Length dependence of ionization potentials of transacetylenes: Internally consistent DFT/$GW$ approach

We follow the evolution of the ionization potential (IP) for the paradigmatic quasi-one-dimensional transacetylene family of conjugated molecules, from short to long oligomers and to the infinite polymer transpolyacetylene (TPA). Our results for short oligomers are very close to experimental available data. We find that the IP varies with oligomer length and converges to the given value for TPA with a smooth, coupled inverse-length-exponential behavior. Our prediction is based on an “internally consistent” scheme to adjust the exchange mixing parameter $\alpha$ of the PBEh hybrid density functional, so as to obtain a description of the electronic structure consistent with the quasiparticle approximation for the IP. This is achieved by demanding that the corresponding quasiparticle correction, in the $GW$@PBEh approximation, vanishes for the IP when evaluated at $\mathrm{PBEh}({\alpha }^{\text{ic}}$). We find that ${\alpha }^{\text{ic}}$ is also system-dependent and converges with increasing oligomer length, enabling the dependence of the IP and other electronic properties to be identified.

Figure 1

Schematic representation of the infinite polymer unit cell (TPA, top) and a finite model oligomer (OTA8, bottom), built by repetition of the unit cell. The hydrogen atoms added to saturate the oligomer chain and the resulting C-H distance are highlighted in red.

Figure 2

Discrete DFT energy level spectra (obtained with the PBE functional) of transacetylene oligomers (OTAs), compared to the density of states (DOS) of the 1D infinite polymer (TPA), calculated explicitly for the periodic model; here we use a Gaussian broadening of 0.05 eV for the DOS. Spectra aligned at the average of the core levels ($C_{1s}$) with those of OTA(50).

Figure 3

Top panel: Evolution of the first ionization potential of OTA series as a function of (a) chain length and (b) inverse chain length, up to $n=30$ double bonds, calculated through the $\mathrm{\Delta }\mathrm{SCF}$ approach with the different methods PBE, PBE0, and HF. Bottom panel: Comparison of the negative HOMO energy and the $\mathrm{\Delta }\mathrm{SCF}$ approach obtained with the PBE functional, in the (c) chain length and (d) inverse chain length representation, up to $n=80$ double bonds. The lines are just guides for the eye.

Figure 4

Length ($l$) dependence of the self-energy correction on the HOMO energy ${\mathrm{\Delta }}_{\mathrm{HOMO}}^{\text{qp}}$ of the OTA chains, defined as the difference between the quasiparticle and the KS energy; results for the PBE functional. The solid line is a fit with ${\mathrm{\Delta }}_{\mathrm{HOMO}}^{\text{qp}}=(1.6l^{-1/2}+0.286)\mathrm{eV}$.

Figure 5

Evolution of the Kohn-Sham and quasiparticle HOMO energy of transacetylene oligomers with an increase of the mixing parameter $\alpha$ of PBEh. The parameter that satisfies the internal consistency criterion, ${\alpha }^{\text{ic}}$, indicated in the central panel, corresponds to the crossing point between the curves calculated with $\mathrm{PBEh}(\alpha$) and $G_0{W}_0$@$\mathrm{PBEh}(\alpha$).

Figure 6

Chain length dependence of the first ionization potential of transacetylene oligomers obtained through $G_0{W}_0$ on different levels of mean-field methods: PBE (circles), PBE0 (solid squares), and internally consistent PBEh (stars); Hartree-Fock (empty squares). The lines are just guides for the eye. Included are also the experimental results (solid triangles) for small oligomers. Inset: difference between the calculated and experimental IP values for the small oligomers; same symbols as for the IP plots.

Figure 7

Size dependence of the internally consistent $\alpha$-parameter of the PBEh functional obtained for transacetylene oligomers. We see that it decreases exponentially with chain length, as indicated by the dashed curve: the line is a fit with ${\alpha }^{\text{ic}}\left(\mathit{l}\right)=0.106e^{-0.694\mathit{l}}+0.755$.

Figure 8

Comparison between the energy level spectra calculated at the $\mathrm{PBEh}({\alpha }^{\text{ic}}$) and the $G_0{W}_0$@$\mathrm{PBEh}({\alpha }^{\text{ic}}$) levels (with the internally consistent mixing parameter) for the $n=8$, 15, and 30 acetylene oligomers, close to the HOMO-LUMO gap energy window.

Figure 9

Comparison of $\mathrm{PBEh}({\alpha }^{\text{ic}}$) (ic-PBE) with PBE: (a) Deviation of the Kohn-Sham (KS) eigenvalues with respect to the $G_0{W}_0$ quasiparticle spectrum plotted over the KS energies, for oligomer lengths $n=2$, 6, and 15 double bonds. (b) Spectra obtained for the longer oligomer $n=30$ double bonds by the different methodologies. In (a), filled symbols indicate occupied states, while empty symbols indicate unoccupied states.