Polarizabilities of Oxazoles: Ab Initio Calculations and Simple Models

Ab initio, electron correlated calculations of the equilibrium geometries, dipole moments, and static dipole polarizabilities are reported for 10 hete...
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8752

J. Phys. Chem. 1996, 100, 8752-8757

Polarizabilities of Oxazoles: Ab Initio Calculations and Simple Models N. El-Bakali Kassimi, Robert J. Doerksen, and Ajit J. Thakkar* Chemistry Department, UniVersity of New Brunswick, Fredericton, NB, Canada E3B 6E2 ReceiVed: NoVember 22, 1995; In Final Form: February 21, 1996X

Ab initio, electron correlated calculations of the equilibrium geometries, dipole moments, and static dipole polarizabilities are reported for 10 heteroaromatic five-membered rings: furan, oxazole, isoxazole, all four oxadiazoles, both oxatriazoles, and oxatetrazole. Our geometries and dipole moments agree well with available experimental microwave determinations. The polarizabilities are expected to be accurate to within 5%. Structural isomerism affects the dipole moments strongly but the dipole polarizabilities are rather insensitive to it. Uncoupled Hartree-Fock calculations indicate that as much as half the polarizability comes from the σ-electrons. Simple empirical formulas based upon atom- and bond-additive models correlate the calculated polarizabilities of 20 five-membered heteroaromatic rings (10 azoles and 10 oxazoles) quite well.

Introduction Polarizabilities are important because they determine long range intermolecular induction and dispersion forces, low-energy electron-molecule scattering cross sections, Langevin capture cross sections in ion-neutral collisions, Verdet constants, and various phenomena such as collision-induced spectral line shifts.1-3 We have begun a systematic study of the polarizabilities of heteroaromatic molecules4,5 to build a sizeable body of accurate polarizability data for such molecules, examine the utility and limits of simple models for these polarizabilities, and study the connection between polarizability and aromaticity. We have previously studied azines4 and azoles5smolecules obtained by aza substitution of benzene and pyrrole respectively. In this paper, we examine oxazoles6sfive-membered heteroaromatic rings obtained by aza substitution of furan. Figure 1 shows all possible oxazoles: furan (1), 1,2-oxazole or isoxazole (2), 1,3-oxazole or just oxazole (3), 1,2,3-oxadiazole (4), 1,2,5-oxadiazole or furazan (5), 1,2,4-oxadiazole (6), 1,3,4oxadiazole (7), 1,2,3,5-oxatriazole (8), 1,2,3,4-oxatriazole (9), and oxatetrazole (10). Accurate gas-phase structures and dipole moments of some oxazoles have been determined by microwave spectroscopy7-17 and by electron correlated, ab initio calculations.18-26 There have been three measurements and three ab initio calculations of the polarizability of furan at optical frequencies.20,27-30 There is also an ab initio calculation of the static polarizability of 1,3-oxazole.23 However, nothing is known about the polarizabilities of the other eight oxazoles. We report systematic, electron correlated, ab initio calculations of the equilibrium geometries, dipole moments, and static dipole polarizabilities of all 10 oxazoles. Uncoupled Hartree-Fock calculations are used to determine the fraction of the mean polarizability that arises from the π-electrons. Additive and other simple models for the polarizability of the oxazoles and the azoles are discussed, and empirical formulas are given that fit the polarizabilities of the azoles and oxazoles quite well. Computational Methods The computational techniques used were the same as in our azole work.5 Hence, only a concise summary is given here; fuller details can be found in ref 5. We use the finite-field method.31,32 We computed the necessary field-dependent ener* Fax 506-453-4981; e-mail: [email protected]. X Abstract published in AdVance ACS Abstracts, April 15, 1996.

S0022-3654(95)03425-3 CCC: $12.00

Figure 1. Oxazoles with MP2/C dipole moments shown to scale, with the arrowhead pointing to the negative end.

gies with GAUSSIAN-9033 using both the Hartree-Fock (HF) self-consistent-field (SCF) and valence, second-order, MøllerPlesset (MP2) perturbation theory methods32,34 to obtain coupled Hartree-Fock (CHF) and coupled MP2 properties. We also use the less accurate uncoupled Hartree-Fock (UCHF) method,35 as implemented in our program,36 in which the field is treated by perturbation theory using the field-free Hartree-Fock wave function as the unperturbed function. UCHF polarizabilities have interpretative utility since they can be partitioned uniquely into contributions from each of the occupied molecular orbitals. Reliable polarizabilities can be calculated only if the basis set includes enough diffuse sp-type and polarization functions.32,37 We used the same basis sets of Gaussian-type functions (GTF) as in our earlier work.5 The smaller [3s2p2d/ 2s2p] set, denoted B, and the larger [5s3p2d/3s2p] set, denoted © 1996 American Chemical Society

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J. Phys. Chem., Vol. 100, No. 21, 1996 8753

TABLE 1: Comparison between Calculated MP2/6-31G* and Microwave (MW) rs Structures (Bond Lengths in picometers and Bond Angles in degrees) furan O1C2 C2C3 C3C4 HC2 HC3 C5O1C2 O1C2C3 C2C3C4 HC2C3 HC3C4 a

1,2,5-oxadiazole

MP2

MWa

136.5 136.5 142.7 108.0 108.0 106.6 110.5 106.2 134.0 127.5

136.2 136.1 143.0 107.5 107.7 106.6 110.7 106.1 133.4 127.8

MP2

1,3,4-oxadiazole

MWb

MP2 MWc

O1N2 N2C3 C3C4

137.3 137.3 O1C2 132.8 130.3 C2N3 140.3 142.1 N3N4 HC2 HC3 108.0 107.6 N5O1N2 112.1 111.2 C5O1C2 O1N2C3 104.7 105.5 O1C2N3 N2C3C4 109.2 108.9 C2N3N4 HC2N3 HC3C4 130.5 130.2

136.0 130.1 140.3 107.9

134.8 129.7 139.9 107.5

101.4 113.6 105.7 128.5

102.0 113.4 105.6 128.5

Reference 9. b Reference 11. c Reference 7.

TABLE 2: Comparison between Calculated MP2/6-31G* and Microwave (MW) rs Structures (Bond Lengths in picometers and Bond Angles in degrees) 1,2-oxazole O1N2 N2C3 C3C4 C4C5 C5O1 HC3 HC4 HC5 C5O1N2 O1N2C3 N2C3C4 C3C4C5 C4C5O1 HC3C4 HC4C3 HC5C4 a

1,3-oxazole a

MP2

MW

139.0 132.7 141.3 136.3 135.2 108.2 107.9 108.0 109.2 105.1 112.3 103.5 110.0 129.3 128.6 134.5

139.9 130.9 142.5 135.6 134.4 107.7 107.4 107.5 108.8 105.3 112.3 103.0 110.6 129.1 128.5 133.4

O1C2 C2N3 N3C4 C4C5 C5O1 HC2 HC4 HC5 C5O1C2 O1C2N3 C2N3C4 N3C4C5 C4C5O1 HC2N3 HC4N3 HC5C4

MP2

MWb

136.0 130.2 138.9 136.0 136.9 108.0 108.0 107.8 104.0 114.8 103.8 109.4 108.0 128.6 121.7 135.5

135.7 129.2 139.5 135.3 137.0 107.5 107.5 107.3 103.9 115.0 103.9 109.0 108.1 127.9 121.9 135.0

Reference 8. b Reference 10.

C, differ only in their atomic substrates. Set B builds on Scha¨fer et al.’s split-valence set,38 whereas the larger set C builds upon Thakkar et al.’s double-ζ basis set39 augmented by diffuse sand p-type GTF. Both basis sets contain two sets of polarization functionssa compact set optimized for electron correlation by Dunning40 and a diffuse set optimized by us for polarizabilities.5 Full details are given in ref 5; here we need only add that the exponent of the diffuse d-GTF for O was chosen to be 0.25 on the basis of preliminary calculations on furan using basis B without the polarization functions for correlation. Equilibrium Geometries A set of consistent equilibrium geometries for all 10 oxazoles determined by a uniform technique is a prerequisite to systematic

ab initio calculations of the polarizabilities. However, neither experimental gas-phase determinations nor electron correlated calculations of equilibrium structures are available for more than five oxazoles. Therefore, we calculated a set of MP2 geometries for all 10 oxazoles using the 6-31G* basis set.34,41 Such calculations generally result32,34 in equilibrium geometries accurate to within 1%. Constraints of C2V symmetry were imposed for furan, 1,2,5-oxadiazole, 1,3,4-oxadiazole, and oxatetrazole, whereas Cs symmetry was imposed on the other six oxazoles. Our MP2 geometries are compared with substitution rs structures7-11 obtained by microwave spectroscopy in Table 1 for furan, 1,2,5-oxadiazole, and 1,3,4-oxadiazole and Table 2 for 1,2-oxazole and 1,3-oxazole. There is generally good agreement between the calculations and experiment. The average discrepancy is 0.7 pm for bond lengths and 0.3° for bond angles. All but three of the 29 MP2 bond lengths agree with experiment within 1.2 pm. None of the 29 MP2 bond angles disagrees with experiment by more than 1.1°. Egregious discrepancies arise in 1,2,5-oxadiazole and 1,2-oxazole where the calculated N2dC3 bond lengths are too long by 2.5 and 1.8 pm, respectively, but the computed C3C4 bond lengths are too short by 1.8 and 1.2 pm, respectively. A comparison with previously reported ab initio equilibrium geometries is necessary. Simandiras et al.18 and Palmer et al.22 reported MP2 geometries for furan using a double-ζ plus polarization (DZP) basis set; their geometries are identical with each other and differ from ours by less than 1.0 pm in the bond lengths and 0.5° in the bond angles. Shaffer and Wierschke19 reported a MP2/6-31G* geometry for 1,3-oxazole which is incorrect. We tried to reproduce their result using various initial points for the optimization but were unable to do so even when we used their result as the starting point. After we had completed this work, Kraka et al.23 published a MP2/6-31G* geometry for 1,3-oxazole which agrees with ours and thus confirms the incorrectness of Shaffer and Wierschke’s result.19 Very recently, El-Azhary has published a string of papers reporting MP2/6-31G** geometries for 1,2,5-oxadiazole,21 1,2oxazole,24 1,3-oxazole,25 and 1,3,4-oxadiazole.26 His results are virtually identical with ours; the sole difference is that his CH bonds are consistently and correctly about 0.5 pm shorter than ours because the 6-31G** basis set has p-GTF on the hydrogen atoms, whereas the 6-31G* basis set does not. SCF geometries have been published for furan,20,42-45 1,2-oxazole,46 and 1,3oxazole,44,47 but they have been superseded by the MP2 calculations cited above. There are also density functional theory (DFT) calculations24,48 of the geometry of 1,2-oxazole; the results are sensitive to the functionals used. Table 3 lists our MP2 geometries for the other five oxazoles. We are unaware of any experimental gas-phase geometries with which they can be compared, although microwave spectroscopy

TABLE 3: MP2/6-31G* Bond Lengths (picometers) and Bond Angles (degrees) 1,2,4-oxadiazole O1N2 N2C3 C3N4 N4C5 C5O1 HC3 HC5 C5O1N2 O1N2C3 N2C3N4 C3N4C5 N4C5O1 HC3N4 HC5N4

139.6 131.9 137.4 130.3 134.7 108.1 108.1 106.3 103.0 115.4 101.2 114.2 123.6 129.0

1,2,3-oxadiazole O1N2 N2N3 N3C4 C4C5 C5O1 HC4 HC5 C5O1N2 O1N2N3 N2N3C4 N3C4C5 C4C5O1 HC4N3 HC5C4

142.8 128.5 137.0 136.3 134.2 107.9 107.9 106.4 107.9 110.0 107.4 108.3 122.0 134.7

1,2,3,4-oxatriazole O1N2 N2N3 N3N4 N4C5 C5O1

141.9 127.8 138.1 130.4 133.5

HC5 C5O1N2 O1N2N3 N2N3N4 N3N4C5 N4C5O1

107.9 103.8 107.1 111.7 103.8 113.5

HC5N4

128.1

1,2,3,5-oxatriazole

oxatetrazole

O1N2 N2N3 N3C4 C4N5 N5O1 HC4

141.4 128.7 136.1 132.6 135.0 107.9

O1N2 N2N3 N3N4

137.7 128.3 138.0

N5O1N2 O1N2N3 N2N3C4 N3C4N5 C4N5O1 HC4N3

109.5 106.7 107.5 112.6 103.7 124.3

N5O1N2 O1N2N3 N2N3N4

107.5 107.2 109.1

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TABLE 4: Dipole Moment Magnitudes µ (debye) and Orientation Angles with Respect to an Inertial Axis, θa or θb (degrees). All Calculated Values Are at the MP2/6-31G* Geometry SCF furan 1,2-oxazole 1,3-oxazole 1,2,3-oxadiazole 1,2,5-oxadiazole 1,2,4-oxadiazole 1,3,4-oxadiazole 1,2,3,5-oxatriazole 1,2,3,4-oxatriazole oxatetrazole a

µ µ θa µ θa µ θb µ µ θa µ µ θb µ θa µ

MP2

6-31G*

B

C

6-31G*

B

C

expt

0.86 3.48 32.6 1.52 26.6 3.90 32.2 3.99 1.48 18.6 3.37 2.31 8.6 3.23 2.9 0.52

0.79 3.34 31.9 1.48 21.7 3.67 33.5 3.79 1.47 21.9 3.38 2.12 8.6 3.15 2.0 0.61

0.82 3.41 32.4 1.47 22.1 3.74 32.7 3.90 1.50 20.2 3.40 2.21 7.9 3.18 2.8 0.56

0.68 3.00 30.3 1.63 27.6 3.69 35.2 3.37 1.07 29.2 3.20 2.10 19.1 3.08 1.5 0.72

0.62 2.88 29.4 1.63 25.6 3.50 36.5 3.14 1.04 33.9 3.20 1.90 20.8 2.97 2.6 0.80

0.65 2.97 30.2 1.64 26.0 3.62 35.6 3.31 1.07 30.6 3.25 2.04 19.3 3.04 1.5 0.73

0.661 ( 0.006a 2.90 ( 0.15b 47 ( 2b 1.50 ( 0.03c 24.8 ( 1c 3.38 ( 0.04d 1.2 ( 0.3e 9 ( 12e 3.04 ( 0.04f

Reference 12. b Reference 15. c Reference 17. d Reference 13. e Reference 16. f Reference 14.

has been carried out16 on 1,2,4-oxadiazole. SCF calculations49 in a small 3-21G basis set34,50 suggest that 1,2,3-oxadiazole is less stable than a formyldiazomethane structure. Our MP2/631G* geometry for 1,2,3-oxadiazole differs from Nguyen et al.’s SCF/3-21G structure49 by more than 4 pm for the O-N and N-N bond lengths. So we calculated the MP2/6-31G* geometry of formyldiazomethane; our O-N and N-N distances respectively turned out to be 8.5 pm shorter and 3.5 pm longer than theirs!49 Frequency calculations were performed to verify that our MP2/6-31G* geometries for both 1,2,3-oxadiazole and formyldiazomethane are true minima. Using MP2/6-31G* geometries for both the forms, we find the formyldiazomethane structure to be more stable than the cyclic structure by 10.6, 3.2, and 2.1 kcal mol-1, respectively, at the SCF, MP2, and QCISD(T) (quadratic configuration interaction with single and double substitutions perturbatively corrected for triple substitutions51) levels. Although the cyclic form is not the lowest energy structure, we examine its polarizability together with those of the hypothetical oxatriazoles and oxatetrazole so that trends in the 10 oxazoles can be studied. Dipole Moments Table 4 summarizes our SCF and MP2 dipole moments calculated at the MP2/6-31G* geometries using three basis sets (6-31G*, B and C). The table also lists gas-phase experimental values obtained from Stark shifts probed by microwave spectroscopy. The magnitude µ of the MP2/C dipole moments ranges from 0.65 D for furan to 3.62 D for 1,2,3-oxadiazole. The dipole moment vectors of the four oxazoles with C2V symmetry necessarily coincide with the symmetry axis. The dipole moments of the other six oxazoles with Cs symmetry lie in the molecular plane, and Table 4 specifies their orientation by listing θ, the acute angle between the dipole moment and an in-plane inertial axis. The orientations, polarities, and relative magnitudes of the MP2/C dipole moments of all 10 oxazoles are most easily grasped from Figure 1. Basis set effects are relatively small, but electron correlation effects are much larger. For example, the MP2/6-31G* and MP2/C values of µ differ by an average of merely 0.04 D and a maximum of 0.07 D. By contrast, the SCF/C and MP2/C values of µ differ by an average of 0.26 D and a maximum of 0.59 D. Similarly, the MP2/6-31G* and MP2/C dipole moment orientations differ by an average of 0.5° and a maximum of 1.6° whereas the effect of electron correlation has average and maximum values of 5.4° and 11° respectively in basis C. The MP2 values of µ, in any of the three basis sets, differ from the experimental values by an average of 0.1 D with a

TABLE 5: Uncoupled Hartree-Fock (UCHF), Coupled Hartree-Fock (CHF), and Finite-Field MP2 Polarizabilitiesa in Basis Sets B and C UCHF R j ∆1R ∆2R 1,2-oxazole R j ∆1R ∆2R 1,3-oxazole R j ∆1R ∆2R 1,2,3-oxadiazole R j ∆1R ∆2R 1,2,5-oxadiazole R j ∆1R ∆2R 1,2,4-oxadiazole R j ∆1R ∆2R 1,3,4-oxadiazole R j ∆1R ∆2R 1,2,3,5-oxatriazole R j ∆1R ∆2R 1,2,3,4-oxatriazole R j ∆1R ∆2R oxatetrazole R j ∆1R ∆2R

furan

CHF

MP2

B

C

B

C

B

C

44.75 16.36 16.39 40.61 16.43 16.75 39.61 15.03 15.05 37.94 17.01 17.12 37.78 17.16 17.45 35.30 14.52 14.74 35.21 13.97 13.98 34.54 16.59 16.73 32.94 14.90 14.97 31.21 15.63 15.87

45.46 15.92 15.96 41.42 16.45 16.70 40.32 14.94 14.96 38.93 17.13 17.22 38.69 17.52 17.79 36.09 14.80 14.95 36.01 14.28 14.30 35.62 17.08 17.20 34.01 15.27 15.33 32.56 16.05 16.26

46.57 20.67 20.90 40.90 18.75 19.24 40.78 18.60 18.95 36.90 17.37 17.55 36.34 17.16 17.21 35.19 16.45 17.21 35.42 16.37 17.09 32.08 15.07 15.13 31.32 14.63 14.98 27.73 12.76 12.76

47.62 19.65 19.86 42.18 18.60 18.92 41.86 18.12 18.40 38.51 17.59 17.78 37.85 17.69 17.74 36.43 16.72 17.16 36.67 16.68 17.15 33.80 15.87 15.89 33.02 15.35 15.58 29.84 13.65 13.65

47.06 21.11 21.59 42.06 19.44 20.15 41.67 19.28 19.97 37.51 17.64 17.97 37.23 17.48 17.57 36.90 17.27 18.53 36.35 16.51 17.77 32.72 15.13 15.24 32.30 14.71 15.66 28.04 12.51 12.60

48.51 20.13 20.58 43.84 19.49 19.94 43.14 18.92 19.48 39.82 18.39 18.74 39.43 18.37 18.44 38.69 17.80 18.50 38.16 17.13 17.90 35.33 16.57 16.67 34.79 15.99 16.66 31.29 14.16 14.26

a All calculated values are at the MP2/6-31G* geometry and are given in atomic units related to SI units by 4π0a03 ≈ 1.648 78 × 10-41 F m2.

maximum discrepancy of 0.2 D. The average deviation of the MP2/C values of θ from their experimental counterparts is 13° and the maximum discrepancy is 22° for 1,2,4-oxadiazole although the latter experimental θ is very shaky.16 Since basis set effects are small, the discrepancies between our best calculations (MP2/C) and experiment for µ and θ must largely be due to the effects of molecular vibration and higher order electron correlation. Polarizabilities We chose coordinate systems with the molecules in the yz plane, the center-of-mass at the origin, and the positive z axis

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J. Phys. Chem., Vol. 100, No. 21, 1996 8755

Figure 2. Percentage differences between the mean polarizability computed by various methods and its counterpart calculated at the CHF level in basis B. The numbering system for the molecules is as in Figure 1.

Figure 3. Percentage differences between the polarizability anisotropy ∆1R computed by various methods and its counterpart calculated at the CHF level in basis B. The numbering system for the molecules is as in Figure 1.

passing through the oxygen. For the four oxazoles with C2V symmetry, this choice is a principal axis system that diagonalizes the polarizability tensor. For the six Cs oxazoles, the x axis is a principal axis but the y and z axes are not, and the polarizability tensor is block diagonal with 2 × 2 and 1 × 1 blocks for the in-plane and out-of-plane components, respectively. Table 5 lists three quantities that are invariant to the choice of coordinate system. One is the mean polarizability given by the trace invariant:

TABLE 6: Comparison of Polarizabilities for Furana

1 1 R j ) (Rxx + Ryy + Rzz) ) (R1 + R2 + R3) 3 3

(1)

in which R1 ) Rxx e R2 e R3 are the eigenvalues of the polarizability tensor. A measure of the polarizability anisotropy is the difference between the mean in-plane and out-of-plane components; it is given in our coordinate system by

1 1 ∆1R ) R| - R⊥ ) (Ryy + Rzz) - Rxx ) (R2 + R3) - R1 2 2 (2) Another invariant, related to the Kerr effect, is given by

∆2R )

[

]

(R1 - R2)2 + (R2 - R3)2 + (R3 - R1)2 2

1/2

(3)

Whenever these anisotropies are close to each other, the inplane anisotropy given by

∆3R ) [(∆2R)2 - (∆1R)2]1/2 ) (x3/2)|R3 - R2|

(4)

is particularly useful. Atomic units52 for polarizability, related to SI units by 4π0a03 ≈ 1.648 78 × 10-41 F m2, are used in this work. Figure 2 shows percent differences between various computed mean polarizabilities and their CHF/B counterparts. It illustrates that the mean polarizability consistently increases as the basis set is enlarged from B to C. This increase averages 2.6, 4.2, and 6.0% at the UCHF, CHF, and MP2 levels, respectively, and does not exceed 12%. Inclusion of the inductive53 contribution in the CHF scheme changes the corresponding UCHF/B values by an average of 3.8% and a maximum of 11%. Note that a naive scaling scheme to correct UCHF polarizabilities for coupling effects will not work since the inductive contribution can be either positive or negative as is evident from Figure 2. Electron correlation consistently increases the mean

method

R j

CHF/4-31G+(dp)b CHF/6-31G+(d,p)c CHF/6-31G+(3d,3p)c CHF/Cd MP2/6-31G+(d,p)c MP2/6-31G(+sd+sp)e MP2/Cd experimentf experimentg experimente

42.1 42.9 46.3 47.6 43.6 47.9 48.5 48.8 49.1 ( 2.2 49.1 ( 0.5

∆1R

∆2R

18.4 18.6 19.7 19.0 21.6 20.1 18.8 15.2 ( 4.2 21.4 ( 3.2

19.7 18.4 18.6 19.9 19.1 22.0 20.6 20.7 15.3 ( 4.4 22.0 ( 3.2

a All values are in atomic units; 4π a 3 ≈ 1.648 78 × 10-41 F m2 0 0 is the conversion factor to SI units. b At the SCF/4-31G* geometry.30 c At the SCF/6-31G** geometry.20 d At the MP2/6-31G* geometry, this work. e At a wavelength of 632.8 nm.29 f At a wavelength of 589.3 nm in carbon tetrachloride solvent.27 g At a wavelength of 632.8 nm in cyclohexane solvent.28

polarizabilities by an average of 2.4 and 4.1% in basis sets B and C, respectively; in no case is the correlation effect greater than 6.2%. Notice that the CHF/C polarizabilities are similar to and often larger than the MP2/B ones showing once again that it is usually better to enlarge the basis set to a reasonable size before including electron correlation. Figure 3 shows percent differences between various values of the polarizability anisotropy ∆1R and their CHF/B counterparts. Basis set and electron correlation effects on ∆1R are similar to those on the mean polarizability. However, the effect of coupling is significantly greater for the UCHF anisotropies than it is for the UCHF mean polarizabilities; the inductive contributions in basis B average 13% and can be as great as 26%. The effects of basis set choice, coupling and correlation on the anisotropy ∆2R are very similar to those for ∆1R (cf. Table 5). The polarizability of furan has been studied quite often. Table 6 contains a comparison of our results for furan with previous work. Given the differences in basis sets and geometries, the CHF static polarizabilities of Hinchliffe and Soscu´n,20 and Keshari et al.30 are as close as can be expected to our CHF/C values. Coonan et al.’s calculated value29 pertains to 632.8 nm, and their zero-frequency value would be lower still. The MP2/ 6-31G+(d,p) mean polarizability of Hinchliffe and Soscu´n20 seems even less accurate than their own CHF/6-31G+(3d,3p) value because the former basis set is quite inadequate for polarizabilities. Our MP2/C static polarizabilities are more reliable than all the previous calculations because our basis set

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TABLE 7: MP2/C Polarizabilities for the Oxazoles at the MP2/6-31G* Geometrya molecule

R1

R2

R3

φ3

R j

∆1R

∆2R

∆3R

%π(⊥)

%π(|)

furan 1,2-oxazole 1,3-oxazole 1,2,3-oxadiazole 1,2,5-oxadiazole 1,2,4-oxadiazole 1,3,4-oxadiazole 1,2,3,5-oxatriazole 1,2,3,4-oxatriazole oxatetrazole

35.09 30.85 30.53 27.56 27.18 26.82 26.74 24.28 24.13 21.84

52.74 47.91 46.77 43.86 44.63 41.72 40.87 39.82 37.43 35.06

57.69 52.77 52.13 48.04 46.48 47.52 46.87 41.88 42.82 36.96

0.0 72.4 127.3 74.8 0.0 35.0 0.0 118.1 72.2 0.0

48.51 43.84 43.14 39.82 39.43 38.69 38.16 35.33 34.79 31.29

20.13 19.49 18.92 18.39 18.37 17.80 17.13 16.57 15.99 14.16

20.58 19.94 19.48 18.74 18.44 18.50 17.90 16.67 16.66 14.26

4.29 4.21 4.64 3.62 1.60 5.02 5.20 1.78 4.67 1.65

64.1 59.8 60.3 57.3 56.1 54.8 55.6 52.9 52.1 49.5

51.1 48.3 48.9 48.3 47.4 45.0 45.7 46.7 44.2 44.6

a All polarizability data is in atomic units that are related to SI units by 4π a 3 ≈ 1.648 78 × 10-41 F m2. The R , R R are eigenvalues of the 0 0 1 2, 3 polarizability tensor; the 1-axis is perpendicular to the molecular plane, and counterclockwise rotation by φ3 degrees brings the 3-4 bond into coincidence with the 3-axis. The UCHF π-fractions of the out-of-plane and in-plane polarizabilities are in the last two columns.

C is significantly better. It has a more accurate double-ζ substrate and has polarization functions that are better tailored for the response properties of aromatic molecules. A comparison with experimental values27-29 is more problematic. Le Fe`vre et al.27 reported polarizabilities for furan at a wavelength of 589.3 nm based on depolarization ratios, refraction, and dielectric polarization in a carbon tetrachloride solvent. Dennis et al.28 reported polarizabilities at a wavelength of 632.8 nm obtained from experimental values of the molar refraction, molar Kerr and Cotton-Mouton constants, dipole moment and anisotropic magnetizability of furan in cyclohexane solvent. Coonan et al.29 reported polarizabilities at a wavelength of 632.8 nm obtained from experimental values of the refractive index, Rayleigh depolarization ratio, and the temperature dependence of the vapor-phase molar Cotton-Mouton constant, dipole moment and anisotropic magnetizability of furan. All three of these experimental determinations of the polarizability of furan are for an optical frequency rather than zero frequency as in our calculations. Moreover, two27,28 of these experimental polarizabilities include solvent effects, whereas our computations are for an isolated furan molecule. Our neglect of vibrational effects54 also clouds the comparison of our polarizability with the vapor-phase experiments.29 Given all these caveats, our MP2/C polarizabilities agree with the experimental values in Table 6 better than could be reasonably expected. There are virtually no data in the literature with which our polarizabilities for the other oxazoles can be compared. Kraka et al.23 have reported mean polarizabilities for 1,3-oxazole. Their best calculated result, an MP2 value of 42.75 computed in a 6-31G(+sd+sp) basis set at the experimental geometry, is in good agreement with our MP2/C value of 43.14. Table 7 summarizes our MP2/C polarizabilities; it lists the eigenvalues, the mean, three anisotropy invariants, and the counterclockwise rotation angle φ3 that brings the bond between atoms 3 and 4 into coincidence with the in-plane 3-axis of polarizability. As in our work on the azoles,5 we expect these results to be accurate to within 5%. The in-plane components are substantially larger than the out-of-plane component because of the electron delocalization in the ring. The mean polarizability of each oxazole is smaller than that of the corresponding azole5 by roughly 5 ( 1 atomic units because the polarizability of an oxygen atom is smaller than that of a nitrogen and hydrogen atom pair. As in the azines4 and azoles,5 each azasubstitution lowers all the polarizability components because the polarizability of a nitrogen atom is smaller than that of a carbon and hydrogen atom pair. Like the azines and azoles, the oxazoles have an electron distribution that is close to that of an oblate symmetric top, and hence the in-plane anisotropy ∆3R is smallsno more than 14% of the mean polarizability. The mean polarizability is very similar for all oxazoles with the same degree of aza substitution. A similar observation for the azoles was rationalized with an additive atom model.5 Thus,

it is natural to ask how well the polarizability of the 10 azoles5 and 10 oxazoles can be approximated by a single additive atom model:

R j ≈ anH + bnC + cnN + dnO

(5)

The constraints nC + nN + nO ) 5 and nH ) nC + 1 - nO reduce eq 5 to a three-parameter model:

R j ≈ (6c - b) + (a + b - c)nH + (b - 2c + d)nO

(6)

We used linear regression to estimate the parameters in eq 6 and find that the model

R j ≈ 30.855 + 4.505nH - 0.5nO

(7)

fits the mean polarizabilities of all 20 azoles and oxazoles with an absolute error that averages 1.1% and does not exceed 3.2%. The smallness of the coefficient of nO suggests that this term can be omitted without loss of much accuracy. Indeed, we find that

R j ≈ 30.5 + 4.5nH

(8)

is almost as good; it leads to an absolute error that averages 1.2% and does not exceed 3.5%. This remarkable correlation has a simple chemical interpretation. The mean polarizability of oxatetrazole is ≈30.5 atomic units (cf. Table 7), and 4.5 atomic units is the average polarizability increase that accompanies the replacement of a nitrogen atom by a C-H group, and the replacement of the oxygen atom by a N-H group. A slightly more elaborate model5 is to assume additive contributions from connections (i.e., bonds without regard to bond order). Thus, for azoles and oxazoles, we write

R j ≈ anNH + bnCH + cnCC + dnCN + enNN + fnCO + gnNO (9) where nAB is the number of connections between atoms of type A and B. The constraints nCN + nNO + nCO + nCC + nNN ) 5, 2nCH ) nCN + nCO + 2nCC and 2nNH + nNO + nCO ) 2 reduce eq 9 to a five-parameter model: R j ≈ B1 + B2nCC + B3nCN + B4nNN + B5nNO

(10)

We estimated the parameters in eq 10 by linear regression and found that

R j ≈ 31.8 + 0.752nNN + 2.804nCN + 5.57nCC - 1.49nNO (11) leads to an average absolute error of 0.49% and a maximum error of 1.5%. Different sorts of models can be obtained by use of molecular parameters other than the atom and connection numbers. Any measure of molecular volume can be expected to be a good parameter to try. We chose to add to eq 6 a term in A3/2 where A is the area enclosed by the five-membered ring; thus, we write

Polarizabilities of Oxazoles

R j ≈ C1 + C2nH + C3nO + C4A3/2

J. Phys. Chem., Vol. 100, No. 21, 1996 8757

(12)

Linear regression with eq 12 leads to

R j ≈ -6.937 + 3.471nH - 1.78nO + 1.0881A3/2 (13) which fits the mean polarizabilities of all 20 azoles and oxazoles with an average absolute error of only 0.44% and a maximum error of 1.27%. Clearly eq 13 summarizes the data more efficiently than eq 11 does. Similar models for the polarizability anisotropy ∆1R are somewhat less accurate. Thus the additive atom model analogous to eq 7:

∆1R ≈ 15.75 + 1.54nH - 1.1nO

(14)

has an absolute error that averages 1.6% and does not exceed 3.6%. The additive connection model analogous to eq 11:

∆1R ≈ 13.664 + 0.633nNN + 1.523nCN + 2.205nCC - 0.4nNO (15) fits the mean polarizabilities of all 20 azoles and oxazoles with an average absolute error of only 1.0%, but the maximum error is 4.2%. Since a volume does not contain any information about the molecular anisotropy, it is not surprising that an analogue of eq 12 obtained by adding a term in A3/2 to eq 14 provides no improvement. UCHF polarizabilities can be separated into σ and π contributions. Table 7 lists the UCHF fractional π-electron contributions to the in-plane and out-of-plane polarizabilities. Observe that the σ-skeleton contributes almost half of the in-plane polarizability, and at least a third of the out-of-plane polarizability. Aza substitution significantly reduces the fractional contribution of the π-electrons to the transverse polarizability. This may be so because the more electronegative nitrogen binds its π-orbital more tightly than a carbon atom does. Various aromaticity indexes based on polarizabilities have been proposed. Bulgarevich et al.55 suggested that the π-contribution to the in-plane polarizability divided by the number of endocyclic bonds is a useful aromaticity index. Lazzeretti and Tossell56 suggested the mean polarizability, and Archibong and Thakkar4 suggested the excess of in-plane over out-of-plane polarizability as measured by ∆1R. Table 7 shows that all three indexes predict a decrease in aromaticity with increasing aza substitution, just as in the azines4 and azoles.5 They also predict that all oxazoles with the same number of nitrogen atoms have roughly the same degree of aromaticity. This work has provided the first systematic set of static dipole polarizabilities for the oxazoles. They are expected to be reliable to within 5%. We think that these polarizabilities, together with those we reported earlier for the azines4 and azoles,5 will be useful in detailed explorations of interesting issues such as aromaticity and simple models of polarizability. Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. References and Notes (1) Buckingham, A. D. AdV. Chem. Phys. 1967, 12, 107. (2) Miller, T. M.; Bederson, B. AdV. At. Mol. Phys. 1977, 13, 1. (3) Miller, T. M.; Bederson, B. AdV. At. Mol. Phys. 1988, 25, 37. (4) Archibong, E. F.; Thakkar, A. J. Mol. Phys. 1994, 81, 557. (5) El-Bakali Kassimi, N.; Doerksen, R. J.; Thakkar, A. J. J. Phys. Chem. 1995, 99, 12790. (6) Katritzky, A. R. Handbook of Heterocyclic Chemistry; Pergamon: New York, 1985. (7) Nygaard, L.; Hansen, R. L.; Nielsen, J. T.; Rastrup-Andersen, J.; Sørensen, G. O.; Steiner, P. A. J. Mol. Struct. 1972, 12, 59. (8) Stiefvater, O. L. J. Chem. Phys. 1975, 63, 2561.

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