J. Phys. Chem. A 2010, 114, 10917–10921
10917
Electronic Structure Calculations on Helical Conducting Polymers Juan D. Ripoll,† Andrei Serna,‡ Doris Guerra,† and Albeiro Restrepo*,† Grupo de Quı´mica-Fı´sica Teo´rica, Instituto de Quı´mica, UniVersidad de Antioquia, AA 1226 Medellı´n, Colombia, and Departamento de Ciencias Quı´micas, Facultad de Ecologı´a y Recursos Naturales, UniVersidad Andre´s Bello, AVenida Repu´blica 275, Santiago, Chile ReceiVed: April 22, 2010; ReVised Manuscript ReceiVed: September 11, 2010
We present a study of the electronic structure and derived properties of polyfurane (PFu), polypyrrol (PPy), and polythiophene (PTh). Two spatial arrangements are considered: trans chain (tc-PFu, tc-PPy, tc-PTh) and cis R-helical (R-PFu, R-PPy, R-PTh). Even at the small sizes considered here, helical conformations appear to be stable. Band gaps of pure, undoped oligomers fall into the semiconductor range. Density of states (DOS) analysis suggest dense valence and conduction bands. Bond length alternation analysis predicts almost complete delocalization of the π clouds in all spatial arrangements. Doping with electron donors or electronwithdrawing impurities reduces all band gaps close to the metallic regime in addition to increasing the DOS for the valence and conduction bands. Introduction Helical polymers are a common occurrence in nature, playing pivotal roles in biological, chemical, and physical processes. A widely known example is the DNA double helix in charge of storing and passing genetic information. An excellent review on the synthesis and uses of helical polymers was given by Nakano and Okamoto.1 There is evidence of traces of helices and super helices in bulk samples of substituted polypyrrole and polythiophene produced by electrochemical deposition.2-4 There is also a synthetic route designed to produce helical polyacetylene.5 An earlier work predicted at semiempirical levels that polypyrrol (PPy), polythiophene (PTh), and their derivatives would exist as stable helices.6 A recent review discussing the existence and properties of chiral polymers has been given by Kane-Maguire and Wallace.7 The quest for miniaturization of electronic circuits has made conducting polymers very interesting from a technological point of view because of, among other factors, the possibility of being used as conducting wires in molecular circuits. Electrical conduction along a one-dimensional molecular helix is expected to induce an interior magnetic field8 (Figure 1); therefore, molecular conducting helices are thought to behave as molecular solenoids. Tomakazu and Nishide8 argue that molecular solenoids have the potential to become a new class of very high density data storage materials for electronic devices. Both molecular wires and solenoids built from conducting polymers can be designed via chemical synthesis with taylored properties used to satisfy specific needs. Several studies proposing molecular solenoids can be found in recent literature. The most studied targets are carbon nanotubes (CNTs),9-13 for which quantum treatments of their magnetic properties have been published.11,14-16 Margan´ska and co-workers11 report that application of a magnetic field parallel to the tube axis can change the conducting properties of the CNT from metallic to semiconducting and vice versa. It has been shown that loop currents for small molecules placed * To whom correspondence should be addressed. E-mail: albeiro@ matematicas.udea.edu.co. † Universidad de Antioquia. ‡ Universidad Andre´s Bello.
between the tip of a scanning tunneling microscope and the substrate can be largely amplified if the energy of the injected electrons is resonant with degenerate eigenstates of the isolated molecule,17,18 a statement that Tsuji et al.10 proposed to be equally applicable to CNTs. DNA19 and π-conjugated helical polymers8 have also attracted attention as potential molecular solenoids. Accurate models for free electrons on helical potentials20 and for the scattering of electrons through helical potentials21 have been developed recently. In this work, we focus in predicting the electronic structure and derived properties of cis R-helical and trans chain configurations of pure and doped polyfuran (PFu), polypyrrole, and polythiophene. Our results lead us to speculate that helical polyfuran, polypyrrole, and polythiophene and their derivatives have the potential to be used as molecular solenoids. Computer Methods In order to achieve a compromise between accuracy and computational demands for calculations on the oligomers, relaxed scans of the torsional angle for 2,2′-bipyrrole were performed at several levels of theory (Figure 2). Maxima and minima of the scans were taken as initial geometrical guesses for optimization of stationary points at several levels of theory. We selected B3LYP/6-31G* as our model chemistry (see justification below). Local maxima and minima in the potential energy surface (PES) for internal rotation of 2,2′-bipyrrole and for the trans chain and cis R-helical conformations of PFu, PPy, and PTh decamers were fully optimized (no geometrical
Figure 1. Electrical conduction along a one-dimensional molecular helix is expected to induce an interior magnetic field.8
10.1021/jp1077642 2010 American Chemical Society Published on Web 09/28/2010
10918
J. Phys. Chem. A, Vol. 114, No. 41, 2010
Ripoll et al. Results and Discussions
Figure 2. Relaxed scans for internal rotation in 2,2′-bipyrrole.
restrictions) and characterized as minima or saddle points by counting the number of negative eigenvalues of the Hessian matrix. Single-point energy calculations were attempted on larger helices constructed by propagating the geometrical features of the helical decamers. Such calculations were used to extract and analyze the wave function of extended helices. All calculations in this work were performed using the Gaussian03 suite of programs.22 The following variables were analyzed: relative stability; bond length alternation (BLA), defined as the difference between the average long and short C-C distances in the conjugated π system;23 ionization potential (IP) and electron affinity (EA); band gap; and density of states (DOS). The effect of doping was simulated by placing 2X atoms (X ) Li, Na, F, Cl) along the symmetry axis of the helices. In this way, we account for the effects of electron-donor and electron-withdrawing dopants while ensuring conservation of the total spin multiplicity of the helix and avoiding open shell systems. We used an approximate DOS model via the function g(E):24
g(E) )
1 ∆√π
[(
∑ exp i
E - εi ∆
)] 2
(1)
where ∆ is the width of the Gaussian function, set to 0.5 eV in our calculations.
Selection of the Level of Theory. The rotational barriers of the dimers joint at the 2,2′ positions have been extensively studied because of the importance of the proper description of the dimer for the posterior application of model chemistries to the oligomers.25,27-31 It has been determined that the PESs for the rotations of 2,2′-bipyrrole and 2,2′-bithiophene contain a total of five stationary points each, out of which only two are well-defined minima. The other three correspond to transition states for intramolecular rotation. It has also been found that the minima lie at out-of-plane conformations in the vicinities of 45° (syn-gauche conformation) and 145° (anti-gauche conformation), the syn-gauche conformation being higher in energy. The rotational PES for 2,2′-bifuran contains only four points, the trans conformation being the most stable (there is no anti-gauche). Our target is a computation model that correctly reproduces the conformational features of the 2,2′-bipyrrole rotational profile at the lowest possible computational expense. A summary of our calculations at selected levels compared against the literature for the internal rotation of 2,2′-bipyrrole is shown in Table 1. Figure 2 depicts the rotational profiles. Our results are in complete agreement with the previous literature. If we take as reference the sophisticated benchmark values (MP2 + ∆CCSD (T)) obtained by Sancho-Garcı´a and Karpfen,25 it is clear that B3LYP performs consistently better than MP2 with all the studied basis sets; it is also seen that increasing the size of the basis set does not make significant improvements on the energy values as to justify the high computational cost. Therefore, we select the B3LYP/6-31G* methodology to calculate the oligomers. Additional support for this choice of methodology comes from previous reports that conclude that hybrid functionals (specifically, B3LYP) are adequate for the prediction of band gaps in conjugated polymers.26 Geometries, and Other Properties. The trans chain conformations for the decamers show little geometric distortion from the constituting monomers. Full geometry optimizations for the three decamers studied here lead to stable cis R-helical conformations. Relative stabilities (cis R-helical, trans chain) are 15.3, 18.3, and 25.8 kcal/mol for PPy, PFu, and PTh, respectively, in favor of the trans decamer chain conformations. Table 2 and Figure 3 show the main geometrical features of the helical conformations. PFu is the more tightly coiled helix, and PTh has the largest pitch. Table 3 shows several calculated parameters. A few observations are worth pointing out from Table 3: (i) Despite that our model comprises only 10 monomeric units with specific conformations (trans chains and cis R-helices) during the
TABLE 1: Torsional Energies (kcal/mol) for 2,2′-Bipyrrolea method
cis
syn-gauche
trans
anti-gauche
trans
this work
b3lyp
mp2
b3lyp
mp2
b3lyp
mp2
b3lyp
mp2
b3lyp
mp2
6-31G* 6-31G** 6-311G* 6-311G** 6-311+G**
3.21 3.19 3.52 3.31 3.07
4.49 4.37 4.86 4.49 4.68
1.61 1.65 1.59 1.63 1.61
1.32 1.35 1.33 1.38 1.24
2.83 2.92 2.50 2.66 2.43
1.53 1.61 1.35 1.49 1.23
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.39 0.38 0.58 0.46 0.31
1.41 1.35 1.63 1.38 1.66
6-31G*28 6-31G*31 6-311++G**28 MP2+∆CCSD (T)25
3.20 3.21 3.00
4.50 4.48 4.90
1.40 1.60 1.40
Literature 1.00 2.90 1.32 2.81 1.00 2.40
1.60 1.53 1.50
0.00 0.00 0.00
0.00 0.00 0.00
0.40 0.40 0.30
1.41 1.41 1.80
a
3.29
1.50
All energies are relative to the global minimum, anti-gauche conformation.
2.13
0.00
0.43
Helical Conducting Polymers’ Electronic Structure
J. Phys. Chem. A, Vol. 114, No. 41, 2010 10919
TABLE 2: Geometrical and Energetical Parameters for Cis r-Helical PFu, PPy, PTh Decamersa helix
pitch Å
X-X distance Å
diameter Å
∆E kcal/mol
extrapolated Eg eV
PFu PPy PTh
6.25 13.80 20.1
2.72 3.03 3.33
12.09 7.43 15.28
18.3 15.3 25.8
2.64 3.79 2.25
a For definition of geometrical variables, see Figure 3. ∆E: difference in energy in favor of the trans chain configuration. Eg: band gap extrapolated to infinitely long helices. All calculations at the B3LYP/6-31G* level.
Figure 4. Delocalized π cloud for R-polyfuran.
TABLE 4: Effect of Doping on the Frontier Orbitals for Helical r-PFu, r-PPy, r-PTh Decamersa
Figure 3. Geometrical parameters for cis R-polyfuran calculated at the B3LYP/6-31G* level.
TABLE 3: B3LYP/6-31G* Calculated Properties for Trans Chain and r-Helical PFu, PPy, PTh Decamers oligomer
IPa (eV)
EAa (eV)
Eg (eV)
BLA (Å)
PFu chain PFu helical PPy chain PPy helical PTh chain PTh helical PFu experimental PPy experimental PTh experimental
4.44 4.25 3.97 4.33 4.73 4.68 4.0032,33
1.74 1.50 0.68 0.49 2.34 2.20
2.70 2.75 3.29 3.84 2.39 2.48 2.3532,34 2.8532,35 2.032,36
0.04 0.05 0.04 0.05 0.06 0.05
a
5.0032,33
oligomer
HOMOb (eV)
LUMOb (eV)
Egb (eV)
R-PFu R-PFu + 2Li R-PFu + 2Na R-PFu + 2F R-PFu + 2Cl R-PPy R-PPy + 2Li R-PPy + 2Na R-PPy + 2F R-PPy + 2Cl R-PTh R-PTh + 2Li R-PTh + 2Na R-PTh + 2F R-PTh + 2Cl
-4.25 -2.15 -2.10 -4.33 -4.27 -4.33 -2.69 -2.69 -4.54 -4.57 -4.68 -2.72 -2.56 -5.06 -5.06
-1.50 -1.58 -1.52 -3.76 -3.70 -0.49 -2.29 -2.29 -4.08 -4.11 -2.20 -2.45 -2.29 -4.73 -4.73
2.75 0.57 0.58 0.57 0.57 3.84 0.40 0.40 0.46 0.46 2.48 0.27 0.27 0.33 0.33
Egc (eV) 0.08 0.22 0.22 0.22 0.19 0.46 0.14 0.14 0.60 0.14 0.03 0.14
a The dopant atoms were placed along the symmetry axis at 1/3 and 2/3 of the length of the helix. b Decamer. c Extrapolation to infinitely long helices (see text).
Calculated from Koopmans’ theorem.
optimizations and that the experimental measurements are made on amorphous bulk polymers, there is a fair agreement between experiment and our calculations. (ii) No calculated band gap is larger than 3.84 eV, which would make all the conformations closely fall into the semiconductor range. (iii) Chain conformations show smaller band gaps than the helical counterparts. This difference can be explained by the smaller p orbital overlap because of nonplanarity in the helical conformation. (iv) Average bond length alternation is ≈0.05 Å and makes no difference between chain and helical conformations. This observation shows that the π cloud is completely delocalized, favoring conduction. A picture of the delocalized π cloud for R-helical polyfuran is shown in Figure 4. The energy difference among conformations (chain vs helical) may appear a little high; however, there are mitigating factors (could be used for all the other properties as well): (i) The properties calculated here refer only to one type of helical and chain conformations, two idealized models. We compare them against experimental measurements, which include contributions from many different structures to the bulk polymer. (ii) To simplify our calculations, we constructed the chain conformation in an all-trans fashion and the helical conformation in an allcis fashion. (iii) The somewhat large energy differences do not change the fact that the helical conformations have already been observed in bulk samples of PPY and PTh.2-4
The choice of 10 monomeric units to treat the helical conformations is arbitrary; therefore, once the B3LYP/6-31G* calculations predicted decamer cis R-helices to be well-defined minima within the corresponding PES, we studied the evolution of properties with the size of the helices by propagating the geometrical features of the decamers. Figure 8 shows the effect that the size of the helices have on the vertical ionization potentials; linear trends are observed for all oligomers: extrapolating for infinitely long helices leads to 4.53, 4.67, 4.75 eV as IPs for cis R-helical PFu, PPy, PTh, respectively. Figure 9 shows how the size of the helices affects the predicted band gaps. Nonlinear trends are observed for all oligomers, extrapolation for infinitely long helices leads to 2.64, 3.70, and 2.25 eV as Egs for cis R-helical PFu, PPy, and PTh, respectively. The calculated IPs and Egs for all extended helices are not too far away from either the calculated values for the decamer or the bulk experimental values (Table 3). Doping. We simulated doping by placing 2X atoms (X ) Li, Na, F, Cl) along the symmetry axis of the helices at 1/3 and 2/3 of the length of the helix. A summary of the doping effect on the helical band gaps and in the HOMO and LUMO orbitals is presented in Table 4. First, we notice that all pure helices exhibit very negative LUMOs, resulting in considerably high electron affinities (Tables 3, 4). Electron-donor dopants (Li, Na) seem to destabilize the HOMO by a great deal, and electronwithdrawing dopants (F, Cl) have a strong stabilizing effect on
10920
J. Phys. Chem. A, Vol. 114, No. 41, 2010
Figure 5. Pure and doped R-PFu decamer densities of states calculated at the B3LYP/6-31G* level.
Ripoll et al.
Figure 8. Vertical ionization potential (IP) as a function of the inverse of the number of monomers in the undoped helices (1/n). Extrapolation to infinitely long helices (1/n ) 0) yields 4.53, 4.67, 4.75 eV as IPs for helical PFu, PPy, PTh, respectively. All calculations at the B3LYP/ 6-31G* level.
Figure 6. Pure and doped R-PPy decamer densities of states calculated at the B3LYP/6-31G* level. Figure 9. Band gap (Eg) as a function of the inverse of the number of monomers in the undoped helices (1/n). Extrapolation to infinitely long helices (1/n ) 0) yields 2.64, 3.70, and 2.25 eV as Eg’s for helical PFu, PPy, and PTh, respectively. All calculations at the B3LYP/631G* level.
donors (Li, Na) increase the DOS in the conduction bands, leaving the valence bands unchanged, whereas electron withdrawers (F, Cl) increase the DOS of the valence bands, leaving the conduction band unchanged. We intentionally avoid open shell calculations that would result from odd numbers of dopants because unrestricted DFT methods are prone to spin contamination, which would require treating the Kohn-Sham orbitals on a different footing from the Hartree-Fock orbitals.37-40 This is a complex issue that exceeds the scope of this report and therefore will be treated in future projects. Figure 7. Pure and doped R-PTh decamer densities of states calculated at the B3LYP/6-31G* level.
the LUMO. We point out that all band gaps are reduced to the vicinities of the metallic regime, regardless of the nature of the dopant. We include density of states graphics produced for all optimized decamer configurations in Figures 5-7. In addition to reducing the band gaps for all helices, only the valence and conduction bands seem to be affected by the dopants. Electron
Conclusions and Perspectives We present a computational study of the electronic structure and derived properties of polyfurane, polypyrrol and polythiophene at the B3LYP/6-31G* level in trans chain and pure and doped cis R-helical conformations. Helices formed by joining 10 monomer units at the 2,2′ positions are predicted to be stable. Extrapolated band gaps for undoped infinitely long oligomers are calculated to be 2.64, 3.70, and 2.25 eV for cis R-PFu, R-PPy, and R-PTh, respectively. There are dense valence and conduction bands, as predicted by DOS analysis, and almost
Helical Conducting Polymers’ Electronic Structure complete delocalization of the π clouds in all spatial arrangements. Doping with electron donors or electron withdrawing agents lowers all band gaps close to the metallic regime. Electron-donor dopants destabilize the HOMO and increase the DOS of the conduction bands, while electron-withdrawing dopants stabilize the LUMO and increase the DOS for the valence bands, regardless of the identity of the dopants. Our results lead us to speculate that helical polyfuran, polypyrrole, and polythiophene and their derivatives have the potential to be used as molecular solenoids. Supporting Information Available: Cartesian coordinates for all optimized R-helices and trans chain structures reported here. This material is available free of charge via the Internet at http://pubs.acs.org. Acknowledgment. The authors thank professor Wagner De Almeida, Universidade Federal de Minas Gerais, for recovering some old results for the rotational barriers of the dimers and graciously forwarding the data to us. J.D.R. thanks Mr. Jose Rafael Ripoll for his valuable help in producing some of the artwork in this paper. We are in debt to Dr. John Fredy Pe´rez, who wrote and provided the program to calculate the densities of states. References and Notes (1) Nakano, T.; Okamoto, Y. Chem. ReV. 2001, 101, 4013. (2) Yang, R.; Dalsin, M.; Evans, D.; Christensen, L.; Hendrickson, L. J. Phys. Chem. 1988, 93, 511. (3) Caple, G.; Wheeler, B.; Swift, R.; Porter, T.; Jeffers, S. J. Phys. Chem. 1990, 94, 5639. (4) Yang, R.; Evans, D.; Christensen, L.; Hendrickson, L. J. Phys. Chem. 1990, 94, 6117. (5) Piao, G.; Akagi, K.; Shirakawa, H.; Kyotani, M. Curr. Appl. Phys. 2001, 1, 121. (6) Cui, C.; Kertesz, M. Phys. ReV. B 1989, 40, 9661. (7) Kane-Maguire, L.; Wallace, G. Chem. Soc. ReV. 2010, 39, 2545. (8) Iwasaki, T.; Nishide, H. Curr. Org. Chem. 2005, 9, 1665. (9) Tsuji, N.; Takajo, S.; Aoki, H. Int. J. Mod. Phys. B 2007, 8-9, 1198. (10) Tsuji, N.; Takajo, S.; Aoki, H. Phys. ReV. B. 2007, 75, 153406. (11) Margan´ska, M.; Szopa, M.; Zipper, E. Phys. ReV. B 2005, 72, 115406. (12) Kawakami, T.; Kitagawa, Y.; Matsuoka, F.; Yamashita, Y.; Isobe, H.; Nagao, H.; Yamaguchi, K. Int. J. Quantum Chem. 2001, 85, 619. (13) Seifert, G.; Ko¨hler, T.; Frauenheim, T. Appl. Phys. Lett. 2000, 77, 1313. (14) Ando, T. J. Phys. Soc. Jpn. 2005, 74, 777. (15) Lu, J. Phys. ReV. Lett. 1995, 74, 1123. (16) Ajiki, T.; Ando, J. J. Phys. Soc. Jpn. 1993, 62, 2470.
J. Phys. Chem. A, Vol. 114, No. 41, 2010 10921 (17) Nakanishi, S.; Tsukada, M. Phys. ReV. Lett. 2001, 87, 126801. (18) Tsukada, M.; et al. J. Phys. Soc. Jpn. 2005, 74, 1079. (19) Tagami, K.; Tsukada, M.; Wada, Y.; Iwasaki, T.; Nishide, H. J. Chem. Phys. 2003, 119, 7491. (20) Kristic´, V.; Rikken, G. Chem. Phys. Lett. 2002, 364, 51. (21) Yeganeh, S.; Ratner, M.; Medina, E.; Mujica, V. J. Chem. Phys. 2009, 131, 014707. (22) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, ReVision B.05 Gaussian, Inc.: Pittsburgh, PA, 2003. (23) Kersetz, M.; Choi, C.; Yang, S. Chem. ReV. 2005, 105, 3448. (24) Santos, J.; Contreras, R.; Chamorro, E.; Fuentealba, P. J. Chem. Phys. 2002, 116, 4311. (25) Sancho, J.; Karpfen, A. Chem. Phys. Lett. 2005, 411, 321. (26) Yang, S.; Olishevski, P.; Kertesz, M. Synth. Met. 2004, 141, 171. (27) Padilla, L.; Toro-Labbe, A. J. Mol. Struct. (THEOCHEM) 1995, 330, 223. (28) Karpfen, A.; Choi, C.; Kertesz, M. J. Phys. Chem. A 1997, 101, 7426. (29) Aleman, C.; Domingo, V.; Fajari, L.; Julia, L.; Karpfen, A. J. Org. Chem. 1998, 63, 1041. (30) Gatti, C.; Frigerio, G.; Benincori, T.; Brenna, E.; Sannicolo, F.; Zotti, G.; Zecchin, S.; Schiavon, G. Chem. Mater. 2000, 12, 1490. (31) Duarte, H.; Duani, H.; De Almeida, W. Chem. Phys. Lett. 2003, 369, 114. (32) Selzner, U.; Lagowski, J.; Pickup, P.; Poirier, R. Synth. Met. 1998, 96, 177. (33) Bredas, J. In Handbook of Conducting Polymers; Skotheim, T. A., Ed.; Marcel Dekker: New York, 1986, p 859. (34) Glenis, S.; Benz, M.; LeGoff, E.; Schindler, J.; Kannewurf, C.; Kanatzidis, M. J. Am. Chem. Soc. 1993, 115, 12519; Chem. Phys. Lett. 2003, 369, 114. (35) Zotti, G.; Martina, S.; Wegner, G.; Schlu¨ter, A. AdV. Mater. 1992, 4, 798. (36) Heeger, A. In Conjugated Polymers and Related Materials; Salaneck, E. R., Lundstro¨m, I., Rånby, B., Eds.; Oxford University Press: Oxford, 1993, p 28. (37) Cohen, A.; Tozer, D.; Handy, N. J. Chem. Phys. 2007, 126, 214104. (38) Wang, J.; Becke, A.; Smith, V. J. Chem. Phys. 1995, 102, 3477. (39) Grafenstein, J.; Cremer, D. Mol. Phys. 2001, 99, 981–989. (40) Wittbrodt, J.; Schlegel, H. J. Chem. Phys. 1996, 105, 6574.
JP1077642
