diagnostic or constructive tool in solids. This indicates that the energy curvature is not a good measure of functional performance.
In our work, we consider two distinct examples of finite systems, namely nanocrystals and 1D molecular chains. In both cases, the systems studied are constructed from periodically repeated motives, either based on a unit-cell of a periodic crystal or the repeat unit in case of molecular chains. Within the sys-tem itself, the external (ionic) potential thus preserves translational periodicity, but the system considered is finite. We calculate the curvature upon charge removal and addition and find that if the excess charge delocalizes over the whole system, the energy curvature has distinctly di↵erent asymptotic behavior for 3D (nanocrystals) and quasi-1D (molecular chains) systems. Performing a detailed analysis of the expression for the curvature we connect it directly to the self-interaction term of the highest occupied state and demonstrate that the scaling behavior is consistent with electrostatic considerations. This behavior is illustrated in Figure 3.2.
For computations on periodic systems using large reference cell, we find that the energy curvature scales linearly with volume, i.e. with the number of unit cells constituting the reference cell. The rate of change of the energy curvature with volume of the reference cell is found to be a finite constant value for the approximate functional studied and to depend on the material itself. In addition, we demonstrate that if the excess charge associated with the curvature of charge removal or addition tends to delocalize over the entire system this rate of change of the curvature with system size should be zero for the exact exchange-correlation functional (as the curvature should vanish for any volume of the reference cell) and therefore may serve as a new useful measure of functional error in periodic solids.
For practical use, we show that the rate of change of the energy curvature can be obtained not only through computations with increasingly large periodic cells but preferably – and more efficiently – by considering changes in the band edge position with dense k-point sampling. Most importantly, we demonstrate that the di↵erence between the scaling of the curvature in periodic and finite systems stems from the treatment of a compensating background charge inher-ently present in calculations with periodic boundary condition and that such the treatment removes part of the self-interaction.
This work has been published as:
Vojtˇech Vlˇcek, Helen R. Eisenberg, Gerd Steinle-Neumann, Leeor Kronik, Roi Baer, Deviations from piecewise linearity in the solid-state limit with ap-proximate density functionals, J. Chem. Phys.142, 034107 (2015).
I have performed the calculations presented in the paper, worked on the theoretical developments shown and written the first draft of the paper jointly with Dr. Helen R. Eisenberg.
3.3 Spontaneous charge carrier localization in extended one-dimensional systems
In Section 3.2 (and in more detail in Chapter 5 we have shown that while the deviation from piecewise linearity can serve as a tool for determining the free parameters in the hybrid functionals, it will ultimately fail for very large and
0 2 4
0.00 0.01 0.02
C [e V]
a
0/L
2d
L
Figure 3.2: Deviation from piecewise linearity (i.e. energy curvature C) for charge removal from alkane chains of di↵erent lengthL and diameterd (indic-ated in the inset of the plot) is shown agains the inverse length of the molecule.
The calculations were performed with localized bases cc-PVDZ and STO-3G shown by diamonds and triangles respectively. The black line represents an analytical expression based on self-interaction of homogeneous electron gas in a 1D system which is described in detail in Chapter 5.
infinite systems, since the hybrid functionals will collapse to their (semi)local form, which satisfies zero deviation. The solution proposed to this conundrum discussed in detail in Chapter 5 is based on the assumption that in a very large or infinite system, the excess charge (corresponding to quasiparticle, i.e. quasihole or quasielectron) would obey the symmetry of the underlying ionic lattice. In Chapter 6, however, we describe the possibility of a spontaneous charge carrier localization that would lead to energy curvature which remains finite even in the limit of infinite systems.
Here we exclude charge carrier localization in extended atomic systems due to disorder, point defects or distortions of the ionic lattice, since it is our goal to investigate the limiting behavior of an infinite periodic system. Hence we study only perfectly ordered structures constructed in a similar fashion as in Chapter 5: We consider 1D molecular chains of conjugated polymers (trans-polyacetylene and polythiophene illustrated in Figure 3.3) which we construct as completely planar and ideally periodic within the chain boundaries.
First we analyze the dependence of the ionization potentials estimated as the negative of the highest occupied eigenstate energy "H, which is (in principle) exact in DFT (cf. Sections 2.2.1 and 2.2.2), and corresponds to the lowest energy of formation of a hole in the system. It has to be noted here that for the class of optimally tuned range-separated hybrid functionals, denoted as BNL*, such equivalence is enforced by tuning for the zero energy curvature condition (cf. Chapter 5). Our results clearly indicate that when the theory accounts for the presence of non-local exchange (either by using the BNL* functional or the Hartree-Fock method) the ionization potential converges to a constant value
3.3. SPONTANEOUS CHARGE CARRIER LOCALIZATION
Figure 3.3: An example of a planar polythiophene molecule is shown with 4 repeat units (containing two thiophene rings each). Yellow, black and white spheres represent sulphur, carbon and hydrogen atoms, respectively.
which becomes independent of system size. Similar behavior is observed for other observables, namely optical absorption peaks and exciton binding energies estimated from TD DFT calculations. This energy stabilization occurs on length scales of several nanometers.
Further analysis of the DFT results reveal that the rapid stabilization of the ionization potential and its independence on the system length is associated with localization of the hole density. Based on the hole distribution along the backbone of the polymer, we calculate the hole characteristic size (for definition and procedure see Chapter 6), which shows two types of regimes illustrated in Figure 3.4: (i) for small systems, the strong quantum confinement dominates and the hole size increases linearly with system length; (ii) for large systems, the hole localizes and its size is independent of polymer length. We also present the connection between the hole localization and experimentally observed polaron formation in conjugated polymers.
In order to confirm that our results are not an artifact of the DFT cal-culations, i.e. that the phenomenon is observed even when the theoretical de-scription does not rely on the mean field approximation, we perform additional computations using many body perturbation theory in the G0W0 approxima-tion introduced in Secapproxima-tion 2.4.3. Since the systems of interest are of significant size, we employ a stochastic GW approach recently developed by Neuhauser et al. (see Chapter 6 for reference and Chapter 7 for a detailed description of the method). This approach allows us to calculate the quasiparticle energies for all chains considered, representing thus the largest ever accomplished G0W0 com-putations for polymers, and confirm that the hole energies become independent of the length of the polymer in close agreement with the results obtained with BNL* functional.
This work has been accepted for publication:
Vojtˇech Vlˇcek, Helen R. Eisenberg, Gerd Steinle-Neumann, Daniel Neuhauser, Eran Rabani, Roi Baer,Spontaneous charge carrier localization in extended one-dimensional systems, Phys. Rev. Lett. (in press)
I have performed all the calculations presented in the manuscript, designed the analysis procedure and performed it, made the implementations necessary for the stochastic GW approach and written the first draft of the paper.
Figure 3.4: Di↵erence in the densities of a neutral system and a cation of a trans-polyacetylene chain with 50 repeat units. Density di↵erences are computed by Hartree-Fock and DFT calculations on various levels: LDA, B3LYP, and the optimally tuned range-separated hybrid functional BNL* (top panel). The isosurface plots present both positive (yellow) and negative (aqua) values due to the redistribution of the density upon ionization. While results obtained with LDA and B3LYP functionals lead to complete delocalization of the hole densities, BNL* and HF shown localization of the hole in the center of the polymer chain. HF results, however, su↵er from spurious density fluctuations along the whole backbone of the polymer (for details see Chapter 6). The lower panel shows cumulative hole distribution ⇢ obtained by partial integration of the hole density along the polymer axis z. The distributions curves obtained for di↵erent methods are distinguished by colors in the graph. The narrow change of the integral value for in BNL* and HF results indicate the presence of a localized hole. For comparison a hole distribution obtained with BNL*
for a shorter chain with 40 repeat units is shown by the black dashed line and indicates that the hole distribution no longer changes with increasing system size.