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Charge self-localization in p-conjugated polymers by long range corrected hybrid functionals† Nan Shao and Qin Wu* We systematically investigate the capability of hybrid functionals for describing charge self-localization in conjugated polymers, using the critical test that the spatial extension of a localized charge should be polymer length independent. We first compare the new long-range corrected (LRC) hybrids with conventional global hybrids and find that the former has a clear and important advantage over the latter in being significantly less spin contaminated. We then focus on LRC hybrids and investigate in detail the dependence of charge localization on the range parameter. We show that this parameter consistently needs to be about 0.2 bohr1 or larger to produce self-localized charges across different polymers. We introduce a new measure to determine the charge localization length, and then consider how properties
Received 25th October 2013, Accepted 3rd February 2014 DOI: 10.1039/c3cp54515f
related to localized charges converge with the polymer length and how they depend on the range parameter. These properties include the reorganization energy in the Marcus theory for electron transfer and the lowest excitation energy of a polaron. We discuss parameter tuning to experimental results and also suggest 0.2 bohr1 without tuning for exploratory studies based on the preference for least
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spin contaminations.
I. Introduction One of the reasons that p-conjugated polymers have attracted so much interest as potential electronic materials is the enormous design possibilities.1–3 Theory and computation play an important role in molecular design, and its significance increases with better, more accurate and predictive methods. Although density functional theory (DFT) methods have been very successful in many studies of organic molecules, they are less so when it comes to charges in conjugated polymers, where the charge is known to be self-localized and form a polaron.4–6 All local and semi-local DFT methods notoriously over-delocalize charges, leading to the term ‘‘delocalization error’’ being coined,7 whose rigorous definition can be derived from the concave deviation in the fractional-charge energy curve. Under the same criterion, exact exchange (i.e. Hartree–Fock exchange) is shown to have localization error, i.e. convex deviation.8–10 Interestingly, the second-order Møller–Plesset perturbation theory (MP2) is shown to correct such errors in Hartree–Fock,41 and promising results for polarons in conjugated polymers were obtained with MP2.14,42,43 Within DFT, however, a functional free of either delocalization or localization errors is not yet available, though
Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA. E-mail: [email protected] † Electronic supplementary information (ESI) available: The reorganization energies and excitation energies calculated for five polymers. See DOI: 10.1039/ c3cp54515f
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some very good approximations exist.11–14 On the empirical level, simple hybrid functionals, which linearly mix exact exchange with semilocal exchange functionals and thus retain computational efficiency, often show much smaller errors.15,16 However, the mixing ratio in these functionals is often not optimized to describe charge localization in conjugated polymers. For instance, B3LYP, which has 20% exact exchange, was shown to over-delocalize the charge defect in polyacetylene.42 It was pointed out some time ago that 50% of exact exchange is needed for charge localization in polymers;14 but later the same authors observed that even 50% of exact exchange does not produce fully localized charges.15 Recent studies saw another class of efficient hybrid functionals, i.e. rangeseparated hybrids (RSH), where the ratio of exact exchange actually changes with different electron interaction distances.17 For molecules in chemistry, it is known that exact exchange at the long range is particularly important because it provides the correct asymptotic decay for the exchange–correlation potential that is missing in semilocal functionals.12,13 Therefore such hybrid functionals are called long-range corrected (LRC) functionals, and a range parameter in these functionals determines how fast the exact exchange mixes in as the electron interaction distance increases. We have shown that LRC hybrid functionals can be effective in describing charge self-localization in oligofluorenes in a previous work.19 The same conclusion was also reached in other reports16,20 for other polymers. However, to the best of our knowledge, there has been no systematic study on how the range parameter in LRC functionals affects the charge localization across different conjugated polymers.
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In this work we systematically investigate the capability of hybrid functionals for describing charge self-localization in conjugated polymers. Specifically we will try to answer the following questions: (1) How do we decide that a charge is fully localized in a polymer and subsequently its charge localization length? (2) Do LRC hybrids have other advantages over conventional hybrids besides the asymptotic behavior? (3) Is there a threshold value of the range parameter in LRC functionals for them to produce charge self-localization? (4) How do the charge localization and its associated properties change with the amount of exact exchange mixed in? Definitive answers to these questions have important practical usages, but are hard to obtain theoretically. The purpose of this work is to learn something through empirical testing that we hope will be useful for future work. For testing purposes, it is important to have a standard that all systems follow. The essential property that we will test here is charge self-localization; thus the first thing is to specify the exact conditions of charge localization. Here we adopt the intuitive criterion that the spatial extension of a localized charge does not depend on the polymer length. That is to say, when a charge is injected into a series of oligomers of increasing lengths, while the charge spreads over the whole oligomer when the molecule is short, at some point, the spatial extension of the charge does not change anymore. Neither do properties that are associated with a localized charge, such as the polaron excitation energy. This test is commonly used in experiments to probe polaron lengths. While easy to understand, it requires repeating experiments or calculations on the same oligomer with increasing length until properties associated with the charge saturate. It is therefore more stringent than only calculating a single long oligomer and showing a centralized charge distribution, because the peak and width of the charge distribution can change when the oligomer length increases if the charge is not truly localized, which has been shown previously.15 Because our focus is on how charge localization depends on the amount of exact exchange in different hybrid functionals, we fixed the semilocal part of the exchange functional to be PBE21 for all our tests. Choosing PBE also has the advantage that its range-separated forms are available.22 We then added a full PBE correlation functional in all calculations (details appear in the next section). Therefore, the only parameters that we are testing are the linear mixing ratio in the conventional hybrids, and the range parameter in the LRC hybrids. We will compare our results with those from some named functionals as used in other reports16,20 and show that the amount of exact exchange is indeed the most important factor for charge localization. We note here that the conventional and LRC hybrids should both be regarded as global hybrid functionals. There are also proposals of local hybrids,18 but those are still early in development stage and will not be considered in this work. To make sure that our results are not limited to a single system, we choose five representative and well studied conjugated polymers:29 polythiophene (PT), polyphenylene (PP), polypyrrole (PPy), poly ( p-phenylenevinylene) (PPV), and poly (2,5-thienylenevinylene) (PTV) (see Scheme 1). We will study
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Scheme 1 Five conjugated polymers whose cations are investigated in this study.
their cations only since semiconducting polymers are commonly used as hole-transport materials. In the rest of this paper, we will first present the computational details (Section II), followed by the results (Section III) of charge and spin distributions, the reorganization energies and the excitation energies. In Section IV we will discuss the determination of localization length and the convergence of associated results. We will present conclusions in Section V.
II. Computational details All calculations are done with the package Q-Chem 4.023 in the unrestricted Kohn–Sham formalism. The basis set 6-31G(d) is used throughout. The exact definitions of the exchange– correlation functionals follow eqn (1) and (2), where we use hPBE and LRC-oPBE to denote the conventional hybrid functionals and the long-range corrected ones, respectively. Expressions for PBE SR-PBE EPBE are from ref. 12, 21–25, while EHF x , Ec , and Ex x is simply the Hartree–Fock exchange, and ELR-HF is the same exchange with x the Coulomb operator 1/r12 replaced by erf(or12)/r12, where erf is the standard error function and r12 is the interelectronic distance. The two parameters under test are a in E hPBE and o in E LRC-oPBE , which xc xc has a unit of distance inverse. PBE = (1 a)EPBE + aEHF EhPBE xc x x + Ec
(1)
ELRC-oPBE = ESR-PBE + ELR-HF + EPBE xc x x c
(2)
To test the range parameter o, we have applied o = 0.1, 0.2, 0.3, and 0.4 bohr1 in the LRC functionals for all polymers. We have also tested the conventional hybrids with a = 0.25, 0.50, 0.75 and 1.00 for selected systems. To get a quantitative idea of how exact exchange is included in LRC hybrids, we plot erf(or12), i.e., the ratio of exact exchange, as a function of o at a few r12 values, shown in Fig. 1. It is clear that larger o values lead to more exact exchange in all distances, but the increase is much faster for the long range (larger r12) than for the short range. For conventional hybrids, the ratio of exact exchange is a constant for all r12 values. We optimize geometries with every new functional, and report charge density and spin density Mulliken populations. The linear response time-dependent (TD) DFT method has been applied for the calculation of excitation energies at the ground-state optimized geometry.
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Paper Table 1
+
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4T 6T+ 8T+ 10T+ 12T+ 14T+ 16T+
Fig. 1 The error function plotted against the range parameter for selected distances.
III. Results 1. Long-range corrected vs. conventional: polythiophene We first compare the conventional hybrids with long-range corrected hybrids using oligo-thiophene as an example. In Fig. 2, we plot the charge distribution on each monomer along the oligothiophene chains with increasing lengths, ranging from 4 to 16 monomers at increments of 2 and aligned at the center. Two conventional hybrids (a = 0.50 and 0.75) and two LRC hybrids (o = 0.1 and 0.2 bohr1) are chosen and their performance compared. The a = 0.50 conventional hybrid is equivalent to the Becke half-and-half functional26 that has been tested before.15 As seen previously, we see the same kind of behavior where even though a peaked charge distribution is present in all the chain lengths, the peak height decreases as the chain length increases, indicating that charge is not really localized. The same pattern is also seen with the LRC hybrid of o = 0.1 bohr1. However, the trend changes when we increase a to 0.75 and o to 0.2 bohr1, where the peak height remains unchanged for chains longer than a
hS2i values for all the data points in Fig. 2
o = 0.1 bohr1
o = 0.2 bohr1
a = 0.50
a = 0.75
0.767 0.778 0.786 0.790 0.791 0.790 0.788
0.808 0.837 0.853 0.859 0.857 0.858 0.858
0.858 0.899 0.926 0.939 0.940 0.928 0.923
1.031 1.240 1.503 1.805 2.135 2.280 2.589
certain length. For really long oligomers, the charge shifts its location along the chain, but the shape of the distribution curve remains the same, as it should be for a truly localized charge. Increasing the parameter values will further localize the charge, a point we will explore more in next subsection. But first we would like to compare the quality of the solutions provided by conventional and LRC hybrids. For that, we look at the spin contamination in the single determinant. In Table 1, we list the hS2i values for all the data points in Fig. 2. Because the charged molecule is a doublet, its exact wavefunction has hS2i = 0.75. However, for calculations done in the spin-unrestricted formalism, the resulting single determinant wavefunction formed from Kohn–Sham orbitals necessarily has some spin contamination.27 Nonetheless, excessive spin contamination should be avoided. The contaminations seen in the a = 0.75 conventional hybrid is undoubtedly due to its high percentage of exact exchange, which makes the results similar to those of unrestricted Hartree–Fock (UHF) calculations. UHF is known to be unreliable for open-shell conjugated systems because of spin contamination.44 Spin contamination in LRC hybrids also increases as more exact exchange is used (with larger o values). However, with o = 0.2 bohr1, charge self-localization is realized and spin contaminations are modest. From this point of view, LRC hybrids have a clear advantage over the conventional hybrids. Therefore, we will only use LRC hybrids in the following tests.
Fig. 2 Charge and spin distribution of nT+ with length from 4 to 16 plotted and specified in different colors. Range parameter o = 0.1 and 0.2 bohr1, and a = 0.50 and 0.75 in conventional hybrids.
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Fig. 3 Charge and spin distribution of P, Py, PV and TV with length from 4 to 16 plotted and specified in different colors. Range parameter o = 0.1 and 0.2 bohr1.
Though we have chosen o = 0.1 and 0.2 bohr1 to contrast the delocalization and localization behaviors, the actual localization may occur with o value between these two numbers. We have tested additional functionals with o = 0.12, 0.14, 0.16 and 0.18 bohr1 and found that the charge self-localization is present for o = 0.16 and 0.18 bohr1. (All results can be found in ESI.†) Therefore the threshold value is probably o = 0.15 bohr1 for polythiophene. However, our purpose in this work is not to pinpoint the o value to such a fine degree for each polymer. Instead, we will probe the more general trend across different polymers; hence we will focus on results for o = 0.1 and 0.2 bohr1, and later also 0.3 and 0.4 bohr1.
populations of all monomers and their positions in the chain to characterize the localization. To find out if the qualitative difference between o = 0.1 bohr1 and o = 0.2 bohr1 is only limited to PT, we have done the same calculations for PP, PPy, PPV and PTV, and the results are collected in Fig. 3. It is clear that a qualitative difference in saturation between results of o = 0.1 bohr1 and o = 0.2 bohr1 can be seen in all systems. For the same calculation, there is also a difference between spin and charge distributions in that the charge distribution can be non-continuous towards both ends of the oligomer. This is a boundary effect resulting from Coulomb interaction in a finite chain. For an infinite polymer, the charge and spin distribution should overlap.
2. Charge density vs. spin density Because of the modest size basis sets, the commonly used Mulliken population analysis should provide a good description of the charge and spin density distributions. We report the
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3. Reorganization energies and polaron excitation energies With truly localized charges, properties that are solely associated with this local charge would saturate as well, i.e. converge as a
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function of the chain length. In this section, we will examine two such properties: the reorganization energy, and the polaron excitation energy. The reorganization energy is an important parameter for describing charge transfer processes. In a condensed medium, it includes both internal and external parts, where the latter is due to polarization of the environment. In this work, we only study the internal part.28 Assuming a self-exchange process for charge transfer, we can calculate the internal reorganization energy as l = (E[1J0] + E[0J1]) (E[1J1] + E[0J0]),
(3)
where the notation E[AJB] means the energy of the A state at the optimized geometry of the B state, and A or B represents the charge. Results of reorganization energies as a function of the chain length of five polymers with different range parameters are plotted in Fig. 4, and the tabulated values are given in the ESI.† It is clear that with o = 0.1 bohr1, the reorganization energy decreases continuously as the chain length increases. With o Z 0.2 bohr1 the reorganization energy becomes constant for longer chains, though non-monotonic change can be seen for short chains. Overall, for a given molecule, the reorganization energy increases with the o value. The electronic absorption spectra of polarons in conjugated polymers have a clear two-band feature that facilitates comparative studies from both theory and experiments.4,29 Hence, experimental results for oligomers with definitive structures and charges can be used as a calibration tool for computation methods if the solvation and counterion effects are known experimentally and are properly accounted for computationally. We have used such a combined experiments and theory approach to the polyfluorene anions.19 An extensive study has also been
carried out by Salzner38,39 for polythiophene. In this work, we have aimed at the intrinsic properties of the functional and left out the solvation and counterion effects in our calculations; therefore we will not make quantitative comparison with those results. Instead, we focus on the evolution of the first polaron band as predicted purely from theory. In Fig. 5 we plot energies of the lowest transition that has significant oscillator strength as calculated by TDDFT. (Tabulated energy and strength values, as well as a plot of strengths, can be found in the ESI.†) Except for a few cases in the shortest (n = 2) oligomers, such a transition is also the lowest transition overall. As seen in Fig. 5, the excitation energy decreases monotonically as the oligomer length increases for all systems and all o values. However, the decrease levels off for longer chains with o Z 0.2 bohr1, but still continues for o = 0.1 bohr1. For a given molecule at any size, this energy increases with the o value. Trends in the oscillator strength are not as clear. In general, the strengths increase with the oligomer length, but they show no obvious convergences. For short oligomers, we see increased strengths as o grows, which agrees with the finding that larger amounts of HF exchange in the XC functional lead to larger oscillator strengths.45 However, such correspondence is lost in all these systems when the chain length becomes sufficiently large.
IV. Discussion Our interest in studying charge localization in semiconducting polymers is ultimately to study the charge transfer/transport mechanism in these materials. This is driven by the fact that a critical bottleneck in enhancing the efficiency of organic photovoltaics is to improve the charge mobility in the polymer.30
Fig. 4 Reorganization energies as defined by eqn (3) for PT+, PP+, PPy+, PPV+ and PTV+ as a function of the oligomer length.
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Fig. 5
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First excitation energies as calculated with TDDFT for PT+, PP+, PPy+, PPV+ and PTV+ as a function of the oligomer length.
Because of the charge localization and the amorphous structure of the polymer domain, ‘charge hopping’ is accepted as the microscopic mechanism in the charge transport process.31 Hence when simulating the charge transport in the bulk materials, a kinetic Monte Carlo strategy is often adopted, where the Marcus theory for charge transfer serves as a model to calculate the rates of elementary hopping steps.32 Important parameters in such simulations include the charge localization length and the reorganization energy. In the first subsection, we will describe our approach to calculating the localization length from the charge distribution results. We will then discuss the convergence behaviors of the reorganization energy and the excitation energy as a function of the oligomer length. The dependence of these energies on the range parameter will also be addressed. 1. Determination of charge localization length We consider the charge localization length as the number of consecutive repeating units on the polymer molecule that holds the majority of the charge distribution. To reach an unambiguous measure by which the charge localization length can be determined, we turn to the spin density distribution because the Mulliken charges show non-continuous distributions, as explained earlier. To measure the charge localization length from the spin distribution, we borrow the idea of the full width at half maximum (FWHM). However, instead of visually measuring it from the distribution curve, which requires plotting and may give a fractional number of repeating units, we use a measure in which we add the spin density monomer by monomer beginning from highest until we have the accumulated spin density population that exceeds 0.75. The number 0.75 is chosen because that is the area covered by FWHM in a normal distribution. With this quantitative measure,
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the charge localization length in our definition means the smallest number of repeating units that contains at least 75% of the charge distribution. For a distribution that self-localizes in the long chain, this number increases initially with oligomer length but then levels off. This is indeed the case for LRC hybrids with o greater than or equal to 0.2 bohr1, as shown in Fig. 6. On the other hand, the localization length continues to grow with o = 0.1 bohr1, indicating that the charge is not truly localized. It is interesting to note that for any polymer, the localization length between o = 0.2 bohr1 and o = 0.4 bohr1 differs by only one unit. In a closely related work,20 two LRC functionals, CAM-B3LYP33 and LC-wPBE,34 are used in the localization study of positively charged oligophenylene vinylene (PV). We note that for both functionals, the range parameter is 0.33 bohr1. However, CAM-B3LYP also has a global damping factor of exact exchange so that even at the long-range limit, there is only 60% of exact exchange. This work,20 using the FWHM measure of the bond length alternation, concludes the localization length (‘‘characteristic size’’) for PV+ is close to 3 by LC-wPBE, in agreement with our o = 0.3 bohr1 results. CAM-B3LYP gives a slightly greater number, which can be expected because of the reduced amount of exact exchange. Therefore, these results support the idea that the amount of exact exchange is the single most important factor in determining charge localization length in conjugated polymers. Though a characteristic size is also assigned to the B3LYP and PBE results in the same work, it is drawn upon calculations for a single oligomer with 10 units.20 Our study in this work suggests that those results may change with increasing oligomer lengths. Finally, to make sure that our observation that the localization length will keep changing with
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Fig. 6
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Charge localization length of PT+, PP+, PPy+, PPV+ and PTV+ as determined by the 75% spin population rule.
o = 0.1 bohr1 holds for even longer oligomers, we have extended the calculations for polyphenylene to 18, 20, 22 and 24 monomers and the results can be found in the ESI.† We do not see converging charge or spin distributions. Therefore, we are more confident to conclude that LRC hybrids with o = 0.1 bohr1 do not produce charge self-localization. 2. Comparing the convergence of reorganization energies and excitation energies We expect properties due to a localized charge to saturate with the chain length. That is indeed the case for the reorganization energy and the lowest polaron excitation energy as shown in Fig. 4 and 5, where the results for o = 0.1 bohr1 decrease continuously as the chain length increase, while for o = 0.2, 0.3, 0.4 bohr1 they converge at long oligomer lengths. Both energies increase in magnitude as the charge becomes more localized (greater o value) and the increases are quite significant, which is different from the localization length. However, there are differences in chain length dependence between the reorganization energy and the excitation energy because the former is a ground state property calculated from two different geometries, while the latter involves electronic response kernel and the virtual states. In particular, the excitation energy decays monotonically as a function of the chain length, which is a consequence of the HOMO–LUMO gap decreasing in the same pattern. The reorganization energy has a less clear trend as a function of the chain length and shows a much smaller change from the shortest chain to the longest chain. But in general it tends to increase initially as the oligomer grows, which can be attributed to the increase of the number of bonds and angles involved in the reorganization as the molecule becomes larger.
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Such a trend is soon balanced by the localized charge distribution, which leads to saturation of the results in longer chains. The fact that the reorganization energy and excitation energies both increase with the range parameter might make it tempting to converge these results with respect to o. However, such convergence has no physical basis, but is rather an artifact of the functional. In fact, the convergence can be expected when full exact exchange is included (when o goes to infinity), at which limit the calculation is simply unrestricted Hartree–Fock plus DFT correlation, known to be a bad choice. A non-empirical approach to tune the range parameter is to utilize the exact ionization potential conditions in DFT.35,36 However, this approach has been shown to be problematic when used for neutral polymer molecules with increasing lengths.37 Moreover, for the charged polymers that we are interested in here, there is an additional problem in the uncertainty of where the second charge occurs, i.e. bipolaron or side-by-side polarons.19 At the current stage we prefer to treat the range parameter empirically, where we fit this single parameter o to reproduce known experimental results and then use the same parameter throughout a complete study. The polaron excitation energy is well suited for such fitting. It has clear experimental signals.19 The calculated results all converge nicely at long chain length, and they also have sufficient sensitivity towards the range parameter. However, proper fitting would require accounting for possible external effects in experiments, such as solvent and counterions, which have been proven to help charge localization.20,38,39 In the case of no experimental data, we recommend using o = 0.2 bohr1 in the LRC hybrid for exploratory studies. Such a functional can fully localize, and is unlikely to over-localize, charges in a
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conjugated polymer; at the same time, it has the least spin contamination, minimizing the risk of getting a qualitatively incorrect picture.44 Quantitatively energy values in the range of a few tenths of eV can be obtained by fine-tuning the range parameter as shown here, and by testing different semilocal functionals, which is not explored in this work.
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V. Conclusions Hybrid functionals can effectively describe charge selflocalization in conjugated polymers and the results critically depend on the amount of exact exchange being included. Using the criterion that the spatial distribution of a localized charge is polymer length independent, we find that conventional hybrids need about 75% of exact exchange, while LRC hybrids require the range parameter to be only 0.2 bohr1 or larger. The modest spin contaminations in the LRC hybrids make them a clear favorite over conventional hybrids with a high percentage of exact exchange. Systematic investigation of charge localization in five different polymers and its dependence on the range parameter points to some common features. First, o has to be sufficiently large to have charge localization. Second, the threshold value seems to be about 0.2 bohr1 for all polymers. Third, further increasing o will make the charge more localized. Fourth, the localization length changes by only one monomer with o values but the reorganization energy and polaron excitation energy can change significantly. The energetic variances point to the necessity of tuning the range parameter, while the commonalities suggest that a single range parameter may work for all systems. Unfortunately, this parameter cannot be determined non-empirically, but may be fitted to experimental results. We have in this work only considered the intrinsic charge localization effects in stand-alone polymer molecules. It represents the most fundamental level in multi-scale modeling of charge transport in polymer electronic materials.40 Our results for the charge localization length provide some quantitative guidance in building such models. Our next step will be to address the uncertainty in the reorganization energy by choosing the range parameter. We have noted that fitting to experimental polaron excitation energies can be an effective approach to determine the range parameter. However, such fitting would require a proper account of external effects that may be important in experimental conditions, such as solvent and counterions. In future work, we will investigate how the charge localization and its associated properties change in the presence of those effects.
Acknowledgements All calculations were performed on the computer clusters at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-AC02-98CH10886.
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25 M. A. Rohrdanz, K. M. Martins and J. M. Herbert, J. Chem. Phys., 2009, 130, 054112. 26 A. D. Becke, J. Chem. Phys., 1993, 98, 5648–5652. 27 J. A. Pople, P. M. W. Gill and N. C. Handy, Int. J. Quantum Chem., 1995, 56, 303–305. 28 G. R. Hutchison, M. A. Ratner and T. J. Marks, J. Am. Chem. Soc., 2005, 127, 2339–2350. 29 Y. Furukawa, J. Phys. Chem., 1996, 100, 15644–15653. ´chet, J. Am. Chem. Soc., 2011, 30 P. M. Beaujuge and J. M. J. Fre 133, 20009–20029. 31 A. Troisi, Chem. Soc. Rev., 2011, 41, 2347–2358. 32 R. A. Marcus, J. Chem. Phys., 1956, 24, 966–978. 33 T. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett., 2004, 393, 51–57. 34 Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai and K. Hirao, J. Chem. Phys., 2004, 120, 8425–8433. 35 T. Stein, L. Kronik and R. Baer, J. Am. Chem. Soc., 2009, 131, 2818–2820.
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36 R. Baer, E. Livshits and U. Salzner, Annu. Rev. Phys. Chem., 2010, 61, 85. ¨rzdo ¨rfer, J. S. Sears, C. Sutton and J. L. Bre ´das, J. Chem. 37 T. Ko Phys., 2011, 135, 204107. 38 U. Salzner, J. Chem. Theory Comput., 2007, 3, 1143–1157. 39 U. Salzner, J. Phys. Chem. A, 2008, 112, 5458–5466. ¨hle, J. Kirkpatrick and D. Andrienko, J. Chem. Phys., 40 V. Ru 2010, 132, 134103. ´nchez and W. Yang, J. Chem. Theory 41 A. J. Cohen, P. Mori-Sa Comput., 2009, 5, 786–792. 42 B. Champagne and M. Spassova, Phys. Chem. Chem. Phys., 2004, 6, 3167–3174. 43 V. Monev, M. Spassova and B. Champagne, Int. J. Quantum Chem., 2005, 1044, 354–366. 44 V. M. Geskin, J. Cornil and J. L. Bredas, Chem. Phys. Lett., 2005, 403, 228–231. 45 M. Miura, Y. Aoki and B. Champagne, J. Chem. Phys., 2007, 127, 084103.
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Charge self-localization in p-conjugated polymers by long range corrected hybrid functionals† Nan Shao and Qin Wu* We systematically investigate the capability of hybrid functionals for describing charge self-localization in conjugated polymers, using the critical test that the spatial extension of a localized charge should be polymer length independent. We first compare the new long-range corrected (LRC) hybrids with conventional global hybrids and find that the former has a clear and important advantage over the latter in being significantly less spin contaminated. We then focus on LRC hybrids and investigate in detail the dependence of charge localization on the range parameter. We show that this parameter consistently needs to be about 0.2 bohr1 or larger to produce self-localized charges across different polymers. We introduce a new measure to determine the charge localization length, and then consider how properties
Received 25th October 2013, Accepted 3rd February 2014 DOI: 10.1039/c3cp54515f
related to localized charges converge with the polymer length and how they depend on the range parameter. These properties include the reorganization energy in the Marcus theory for electron transfer and the lowest excitation energy of a polaron. We discuss parameter tuning to experimental results and also suggest 0.2 bohr1 without tuning for exploratory studies based on the preference for least
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spin contaminations.
I. Introduction One of the reasons that p-conjugated polymers have attracted so much interest as potential electronic materials is the enormous design possibilities.1–3 Theory and computation play an important role in molecular design, and its significance increases with better, more accurate and predictive methods. Although density functional theory (DFT) methods have been very successful in many studies of organic molecules, they are less so when it comes to charges in conjugated polymers, where the charge is known to be self-localized and form a polaron.4–6 All local and semi-local DFT methods notoriously over-delocalize charges, leading to the term ‘‘delocalization error’’ being coined,7 whose rigorous definition can be derived from the concave deviation in the fractional-charge energy curve. Under the same criterion, exact exchange (i.e. Hartree–Fock exchange) is shown to have localization error, i.e. convex deviation.8–10 Interestingly, the second-order Møller–Plesset perturbation theory (MP2) is shown to correct such errors in Hartree–Fock,41 and promising results for polarons in conjugated polymers were obtained with MP2.14,42,43 Within DFT, however, a functional free of either delocalization or localization errors is not yet available, though
Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA. E-mail: [email protected] † Electronic supplementary information (ESI) available: The reorganization energies and excitation energies calculated for five polymers. See DOI: 10.1039/ c3cp54515f
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some very good approximations exist.11–14 On the empirical level, simple hybrid functionals, which linearly mix exact exchange with semilocal exchange functionals and thus retain computational efficiency, often show much smaller errors.15,16 However, the mixing ratio in these functionals is often not optimized to describe charge localization in conjugated polymers. For instance, B3LYP, which has 20% exact exchange, was shown to over-delocalize the charge defect in polyacetylene.42 It was pointed out some time ago that 50% of exact exchange is needed for charge localization in polymers;14 but later the same authors observed that even 50% of exact exchange does not produce fully localized charges.15 Recent studies saw another class of efficient hybrid functionals, i.e. rangeseparated hybrids (RSH), where the ratio of exact exchange actually changes with different electron interaction distances.17 For molecules in chemistry, it is known that exact exchange at the long range is particularly important because it provides the correct asymptotic decay for the exchange–correlation potential that is missing in semilocal functionals.12,13 Therefore such hybrid functionals are called long-range corrected (LRC) functionals, and a range parameter in these functionals determines how fast the exact exchange mixes in as the electron interaction distance increases. We have shown that LRC hybrid functionals can be effective in describing charge self-localization in oligofluorenes in a previous work.19 The same conclusion was also reached in other reports16,20 for other polymers. However, to the best of our knowledge, there has been no systematic study on how the range parameter in LRC functionals affects the charge localization across different conjugated polymers.
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In this work we systematically investigate the capability of hybrid functionals for describing charge self-localization in conjugated polymers. Specifically we will try to answer the following questions: (1) How do we decide that a charge is fully localized in a polymer and subsequently its charge localization length? (2) Do LRC hybrids have other advantages over conventional hybrids besides the asymptotic behavior? (3) Is there a threshold value of the range parameter in LRC functionals for them to produce charge self-localization? (4) How do the charge localization and its associated properties change with the amount of exact exchange mixed in? Definitive answers to these questions have important practical usages, but are hard to obtain theoretically. The purpose of this work is to learn something through empirical testing that we hope will be useful for future work. For testing purposes, it is important to have a standard that all systems follow. The essential property that we will test here is charge self-localization; thus the first thing is to specify the exact conditions of charge localization. Here we adopt the intuitive criterion that the spatial extension of a localized charge does not depend on the polymer length. That is to say, when a charge is injected into a series of oligomers of increasing lengths, while the charge spreads over the whole oligomer when the molecule is short, at some point, the spatial extension of the charge does not change anymore. Neither do properties that are associated with a localized charge, such as the polaron excitation energy. This test is commonly used in experiments to probe polaron lengths. While easy to understand, it requires repeating experiments or calculations on the same oligomer with increasing length until properties associated with the charge saturate. It is therefore more stringent than only calculating a single long oligomer and showing a centralized charge distribution, because the peak and width of the charge distribution can change when the oligomer length increases if the charge is not truly localized, which has been shown previously.15 Because our focus is on how charge localization depends on the amount of exact exchange in different hybrid functionals, we fixed the semilocal part of the exchange functional to be PBE21 for all our tests. Choosing PBE also has the advantage that its range-separated forms are available.22 We then added a full PBE correlation functional in all calculations (details appear in the next section). Therefore, the only parameters that we are testing are the linear mixing ratio in the conventional hybrids, and the range parameter in the LRC hybrids. We will compare our results with those from some named functionals as used in other reports16,20 and show that the amount of exact exchange is indeed the most important factor for charge localization. We note here that the conventional and LRC hybrids should both be regarded as global hybrid functionals. There are also proposals of local hybrids,18 but those are still early in development stage and will not be considered in this work. To make sure that our results are not limited to a single system, we choose five representative and well studied conjugated polymers:29 polythiophene (PT), polyphenylene (PP), polypyrrole (PPy), poly ( p-phenylenevinylene) (PPV), and poly (2,5-thienylenevinylene) (PTV) (see Scheme 1). We will study
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Scheme 1 Five conjugated polymers whose cations are investigated in this study.
their cations only since semiconducting polymers are commonly used as hole-transport materials. In the rest of this paper, we will first present the computational details (Section II), followed by the results (Section III) of charge and spin distributions, the reorganization energies and the excitation energies. In Section IV we will discuss the determination of localization length and the convergence of associated results. We will present conclusions in Section V.
II. Computational details All calculations are done with the package Q-Chem 4.023 in the unrestricted Kohn–Sham formalism. The basis set 6-31G(d) is used throughout. The exact definitions of the exchange– correlation functionals follow eqn (1) and (2), where we use hPBE and LRC-oPBE to denote the conventional hybrid functionals and the long-range corrected ones, respectively. Expressions for PBE SR-PBE EPBE are from ref. 12, 21–25, while EHF x , Ec , and Ex x is simply the Hartree–Fock exchange, and ELR-HF is the same exchange with x the Coulomb operator 1/r12 replaced by erf(or12)/r12, where erf is the standard error function and r12 is the interelectronic distance. The two parameters under test are a in E hPBE and o in E LRC-oPBE , which xc xc has a unit of distance inverse. PBE = (1 a)EPBE + aEHF EhPBE xc x x + Ec
(1)
ELRC-oPBE = ESR-PBE + ELR-HF + EPBE xc x x c
(2)
To test the range parameter o, we have applied o = 0.1, 0.2, 0.3, and 0.4 bohr1 in the LRC functionals for all polymers. We have also tested the conventional hybrids with a = 0.25, 0.50, 0.75 and 1.00 for selected systems. To get a quantitative idea of how exact exchange is included in LRC hybrids, we plot erf(or12), i.e., the ratio of exact exchange, as a function of o at a few r12 values, shown in Fig. 1. It is clear that larger o values lead to more exact exchange in all distances, but the increase is much faster for the long range (larger r12) than for the short range. For conventional hybrids, the ratio of exact exchange is a constant for all r12 values. We optimize geometries with every new functional, and report charge density and spin density Mulliken populations. The linear response time-dependent (TD) DFT method has been applied for the calculation of excitation energies at the ground-state optimized geometry.
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Paper Table 1
+
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4T 6T+ 8T+ 10T+ 12T+ 14T+ 16T+
Fig. 1 The error function plotted against the range parameter for selected distances.
III. Results 1. Long-range corrected vs. conventional: polythiophene We first compare the conventional hybrids with long-range corrected hybrids using oligo-thiophene as an example. In Fig. 2, we plot the charge distribution on each monomer along the oligothiophene chains with increasing lengths, ranging from 4 to 16 monomers at increments of 2 and aligned at the center. Two conventional hybrids (a = 0.50 and 0.75) and two LRC hybrids (o = 0.1 and 0.2 bohr1) are chosen and their performance compared. The a = 0.50 conventional hybrid is equivalent to the Becke half-and-half functional26 that has been tested before.15 As seen previously, we see the same kind of behavior where even though a peaked charge distribution is present in all the chain lengths, the peak height decreases as the chain length increases, indicating that charge is not really localized. The same pattern is also seen with the LRC hybrid of o = 0.1 bohr1. However, the trend changes when we increase a to 0.75 and o to 0.2 bohr1, where the peak height remains unchanged for chains longer than a
hS2i values for all the data points in Fig. 2
o = 0.1 bohr1
o = 0.2 bohr1
a = 0.50
a = 0.75
0.767 0.778 0.786 0.790 0.791 0.790 0.788
0.808 0.837 0.853 0.859 0.857 0.858 0.858
0.858 0.899 0.926 0.939 0.940 0.928 0.923
1.031 1.240 1.503 1.805 2.135 2.280 2.589
certain length. For really long oligomers, the charge shifts its location along the chain, but the shape of the distribution curve remains the same, as it should be for a truly localized charge. Increasing the parameter values will further localize the charge, a point we will explore more in next subsection. But first we would like to compare the quality of the solutions provided by conventional and LRC hybrids. For that, we look at the spin contamination in the single determinant. In Table 1, we list the hS2i values for all the data points in Fig. 2. Because the charged molecule is a doublet, its exact wavefunction has hS2i = 0.75. However, for calculations done in the spin-unrestricted formalism, the resulting single determinant wavefunction formed from Kohn–Sham orbitals necessarily has some spin contamination.27 Nonetheless, excessive spin contamination should be avoided. The contaminations seen in the a = 0.75 conventional hybrid is undoubtedly due to its high percentage of exact exchange, which makes the results similar to those of unrestricted Hartree–Fock (UHF) calculations. UHF is known to be unreliable for open-shell conjugated systems because of spin contamination.44 Spin contamination in LRC hybrids also increases as more exact exchange is used (with larger o values). However, with o = 0.2 bohr1, charge self-localization is realized and spin contaminations are modest. From this point of view, LRC hybrids have a clear advantage over the conventional hybrids. Therefore, we will only use LRC hybrids in the following tests.
Fig. 2 Charge and spin distribution of nT+ with length from 4 to 16 plotted and specified in different colors. Range parameter o = 0.1 and 0.2 bohr1, and a = 0.50 and 0.75 in conventional hybrids.
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Fig. 3 Charge and spin distribution of P, Py, PV and TV with length from 4 to 16 plotted and specified in different colors. Range parameter o = 0.1 and 0.2 bohr1.
Though we have chosen o = 0.1 and 0.2 bohr1 to contrast the delocalization and localization behaviors, the actual localization may occur with o value between these two numbers. We have tested additional functionals with o = 0.12, 0.14, 0.16 and 0.18 bohr1 and found that the charge self-localization is present for o = 0.16 and 0.18 bohr1. (All results can be found in ESI.†) Therefore the threshold value is probably o = 0.15 bohr1 for polythiophene. However, our purpose in this work is not to pinpoint the o value to such a fine degree for each polymer. Instead, we will probe the more general trend across different polymers; hence we will focus on results for o = 0.1 and 0.2 bohr1, and later also 0.3 and 0.4 bohr1.
populations of all monomers and their positions in the chain to characterize the localization. To find out if the qualitative difference between o = 0.1 bohr1 and o = 0.2 bohr1 is only limited to PT, we have done the same calculations for PP, PPy, PPV and PTV, and the results are collected in Fig. 3. It is clear that a qualitative difference in saturation between results of o = 0.1 bohr1 and o = 0.2 bohr1 can be seen in all systems. For the same calculation, there is also a difference between spin and charge distributions in that the charge distribution can be non-continuous towards both ends of the oligomer. This is a boundary effect resulting from Coulomb interaction in a finite chain. For an infinite polymer, the charge and spin distribution should overlap.
2. Charge density vs. spin density Because of the modest size basis sets, the commonly used Mulliken population analysis should provide a good description of the charge and spin density distributions. We report the
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3. Reorganization energies and polaron excitation energies With truly localized charges, properties that are solely associated with this local charge would saturate as well, i.e. converge as a
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function of the chain length. In this section, we will examine two such properties: the reorganization energy, and the polaron excitation energy. The reorganization energy is an important parameter for describing charge transfer processes. In a condensed medium, it includes both internal and external parts, where the latter is due to polarization of the environment. In this work, we only study the internal part.28 Assuming a self-exchange process for charge transfer, we can calculate the internal reorganization energy as l = (E[1J0] + E[0J1]) (E[1J1] + E[0J0]),
(3)
where the notation E[AJB] means the energy of the A state at the optimized geometry of the B state, and A or B represents the charge. Results of reorganization energies as a function of the chain length of five polymers with different range parameters are plotted in Fig. 4, and the tabulated values are given in the ESI.† It is clear that with o = 0.1 bohr1, the reorganization energy decreases continuously as the chain length increases. With o Z 0.2 bohr1 the reorganization energy becomes constant for longer chains, though non-monotonic change can be seen for short chains. Overall, for a given molecule, the reorganization energy increases with the o value. The electronic absorption spectra of polarons in conjugated polymers have a clear two-band feature that facilitates comparative studies from both theory and experiments.4,29 Hence, experimental results for oligomers with definitive structures and charges can be used as a calibration tool for computation methods if the solvation and counterion effects are known experimentally and are properly accounted for computationally. We have used such a combined experiments and theory approach to the polyfluorene anions.19 An extensive study has also been
carried out by Salzner38,39 for polythiophene. In this work, we have aimed at the intrinsic properties of the functional and left out the solvation and counterion effects in our calculations; therefore we will not make quantitative comparison with those results. Instead, we focus on the evolution of the first polaron band as predicted purely from theory. In Fig. 5 we plot energies of the lowest transition that has significant oscillator strength as calculated by TDDFT. (Tabulated energy and strength values, as well as a plot of strengths, can be found in the ESI.†) Except for a few cases in the shortest (n = 2) oligomers, such a transition is also the lowest transition overall. As seen in Fig. 5, the excitation energy decreases monotonically as the oligomer length increases for all systems and all o values. However, the decrease levels off for longer chains with o Z 0.2 bohr1, but still continues for o = 0.1 bohr1. For a given molecule at any size, this energy increases with the o value. Trends in the oscillator strength are not as clear. In general, the strengths increase with the oligomer length, but they show no obvious convergences. For short oligomers, we see increased strengths as o grows, which agrees with the finding that larger amounts of HF exchange in the XC functional lead to larger oscillator strengths.45 However, such correspondence is lost in all these systems when the chain length becomes sufficiently large.
IV. Discussion Our interest in studying charge localization in semiconducting polymers is ultimately to study the charge transfer/transport mechanism in these materials. This is driven by the fact that a critical bottleneck in enhancing the efficiency of organic photovoltaics is to improve the charge mobility in the polymer.30
Fig. 4 Reorganization energies as defined by eqn (3) for PT+, PP+, PPy+, PPV+ and PTV+ as a function of the oligomer length.
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Fig. 5
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First excitation energies as calculated with TDDFT for PT+, PP+, PPy+, PPV+ and PTV+ as a function of the oligomer length.
Because of the charge localization and the amorphous structure of the polymer domain, ‘charge hopping’ is accepted as the microscopic mechanism in the charge transport process.31 Hence when simulating the charge transport in the bulk materials, a kinetic Monte Carlo strategy is often adopted, where the Marcus theory for charge transfer serves as a model to calculate the rates of elementary hopping steps.32 Important parameters in such simulations include the charge localization length and the reorganization energy. In the first subsection, we will describe our approach to calculating the localization length from the charge distribution results. We will then discuss the convergence behaviors of the reorganization energy and the excitation energy as a function of the oligomer length. The dependence of these energies on the range parameter will also be addressed. 1. Determination of charge localization length We consider the charge localization length as the number of consecutive repeating units on the polymer molecule that holds the majority of the charge distribution. To reach an unambiguous measure by which the charge localization length can be determined, we turn to the spin density distribution because the Mulliken charges show non-continuous distributions, as explained earlier. To measure the charge localization length from the spin distribution, we borrow the idea of the full width at half maximum (FWHM). However, instead of visually measuring it from the distribution curve, which requires plotting and may give a fractional number of repeating units, we use a measure in which we add the spin density monomer by monomer beginning from highest until we have the accumulated spin density population that exceeds 0.75. The number 0.75 is chosen because that is the area covered by FWHM in a normal distribution. With this quantitative measure,
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the charge localization length in our definition means the smallest number of repeating units that contains at least 75% of the charge distribution. For a distribution that self-localizes in the long chain, this number increases initially with oligomer length but then levels off. This is indeed the case for LRC hybrids with o greater than or equal to 0.2 bohr1, as shown in Fig. 6. On the other hand, the localization length continues to grow with o = 0.1 bohr1, indicating that the charge is not truly localized. It is interesting to note that for any polymer, the localization length between o = 0.2 bohr1 and o = 0.4 bohr1 differs by only one unit. In a closely related work,20 two LRC functionals, CAM-B3LYP33 and LC-wPBE,34 are used in the localization study of positively charged oligophenylene vinylene (PV). We note that for both functionals, the range parameter is 0.33 bohr1. However, CAM-B3LYP also has a global damping factor of exact exchange so that even at the long-range limit, there is only 60% of exact exchange. This work,20 using the FWHM measure of the bond length alternation, concludes the localization length (‘‘characteristic size’’) for PV+ is close to 3 by LC-wPBE, in agreement with our o = 0.3 bohr1 results. CAM-B3LYP gives a slightly greater number, which can be expected because of the reduced amount of exact exchange. Therefore, these results support the idea that the amount of exact exchange is the single most important factor in determining charge localization length in conjugated polymers. Though a characteristic size is also assigned to the B3LYP and PBE results in the same work, it is drawn upon calculations for a single oligomer with 10 units.20 Our study in this work suggests that those results may change with increasing oligomer lengths. Finally, to make sure that our observation that the localization length will keep changing with
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Fig. 6
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Charge localization length of PT+, PP+, PPy+, PPV+ and PTV+ as determined by the 75% spin population rule.
o = 0.1 bohr1 holds for even longer oligomers, we have extended the calculations for polyphenylene to 18, 20, 22 and 24 monomers and the results can be found in the ESI.† We do not see converging charge or spin distributions. Therefore, we are more confident to conclude that LRC hybrids with o = 0.1 bohr1 do not produce charge self-localization. 2. Comparing the convergence of reorganization energies and excitation energies We expect properties due to a localized charge to saturate with the chain length. That is indeed the case for the reorganization energy and the lowest polaron excitation energy as shown in Fig. 4 and 5, where the results for o = 0.1 bohr1 decrease continuously as the chain length increase, while for o = 0.2, 0.3, 0.4 bohr1 they converge at long oligomer lengths. Both energies increase in magnitude as the charge becomes more localized (greater o value) and the increases are quite significant, which is different from the localization length. However, there are differences in chain length dependence between the reorganization energy and the excitation energy because the former is a ground state property calculated from two different geometries, while the latter involves electronic response kernel and the virtual states. In particular, the excitation energy decays monotonically as a function of the chain length, which is a consequence of the HOMO–LUMO gap decreasing in the same pattern. The reorganization energy has a less clear trend as a function of the chain length and shows a much smaller change from the shortest chain to the longest chain. But in general it tends to increase initially as the oligomer grows, which can be attributed to the increase of the number of bonds and angles involved in the reorganization as the molecule becomes larger.
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Such a trend is soon balanced by the localized charge distribution, which leads to saturation of the results in longer chains. The fact that the reorganization energy and excitation energies both increase with the range parameter might make it tempting to converge these results with respect to o. However, such convergence has no physical basis, but is rather an artifact of the functional. In fact, the convergence can be expected when full exact exchange is included (when o goes to infinity), at which limit the calculation is simply unrestricted Hartree–Fock plus DFT correlation, known to be a bad choice. A non-empirical approach to tune the range parameter is to utilize the exact ionization potential conditions in DFT.35,36 However, this approach has been shown to be problematic when used for neutral polymer molecules with increasing lengths.37 Moreover, for the charged polymers that we are interested in here, there is an additional problem in the uncertainty of where the second charge occurs, i.e. bipolaron or side-by-side polarons.19 At the current stage we prefer to treat the range parameter empirically, where we fit this single parameter o to reproduce known experimental results and then use the same parameter throughout a complete study. The polaron excitation energy is well suited for such fitting. It has clear experimental signals.19 The calculated results all converge nicely at long chain length, and they also have sufficient sensitivity towards the range parameter. However, proper fitting would require accounting for possible external effects in experiments, such as solvent and counterions, which have been proven to help charge localization.20,38,39 In the case of no experimental data, we recommend using o = 0.2 bohr1 in the LRC hybrid for exploratory studies. Such a functional can fully localize, and is unlikely to over-localize, charges in a
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conjugated polymer; at the same time, it has the least spin contamination, minimizing the risk of getting a qualitatively incorrect picture.44 Quantitatively energy values in the range of a few tenths of eV can be obtained by fine-tuning the range parameter as shown here, and by testing different semilocal functionals, which is not explored in this work.
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V. Conclusions Hybrid functionals can effectively describe charge selflocalization in conjugated polymers and the results critically depend on the amount of exact exchange being included. Using the criterion that the spatial distribution of a localized charge is polymer length independent, we find that conventional hybrids need about 75% of exact exchange, while LRC hybrids require the range parameter to be only 0.2 bohr1 or larger. The modest spin contaminations in the LRC hybrids make them a clear favorite over conventional hybrids with a high percentage of exact exchange. Systematic investigation of charge localization in five different polymers and its dependence on the range parameter points to some common features. First, o has to be sufficiently large to have charge localization. Second, the threshold value seems to be about 0.2 bohr1 for all polymers. Third, further increasing o will make the charge more localized. Fourth, the localization length changes by only one monomer with o values but the reorganization energy and polaron excitation energy can change significantly. The energetic variances point to the necessity of tuning the range parameter, while the commonalities suggest that a single range parameter may work for all systems. Unfortunately, this parameter cannot be determined non-empirically, but may be fitted to experimental results. We have in this work only considered the intrinsic charge localization effects in stand-alone polymer molecules. It represents the most fundamental level in multi-scale modeling of charge transport in polymer electronic materials.40 Our results for the charge localization length provide some quantitative guidance in building such models. Our next step will be to address the uncertainty in the reorganization energy by choosing the range parameter. We have noted that fitting to experimental polaron excitation energies can be an effective approach to determine the range parameter. However, such fitting would require a proper account of external effects that may be important in experimental conditions, such as solvent and counterions. In future work, we will investigate how the charge localization and its associated properties change in the presence of those effects.
Acknowledgements All calculations were performed on the computer clusters at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-AC02-98CH10886.
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