Pyrrole-Based Hydrazone Organic Nonlinear Optical Crystals and Their Polymorphs O-Pil Kwon,*,†,‡ Mojca Jazbinsek,‡ Hoseop Yun,§ Jung-In Seo,| Eun-Mi Kim,† Yoon-Sup Lee,| and Peter Gu¨nter‡
CRYSTAL GROWTH & DESIGN 2008 VOL. 8, NO. 11 4021–4025
Department of Molecular Science and Technology, Ajou UniVersity, Suwon 443-749, Korea, Nonlinear Optics Laboratory, ETH Zurich, CH-8093 Zurich, Switzerland, DiVision of Energy Systems Research and Department of Chemistry, Ajou UniVersity, Suwon 443-749, Korea, and Department of Chemistry and School of Molecular Science (BK 21), Korea AdVanced Institute of Science and Technology (KAIST), Daejeon 305-701, Korea ReceiVed February 26, 2008; ReVised Manuscript ReceiVed August 5, 2008
ABSTRACT: A series of new organic nonlinear optical nitrophenylhydrazone crystals based on pyrrole with various substituents (X: -H, -CH3, -CF3) on the hydrazone group (–CXdN–NH–) has been synthesized and their polymorphs have been investigated. The chromophore 2-((2-(4-nitrophenyl)hydrazono)methyl)-1H-pyrrole (HNPH) exhibits two polymorphs with large macroscopic nonlinearity, with a powder second harmonic generation activity of 420 times and 210 times that of urea at 1.9 µm for the anisotropic polar HNPH-I with Pn space group and the isotropic nonpolar HNPH-II polymorph with P212121 space group, respectively. The microscopic and macroscopic nonlinearities in the crystallographic system have been analyzed theoretically using finite field calculations. Introduction 1,2
Organic nonlinear optical materials have been intensively studied due to their promising applications in integrated photonics3 and THz wave generation and detection.4 Efficient second-order nonlinear optical materials require optimization of acentric orientation of the molecules in the crystalline lattice or in a polymer matrix. Compared with poled polymers, organic nonlinear optical crystals exhibit superior long-term temporal stability of the polar order of chromophores, higher chromophore density and superior photochemical stability.5 However, in crystals efficient acentric packing of highly polar molecules is difficult to obtain and cannot be predicted solely based on the molecular structure yet. In order to achieve optimal noncentrosymmetric crystalline packing, many different crystal engineering approaches have been investigated such as the use of molecular asymmetry, strong Coulomb interactions, nonrodshaped π-conjugated cores, supramolecular synthetic approaches, cocrystallizations, and octupolar symmetry.1,6-9 Among the crystal engineering approaches, nonrod-shaped π-conjugated cores based on hydrazone have been shown to have a high tendency to form noncentrosymmetric crystals with large macroscopic quadratic nonlinearities.8,9 In contrast to the well-known stilbene derivatives, hydrazone derivatives generally exhibit a bent and nonrod-shaped conformation because of the nonrigid nitrogen-nitrogen single bond (–CHdN–NH–). Most of the 4-nitrophenylhydrazone derivatives exhibit very strong second harmonic generation (SHG) signals and good crystal characteristics.8 For example, 4-dimethylaminobenzaldehyde4-nitrophenylhydrazone (DANPH)8a and 3,4-dihydroxybenzaldehyde-4-nitrophenylhydrazone (3,4-DHNPH)8b based on an aromatic benzaldehyde, exhibit very strong SHG signals comparable to that of N,N-dimethylamino-N′-methylstilbazolium p-toluenesulfonate (DAST),10 which is the state-of-the-art * To whom correspondence should be addressed. E-mail: opilkwon@ ajou.ac.kr;
[email protected]. † Department of Molecular Science and Technology, Ajou University. ‡ Nonlinear Optics Laboratory, ETH Zurich. § Division of Energy Systems Research and Department of Chemistry, Ajou University. | Korea Advanced Institute of Science and Technology.
organic nonlinear optical crystal salt with the second-order nonlinear optical susceptibility of χ(2)(-2ω,ω,ω) ) 2020 pm/V at 1.3 µm and 420 pm/V at 1.9 µm. The heteroaromatic thiophene-based hydrazone derivative 5-(methylthio)thiophenecarboxaldehyde-4-nitrophenylhydrazone (MTTNPH)9a also shows the same order of powder SHG efficiency as DANPH. Five-membered heteroaromatic pyrrole is a more efficient π-conjugation bridge for enhancing molecular nonlinearity compared to the heteroaromatic thiophene.11 However, acentric nitrophenylhydrazone crystals based on heteroaromatic pyrrole have not yet been reported. In this work, we investigate a series of pyrrole-based nonrod-shaped 4-nitrophenylhydrazone crystals and their polymorphs having a large macroscopic nonlinearity. The microscopic and macroscopic nonlinearities in the crystallographic system were also investigated theoretically using quantum chemical calculations based on density functional theory (DFT). Experimental Section Synthesis and General Characterization. The investigated pyrrolebased hydrazone chromophores 2-((2-(4-nitrophenyl)hydrazono)methyl)-1H-pyrrole (HNPH), 2-(1-(2-(4-nitrophenyl)hydrazono)ethyl)1H-pyrrole (MNPH), and 2-(2,2,2-trifluoro-1-(2-(4-nitrophenyl)hydrazono)ethyl)-1H-pyrrole (TFNPH) were synthesized by a condensation of 4-nitrophenylhydrazine with the corresponding aldehyde or ketone in the presence of aceticacid catalyst.8,9 The materials were purified by recrystallization in methanol solution. UV/vis/near-infrared absorption spectra were recorded by a JASCO V-570 spectrometer. Differential scanning calorimetry (DSC) was carried out using TA Instruments Q10 calorimeter (10 °C/min scan rate). HNPH: 1H NMR (CDCl3, δ): 6.248-6.270 (1H, m), 6.101-6.422 (1H, m), 6.925 (1H, s), 7.002-7.025 (2H, d, J ) 9.2 Hz), 7.689 (1H, s), 7.832 (1H, s), 8.136-8.160 (2H, d, J ) 9.6 Hz), 8.964 (1H, s). MNPH: 1H NMR (CDCl3, δ): 1.594 (2H, s), 2.264 (3H, s), 6.261-6.935 (1H, m), 7.094-7.117 (2H, d, J ) 9.2 Hz), 7.270 (2H, s), 7.634 (1H, s), 8.176-8.119 (2H, d, J ) 9.2 Hz), 9.121 (1H, s). TFNPH: 1H NMR (CDCl3, δ): 6.488-6.510 (1H, m), 6.827-6.846 (1H, m), 7.008-7.071 (1H, m), 7.215-7.238 (2H, d, J ) 9.2 Hz), 8.197-8.219 (2H, d, J ) 8.8 Hz), 8.727 (1H, s), 9.005 (1H, s). X-ray Crystal Structures. HNPH-I (plate): CCDC-676398, µ(Mo KR) ) 0.072 mm-1. Of 9674 reflections collected in the θ range
10.1021/cg800218u CCC: $40.75 2008 American Chemical Society Published on Web 09/20/2008
4022 Crystal Growth & Design, Vol. 8, No. 11, 2008 Scheme 1. Chemical Structures of the Investigated Chromophores and Their Abbreviations
Kwon et al. Table 1. Physical, Chemical, and Structural Data of the Investigated Pyrrole-Based Hydrazone Derivativesa point λmax (nm) Tm (°C) βmax (10-30 esu) powder SHG group
2.0°-27.5° using ω scans on a Rigaku R-axis Rapid S diffractometer, 4463 were unique reflections (Rint ) 0.0384, completeness ) 96.4%). The structure was solved and refined against F2 using SHELX97,12 308 variables, wR2) 0.2046, R1 ) 0.0621 (2684 reflections having Fo2 > 2σ(Fo2)), GOF ) 0.993, and max/min residual electron density 0.286/ -0.230 eÅ-3. HNPH-II (needle): CCDC-676399, µ(Mo KR) ) 0.072 mm-1. Of 7508 reflections collected in the θ range 2.0°-27.5° using ω scans on a Rigaku R-axis Rapid S diffractometer, 2477 were unique reflections (Rint ) 0.0217, completeness ) 97.7%). The structure was solved and refined against F2 using SHELX97,12 155 variables, wR2) 0.1927, R1) 0.0536 (1833 reflections having Fo2 > 2σ(Fo2)), GOF ) 1.055, and max/min residual electron density 0.459/-0.339 eÅ-3. Computational Details. The microscopic nonlinear optical properties, that is, first hyperpolarizability βijk tensor and the maximal first hyperpolarizability βmax (hyperpolarizability component along the charge-transfer direction) were calculated by quantum chemical calculations in the same manner as in ref 13 by using the hybrid functional B3LYP14 with the 6-311+G(d) basis set. The optimized (OPT) molecules and experimental (EXP) molecules, determined by the X-ray diffraction analysis, were analyzed by the finite field (FF) method. Powder SHG Measurements. For screening the macroscopic nonlinearity of the pyrrolic hydrazone crystals the Kurtz and Perry powder test15 was performed at a fundamental wavelength of 1.9 µm by measuring the reflected SHG efficiency.7 We measured the powder second-harmonic generation (SHG) activity of the hydrazone crystals relative to DAST (with 1000 times higher second harmonic signal than that of urea at 1.9 µm).10
HNPH-Ib
426
193
HNPH-IIc
426
188
MNPH TFNPH
420 380
210 125
84.5d (68.6f, 80.7g)e 84.5d (68.0)e 88.3d 77.7d
420
m
210
222
60 50
a The measured powder SHG efficiency at a fundamental wavelength of 1.9 µm is given relative to that of urea powder; λmax is the wavelength of the maximum absorption in methanol solution and Tm is the melting temperature. The maximal first hyperpolarizability βmax was calculated by the finite-field method for the optimized (OPT) and experimental (EXP) molecular structures at B3LYP/6-311+G* level; the full hyperpolarizability tensors βijk are listed in the Supporting Information. b Plate phase. c Needle phase. d OPT. e EXP. f HNPH-I-O1. g HNPH-I-O3.
Results and Discussion The chemical structures of the investigated nonrod-shaped chromophores are shown in Scheme 1, together with their abbreviations. They consist of the nonrod-shaped 4-nitrophenylhydrazone based on heteroaromatic pyrrole with different substituents (X: –H, –CH3, –CF3) on the hydrazone group (–CXdN–NH–). Absorption Properties. The absorption spectra of the hydrazone chromophores were measured in methanol solution. The wavelengths of maximum absorption λmax are given in Table 1. The wavelength of maximum absorption is λmax ) 426 nm for HNPH, λmax ) 420 nm for MNPH and λmax ) 380 nm for TFMPH in methanol. The TFMPH chromophore having -CF3 group shows a significant bathochromic shift of the wavelength of maximum absorption λmax owing to the influence of the -CF3 substituents on the π-conjugated part. The first hyperpolarizability βmax of the optimized (OPT) molecules obtained from quantum chemical calculations exhibits a similar behavior, the lowest values in TFMPH chromophore (see Table 1). This result is in accordance with the generally observed (and calculated) red shift of λmax for the molecules with a higher nonresonant molecular nonlinearity (i.e., nonlinearity-transparency tradeoff).1 Bulk Crystals and Powder SHG Measurements. Single crystals of the investigated pyrrolic hydrazone derivatives were grown from methanol solution by the rapid cooling method. For the HNPH chromophore we found concomitant polymorphs:
Figure 1. Molecular structures (top and side view) in HNPH-I (a) and -II (b) crystals and photograph of the crystals grown by rapid cooling method in methanol. Hydrogen atoms are omitted for clarity.
greenish plate (HNPH-I) of a realtively large size with 6 × 3 × 1 mm3 and dark blue needle (HNPH-II) with a length of up to 12 mm as shown in Figure 1. In the Kurtz and Perry powder test,15 all pyrrolic hydrazone crystals grown from the solution exhibit strong second harmonic generation activities of 420, 210, 60 and 50 times as that of urea for HNPH-I, HNPH-II, MNPH, and TFNPH, respectively (see Table 1). The powder SHG signal at 1.9 µm of HNPH-I is slightly higher than that of the promising hydrazone 3,4-DHNPH crystals studied previously.8b X-ray Crystal Structures. For second-order nonlinear optical applications, HNPH-I and -II crystals appear the most attractive ones due to their large macroscopic nonlinearity and the large crystal size. We therefore investigated their crystal structures in more detail by X-ray crystallography; the results are given in Table 2. The HNPH-I and -II polymorphs have noncentrosymmetric structures, monoclinic with space group symmetry Pn, and orthorhombic with a space group symmetry P212121, respectively.
Pyrrole-Based Hydrazone Organic NLO Crystals
Crystal Growth & Design, Vol. 8, No. 11, 2008 4023
Table 2. Summary of Crystallographic Data for the HNPH-I and HNPH-II Crystals
formula morphology formula weight crystal system space group point group a (Å) b (Å) c (Å) R (deg) β (deg) γ (deg) V (Å3) Z
HNPH-I
HNPH-II
C22H20N8O4 plate 460.46 monoclinic Pn m 11.0487(12) 5.8345(6) 17.495(2) 90 95.648 (3) 90 1122.3(2) 2
C11H10N4O2 needle 230.22 orthorhombic P212121 222 6.0209(4) 11.8038(8) 15.6375(12) 90 90 90 1111.36(13) 4
Table 3. Dihedral Angles (deg) for the HNPH-I-O1 (EXP), HNPH-I-O3 (EXP), HNPH-II (EXP), and HNPH (OPT) Conformers
parameter
HNPHI-O1 (EXP)
HNPHI-O3 (EXP)
HNPH-II (EXP)
HNPH (OPT)
θ′C-C(∠N2-C1-C6-N7) θ′′C-C (∠C5-C1-C6-N7) θCdN (∠C1-C6-N7-N8) θ′N-C (∠N7-N8-C9-C10) θ′′N-C (∠N7-N8-C9-C14) θN-N (∠C6-N7-N8-C9)
-5.5 2.7 3.1 -4.2 4.5 2.0
3.6 -1.4 -2.8 6.5 -7.2 7.1
8.2 -8.9 4.1 -4.0 1.4 8.8
0.0 0.0 0.0 0.0 0.0 0.0
Figure 1 shows the molecular structures in the HNPH-I and -II crystals as determined by single crystal X-ray analysis. In the HNPH-I phase, two conformers (HNPH-I-O1 and -O3) were observed, while only one conformer was observed in the HNPHII phase. The experimental (EXP) HNPH-I-O1, HNPH-I-O3, and HNPH-II molecules having a different conformation (or planarity of the conjugation system, see Table 3) in the crystal structure exhibit a different microscopic nonlinearity as listed in Table 1. Generally, nearly planar π-conjugated bridges are more efficient for π-electron delocalization and lead to higher microscopic molecular nonlinearities.1 However, recently a large microscopic nonlinearity of nonlinear optical molecules having nonplanar π-conjugated bridges have been reported.16,17 One example is charge-transfer molecules based on twisted π-electron system.16 The twist of the π-conjugation bridge reduces the overlap between the orbitals of electron donor and acceptor and leads to highly charge-separated zwitterionic structure (i.e., large hyperpolarizability).16a Another example is zwitterionic σ-bonded molecules, in which electron donor and acceptor groups are linked by a σ-bond.17 The σ-bond in the π-σ-π electron system breaks the conjugation and leads to enforce zwitterionic behavior.17c All three HNPH conformers (HNPHI-O1, HNPH-I-O3 and HNPH-II) in the crystalline state exhibit a nonplanar twisted structure (see Table 3). The difference of molecular nonlinearity of the HNPH conformers in the crystalline state may be attributed to both the contributions of the twisted π-electron system and the σ-bond in the π-σ-π electron system: as listed in Table 3, the dihedral twisting of the C)N bond (θCdN), the C-C bond (θC-C) and the N-C bond (θC-N) is related to the contribution of twisted π-electron system and the dihedral twisting of the C-N bond (θN-N) is related to the contribution of the σ-bond in the π-σ-π electron system. The crystal-packing diagram of the HNPH-I phase is shown in Figure 2. HNPH-I crystal exhibits a noncentrosymmetric
Figure 2. Crystal packing diagram of the Pn phase of HNPH-I crystal projected along the symmetry b-axis: (a) Molecules in the unit cell and with respect to the Cartesian xyz system, in which the effective first hyperpolarizability has been evaluated; hydrogen atoms are omitted for clarity. (b) Schematic illustration of the intermolecular interactions of three acentric polymer-like chains. Molecules in a polymer-like chain are linked with strong hydrogen bonds of N-O · · · H-N with distances of about 2.13 and 2.21 Å (thick dotted lines). The polymer-like chains are linked with weak hydrogen bonds of N-O · · · H-C with distances of about 2.50 and 2.70 Å (thin dotted lines).
Figure 3. Crystal packing diagram of the P212121 phase of HNPH-II crystal projected along the a-axis. Molecules in a polymer-like chain are linked with weak hydrogen bonds of N-O · · · H-N with distances of about 2.51 Å (thick dotted lines). These chains are linked with weak hydrogen bonds of N-O · · · H-C with distances of about 2.25 Å (thin dotted lines).
packing of the chromophores, monoclinic with space group symmetry Pn. HNPH molecules form zigzag-shaped polymerlike chain with strong hydrogen bonds of N-O1 · · · H-N and N-O3 · · · H-N with distance of about 2.21 and 2.13 Å, respectively (see Figure 2b). The zigzag-shaped chains are linked with weak hydrogen bonds of N-O2 · · · H-C and N-O4 · · · H-C with distance of about 2.50 and 2.70 Å, respectively. The crystallographic structure of HNPH-II phase with pointgroup symmetry 222 is shown in Figure 3. Also for this polymorph we can identify polymer-like chains formed by N-O · · · H-N hydrogen bonds; however, in this case these bonds are much weaker, with a distance of about 2.51 Å. The
4024 Crystal Growth & Design, Vol. 8, No. 11, 2008
Kwon et al.
Table 4. Components of the Effective Hyperpolarizability Tensor eff βijk (10-30 esu) in the Cartesian xyz Systema HNPH-I HNPH-II
β111eff
β221eff
β331eff
β113eff
β223eff
β333eff
37.9 β123eff 9.5
-5.0
-3.2
0
-11.7
-5.3
a The Cartesian xyz system is for HNPH-I defined so that the x-axis is rotated by 38° from the crystallographic a-axis toward the c-axis around the symmetry y ) b-axis (see Figure 2a), and coincides with the crystallographic system for HNPH-II.
chains are linked with N-O · · · H-C hydrogen bonds with a distance of about 2.25 Å. Macroscopic Nonlinearities in the Crystallographic System. The macroscopic second-order susceptibilities χ(2) can be estimated from the microscopic first hyperpolarizabilities βijk using the oriented gas model,18 giving for example for secondharmonic generation (2) eff χijk (-2ω, ω, ω) ) Nf i2ωfjωfkωβijk (-2ω, ω, ω)
(1)
where N is the number of molecules per unit volume and fiω are the local filed correction factors. The effective βijkeff coefficients in the crystal can be calculated from the hyperpolarizability tensor components βmnp of the molecules as eff βijk )
1 n(g)
n(g) 3
∑ ∑ cos(θims) cos(θjns) cos(θskp)βmnp
(2)
s mnp
where n(g) is the number of equivalent positions in the unit cell, s denotes a site in the unit cell, and θims is the angle between the Cartesian axis i and the molecular axis m.18 The microscopic first-hyperpolarizability tensor elements βijk for the investigated molecules (EXP) were calculated by the finite field method and are given in the Supporting Information. The effective hyperpolarizabilities βijkeff in the crystalline state were obtained by using eq 2 in the same manner as in ref 13, that is, by considering the full three-dimensional character of βijk tensors. It is interesting to note that the space-group of HNPH-II is nonpolar, that is, the still considerable macroscopic SHG efficiency is only due to the off-diagonal component of the effective hyperpolarizability tensor β123eff. In case of HNPH-I crystals, the contribution to the maximal component, the diagonal effective coefficient β111eff comes almost solely from HNPH-I-O3 molecules that pack with a very high order parameter cos3 θI-O3 ) 0.98, where θI-O3 ) 7.3° is the angle between the main charge transfer axis of the HNPH-I-O3 molecule and the polar axis. In one-dimensional model,18 its amplitude can be estimated as β111eff,1D ) 1/2βmax cos3 θI-03 ) 39.4 × 10-30 esu, which is still in a reasonable agreement with the value in Table 4, although the contribution of the HNPH-IO1 conformer has been completely ignored. On the other hand, the off-diagonal components reported in Table 4 (β221eff, β331eff, β223eff, and β333eff) are mostly due to the HNPH-I-O1 chromophores that pack almost normal to the HNPH-I-O3 chromophores in the mirror symmetry plane, as shown in Figure 2a, and make the resulting macroscopic second-order nonlinear optical tensor more isotropic. The measured powder SHG efficiency is proportional to the squared effective β components, averaged over all possible orientations 〈(βeff)2〉.15 This spatial average can be calculated from βijkeff components by taking into account the point-group symmetry of the particular crystal,15 e.g., 〈(βeff)2〉 ) 5(β123eff)2/7 for the point-group symmetry 222. We obtained 〈(βeff)2〉 ) 308 × 10-60 esu2 and 64 × 10-60 esu2 for HNPH-I and HNPH-II,
respectively. Solely based on the calculated molecular nonlinearity, SHG efficiency of the HNPH-I powder should be five times larger as that of the HNPH-II powder. However, the measured SHG efficiencies differ by a factor of 2 only. Since the chromophore density is similar for both polymorphs, this discrepancy may be due to different refractive indices, coherence lengths and eventual phase matching, which are also important for the macroscopic SHG efficiency.15 On the other hand, it is also possible that the molecular nonlinearity in a crystalline state is different for both polymorphs due to different intermolecular interactions (note that the calculated values of βijk correspond to the noninteracting molecules in a gas phase). To further investigate this possibility, we measured the absorption spectrum of a thin layer of HNPH powder. For both polymorphs, three absorption peaks in the range of 400-700 nm are observed, at about 430, 480 and 530 nm for HNPH-I, and at about 450, 500, and 580 nm for HNPH-II. All peaks are red-shifted by about 20-50 nm in HNPH-II with respect to HNPH-I, indicating a possible enhancement of the first hyperpolarizability in the solid state for HNPH-II compared to HNPH-I, which is also in accordance with our powder-test results. A possible increase of the first hyperpolarizability due to hydrogen bonding has been suggested in literatature.19 In our case intrachain hydrogen bonds are in both polymorphs N-O · · · H-N bonds and involve the –NH–N) π-conjugated bridge: these are much stronger in HNPH-I polymorph (distance of 2.13-2.21 Å compared to 2.51 Å in HNPH-II). On the other hand, the strongest interchain bonds are N-O · · · H-C headto-tail hydrogen bonds that involve only the donor and the acceptor part of the molecule: these are stronger in HNPH-II polymorph (distance of 2.25 Å compared to 2.50-2.70 Å in HNPH-I). Our results indicate that the head-to-tail case may be beneficial for the enhancement of the first hyperpolarizability in the solid state of HNPH molecules, however further investigations are needed to support this hypothesis, since also the angles between the chromophores are different in both polymorphs. Calculations in the gas phase result in a lower diagonal effective hyperpolarizability tensor element of HNPH-I crystals (β111eff) 38 × 10-30 esu) compared to the state-of-the-art DAST crystals (β111eff )130 × 10-30 esu, calculated in onedimensional charge-transfer approximation18 of OPT molecules). However, HNPH-I crystals still exhibit a large SHG efficiency (42% of DAST at 1.9 µm), which may be attributed to the almost double chromophore number density N, 3.6 × 1027 m-3 and 1.9 × 1027 m-3 for HNPH-I and DAST respectively, and also intermolecular hyperpolarizability enhancement. Both anisotropic polar HNPH-I and isotropic nonpolar HNPH-II polymorphs appear therefore very promising new materials for nonlinear optical applications. Conclusion We investigated a new series of nonlinear optical crystals based on pyrrolic 4-nitrophenylhydrazone (NPH) derivatives. Four NPH crystals including concomitant HNPH-I and -II polymorphs grown from solution exhibit strong SHG signal in the powder test at 1.9 µm. The monoclinic Pn phase of HNPH-I and orthorhombic P212121 phase of HNPH-II exhibit large SHG efficiency of 420 and 210 times as that of reference urea, respectively, and yield large-size crystals. Therefore, the newly developed pyrrole-based hydrazone crystals are potentially efficient materials for second-order nonlinear optical applications. Acknowledgment. This work has been supported by Swiss National Science Foundation, the new faculty research fund of
Pyrrole-Based Hydrazone Organic NLO Crystals
Ajou University and the Korea Foundation for International Cooperation of Science & Technology (KICOS) through a grant provided by the Korean Ministry of Science & Technology (MOST) (No. 2007-00440). Supporting Information Available: Details of quantum chemical calculations. This material is available free of charge via the Internet at http://pubs.acs.org.
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