Article pubs.acs.org/JPCA
Semiexperimental Equilibrium Structure of the Lower Energy Conformer of Glycidol by the Mixed Estimation Method Jean Demaison,*,† Norman C. Craig,*,‡ Andrew R. Conrad,§ Michael J. Tubergen,*,§ and Heinz Dieter Rudolph*,∥ †
Laboratoire de Physique des Lasers, Atomes et Molécules, Université de Lille I, 59655 Villeneuve d’Ascq Cedex, France Department of Chemistry and Biochemistry, Oberlin College, Oberlin, Ohio 44074, United States § Kent State University, Kent, Ohio 44242, United States ∥ Department of Chemistry, University of Ulm, D-89069 Ulm, Germany ‡
ABSTRACT: Rotational constants were determined for 18O-substituted isotopologues of the lower energy conformer of glycidol, which has an intramolecular inner hydrogen bond from the hydroxyl group to the oxirane ring oxygen. Rotational constants were previously determined for the 13C and the OD species. These rotational constants have been corrected with the rovibrational constants calculated from an ab initio cubic force field. The derived semiexperimental equilibrium rotational constants have been supplemented by carefully chosen structural parameters, including those for hydrogen atoms, from medium level ab initio calculations. The combined data have been used in a weighted least-squares fit to determine an equilibrium structure for the glycidol H-bond inner conformer. This work shows that the mixed estimation method allows us to determine a complete and reliable equilibrium structure for large molecules, even when the rotational constants of a number of isotopologues are unavailable.
1. INTRODUCTION Glycidol, or 2-oxiranemethanol, C3H6O2, is a chiral and bifunctional compound with a great variety of uses.1 The molecule has internal hydrogen bonding expressed in two different conformers. Determining an equilibrium structure for this asymmetric 11 atom molecule from rotational constants for only heavy atom substitution (plus OD) depends on new methods. The microwave spectrum of glycidol has been investigated several times.2−4 Two conformers were confirmed.3 The lowest energy conformer, called H-bond inner (see Figure 1), has an internal hydrogen bond formed between the oxirane ring oxygen atom and the hydroxyl group hydrogen atom from above. This conformer is measured to have 3.6 kJ mol−1 lower energy than the H-bond outer conformer (see Figure 1),3 which has an internal hydrogen bond formed between the hydroxyl group hydrogen atom and the pseudo-π-electrons of the oxirane ring. Recently, the microwave spectrum of the glycidol−water complex was measured, and the rotational constants of the 13C isotopologues of glycidol were determined as part of this study permitting the determination of a partial substitution (rs) structure.4 However, this structure for the glycidol H-bond inner is inaccurate because several atoms of this molecule have small Cartesian coordinates.5 For a medium-sized molecule, such as glycidol, it is not possible to obtain a structure good to 0.001 Å by midlevel quantum chemical (QC) calculations. However, it is now © 2012 American Chemical Society
possible to reduce the errors in structures by using the semiexperimental technique whereby equilibrium rotational constants are derived from experimental ground state rotational constants and rovibrational corrections derived from a quantum chemical cubic force field.6 Nonetheless, the Kraitchman’s equations7 used to determine the Cartesian coordinates of the substituted atoms are quite sensitive to the remaining errors. One way to, at least partially, obviate this difficulty is to use the least-squares technique, which smoothes the errors. However, the determined parameters will be affected by the uncertainty of any fixed parameters. It is still better to use the mixed estimation method where no parameters are fixed, but auxiliary information is added directly to the data matrix for the leastsquares fit.8,9 This auxiliary information, usually called predicate observations, consists in carefully chosen values for the internal coordinates of the unsubstituted atoms, together with their corresponding uncertainties. This approach is relatively easy because QC calculations at a medium level of theory permit determining the internal coordinates of many light atoms with a reasonable accuracy. These favorable predictions are particularly true for the hydrogen atoms. Furthermore, it is advantageous, when possible, to also use predicate observations for the substituted atoms in order to offset the possible harmful Received: June 5, 2012 Revised: July 25, 2012 Published: August 15, 2012 9116
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2. DETERMINATION OF THE ROTATIONAL CONSTANTS The rotational constants of the parent and OD species3 as well as those of the 13C species4 are already known. The ground state rotational spectra of the two 18O species in natural abundance were measured with a minicavity Fourier-transform microwave spectrometer as described in ref 4. The lines, which involve a-, b-, and c-type transitions, are given in Table 1. The lines were fit to rotational constants and quartic centrifugal distortion constants using a Watson Hamiltonian in the asymmetric reduction and the Ir representation. The number of measured transitions, nine, is small compared to the number of fitted parameters. For this reason, the method of predicate observations (mixed estimation) was used in the fitting in which the quartic centrifugal distortion constants of the parent species with an uncertainty of ten percent are input as supplementary data in a weighted least-squares fit. The results are given in Table 2. Although the number of transitions is Table 2. Spectroscopic Parameters of the 18O Isotopologues of Glycidol H-Bond Inner parenta A (MHz) B (MHz) C (MHz) ΔJ (kHz) ΔJK (kHz) ΔK (kHz) δJ (kHz) δK (kHz) Nb σ (kHz)c
Figure 1. Structures of the two most abundant conformers of glycidol.
2.3826(5) −1.427(3) 5.18(1) 0.31558(9) −9.78(1)
18
18
Ooxirane
10104.8466(5) 3992.602(2) 3712.374(2) 2.34(4) −1.45(11) 4.7(3) 0.33(3) −9.7(9) 9 0.2
Ohydroxyl
10186.457(1) 3950.101(4) 3631.952(3) 2.32(7) −1.6(2) 4.6(6) 0.29(5) −10(2) 9 0.3
a
From ref 3; used as predicate observations with an uncertainty of 10%. bNumber of transitions included in the fit. cStandard deviation of the fit.
effect of the errors of the semiexperimental equilibrium rotational constants. The article is organized as follows. Section 2 describes the experimental determination of the ground state rotational constants. Section 3 is devoted to the quantum chemical calculation of the structure and an anharmonic force field of glycidol. Section 4 focuses on the determination of reliable predicate observations. In section 5, the equilibrium structure derived from Kraitchman equations is discussed. In section 6, the semiexperimental structure is presented.
quite small, the system of normal equations is well conditioned in both cases and there are no large correlations between the rotational constants and the centrifugal distortion constants. In other words, although the centrifugal distortion constants are only marginally determined, their uncertainities should not affect the accuracy of the rotational constants.
3. COMPUTATIONAL DETAILS To estimate the predicate observations, the structure of glycidol has been computed at two levels of theory, second-order Møller−Plesset perturbation theory (MP2)10 and Kohn−Sham
Table 1. Microwave Lines for 18O Species of Glycidol H-Bond Inner 18
18
O oxirane
O hydroxyl
J′
Ka′
Kc′
J″
Ka″
Kc″
freq. (MHz)
obs − calcd
freq. (MHz)
obs − calcd
1 1 2 2 2 3 2 2 2
1 1 1 0 1 0 2 2 1
1 0 2 2 1 3 1 0 2
0 0 1 1 1 2 2 2 1
0 0 1 0 1 1 1 1 0
0 0 1 1 0 2 2 2 1
13817.190 14097.456 15129.626 15400.457 15690.139 17235.670 19177.425 19186.840 21241.851
−0.001 0.000 0.000 0.001 0.000 0.000 0.004 −0.002 0.000
13818.379 14136.565 14845.858 15152.162 15482.221 16768.510 19663.536 19675.386 21082.194
0.000 −0.001 0.000 0.000 0.001 0.000 0.012 −0.008 0.001
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Table 3. Structure of Glycidol H-Bond Inner (Distances in Å, Angles in Degrees) least-squares fit method/basis set r(C2−C3) r(C3−O2) r(C3−H3B) r(C3−H3A) r(C2−H2) r(C1−C2) r(C1−H1A) r(C1−H1B) r(C1−O1) r(O1−H4) ∠(O2C3C2) ∠(C3C2H3B) ∠(C3C2H3A) ∠(C3C2H2) ∠(C3C2C1) ∠(C2C1H1A) ∠(C2C1H1B) ∠(C2C1O1) ∠(C1O1H4) τ(O2C3C2H3B) τ(O2C3C2H3A) τ(O2C3C2H2) τ(O2C3C2C1) τ(C3C2C1H1A) τ(C3C2C1H1B) τ(C3C2C1O1) τ(C2C1O1H4) Derived Parameters r(C2−O2) ∠(C2O2C3)
B3LYP/6-311Ga
MP2/cc-pVTZ
MP2/cc-pVQZ
predicate
18c predicatesb
27 predicatesb
1.4611 1.4385 1.0839 1.0828 1.0873 1.5113 1.0972 1.0911 1.4128 0.9640 59.18 119.59 119.28 118.30 122.58 108.64 109.62 112.86 107.35 102.73 −103.05 −102.89 102.07 −151.35 91.36 −27.89 −51.34
1.4595 1.4453 1.0813 1.0806 1.0845 1.5056 1.0946 1.0892 1.4115 0.9658 59.32 119.39 118.62 118.49 121.10 108.70 109.84 111.64 105.17 102.52 −103.00 −102.91 101.25 −151.98 90.15 −28.88 −47.07
1.4566 1.4429 1.0803 1.0798 1.0839 1.5034 1.0935 1.0882 1.4092 0.9638 59.33 119.36 118.72 118.47 121.36 108.69 109.86 111.82 105.74 102.46 −102.90 −102.81 101.24 −151.40 90.68 −28.35 −48.10
1.4595(30) 1.4390(30) 1.0813(20) 1.0806(20) 1.0844(20) 1.5056(30) 1.0946(20) 1.0892(20) 1.4100(20) 0.9628(30) 59.33(30) 119.36(30) 118.72(30) 118.47(30) 121.36(30) 108.69(30) 109.86(30) 111.82(30) 105.74(30) 102.46(50) −102.90(50) −102.81(50) 101.24(50) −151.40(50) 90.68(50) −28.35(50) −48.10(50)
1.4651(19) 1.44272(55) 1.08124(97) 1.08060(97) 1.08451(97) 1.5057(29) 1.09453(97) 1.08915(97) 1.40990(76) 0.9629(14) 58.723(82) 119.32(14) 118.70(14) 118.45(14) 121.40(12) 108.67(14) 109.84(14) 112.137(84) 105.78(14) 102.44(24) −102.84(23) −102.81(24) 101.33(17) −151.39(23) 90.66(23) −29.24(37) −48.04(24)
1.4612(12) 1.44214(64) 1.0813(12) 1.0807(12) 1.0845(12) 1.5064(16) 1.0947(12) 1.0894(12) 1.41019(78) 0.9633(19) 58.963(75) 119.34(18) 118.71(19) 118.47(19) 121.58(10) 108.64(19) 109.83(19) 112.081(70) 105.76(18) 102.50(30) −102.97(29) −102.71(31) 101.02(13) −151.30(29) 90.58(29) −28.48(26) −48.14(31)
1.4318 61.20
1.4375 60.83
1.4352 60.81
1.430(30) 60.81(30)
1.4259(15) 61.423(97)
1.4289(14) 61.184(70)
6-311+G(3df,2pd). bNumber of predicate values used in the fit, see text. cThe parameters determined with the help of a predicate value are underlined. a
equilibrium rotational constants found with the α/2 sums needed for the relationship
density functional theory (DFT)11 using Becke’s threeparameter hybrid exchange functional12 and the Lee−Yang− Parr correlation functional,13 together denoted as B3LYP. Correlation-consistent polarized n-tuple-zeta basis sets ccpVnZ14 with n ∈ {T, Q} were employed. The split-valence basis set 6-311+G(3df,2pd), as implemented in Gaussian03 (revision D.01),15 was also employed with the B3LYP method because it is known to give relatively accurate results with the B3LYP DFT technique.16,17 All computations utilized the Gaussian03 or Gaussian09 program suites.15 The ab initio structures computed at these different levels of theory are given in Table 3. As far as the angles are concerned, the values from MP2/cc-pVTZ (with G03 E.01) and MP2/cc-pVQZ (with G09) are close. However, some B3LYP/6-311+G(3df,2pd) values are significantly different. Quadratic and cubic force constants were computed at the B3LYP/6-311++G** level of theory with Gaussian03 (revision C.02). The B3LYP method was preferred to the MP2 method because it is more affordable from the point of view of computer time and memory. In calculating the vibration− rotation constants (α), the vibration−rotation module of Gaussian03 was used. For each isotopologue, the coordinates of the optimized structure were transformed to the principal axis system of the isotopologue prior to input. Table 4 contains the observed ground state rotational constants3,4 and the
Beg = B0g +
1 2
3N − 6
∑
αi g
i=1
where g is for the a, b, and c principal axes, and i scans the normal modes.
4. DETERMINATION OF THE PREDICATE OBSERVATIONS First, the Atoms in Molecules (AIM) theory18 with its implementation in Gaussian03 by Cioslowski et al.19 was used to calculate the bond ellipticity ε and bond critical point density ρb. The bond ellipticity provides a measure of the extent to which the charge is preferentially accumulated at different angles in a given plane perpendicular to the bond path and, for this reason, is a measure of the π-character of bond. The bond critical point density gives the amount of electron density shared between the two bonded atoms. It is roughly proportional to the bond length. The results, calculated at the B3LYP/6-311+G(3df,2pd) level of theory, are given in Table 5 together with those of oxirane, c-C2H4O, and methanol, CH3OH. The bonds of the three-membered ring of glycidol have a large bond ellipticity as in oxirane and thus have a large π9118
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C1−O1 bonds. As shown in ref 20, the MP2/cc-pVTZ level of theory should be able to deliver an accurate prediction for these bond lengths. For the C1−O1 bond, ρb = 0.267 in glycidol, which is larger than ρb = 0.260 in CH3OH; thus, the C−O bond length is expected to be shorter in glycidol than methanol, for which the value is re(C−O) = 1.417 Å.22 Likewise, for the O−H bond, ρb = 0.373 in glycidol, which is smaller than ρb = 0.379 in CH3OH, indicating the OH bond should be slightly longer in glycidol than in methanol, where re = 0.957 Å.22 The bond critical point densities also allow us to order the C−H bond lengths: C1H1A > C1H1B > C2H > C3H3B > C3H3A. The determination of the predicates has already been discussed in two previous papers.20,23 Briefly, the MP2/ccpVTZ level of theory supplies an accurate prediction for the C−H lengths as well as for the C−C single bond lengths. The accuracy is about 0.002 Å for the C−H bonds and slightly better than 0.003 Å for the C−C bonds. The C1−O1 single bond length in an alcohol may be estimated from an empirical formula using as input the B3LYP/6-311+G(3df,2pd) value or the MP2/cc-pVQZ value.24 Both methods give almost the same result, r(C−O) = 1.410(3) Å. The estimation of the C3−O2 bond length in the ring is much more difficult because of its significantly longer length than the bonds investigated in ref 24 and because of the large π-character of this bond. However, the comparison of the results for oxirane, see Table 5, allows us to estimate an offset, Δr = re − r[ab initio], for the three levels of theory, B3LYP/6-311+G(3df,2pd), MP2/cc-pVTZ, and MP2/ cc-pVQZ. For glycidol, these offsets give a compatible result with r(C3−O2) = 1.439(3) Å. The C2−C3 bond length can be estimated in the same way to give r(C2−C3) = 1.459(3) Å, quite close to the MP2/cc-pVTZ value as predicted in ref 20. The O−H bond was discussed in another paper,22 and it was found that re − r[MP2/cc-pVTZ] = −0.0015(11) Å and that re − r[MP2/cc-pVQZ] = −0.0031(10) Å. It can also be predicted
Table 4. Rotational Constants (MHz) of Glycidol H-Bond Inner normal
13
C1
13
C2
13
C3
18
O2 in the ring
18
O1 OH
OD
a
A B C A B C A B C A B C A B C A B C A B C
ground state
equilibrium
e − ca
10347.86 4102.36 3781.95 10254.29 4065.48 3738.20 10237.80 4095.78 3761.83 10227.43 4037.28 3736.25 10104.85 3992.60 3712.37 10186.45 3950.10 3631.95 10010.30 4056.73 3717.02
10415.45 4161.98 3832.62 10320.48 4123.94 3787.69 10303.05 4155.45 3812.04 10294.57 4094.57 3785.61 10168.78 4051.87 3762.89 10251.86 4007.21 3680.33 10075.10 4116.47 3767.10
−0.05 −0.10 −0.08 0.01 −0.23 −0.16 −0.02 0.30 0.31 −0.25 −0.37 −0.21 0.19 0.31 0.14 0.13 0.04 0.04 −6.55 2.70 1.85
Residuals of the mixed estimation fit, equilibrium − calculated.
character. For this reason, as shown in ref 20, the MP2 method might not be accurate for predicting these bond lengths. However, they are expected to be rather similar to those of oxirane. In particular, for the C2−C3 bond, the bond critical point density is almost the same in glycidol and in oxirane indicating that the C−C bond length is similar in both glycidol and oxirane, where re(C−C) = 1.462 Å.21 The bond ellipticity is quite small for the other bonds, in particular the C1−C2 and
Table 5. Bond Critical Point Density ρb, Bond Ellipticity ε, and Bond Lengths (Å) for Glycidol H-Bond Inner, Oxirane, and Methanol method/basis set Glycidol C2−C3 C3−O2 C2−O2 C3−H3B C3−H3A C2−H C2−C1 C1−H1A C1−H1B C1−O1 O1−H1 Oxiraneb C−C C−O C−H Methanolc C−O O−H C−Hs C−Ha a
ρba
εa
B3LYP/6-311+Ga
MP2/cc-pVTZ
MP2/cc-pVQZ
0.264 0.248 0.253 0.295 0.297 0.293 0.262 0.287 0.291 0.267 0.373
0.244 0.742 0.659 0.035 0.034 0.029 0.071 0.043 0.040 0.019 0.021
1.4611 1.4385 1.4318 1.0839 1.0828 1.0873 1.5113 1.0972 1.0911 1.4128 0.9640
1.4595 1.4453 1.4375 1.0813 1.0806 1.0845 1.5056 1.0946 1.0892 1.4115 0.9658
1.4566 1.4429 1.4352 1.0803 1.0798 1.0839 1.5034 1.0935 1.0882 1.4092 0.9638
0.260 0.256 0.295
0.281 0.655 0.034
1.4630 1.4271 1.0843
1.4622 1.4317 1.0816
1.4588 1.4303 1.0805
1.4608 1.4273 1.0821
0.260 0.379 0.293 0.287
0.006 0.023 0.045 0.048
1.4210 0.9598 1.0878 1.0939
1.4185 0.9595 1.0855 1.0913
1.4174 0.9578 1.0842 1.0897
1.4172 0.9568 1.0857 1.0912
re
B3LYP/6-311+G(3df,2pd). bReference 21. cReference 22. 9119
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ref 20, in particular when the corresponding torsional frequency is low. For glycidol, there should be no problem because the harmonic frequency for torsion around the CCOH bond is 148 cm−1 as computed with the B3LYP/6-311++G** model. Finally, the MP2/cc-VQZ angles were chosen as predicate values with an uncertainty of 0.3° for the bond angles and 0.5° for the torsional angles.
by using offsets from the results of methanol (see Table 5). All methods give compatible results with r(O−H) = 0.963(3) Å. For the bond angles, the mean accuracy is 0.4° for the MP2/ cc-pVTZ level of theory and 0.25° for the MP2/cc-pVQZ level.25 In many cases, the difference between the MP2/ccpVTZ and the MP2/cc-pVQZ angles is small. It is particularly true for the ∠(HCH) and ∠(CCH) angles. However, one notable exception is the ∠(COH) angle as can be seen in Table 6. The MP2/cc-pVTZ value is about 0.4−0.6° smaller than the
5. STRUCTURE USING THE KRAITCHMAN EQUATIONS Common practice in microwave spectroscopy has been to use the Kraitchman substitution method with ground state rotational constants to find structures. A fit to equilibrium rotational constants would be an improvement. Using the available semiexperimental equilibrium rotational constants of Table 4, a partial structure was calculated with Kraitchman’s equations for the substitution method.7 The results are given in Table 7. Inspection of the Cartesian coordinates shows that the c-coordinate of O1 is imaginary, whereas the c-coordinates of C1 and C2 are quite small,
