Electron Delocalization in Polyenes: A Semiexperimental Equilibrium

Article pubs.acs.org/JPCA

Electron Delocalization in Polyenes: A Semiexperimental Equilibrium Structure for (3E)‑1,3,5-Hexatriene and Theoretical Structures for (3Z,5Z)‑, (3E,5E)‑, and (3E,5Z)‑1,3,5,7-Octatetraene Norman C. Craig,*,† Jean Demaison,*,‡ Peter Groner,§ Heinz Dieter Rudolph,∥ and Natalja Vogt∥ †

Department of Chemistry and Biochemistry, Oberlin College, Oberlin, Ohio 44074, United States Laboratoire de Physique des Lasers, Atomes et Molécules, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France § Department of Chemistry, University of Missouri−Kansas City, Kansas City, Kansas 64110-2499, United States ∥ Chemical Information Systems, Universität Ulm, D-89069, Ulm, Germany ‡

S Supporting Information *

ABSTRACT: Electronic structure theory reveals that π-electron delocalization increases with the chain length in polyenes. To analyze quantitatively this effect a semiexperimental equilibrium structure has been determined for trans-hexatriene by the mixed estimation method. For this fit rotational constants for a number of carbon and hydrogen isotopologues as well as a high-level ab initio structure have been used. The accuracy is 0.001 Å for bond lengths and 0.1° for bond angles. For the three isomers of octatetraene, high-level ab initio calculations have given a comparably accurate structure. These structures have been used in comparison with the structure of s-trans-butadiene to show that “CC” bonds increase in length and “C−C” bonds decrease in length as the polyene chain lengthens. These structural effects of π-electron delocalization increase toward the center of polyenes. Most likely, π−π conjugation in the molecules studied plays a large part in their planarity that, in turn, forces the hydrogen atoms of cis fragments in bay regions to be in a close contact. Their distance is indeed shorter than the sum of their van der Waals radii, and they seem to participate in a six-membered ring. compared to butadiene.4 The effect is greater in the center of the molecule where the “CC” bond is longer than the two “CC” bonds at the ends of the molecule. Interaction between the C2H and C5H hydrogen atoms causes distortion in the structure of the carbon backbone, reflected in opening up the interior C−CC bond angles in cHTE, while the molecule remains planar. In the present paper we determine the semiexperimental equilibrium structure of trans-1,3,5-hexatriene (tHTE) for comparison with the structure of cHTE. A schematic structure and the approximate locations of the principal rotation axes of tHTE are shown in Figure 1. tHTE has a lower energy than cHTE. Experimental determinations of the enthalpies of hydrogenation of the two isomers in acetic acid solution gave ΔrH°298 = 4.48 ± 2.30 kJ/ mol for tHTE → cHTE.5 Of course, solution in polar acetic acid may alter the energy difference between the isomers compared to the gas phase. cHTE has a weak dipole moment. From electronic calculations at the CCSD(T)/cc-pwCVQZ(AE) level and zero point vibrational energy (anharmonic) and thermal energy at the B3LYP/cc-pVTZ level of theory, ΔrH°298 = 7.40 kJ/mol. See the section on Computational Details.

1. INTRODUCTION Polyenes are of great importance in biological processes and in organic electronic conductors. Polyenes play essential roles in vision and in photosynthesis. They are also significant parts of biological pigments. When doped, very long polyenes act as conductors of electrons. Better knowledge of the detailed structures of polyenes would contribute to an understanding of their function. Electronic structure theory reveals that π-electron delocalization increases with the chain length in polyenes. The shift of the gap for electronic transitions toward the red with increasing chain length is a manifestation of increasing π-electron delocalization.1 The consequences for structure are that the “CC” bonds lengthen and the “sp2−sp2” single bonds shorten to an increasing degree with chain length. In the shortest polyene, s-trans-1,3-butadiene, these changes have been confirmed for the equilibrium structure.2 The “CC” bond length is longer and the “C−C” bond length is shorter than for localized CC and the C−C bonds, respectively, in s-transbutadiene. The localized values come from high-level ab initio calculations for the conformer of butadiene twisted 90° around the C−C bond.3 In this conformation the two π-bonds are orthogonal and thus do not interact. Increased π-electron delocalization in cis-1,3,5-hexatriene (cHTE) is accompanied by greater blurring between the “CC” and “C−C” bond lengths in the equilibrium structure © 2014 American Chemical Society

Received: October 10, 2014 Revised: November 21, 2014 Published: December 3, 2014 195

dx.doi.org/10.1021/jp510237h | J. Phys. Chem. A 2015, 119, 195−204

The Journal of Physical Chemistry A

Article

Observing rotational transitions for the other two nonpolar isomers would depend on high-resolution infrared spectroscopy. The small B0 and C0 rotational constants, in addition to the other complications cited above, make such experiments also unattractive. We have turned to high-level ab initio calculations to obtain the structures of the three isomers of OTE, informed by the knowledge gained from ab initio calculations done for the two isomers of HTE. We have also computed the energy differences between the isomers of OTE. The trans,trans isomer has the lowest energy. The cis,cis isomer has the highest energy. ΔrH°298(ct-tt)= 7.1 kJ/mol, and ΔrH°298(cc-tt)= 14.9 kJ/mol. The electronic energy differences were computed at the CCSD(T)/ccpVTZ(FC) level of theory. The zero point vibration energies (harmonic) and the thermal energy contributions were computed at the B3LYP/cc-pVTZ level of theory. See the section on Computational Details. Some experimental evidence for the structure of polyenes exists, and a number of theoretical calculations have been reported for the all-trans species. Two recent examples of calculations are cited here. Marian and Gilka computed structures for hexatriene, octatetraene, decapentaene, and the C26 polyene with density functional theory.6 Angeli and Pastore applied the CASSCF method to octatetraene.7 Semiexperimental structures for butadiene and cHTE have already been cited.2,4 An early gas electron diffraction (GED) study led to experimental bond parameters for tHTE.8 An investigation of the crystal structure yielded bond parameters for ttOTE.9 For determining the semiexperimental structure of tHTE, extensive information is available for ground state rotational constants of isotopologues of this substance. The parent tHTE was the first species to be investigated by high-resolution infrared (IR) spectroscopy.10 Work on the 1,1-d2 and cis-1-d1 species 11 and on the 1- 13 C 1 species 12 followed. The investigation of the 1-13C1 species gave the ancillary result that rotational constants of the 2-13C1 and 3-13C1 species could be predicted accurately from theory.12 As part of this work on the 13C isotopic species, a first estimate of the structure of the carbon backbone of tHTE was made with the Kraitchman substitution method.12,13 A final IR investigation of the 2-d1 and 3-d1 species gave their ground state rotational constants.14 In the present work, equilibrium rotational constants have been determined by correcting the ground state rotational constants with vibration−rotation interaction constants computed with cubic force constants obtained by ab initio methods. For two reasons the mixed estimation method is needed for determining the semiexperimental structure of tHTE despite the abundance of structural data from rotational constants. As seen in Figure 1, the a rotational axis passes close to many of the atoms, especially the end carbon atoms. Thus, the perpendicular distances of atoms from this axis are rather small. It is well-known that small coordinates cause errors in structure fitting.15 In addition, rotational constants were not observed for the trans-1-d1 species. In the mixed estimation method a predicate structure for the molecule is found from a high level of theory. The predicate structure and the equilibrium moments of inertia of all the isotopologues, each with appropriate uncertainties, are used concurrently to fit a semiexperimental structure.16,17

Figure 1. Schematic structure for trans-hexatriene with a and b principal rotation axes.

Exploring the effect of further chain lengthening in octatetraene is impractical with the semiexperimental method. Octatetraene (OTE) has three isomers: trans,trans (3E,5E), cis,cis (3Z,5Z), and cis,trans (3Z,5E), as defined with respect to the second and third double bonds. Schematic structures and approximate locations of principal axes are shown for these three isomers in Figure 2. Only the cis,trans isomer, which is polar with a small dipole moment (calculated as 0.04 D), might be investigated by microwave spectroscopy. However, its low yield in syntheses, its low volatility, and its instability at elevated temperatures make such an investigation quite unattractive.

2. COMPUTATIONAL DETAILS To calculate the ab initio structures, the coupled cluster theory including single and double excitations (CCSD) augmented

Figure 2. Schematic structures for the three isomers of octatetraene with a and b principal rotation axes. 196



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Table 1. Rotational Constants for Isotopologues of trans-Hexatriene and Vibration−Rotation Constants from MP2/cc-pVTZ Model B0(obsd)/MHz

0.5α sum/MHz

A B C Δc

26209.298 68 1338.8701 1274.4147 −0.1915

277.4204 7.6434 7.1069

A B C Δc

26166.491 32 1306.6994 1245.0800 −0.1731

277.0165 7.4344 6.9223

A B C Δc

25982.0110 1326.7915 1262.9357 −0.1921

272.3322 7.5396 7.0105

A B C Δc

26088.0386 1337.6410 1273.0207 −0.1937

275.4819 7.5893 7.0539

A B C Δc

24264.8418 1305.0565 1238.7424 −0.0970

255.1671 7.2133 6.7160

A B C Δc

23495.4964 1327.4990 1257.0268 −0.1666

229.1347 7.5650 6.9812

A B C Δc

23840.1738 1337.521 15 1267.1058 −0.2010

241.9395 7.5053 6.9463

A B C Δc

24039.9075 1254.751 45 1193.1650 −0.2330

251.4078 6.9281 6.4459

Be/MHz d0 26486.7190 1346.5136 1281.5216 −0.0460 1-13C1 26443.5078 1314.1338 1252.0023 −0.0270 2-13C1d 26254.3432 1334.3311 1269.946 15 −0.0471 3-13C1d 26363.5206 1345.2303 1280.0746 −0.0474 1-cis-d1 24520.0089 1312.2698 1245.4585 0.0484 2-d1 23724.6311 1335.0640 1264.007 95 −0.0221 3-d1 24082.1133 1345.0263 1274.0521 −0.0542 1,1-d2 24291.3153 1261.6795 1199.610 85 −0.0797

ga

Be(se)b/MHz

residuals of fit/MHz

−0.2937 −0.0272 0.0031

26490.956 1346.534 1281.519 −0.037

−4.963 −0.080 0.034

−0.2932 −0.0266 0.0030

26447.731 1314.153 1252.000 −0.018

−5.077 −0.010 0.034

−0.2911 −0.0270 0.0031

26258.506 1334.351 1269.944 −0.038

−0.179 −0.066 0.060

−0.2923 −0.0272 0.0031

26367.718 1345.250 1280.072 −0.038

−4.345 −0.076 0.044

−0.2719 −0.0265 0.0030

24523.640 1312.289 1245.456 0.058

−3.215 0.132 −0.067

−0.2631 −0.0270 0.0031

23728.030 1335.084 1264.006 −0.013

−4.182 0.016 0.043

−0.2670 −0.0272 0.0031

24085.616 1345.046 1274.050 −0.045

−5.073 −0.046 0.089

−0.2694 −0.0255 0.0029

24294.879 1261.697 1199.609 −0.070

−3.572 −0.150 0.057

a g factor for the electronic contribution to the rotational constant; values for species other than d0 were obtained by scaling d0 values by the ratio of Be values of the dx and d0 species. bThe g correction was applied to the Be with a factor of 1 − g/1836. cInertial defect: Δ = Ic − Ia − Ib. dFrom predictions ref 12. All other ground state rotational constants from experiments.

the structure was calculated at the CCSD(T)_FC/VTZ level of theory, and the small effect of further basis set enlargement, VTZ → VQZ, was estimated at the MP2 level. The core−core and core−valence correlation correction was computed at the MP2 level using the wCVQZ basis set. The resulting reBO estimate was:

with a perturbational estimate of the effects of connected triple excitations, CCSD(T),18 was employed, and the second-order Møller−Plesset (MP2)19,20 level of theory was used to estimate correction terms, whose computation at the CCSD(T) level would have been unnecessarily expensive. The correlationconsistent polarized triple-ζ (cc-pVTZ, abbreviated as VTZ) and quadruple-ζ (cc-pVQZ, abbreviated as VQZ) basis sets21 in the frozen core (FC) approximation, which excludes the carbon 1s electrons from the correlation treatment, were adopted. Optimizations were also performed with the correlationconsistent polarized weighted core−valence triple-ζ, ccpwCVTZ,22 and quadruple-ζ, cc-pwCVQZ,23 basis sets (abbreviated as wCVTZ and wCVQZ, respectively). To calculate an ab initio structure close to the equilibrium structure, two different ways have been used for tHTE. First,

re BO(I) = CCSD(T)_FC/VTZ + MP2(FC)/VQZ − MP2(FC)/VTZ + MP2(AE)/wCVQZ − MP2(FC)/wCVQZ

(1)

where BO means Born−Oppenheimer and AE means that all electrons are correlated. 197



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Table 2. Internal Coordinates of trans-Hexatriene (Distances in Å, Angles in deg) method basis set

MP2 VTZ

MP2 VQZ

MP2 wCVQZ

MP2(ae) wCVQZ

CCSD(T) VTZ(fc)

re(I)a

CCSD(T) wCVQZ(ae)

extrapol.b MP2/VTZ

rSE e MP2/VTZc

rSE e B3LYP/VTZc

r(C1C2) r(C2C3) r(C3C4) r(C1Hcis) r(C1Htrans) r(C2H) r(C3H) ∠(C1C2C3) ∠(C2C3C4) ∠(C2C1Hcis) ∠(C2C1Htrans) ∠(C1C2H) ∠(C3C2H) ∠(C4C3H) ∠(C5C4H) ∠(HcisC1Htrans)

1.3424 1.4464 1.3502 1.0819 1.0797 1.0846 1.0859 123.69 123.68 120.86 121.41 119.40 116.91 118.97 117.35 117.73

1.3402 1.4442 1.3482 1.0810 1.0788 1.0840 1.0854 123.66 123.67 120.84 121.40 119.41 116.93 118.97 117.36 117.77

1.3397 1.4438 1.3477 1.0810 1.0787 1.0839 1.0853 123.65 123.66 120.84 121.40 119.41 116.94 118.98 117.36 117.76

1.3365 1.4406 1.3445 1.0795 1.0772 1.0824 1.0838 123.67 123.69 120.84 121.41 119.40 116.93 118.96 117.35 117.74

1.3455 1.4557 1.3517 1.0844 1.0821 1.0865 1.0876 123.72 123.71 121.03 121.50 119.57 116.71 119.20 117.09 117.48

1.3401 1.4503 1.3465 1.0820 1.0797 1.0844 1.0856 123.71 123.72 121.01 121.49 119.57 116.72 119.19 117.09 117.50

1.3394 1.4501 1.3460 1.0822 1.0798 1.0845 1.0857 123.72 123.71 121.01 121.49 119.57 116.72 119.19 117.10 117.50

1.3389 1.4495 1.3455 1.0819 1.0796 1.0843 1.0855 123.70 123.70 120.99 121.48 119.57 116.73 119.20 117.11 117.53

1.3390(8) 1.4494(7) 1.3461(14) 1.0823(9) 1.0793(9) 1.0844(9) 1.0854(9) 123.6(4) 123.67(8) 121.02(8) 121.45(9) 119.65(9) 116.68(8) 119.19(9) 117.14(11) 117.54(11)

1.3389(9) 1.4495(9) 1.3462(16) 1.0822(10) 1.0792(11) 1.0843(10) 1.0854(10) 123.68(5) 123.69(9) 120.98(9) 121.43(10) 119.64(11) 116.68(10) 119.18(11) 117.13(13) 117.58(13)

a

See eq 1. bMean value of the extrapolation of the B and C rotational constants with the rovibrational correction calculated at the MP2/VTZ level, see text. cLevel of theory used to calculate the cubic force field.

Following the work on tHTE, the rotational structure in a mixture of partly deuterated species was investigated by IR spectroscopy. These included tHTE-1,1-d2 and -cis-1-d1.11 (“cis” is defined with respect to the C2−C3 bond). Bands of the trans-1-d1 species in the spectrum were too badly overlapped to be analyzable. A third investigation was for tHTE-1-13C1.12 Not only were ground state rotational constants obtained for this species, but strong evidence was supplied to show that the ground state rotational constants for the 2-13C1 and 3-13C1 species could be predicted accurately from computed values. At the B3LYP/cc-pVTZ level of theory the ground state A0, B0, and C0 rotational constants were computed with Gaussian03 (G03) and its vibration−rotation supplement. Scale factors for A0, B0, and C0 were found as the obsd/calcd ratios for the normal species. These scale factors were then applied to the calculated ground state rotational constants for the 2-13C1 and 3-13C1 species to obtain the predictions.12 The last investigation was the analysis of rotational structure in bands of tHTE-2-d1 and -3-d1.14 Table 1 lists the available ground state rotational constants for the isotopologues of tHTE.

The structure of tHTE was also calculated at the CCSD(T) _AE level of theory with the cc-pwCVTZ and cc-pwCVQZ basis sets. This gives another estimate: re BO(II) = CCSD(T)_AE/wCVQZ

(2)

For the larger OTE isomers, only the first, cheaper method was used. To compute rovibrational interaction constants the cubic force field was calculated at two different levels of theory: MP2(FC)/VTZ and B3LYP/VTZ where B3LYP is a Kohn− Sham density functional method (DFT)24 using Becke’s threeparameter hybrid exchange functional25 and the Lee−Yang− Parr correlation functional.26 Most ab initio calculations were executed with Gaussian09 (G09),27 but Molpro28,29 was used for the CCSD(T) optimizations with the wCVQZ(AE) basis set for tHTE and with the VTZ basis set for the OTE isomers. The G09 calculations were done on the Ohio Supercomputer, a few G03 calculations were done at Oberlin, and the Molpro calculations were done on a computer cluster in Ulm. The largest calculation is the CCSD(T)_AE/wCVQZ for tHTE structure optimization performed with Molpro. It required about 609 h on 12 CPU cores. For the CCSD(T)_FC/ccpVTZ calculation for ctOTE, 83 h were needed for 12 cores. Calculations for the higher symmetry tt and cc isomers took less time.

4. EQUILIBRIUM STRUCTURE OF TRANS-HEXATRIENE 4.1. Ab Initio Structure. The ab initio structure of tHTE was computed with the constraint of C2h point group symmetry using eqs 1 and 2. The T1 diagnostic calculated at the CCSD(T)_AE/wCVQZ level of theory is only 0.010. This small value indicates that the nondynamic electron correlation is probably not important.30 Although this diagnostic is not always reliable,31 it is, however, probable that the structure computed at the CCSD(T) level is reliable. Correlating all the electrons at the MP2/wCVQZ level leads to the expected shortening of the bond lengths, 0.0032 Å for the CC bonds and 0.0015 Å for the CH bonds. The bond angles are not significantly affected. Upon going from VTZ to VQZ, the CC bonds are shortened by 0.002 Å and the CH bonds by less than 0.001 Å. The change of the bond angles is almost negligible (less than 0.04°). Thus, it may be concluded that for bond angles and the CH bond lengths convergence is almost achieved with the VQZ basis set. The structure has also been

3. GROUND STATE AND EQUILIBRIUM ROTATIONAL CONSTANTS FOR TRANS-HEXATRIENE Ground state rotational constants for tHTE and a number of isotopologues came from the analysis of rotational structure in high-resolution IR spectra. The standard microwave method does not apply to these essentially nonpolar species. An investigation of tHTE itself with a resolution of 0.0015 cm−1 demonstrated the feasibility of analyzing the rotational structure for this rather large molecule with two small rotational constants.10 The one rotational constant of reasonable size for this near-prolate top (κ = −0.995) helped make the analysis feasible despite the formidable density of the spectra, which were far from being baseline resolved. 198



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α/2 sums). A statistical analysis of published results shows that, in the best cases, the error in α/2 sums is about 2%, though it can be as large as 10%.39 As the rovibrational correction for the A constant at 277 MHz is much larger than the correction for the B and C constants (about 7 MHz), we may conclude that the main source of error comes from the inaccuracy of the semiexperimental equilibrium Ae constant. The outlying values of the inertial defect for the isotopologues 1-cis-d1 and 1,1-d2 have to be noted. We will see below in the fits that the residuals for the Be rotational constants for these two species are slightly larger indicating that these rotational constants are slightly less accurate; see Table 1 and Table S1 of the Supporting Information. To determine 13 independent internal coordinates, the rotational constants of eight isotopologues are available, i.e. only 16 independent data as the molecule is planar. The first attempt to fit a structure to the corresponding moments of inertia gave an unsatisfactory result. The standard deviations of the fitted internal coordinates are large: about 0.010 Å for the bond lengths and about 1° for the bond angles. The condition number at κ = 2194 is extremely large indicating that the system is ill-conditioned.17 Obviously, the large standard deviations are due to the limited accuracy of the semiexperimental rotational constants, but the main problem is that many Cartesian coordinates are small hampering their accurate determination; see Table 3. From the residuals of the fit, it is

optimized at the CCSD(T)_AE level with the wCVTZ and wCVQZ basis sets; see Table S2 of the Supporting Information. Thus, it is possible to check the convergence of the CC bonds with the simple n−3 extrapolation formula32 r(n) = r∞ + An−3

(3)

where n is the cardinal number, i.e., n = T = 3 and n = Q = 4. This formula for extrapolating internal coordinates is purely empirical. Furthermore, it requires a smooth convergence to the complete basis set limit, which is the case here. Moreover, this formula is not expected to be accurate. However, it gives a trend and indicates that the CC bonds lengths are slightly too long with the wCVQZ basis set (about 0.001 Å). This result is in agreement with the observation that the accuracy of the CCSD(T) method with a basis set of quadruple-ζ quality is about 0.002 Å.33 In this paper, Helgaker et al. use the CCSD(T) method with the VQZ basis set, with all electrons correlated. They consider 27 molecules (see their Table 14) and claim an accuracy of 0.16 pm (relative to experiment) at this level of theory (see their Table 15). However, eq 3 is approximate; a more accurate method will be described below in section 4.3. The results of the structure determination are given in Table 2. It is seen that the agreement between the two structures using either eq 1 or eq 2 is almost perfect. 4.2. Semiexperimental Structure. The semiexperimental (SE) equilibrium rotational constants, BeSE, were calculated from the experimental ground-state rotational constants, B0, of the particular molecule using the equation BeSE = B0 + ΔBvib + ΔBel

Table 3. Cartesian Coordinates (in Å) for trans-Hexatriene from the Semiexperimental Structure with the MP2/VTZ Force Field

(4) C1 C2 C3 C4 C5 C6 H1trans H1cis H2 H3 H4 H5 H6cis H6trans

where ΔBvib is the rovibrational correction calculated from the cubic force field with the VIBROT program36 and ΔBel is the electronic correction, which may be obtained from the rotational g tensor.37 To estimate the electronic contribution to rotational constants, the g-tensor was computed using G03 at the B3LYP/6-311+G(3df,2pd) level of theory. The results are gaa = −0.294, gbb = −0.027, and gcc = 0.003. The ground state and SE equilibrium rotational constants are given in Table 1 for the MP2/VTZ force field and in Table S1 of the Supporting Information for the B3LYP/VTZ force field. The ground state inertial defect of the parent species, Δ0 = −0.191 uÅ2 is close to the value found for cHTE, Δ0 = −0.166 uÅ2.4 These significant negative values indicate the presence of at least one lowfrequency out-of-plane vibration,38 which is confirmed by the analysis of the infrared spectra.10 The corresponding semiexperimental equilibrium inertial defect with the rovibrational correction calculated with the MP2/cc-pVTZ force field, Δe = −0.037 uÅ2, is consistent with the planar structure for tHTE. Although Δe is much smaller than Δ0, it is still significantly different from zero indicating that either the rovibrational corrections or the ground state rotational constants are not accurate. For the other isotopologues, Δe has the same order of magnitude but is not constant as can be seen in Table 1. With the rovibrational correction calculated with the B3LYP/VTZ force field similar results are obtained, but the value of Δe is slightly larger (in absolute value) indicating that this force field is slightly less accurate; see Table S1 of the Supporting Information. Although the ground state A rotational constant is the least accurate as is usual for prolate top molecules, the main source of error is certainly the rovibrational corrections applied (the 34,35

a

b

−3.05799(38) −1.86015(52) −0.60208(73) 0.60208(73) 1.86015(52) 3.05799(38) −3.9732(11) −3.1423(14) −1.8015(15) −0.6520(23) 0.6520(23) 1.8015(15) 3.1423(14) 3.9732(11)

0.17940(39) −0.41901(56) 0.30082(77) −0.30082(77) 0.41901(56) −0.17940(39) −0.3926(18) 1.25837(86) −1.50182(86) 1.3851(11) −1.3851(11) 1.50182(86) −1.25837(86) 0.3926(18)

possible to estimate the accuracy of the semiexperimental rotational constants as about 4.2 MHz for Ae, 0.04 MHz for Be, and 0.03 MHz for Ce. The quality of the fit can be considerably improved by using the mixed estimation method.17,40 The predicates are the best ab initio parameters determined in the previous section (CCSD(T)_AE/wCVQZ structure of Table 2) with a conservative uncertainty of 0.002 Å for the bond lengths and 0.2° for the bond angles. These uncertainties come from a statistical analysis of the differences between structural parameters of CCSD(T)/VQZ quality and their experimental equilibrium values.33,41 At each step of the fit, an analysis of the residuals17 permitted checking of the appropriateness of the weights and the compatibility of the rotational constants and the predicate observations. The results are given in the last two columns in Table 2 for the rovibrational corrections calculated 199



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Table 4. Internal Coordinates of trans,trans and cis,cis-Octatetraenes (Distances in Å, Angles in deg) trans,trans

C1C2 = C7C8 C2C3 = C6C7 C3C4 = C5C6 C4C5 C8Hcis C1Htrans = C8Htrans C2H = C7H C3H = C6H C4H = C5H C1C2C3 = C6C7C8 C2C3C4 = C5C6C7 C3C4C5 = C4C5C6 C1C2H = C8C7H C2C1Hcis = C7C8Hcis C2C1Htrans = C7C8Htrans C3C2H = C6C7H C2C3H = C7C6H C3C4H = C6C5H C4C3H = C5C6H C4C5H = C5C4H HC1H = HC8H a

CCSD(T) VTZ 1.3460 1.4544 1.3537 1.4495 1.0844 1.0821 1.0866 1.0876 1.0877 123.77 123.66 123.83 119.53 121.03 121.49 116.70 117.15 119.02 119.20 117.14 117.48

cis,cis re(I)a

CCSD(T) VTZ

re(I)b

1.3406 1.4488 1.3485 1.4441 1.0821 1.0797 1.0845 1.0855 1.0857 123.76 123.67 123.83 119.55 121.01 121.48 116.70 117.14 119.02 119.19 117.15 117.50

1.3461 1.4573 1.3570 1.4540 1.0846 1.0823 1.0842 1.0863 1.0839 122.73 126.53 125.59 118.88 121.03 121.51 118.39 115.58 117.14 117.90 117.28 117.46

1.3407 1.4517 1.3519 1.4484 1.0822 1.0799 1.0822 1.0841 1.0819 122.72 126.53 125.59 118.84 121.02 121.50 118.45 115.58 117.08 117.88 117.33 117.48

See eq 1 and Table S4 of the Supporting Information. bSee eq 1 and Table S5 of the Supporting Information.

Table 5. Internal Coordinates of cis,trans-Octatetraene (Distances in Å, Angles in deg) method basis set

CCSD(T)_FC VTZ

re(I)

C1C2 C2C3 C3C4 C4C5 C5C6 C6C7 C7C8 C1Hcis C1Htrans C2H C3H C4H C5H C6H C7H C8Hcis C8Htrans

1.3459 1.4545 1.3539 1.4519 1.3565 1.4570 1.3461 1.0844 1.0821 1.0866 1.0877 1.0856 1.0862 1.0861 1.0845 1.0846 1.0823

1.3403 1.4490 1.3488 1.4464 1.3512 1.4515 1.3406 1.0820 1.0797 1.0845 1.0857 1.0835 1.0841 1.0841 1.0825 1.0822 1.0799

C1C2C3 C2C3C4 C3C4C5 C4C5C6 C5C6C7 C6C7C8 C2C1Hcis C2C1Htrans HC1H C1C2H C3C2H C2C3H C4C3H C3C4H C5C4H C4C5H C6C5H C5C6H C7C6H C6C7H C8C7H C7C8Hcis C7C8Htrans HC8H

CCSD(T)_FC VTZ

re(I)

123.730 123.681 122.953 126.299 126.185 122.883 121.022 121.493 117.485 119.543 116.727 117.112 119.207 118.392 118.655 115.789 117.912 118.027 115.788 118.223 118.894 121.027 121.503 117.470

123.717 123.697 122.949 126.303 126.196 122.872 121.007 121.489 117.504 119.539 116.743 117.108 119.196 118.346 118.705 115.799 117.898 118.013 115.791 118.281 118.847 121.014 121.499 117.487

two columns of Table 2 are regarded as correct to 0.001 Å and 0.1° with a somewhat larger uncertainty for the C3C4 bond. 4.3. Extrapolation to Infinite Basis Set Size. To check whether convergence is achieved at the CCSD(T)_AE/ccpwCVQZ level of theory, the computed ab initio rotational constants for the normal species were extrapolated to infinite basis set size.42,43 The ab initio equilibrium values of the

at the MP2/VTZ and B3LYP/VTZ levels. Although these two sets of rovibrational corrections are different, the final results are in very good agreement between each other as well as with the best ab initio structures. It has to be noted that the MP2/ VTZ gives slightly more precise results in agreement with the discussion of the inertial defect. The reSE structures in the last 200



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rotational constants of the parent species Xe (X = A, B, C) are calculated first. For this, the ab initio structure was optimized using the CCSD(T) method with the wCVnZ basis sets (n = T, Q), all electrons being correlated. To these Xe(T) and Xe(Q) values, the rovibrational correction ΔX0 = X0 − Xe is added, yielding X0(T) and X0(Q) under the assumption that this rovibrational correction is almost constant for these basis sets, a generally excellent assumption,39 which is further verified in the present case. The g corrections were also made. Then, the rotational constants X0(T) and X0(Q) are extrapolated as a linear function of the computed structural parameters re(T) and re(Q). The intersection of the line with the experimental X0 of the parent species gives the extrapolated re. This calculation was done with the rovibrational correction calculated with the MP2/VTZ and B3LYP/VTZ force fields. The results are given in Table S2 of the Supporting Information, and the mean for B0 and C0 results in the third-from-last column of Table 2. Both force fields give almost identical results, but it is observed that the extrapolation with the A-rotational constant is less accurate, which is not surprising because its rovibrational correction is about 36 times larger. The conclusion of this extrapolation is that convergence is practically achieved at the CCSD(T)_AE/ wCVQZ level of theory. In addition, the extrapolated results are in good agreement with the values of the reSE structures.

0.010 au, as for cHTE. This result can indicate the existence of a weak bond between these two pairs of hydrogen atoms. There are also two ring points with an electron density ρ = 0.0099 au, corresponding to the six-membered rings H2C2C3C4C5H5 and H4C4C5C6C7H7. This electron density is slightly larger than in cHTE where it is 0.0096 au. The torsional vibrations of cHTE and tHTE seem to confirm the existence of this H···H bond. The two lowest frequencies for torsional modes of the trans isomer are 258 and 94 cm−1, whereas for the cis isomer they are 332 and 155 cm−1.10 These two modes are probably torsioning around the single bonds and torsioning around the double bond. The lower frequencies for the trans isomer imply that such torsioning is less constrained for this isomer. Although the AIM calculation indicates the existence of a weak H···H bond, a controversial interpretation is possible. Using the natural bond orbital theory, Weinhold et al.54 have shown in the particular case of cis-2-butene that the repulsive H···H interaction is dominant being only slightly softened by a weak donor−acceptor attraction. As seen in the following discussion, large structural adjustments occur to relieve H···H repulsion in the various cis configurations of the polyenes. Weinhold et al. attribute donor−acceptor interactions between eclipsed C−H bonds and CC bonds as a source of planarity.54

5. EQUILIBRIUM STRUCTURE OF THE OCTATETRAENE ISOMERS The ab initio structures of the three OTE isomers were computed with the constraint of planarity using eq 1. The final results are given in Table 4 for the cis,cis and trans,trans isomers and in Table 5 for the cis,trans form. Details of the results of the calculations of bond parameters with different models are given in Tables S3−S5 of the Supporting Information. For the three isomers the T1 diagnostic is only 0.012, indicating that the CCSD(T) method should be accurate. For the cis,trans isomer, the nonbonded distance in the cis part of the molecule between the hydrogen atoms attached to the carbon atoms C2 and C5 is 2.141 Å, i.e., shorter than the sum of the van der Waals radii for hydrogen atoms, which is 2.4 Å. It is comparable to the value found in cHTE, 2.138 Å,4 and, as for cHTE, can indicate the existence of a weak hydrogen−hydrogen bond between these two hydrogen atoms forming a six-membered ring helping stabilize the planar structure. In the case of the cis,cis isomer of OTE, the distance between the corresponding hydrogen atoms at 2.109 Å is still shorter. This shortening could be considered as evidence of strengthening the hydrogen−hydrogen bond. In the presence of such a short H···H distance, one may wonder whether the molecule is planar. A B3LYP/VTZ calculation and MP2 calculations with triple- and quadruple-ζ basis sets seem to confirm the planarity of the molecule. However, it is known that these levels of theory are not always reliable. For this reason, we have performed a CCSD/6-311++G(3df,3pd) optimization relaxing most dihedral angles; see Table S6 of the Supporting Information. This calculation confirms the planarity of the molecule. As for cHTE,4 the atoms in molecules (AIM) theory44,45 with its implementation in G03 by Cioslowski et al.46−53 was used to characterize the existence of hydrogen−hydrogen bonds in ccOTE. The calculations were performed at the B3LYP/6-311G** level of theory with the ab initio equilibrium structure. Indeed, for the cis fragments of the OTEs, there are bond critical points (BCP) between the atoms H2 and H5 and the atoms H4 and H7, respectively, with an electron density ρ =

6. DISCUSSION 6.1. Comparison of Structures of trans- and cisHexatriene. The high accuracy of the determination of the bond parameters for the two isomers of HTE makes possible a close comparison of bond lengths and bond angles for these two species with those of s-trans-butadiene. The most significant structural parameters for the trans and cis isomers of hexatriene are shown in Figure 3. Data for the cis isomer comes from our previous work.4 Bond lengths are shown in red, and bond angles in blue. For both isomers the effect of πelectron delocalization increases toward the center of the molecule. The central “CC” is longer than the outer “CC”

Figure 3. Comparison of bond lengths and bond angles in the trans and cis isomers of hexatriene. 201



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bonds. More generally, the structural effects of the π-electron delocalization, which are already evident in s-trans-butadiene,2 increase with chain length in polyenes. This effect is seen by comparison with the bond parameters in butadiene. In butadiene the “CC” bond length is 1.338 Å, and the “C− C” bond length is 1.454 Å.2 As shown for the two HTE isomers in Figure 3, all the “CC” bonds are longer than 1.338 Å; all the “C−C” bond lengths are shorter than 1.454 Å. The biggest differences between the two isomers of HTE are the consequence of interaction of the H2 and H5 hydrogen atoms in the cis isomer, which seems to be mainly repulsive even though a weak H---H bond may exist between these two atoms.4 The C2C3C4 bond angle increases from 123.7° in the trans isomer to 126.3° in the cis isomer. The C3C2H2 bond angle increases from 116.8° in the trans isomer to 118.3° in the cis isomer. The C3C4 bond length increases by 0.003 Å in the cis isomer, and the C2C3 bond length increases by 0.002 Å. In addition, the C2H2 bond length decreases from 1.085 Å in the trans isomer to 1.082 Å in the cis isomer. Other CH bond parameters are essentially unchanged between the two isomers, including the cis−trans sensitivity of the two C1H1 bonds. The C1H1trans bond is shorter than the C1H1cis bond, as was also found for butadiene.2 6.2. Comparison of Structures of the Three Isomers of Octatetraene. As for the isomers of HTE, accurate determination of the bond parameters is necessary to making comparisons among the isomers of OTE and other polyenes. Table 2 shows that the bond parameters for the re(I) predicate structure for tHTE differ from those for reSE structure by no more than 0.001 Å and 0.1°. Thus, we regard the calculated predicate values of bond parameters for the isomers of OTE as good to 0.001 Å and 0.1°. Figure 4 compares the bond parameters for the carbon backbones and selected C−H bonds of the three isomers of OTE. Bond lengths are shown in red; bond angles are shown in blue. Upon increasing the length of the polyene from HTE to OTE, the structural parameters reflect increased π-electron delocalization. Focusing on the structure of tHTE in Figure 3 and on the structure of ttOTE in Figure 4, we see that the C C bonds lengthen and the new “C−C” bond shortens. As with tHTE, the structural effects of π-electron delocalization increase toward the center of the molecule. The innermost C−C bond is shortened the most; the innermost CC bond is lengthened the most. With DFT calculations Marian and Gilka confirmed that these structural effects of π-electron delocalization continue to increase gradually with chain length in C10 and C26 polyenes.6 In ccOTE and in the cis half of ctOTE, adjustments in structure reflect the interaction between H2···H5 (and H4··· H7) hydrogen atoms. In comparison with ttOTE the C−CC bond angles in the cis regions increase by almost 3°. One C− C−H bond angle increases by 1.3°. The CC and the C−C bond lengths increase in the cis regions, whereas the C2−H2 and C4−H4 bond lengths and their equivalents in the other half of the cc molecule decrease by 0.004 Å in comparison to ttOTE. The bond angles for the interior C4−C5−H4 and C5− C4−H5 bonds are little changed in the cc molecule. As expected, the two halves of the structure of ctOTE have close similarities with ccOTE and ttOTE, respectively.

Figure 4. Comparison of bond lengths and bond angles in the three isomers of octatetraene.

7. CONCLUSIONS Using the mixed estimation method, a semiexperimental equilibrium structure has been determined for tHTE. Equilibrium rotational constants were found from experimental ground state rotational constants and computed vibration− rotational constants and electronic g factors for eight hydrogen and carbon isotopologues. The structure was determined by fitting concurrently all these moments of inertia and a highlevel ab initio structure, each with appropriate uncertainties. Because of many small Cartesian coordinates for tHTE, it was not possible to obtain a satisfactory equilibrium structure based alone on the experimental rotational constants. For the three isomers of octatetraene, theoretical structures were computed at the level of theory comparable to that used for tHTE. Because the theoretical structure for tHTE agrees with the semiexperimental structure within 0.001 Å and 0.1°, we regard the theoretical structures of the isomers of OTE, determined at the same level of theory, as comparably good. This accuracy is needed to assess the structural effects of π-electron delocalization with an increase in length of polyene chains. In general, compared to HTE “CC” bonds increase in length and “C−C” bonds decrease in length with the effects increasing toward the center of the molecule. As in cHTE, an interaction occurs in OTE between adjacent hydrogen atoms that is dominantly repulsive. The ct−tt energy difference for the OTE isomers is quite close to the c−t energy difference in the isomers of HTE. The cc−tt energy difference for the OTE isomers is, as expected for the two H···H interactions, essentially twice the c−t energy difference in the isomers of HTE. 202



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(12) Craig, N. C.; Tian, H.; Blake, T. A. Analysis of the Rotational Structure in the High-Resolution Infrared Spectrum of transHexatriene-1-13C1: A Semiexperimental Equilibrium Structure for the C6 Backbone of trans-Hexatriene. J. Phys. Chem. A 2012, 116, 3148− 3155. (13) Kraitchman, J. Determination of Molecular Structure from Microwave Spectroscopic Data. Am. J. Phys. 1953, 21, 17−24. (14) Craig, N. C.; Chen, Y.; van Besien, H.; Blake, T. A. Analysis of the Rotational Structure in the High-Resolution Infrared Spectra of trans-Hexatriene-2-d1 and -3-d1. J. Mol. Spectrosc. 2014, 303, 26−35. (15) Gordy, W.; Cook, R. L. Microwave Molecular Spectra; Wiley: New York, 1984; p 698. (16) Demaison, J.; Craig, N. C.; Groner, P.; Écija, P.; Cocinero, E. J.; Lesarri, A.; Rudolph, H. D. Accurate Equilibrium Structures for Piperidine and Cyclohexane. J. Phys. Chem. A, in press. (17) Demaison, J. The Method of Least Squares. In Equilibrium Molecular Structures; Demaison, J., Boggs, J. E., Császár, A. G., Eds.; CRC Press: Boca Raton, FL, 2011; pp 29−52. (18) Purvis, G. D., III; Bartlett, R. J. A Full Coupled-cluster Singles and Doubles Model: the Inclusion of Disconnected Triples. J. Chem. Phys. 1982, 76, 1910−1918. (19) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A Fifth-order Perturbation Comparison of Electron Correlation Theories. Chem. Phys. Lett. 1989, 157, 479−483. (20) Møller, C.; Plesset, M. S. Note on an Approximation Treatment for Many-Electron Systems. Phys. Rev. 1934, 46, 618−622. (21) Dunning, T. H., Jr. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007−1023. (22) Woon, D. E.; Dunning, T. H., Jr. Gaussian Basis Sets for Use in Correlated Molecular Calculations. V. Core-Valence Basis Sets for Boron through Neon. J. Chem. Phys. 1995, 103, 4572−4585. (23) Peterson, K. A.; Dunning, T. H., Jr. Accurate Correlation Consistent Basis Sets for Molecular Core−Valence Correlation Effects: The Second Row Atoms Al−Ar, and the First Row Atoms B−Ne Revisited. J. Chem. Phys. 2002, 117, 10548−10560. (24) Kohn, W.; Sham, L. Self-Consistent Equations Including Exchange and Correlation Effects. J. Phys. Rev. A 1965, 140, 1133− 1138. (25) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652. (26) Lee, C. T.; Yang, W. T.; Parr, R. G. Development of the ColleSalvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B 1988, 37, 785−789. (27) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson., G. A.; et al. GAUSSIAN09, Revision B.01; Gaussian Inc.: Wallingford CT, 2010. (28) Werner, H.-J.; Knowles, P. J.; Manby, F. R.; Schütz, M.; Celani, P.; Knizia, G.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G. MOLPRO; University College Cardiff Consultants Limited: Cardiff, U.K., 2009. (29) Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M. Molpro: a General Purpose Quantum Chemistry Program Package. WIREs Comput. Mol. Sci. 2012, 2, 242−253. (30) Lee, T. J.; Taylor, P. R. A Diagnostic for Determining the Quality of Single-Reference Electron Correlation Methods. Int. J. Quant. Chem. Symp. 1989, 23, 199−207. (31) Sprague, M. K.; Irikura, K. K. Quantitative Estimation of Uncertainties from Wavefunction Diagnostics. Theor. Chem. Acc. 2014, 133, 1544. (32) Puzzarini, C. Extrapolation to the Complete Basis Set Limit of Structural Parameters: Comparison of Different Approaches. J. Phys. Chem. A 2009, 113, 14530−14540. (33) Helgaker, T.; Ruden, T. A.; Jørgensen, P.; Olsen, J.; Klopper, W. Review Commentary. A Priori Calculation of Molecular Properties to Chemical Accuracy. J. Phys. Org. Chem. 2004, 17, 913−933. (34) Clabo, D. A.; Allen, W. D.; Yamaguchi, Y.; Remington, R. B.; Schaefer, H. F., III A Systematic Study of Molecular Vibrational

ASSOCIATED CONTENT

S Supporting Information *

Tables of equilibrium rotational constants for tHTE computed with contributions from the B3LYP/cc-pVTZ model, details of the extrapolation to infinite basis set for the structure for tHTE, internal coordinates for the trans,trans, cis,cis, and cis,trans isomers of OTE computed at various levels of theory, and the structure of ccOTE computed with CCSD/6-311++G(3df,3pd) for use in the AIM calculation. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*(N.C.C.) E-mail: [email protected]. *(J.D.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The Dreyfus Foundation (Senior Scholar Mentor grant) and Oberlin College supported the work of N.C.C. Calculations were made possible by an allocation of computing time at the Ohio Supercomputing Center. We thank Dr. P. I. Dem’yanov from Lomonosov Moscow State University for helpful discussions.

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REFERENCES

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Electron Delocalization in Polyenes: A Semiexperimental Equilibrium

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