3.4 Stochastic GW calculations on large thio-phene polymers
We investigate the phenomenon of spontaneous charge localization presented in Section 3.3 in more detail. For this, we directly aim at describing the ionization potentials, corresponding to the lowest quasihole energy, at the level of many-body perturbation theory within the GW approximation. As in Chapter 6, we rely on the use of the stochastic formulation of the GW approximation (see Chapter 6 and 7 for references and details), which allows us to calculate properties of systems with unprecedented sizes.
First, we present a more detailed derivation of the stochastic approach and describe the key ingredients of the formulation that improve the scaling of the algorithm with respect to system size. In this context, the algorithm described in Chapter 7 represents a modification of the original work of Neuhauseret al. as it provides quasiparticle correction (see Section 2.4.3 and Chapter 7) calculated directly for a given Kohn-Sham eigenstate obtained from the underlying DFT computations.
We then apply the stochastic GW approach to polythiophene polymer chains of increasing sizes and analyze the individual contributions to the quasiparticle shifts. We find that though we employ a “single shot” GW approach start-ing from Kohn-Sham DFT ground state calculation with the LDA functional, which does not exhibit localization of the quasihole (see Chapter 6), the pre-dicted quasiparticle energies become independent of the length of the polymer chain (localization). We further analyze the individual contributions to the self-energy⌃, and find that the major part of the quasiparticle shift is supplied by the exchange part of⌃. While the magnitude of the polarization contribution remains almost constant, the exchange contribution decreases with increasing length of the polymer. For long chains of polythiophene, the ionization poten-tial becomes independent of the systems size and this e↵ect is driven by the exchange part of the self-energy. Our finding further supports the claims made in Section 3.3 and Chapter 6 and we moreover illustrate the strength of the GW approach.
We also investigate the possible localization in systems with higher dimen-sionality: We construct systems of three stacked planar polythiophene chains (see Chapter 7 for illustration) with interplanar distance typical for the con-densed polythiophene phase. It has to be noted that in such systems there is a weak interaction among the individual chains which makes the system e↵ectively two-dimensional. We again calculate the ionization potential for a set of stacked polymers using the stochastic GW method and reach extremely large systems containing up to 1446 valence electrons. By comparing the results for the single polymer chain and the polymer stacks of di↵erent length, we find that the higher dimensionality inhibits localization. By examining the individual contributions to⌃, we observe that while for the small systems, the behavior of the exchange and polarization parts for the stacked polymers is similar to that of the isolated chain, for large systems the exchange part of⌃for the stacks keeps decreasing with system length. The results are shown in Figure 3.5. Based on this obser-vation, we conjecture that either: (i) The localization phenomenon is limited to 1D systems. (ii) Alternatively, the length scale of localization is much smaller in 1D systems and it increases with the dimensionality of the system.
5.0 6.0 7.0 8.0 9.0 10.0
0.0 0.5 1.0 1.5 2.0
10 3 2 1 0.5
experiment
I [eV]
1/M M
BNL*
LDA LDA − stack sGW sGW−stack
Figure 3.5: Ionization potentialsI for polythiophene polymers of di↵erent sizes (given in the number of repeat unitsM - for details see Chapter 6 and Chapter 7). The filled green circles represent DFT calculations with the local dens-ity approximation, used as a starting point for “one-shot” GW computations shown in filled black circles. The calculations for three polythiophene molecules stacked on top of each other are shown in open green and black circles for LDA and GW results, respectively. The calculations with optimally tuned range-separated hybrid functional (BNL*) for an isolated polymer strand are shown in red triangles and experimental results for isolated molecules are presented for comparison. For single polymer chains, we observe thatIbecomes independent of system size whenM >5.5, while localization is not observed for the stacked system considered.
This work will be submitted for publication as:
Vojtˇech Vlˇcek, Eran Rabani, Daniel Neuhauser, Roi Baer, Stochastic GW calculations on large thiophene polymers, to be submitted to J. Chem. Theory Comput.
I have performed all the calculations presented in the manuscript, made all necessary implementations for the stochastic GW approach, and written the first draft of the paper.
Chapter 4
Improved Ground State Electronic Structure and Optical Dielectric
Constants With a
Semi-Local Exchange Functional
Vojtˇ ech Vlˇ cek
1, Gerd Steinle-Neumann
1, Linn Leppert
2, Rickard Armiento
3, Stephan K¨ ummel
24.1 Abstract
A recently published generalized gradient approximation functional within dens-ity functional theory (DFT) has shown, in a few paradigm tests, an improved KS orbital description over standard (semi-)local approximations. The char-acteristic feature of this functional is an enhancement factor that diverges like sln(s) for large reduced density gradientsswhich leads to unusual properties.
We explore the improved orbital description of this functional more thoroughly by computing the electronic band structure, band gaps, and the optical dielec-tric constants in semiconductors, Mott insulators, and ionic crystals. Compared to standard semi-local functionals, we observe improvement in both the band gaps and the optical dielectric constants. In particular, the results are similar to those obtained with orbital functionals or by perturbation theory methods in that it opens band gaps in systems described as metallic by standard (semi-)local density functionals, e.g., Ge,↵-Sn, and CdO.
1Bayerisches Geoinstitut, Universit¨at Bayreuth, 95440 Bayreuth, Germany
2Theoretische Physik IV, Universit¨at Bayreuth, 95440 Bayreuth, Germany
3Department of Physics, Chemistry and Biology (IFM), Link¨oping University, 58183 Link¨oping, Sweden