6.4 Supplementary material
6.4.3 Exciton Energy and Size
log
✓ "th
"
◆◆2
M0. (6.3)
This parameter is also shown in Table 6.5 and we foundMCtP A= 20 (`tP Ac = 4.9 nm) for tPA and a much more localized holeMP TC = 4 (`P Tc = 3.1 nm) for PT. The fact that the IPs converge to their asymptotic values in HF and BNL*
calculations is illustrated also in Figure 6.5 where the derivative of d"/d(M 1) (obtained by finite di↵erences) is shown together with the analytical derivative of our model (Eq. 6.2).
We applied the same polymer size-dependence analysis to IPs calculated us-ing BNL* with local correlation (shown in Figure 6.5). Followus-ing Eqs. (6.2) and (6.3) we find (see Table 6.5) a hole with MCtP A= 20 (`tP Ac = 7.9 nm) for tPAand considerably more localizedMCP T = 5.5 (`T Pc = 4.3 nm) for PT. Addi-tionally, we also employed gradient corrected correlation functionals (PBEc [64]
and LYPc [65]) and found them to yield IP values lower by ⇠0.4 eV but their dependence onM 1 has almost identical shape and the range separation para-meters do not change significantly (<0.2 nm 1) when LYP/PBE correlation is used instead of LDA in the BNL* calculation.
6.4.3 Exciton Energy and Size
We calculated optical absorption spectra and fundamental gaps for selected tPA polymers using adiabatic LDA, BNL* and TDHF approaches. The fundamental (Eg) and optical gaps (Eopt) are reported in Table 6.6. We here provide a more detailed discussion of the LDA and TDHF results as the BNL* results are discussed in the paper itself.
M
0.00 0.05 0.10 0.15 0.20 0.25 50 20 10 8 7 6 5 4
Figure 6.5: The derivative of the ionization potential for (a) tPA and (b) PT with respect to the inverse length of the polymer (given in the number of repeat units M) is shown for results of LDA, BNL* andGW@LDA in green squares, red triangles and black pentagons colors respectively. The analytical derivative of the model fit given in Eq. (6.2) is shown by full line in respective colors.
The ALDA optical gapsEoptALDA are close to available experimental results [26–29] but consistently lower by 0.2-0.3 eV. Furthermore, EoptLDA are larger than the corresponding LDA fundamental gaps ELDAg , and hence EbLDA = EgLDA EoptLDA, the exciton binding energy, is predicted to be negative in LDA.
This corresponds to an unbound exciton (hole and electron do not attract and stay close together). Even for the infinite polymer length the binding energy does not become positive althoughEbLDA!0 in this limit (see the right panel of Figure 6.2). The infinite length asymptotic limit of the LDA optical gap can be obtained from fitting the results to the model of Eq. (6.2), leading to the resultEoptLDA(1) = 1.4 eV. This is 0.5 eV lower than the experimental estimate of the infinite optical gap,Eoptexp(1) = 1.9 eV [30].
The TDHF results exhibit peaks shifting to lower energies with increasing system size and in general very close to the experimental observations available for small polymers. Since HF does not allow for binding an extra electron in the system, i.e. the lowest unoccupied eigenvalue is larger than the vacuum level, fundamental gaps of HF thus coincide with the ionization potentials and the exciton is strongly bound. Extrapolating our results for long chains containing up to 40 repeat units, the TDHF exciton binding energy in an infinite polymer is significantly greater than 3 eV, with the maximum of the optical absorption being at 2.7 eV, based on the model fit using Eq. (6.2). This is however sig-nificantly larger than experiments, where the absorption maximum occurs at 1.9 eV [30].
6.4. SUPPLEMENTARY MATERIAL
M BNL* LDA HF Exp.
Eopt Eg Eopt Eg Eopt Eg Eopt
1 7.74 11.02 7.81 5.59 7.49 10.12 7.65 2 5.95 9.48 5.60 3.85 5.98 8.68 5.92 3 4.92 8.41 4.52 2.99 5.11 7.95 4.95 5 3.82 6.65 3.37 2.16 4.55 7.48 4.02
8 3.03 5.38 2.51 1.63 4.16 7.18
-10 2.72 4.87 2.17 1.44 3.51 6.69 -16† 2.36 4.25 1.60 1.16 3.06 6.85 -40† 1.97 3.77 1.03 0.92 2.77 5.99
-Table 6.6: Fundamental (Eg) and optical gaps (Eopt) given in units of eV, for various tPA polymers containing M repeat units. The two longest polymers (M=16 and 40, indicated by a dagger) were calculated with the 3-21G basis set while the rest of the results were computed with the cc-pvTZ basis set.
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Stochastic GW calculations on large thiophene
polymers
Vojtˇ ech Vlˇ cek
1,2, Eran Rabani
3,4, Daniel Neuhauser
5, Roi Baer
1,67.1 Abstract
We review in detail the stochastic formulation of the GW approximation and discuss the advantages of this formalism over common approaches. We employ stochastic GW for calculations on large thiophene polymers in 1D (as isolated chains) or 2D (as stacked polythiophene molecules), containing up to 1446 elec-trons. We demonstrate the practical calculations of the self-energies and show that with increasing system size the approach becomes e↵ectively less expensive.
leading to approximately linear scaling of the algorithm. For the 1D systems we confirm the presence of a recently discovered spontaneous quasiparticle localiz-ation, for the stacked polythiophene molecules the localization is not observed.
This suggests that the spontaneous quasiparticle localization is either limited to 1D systems, or occurs on much larger lengthscales in higher dimensions.
1Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew Univer-sity of Jerusalem, Jerusalem 91904, Israel
2Bayerisches Geoinstitut, Universit¨at Bayreuth, 95440 Bayreuth, Germany
3Department of Chemistry, University of California and Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A.
4The Sackler Center for Computational Molecular and Materials Science, Tel Aviv Univer-sity, Tel Aviv 69978, Israel
5Department of Chemistry and Biochemistry, University of California, Los Angeles, Cali-fornia 90095, U.S.A.
6On sabbatical in the Department of Chemistry, University of California, Berkeley, Cali-fornia 94720, U.S.A.
7.2. INTRODUCTION
7.2 Introduction
First principles calculations of electronic structure play central role in predicting and understanding behavior of matter at the nanoscale, and their development is one of the thriving fields in theoretical physics and chemistry. Highly accurate methods can be applied only to small systems with few electrons. Density func-tional theory (DFT) [1, 2] has proven to be a reliable tool for the prediction of ground state properties, and with the currently available numerical implement-ations and computational power, calculimplement-ations for extremely large systems with thousands of electrons [3–5] are possible. Nevertheless, DFT is limited to the ground state and, in principle, cannot provide quasiparticle properties which are often required. The only exception is the lowest energy needed to remove an electron from the system (ionization potential) which it should predict exactly, in principle [6, 7]. Nevertheless, DFT falls short in terms of accuracy and yields ionization potential (IP) values with substantial deviation from experiment in many cases [8–12].
Quasiparticle energies are crucial not only from a theoretical perspective, but also for characterizing the possible excited states of systems under external influence; most notably for the prediction and analysis of photoemission spectra that directly characterize the electronic states within the system and ultimately help to design new (opto)electronic devices [13–18]. Theories beyond DFT, based on the many-body perturbation approach, are thus usually needed for the description of quasiparticle states. An established technique is the GW approximation [13, 19–23], which is the central point of this paper. Standard implementations of GW pose significant computational demands [24, 25] and are thus limited to relatively small systems with several tens of electrons. In this paper, we review in detail a recently developed stochastic formulation of GW [26] which overcomes many of the significant limitations and allows us to calculate systems of unprecedented sizes, as it scales approximately linearly with system size. We illustrate the technique on large thiophene polymers.
In a recent study [27] we have shown that in 1D strands of conjugated polymers the ionization potentials become independent of the system length and the associated quasiparticles (holes) spontaneously localize and break the periodicity of the underlying ionic lattice. Our finding has a significant bear-ing on the understandbear-ing of the formation of quasiparticles (QPs) in extended 1D systems and the theory of electronic structure in such systems: While the individual ground state electronic eigenstates span the whole polymer and ul-timately correspond to the Bloch states in an infinite system, the QP size is finite and as such its properties depend solely on its immediate neighborhood.
Such spontaneous charge localization can be a precursor of other phenomena, such as creation of self-trapped states [28–30], which subsequently lead to the formation of polarons, bipolarons or solitons [31–35] that were observed to be created by ionization on very short timescales [36]. Detailed understanding of these processes is thus necessary for further wide technological application of conjugated polymers in electronics [37–40].
In the present paper, we use the example of planar polythiophene (PT) chains with ideal periodic geometry within the polymer chain and investigate this phenomenon in 1D systems of increasing length. To determine whether this behavior may be observed in systems with higher spatial dimensions, we perform calculations on stacked PT molecules (containing up to 20 thiophene
repeat unit
7.2 a 0
Figure 7.1: Example of a system with three parallel stacked molecules of PT with 3 repeat units in each layer (each containing 2 thiophene rings). Yellow spheres denote S, black C and white spheres H positions. The distance between two adjacent layers of PT is 7.2a0.
rings in three layers - see Figure 7.1). The stochastic formulation of the GW approximation [26] is used for systems which contain up to 1446 electrons. The paper is organized as follows: In Section 7.3 we provide a detailed description of the stochastic formulation of the GW approximation and emphasize its major merits. Next we show its computational performance together with the actual results for the PT polymers and a detailed discussion on the role of various contributions to the QP energy (Section 7.4). Our findings are summarized together with the implications for DFT in Section 7.5.