Ionization potentials of the PT chains

In DFT, the ionization potential is taken as a negative of the highest occupied eigenvalue "H. However, calculations with (semi)local density functionals suf-fer from delocalization error, which also leads to spreading of the hole density over the whole system preventing localization [27]. As a consequence the LDA results for "H not only severely underestimate the IPs of the polythiophene chains with respect to experiments, but for large polymers the ionization po-tential shows a dependence of 1/L. For comparison, we also show calculations with optimally tuned range-separated hybrid functional (BNL*) [45, 56–58] in which the delocalization error is minimized by enforcing the IP theorem [10, 59],

E^{N 1} E^N ="H, whereE^N is the total energy ofN-particle system. The

ioniz-ation potentials obtained with BNL* show a completely di↵erent behavior from the LDA results, and for large systems the IPs become independent of system size.

We emphasize that the localization of the hole inherently captures the re-sponse of the system to the excess charge. The density of the hole in the system is thus given as

n(r) =n^N(r) n^{N 1}(r), (7.66) where the superscriptsN andN 1 denote the density evaluated for the neutral system and a cation. n(r) thus does not correspond to the density of the highest occuppied eigenstate | ^H|² which is not necessarily localized [27]. We use the LDA results in which H spans the whole system and calculated the quasiparticle correction from Eq. (7.19) for the highest occupied KS eigenstate

"_H. TheGW estimates of the IPs are shown in Figure 7.4.2. The QP energy is

shifted substantially towards better agreement with experimental data available and becomes independent of the system size, consistent with the BNL* results.

Our results can be fitted by

I(M) =I₁ ⇥e ^↵p

M/M0, (7.67)

5.0

Figure 7.5: Results for PT containing between 1 and 20 thiophene rings (M=0.5 - 10). The full green circles represent the underlying LDA calculations for single PT strands while open green circles show results for the stacked molecules (illustrated in Figure 7.1). The quasiparticle energies calculated with the one-shot stochastic GW method on top of the LDA starting point are shown by full and open black circles for the single PT strands and three stacked layers of PT, respectively. DFT results obtained with optimally-tuned range-separated hybrid functional (BNL*) from Ref. [27] are shown for comparison in open red triangles. Experimental data shown in gray circles were taken from Refs. [53–

55].

shown by dashed lines in Figure 7.4.2. The values ofI₁= 6.4 eV, = 10.1 eV, M0 = 0.25 reproduce the calculated IPs best. The critical size MC is chosen such that forM > MC the di↵erenceI(M) I₁ is smaller than 0.1 eV:

From the results we estimate that the hole energy becomes independent of the polymer size for M>5.5 repeat units, i.e. 11 thiophene rings. This corresponds to a length of 4.2 nm, in close agreement with BNL*. This not only validates our DFT results on spontaneous hole localization in the sytems considered, but also strengthens the optimal tuning procedure employed with the BNL* functional.

Note that in LDA vXC(r) is local and cannot provide an eigenstate energy independent of system size if _H is delocalized. Nevertheless, if LDA is used as a starting point for (even single shot) GW, the QP corrections calculated through Eq. (7.19) lead to qualitative change in the description of the charge removal.

To investigate this further, we evaluated Eqs. (7.36) and (7.37) independ-ently and the individual contributions to the QP shift are shown in Figure 7.4.2.

The polarization part of the self energy⌃P is relatively small (⇠0.5 eV) and de-creases slightly with system size. Our results for single polythiophene strands do

7.4. RESULTS AND DISCUSSION

−2.5 −2.0 −1.5 −1.0 −0.5 0.0

0.0 0.5 1.0 1.5 2.0

10 3 2 1 0.5

∆ε

[eV]

1/M M

Σ_X−v_xc stack Σ

X−v_xc ΣP−v_xc stack Σ_P−v_xc

Figure 7.6: Contributions to the quasiparticle corrections for PT containing between 1 and 40 thiophene rings (M=0.5 - 20), with up to 962 valence electrons, for the single PT chains and between 1 and 20 rings in each layer for the stacked chains. The purple circles show the contribution stemming purely from the exchange part of the self energy ⌃X given in Eq. (7.36). The contribution solely from⌃P (Eq. 7.37) is shown in black circles. The filled circles stand for calculations with a single PT strand while open circles represent the results for the stacked systems.

not indicate a significant change in the behavior of⌃P for long polymer chains.

On the other hand, the magnitude of the ⌃X contribution to the QP correc-tion dominates: For small systems it is large (-2.6 eV for the single thiophene molecule) and it decreases significantly with polymer size. For large systems however, it becomes constant with⌃X ⇠ 1.1 eV.

The results for the IP of the stacked PT molecules are also shown in Figure 7.4.2 where the length of the system is defined as the number of repeat units in a single layer. The LDA results show energies that are lower than the estimates for a single polymer strand and IP values further decrease with increasing length of the system. The quasiparticle correction shifts the values towards higher energies, similar to the single PT strands, but the magnitude of this correction is smaller. The individual contributions to the quasiparticle corrections are shown in Figure 7.4.2 such that they can be compared to the results for single PT strands (like in Figure 7.4.2, the length is expressed by a number of repeat units in a single layer). The major di↵erence between the single and mutiple layer systems is in the behavior of the exchange part of the self energy, which keeps decreasing as a function ofM, even for large systems.

For small systems the general behavior of the ionization potential (aside from the shift to lower energies) is almost identical for a single strand of PT and stacked system. For systems containing more than 3 repeat units in each layer (i.e. 6 PT rings), however, the behavior di↵ers and the rapid stabilization of the ionization potential with increasing system size, associated with the

local-ization of the quasihole in the chains, is not observed. We cannot infer whether the quasiparticle will localize if the system size if further increased, or if the localization phenomenon occurs at much larger lengthscales. It is also possible that for the large stacked system, the starting point dependence becomes more crucial. Clearly, further investigations are necessary.