have to be considered explicitly, and the numerical effort to solve the KS equations can be reduced drastically.
Such potentials are termed pseudopotentials. In contrast to model potentials, they are usually constructed based on anab initioapproach. Especially norm-conserving pseudopotentials [HSC79]
provide very accurate results. Additionally, pseudopotentials can be used to include relativistic effects into a DFT calculation, which becomes relevant for heavy atoms [Kle80].
In the following, I briefly outline the construction of pseudopotentials as described in Refs.
[BHS82, RRKJ90, TM91, PTA+92, KK08], focusing on the method of Ref. [TM91]. First, one solves the atomic, radial KS equation using the DFA for which the pseudopotential should be constructed. Thus, the all-electron-valence orbitalsϕlmae-v(r) are obtained for this particular atom, withldenoting the angular momentum andmthe magnetic quantum number of the valence shell.
Pseudo-valence orbitalsϕlmps-v(r)are then created based on the following approach: Outside a certain radiusrc(l), the radial part ofϕlmae-v(r)andϕlmps-v(r)must be identical. Forr≤rc(l), the radial part of ϕlmps-v(r) is constructed by using a smooth, analytical function such that the norm of the all-electron- and pseudo-valence orbital is the same. Hence, the pseudo-valence orbitals do not exhibit the strong oscillations in the core region which are typically found for the all-electron orbitals. As a consequence, the numerical treatment is significantly simplified.
Thereafter, a screened pseudopotentialvps-scrl (r)is obtained by inversion of the radial KS equa-tion. Due to this step, the valence-state KS eigenvalues of the pseudopotential method agree with the all-electron results. The potential vps-scrl (r) is then unscreened by using the valence-pseudo densitynps-v(r)via
vpsl (r) =vps-scrl (r)−vH[nps-v](r)−vxc[nps-v](r). (A.2) Two things have to be noted here. First, for the ionic pseudopotential vpsl (r) one relies on the approximation that the xc potential can be split linearly into a core and valence part, i.e.,vxc[n](r)≈ vxc[nc](r)+vxc[nv](r). This relation only holds if core and valence densities do not strongly overlap.
In other cases, one can explicitly take the core density into account via a non-linear core-correction to Eq.(A.2) [LFC82].
Second, the ionic pseudopotential is different for different angular momentum components.
Consequently, the general pseudopotential must be constructed by using thel-dependentvpsl (r)in combination with a projection operator for that l-component on the angular part of the orbital, rendering the pseudopotential operator nonlocal [TM91]. In practice, typically the efficient repre-sentation of the nonlocal pseudopotential according to Ref. [KB82] is employed.
For a DFT calculation of, e.g., a molecule, the overall pseudopotential is compiled by adding the pseudopotential of each atom. By construction, the pseudopotential approach exactly reproduces the KS eigenvalue spectrum for a single atom. However, for a system with many atoms, this is only the case if the the pseudopotential is transferable to different chemical environments [HSC79]. In this context, the parametersrc(l)are of special importance, since they determine the radius outside which the pseudo-valence orbitals coincide with the true KS orbitals. Thus, when constructing atomic pseudopotentials, the rc(l) must be chosen small enough to ensure a high transferability, while being large enough to produce smooth pseudo-valence orbitals [KK08].
Directly Evaluated Generation of Pseudopotentials for Semilocal Functionals
In order to create and evaluate pseudopotentials for semilocal functionals effectively, I developed a procedure to test pseudopotentials for different chemical environments directly after their con-struction. For this, pseudopotentials are generated with the atomic radial code of José Luís Martins (available at http://bohr.inesc-mn.pt/~jlm/pseudo.html, latest access on March 27, 2016) based on the approach described previously. The pseudopotential generation requires information regarding the desired state (typically the ground state) of the respective atom, i.e., a specific electronic configuration, as well as a choice for the cutoff radiirc(l)as input.
Subsequent to their construction, pseudopotentials are tested in an automated fashion by the script genPP. Based on a multitude of electronic configurations of excited and ionized states provided by the user,genPPinitializes calculations of the atom in these states using all electrons on the one and the previously designed pseudopotential on the other hand. Afterwards, a summary of the differences between the all-electron and the pseudopotential computation is printed for the following quantities: the eigenvalues of the occupied and some unoccupied valence states, the total energy and the radiusrat which|r·ϕlm(r)|becomes extremal. These numbers are compared for all excitation and ionization configurations, yielding significant, clear indicators for the performance of the pseudopotential constructed with those specific values for therc(l). By a systematic variation of the cutoff radiirc(l)and repetition of the procedure usinggenPP, these indicators can be optimized and pseudopotentials can be tested effectively and reliably.
DOS(arb.units)
-8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 energy (eV)
ultrasoft(5.95, 5.95, 5.95) soft(4.47, 4.47, 4.47) semisoft(3.48, 3.48, 3.48)
optimized(2.39, 2.88, 2.33) semihard(2.08, 2.39, 1.79) hard(1.98, 1.98, 1.00)
Figure A.5:KS DOS of a Pd6cluster obtained with the LSDA. The grey dashed lines gives the DOS obtained in TURBOMOLE (QZVPP) with the position of the ho KS eigenvalue marked with the red tick. The black solid lines show the DOS obtained in PARSEC with an LSDA pseudopotential constructed using the parameters denoted in the graph in the form (rc(s),rc(p),rc(d)). The blue tick marksεhofor the pseudopotential calculation. The pseudopotential was constructed to include relativistic effects [Kle80].
The influence of the cutoff radii on computational results is illustrated in Fig. A.5, which shows the KS DOS of a Pd6 cluster calculated in PARSEC and inTURBOMOLE (taken as a reference
A.4 Pseudopotential Generation
in the following). Several pseudopotentials for Palladium with 10 d-electrons were obtained by using genPP with the LSDA. The optimized cutoff radii were determined as rc(s) =2.39 a.u., rc(p) =2.88 a.u. andrc(d) =2.33 a.u.. The cutoff radii were varied to create harder and softer pseudopotentials as specified in Fig. A.5. It becomes evident that the optimized pseudopotential parameters provide an accurate description in the DOS, while too large or too small cutoff radii only lead to insufficient agreement. Yet, the DOS of Pd6appears as rather robust under variation of therc(l), since the semisoft and semihard pseudopotentials also lead to a sufficiently accurate KS DOS. In Fig. A.6, the observation that the optimized pseudopotential cutoff radii lead to a reliable KS DOS is further verified. It is shown for a Pd7and two Pd13clusters with different geometries, that such a choice for therc(l)results in a DOS that agrees acceptably well with theTURBOMOLE result in terms of shape and absolute position on the energy scale.
DOS(arb.units)
-10.0 -9.0 -8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0
energy (eV)
Pd7
Pd13Cs
Pd13C3ν
Figure A.6:KS DOS of a Pd7 and two Pd13 clusters with different geometries denotedCs
andC3ν (see Ref. [KCO+11]). The grey dashed lines gives the DOS obtained inTURBOMOLE(QZVPP for Pd13Cs and TZVPP for Pd13C3ν and Pd7) with the position of the ho KS eigenvalue marked with the red tick. The black solid lines show the DOS obtained inPARSECwith an LSDA pseudopotential constructed using the optimized parameters. The blue tick marks εho for the pseudopotential calculation. The pseudopotential was constructed to include relativistic effects [Kle80].