Computational chemistry and experimental techniques can be combined into a powerful tool for determining the absolute configuration of chiral molecules.1–10 The conceptually simplest approach is to compare theoretically calculated and experimentally measured specific optical rotations to deduce the absolute stereochemistry. This procedure relies first and foremost on the accuracy of the theoretical model used to compute the optical rotation. Theoretical predictions of optical rotations rely on quantum mechanics11 and require advanced quantum chemical methodologies. Despite tremendous advances in this field, the computer time required for optical rotation calculations remains a bottleneck for routine applications. It is therefore important to strike a reasonable balance between accuracy and efficiency when choosing the theoretical model.
From a computational point of view, density functional theory (DFT) is the simplest electronic structure method that incorporates electron correlation effects. Stephens and coworkers7, 9, 10, 12–15 have assessed the accuracy of the B3LYP exchange-correlation functional16, 17 in optical rotation calculations at the molecular equilibrium geometry. For a test set of 30 chiral molecules with absolute values of experimental sodium D-line specific optical rotations ranging from ∼10–1200° (dm g/cm3)−1 a mean absolute deviation of 20° (dm g/cm3)−1 was found.13 For small-angle rotations [absolute value <100° (dm g/cm3)−1], the wrong sign was predicted for about 12% of 65 molecules at the sodium D-line.10 Recently, Crawford and Stephens15 compared B3LYP and coupled cluster singles and doubles (CCSD)18 calculations of sodium D-line optical rotations to experimental results for 13 molecules and found that the accuracies of the two models are almost identical (excluding the problematic case of norbornenone) when comparing with liquid-phase experimental data. However, a number of other studies have demonstrated that, at shorter wavelengths than the sodium D-line (589.3 nm), B3LYP and CCSD often differ substantially due to the generally more accurate prediction of excitation energies and rotatory strengths by the latter. For example, at 355 nm, B3LYP predicts specific rotations of methylthiirane, methyloxirane, and fluoro-oxirane to be 131.2,19 18.8,19 and 58.1°20 (dm g/cm3)−1, respectively, whereas CCSD gives −96.7,19 −33.3,19 and 0.5°20 (dm g/cm3)−1 (the aug-cc-pVTZ basis set was used in all cases).
In addition to the purely electronic response at the equilibrium geometry, vibrational, conformational, and solvent effects are often crucial for reliable predictions of optical rotation. Vibrational effects are generally non-neglible21–26 and have been shown to account for as much as 20% of the optical rotation of rigid molecules.22 In the case of methyloxirane, vibrational effects must be taken into account to reproduce the experimental gas phase optical rotation to within a few degrees.27–30 For conformationally flexible molecules, a simple Boltzmann procedure has been applied to average the electronic contributions from different conformations.5, 25, 31–38 Comparing to a rigorous vibrational averaging procedure involving all conformations of 3-chloro-1-butene, Crawford and Allen26 have shown that the simple approach benefits from cancellation of errors. Solvent effects on optical rotations have been modeled by means of continuum methods, both with20, 39 and without14, 28 vibrational contributions. The major problem with the continuum approach is that the chiral solvent configuration induced by the solute may contribute to or even dominate the observed optical rotation.40 It is therefore not surprising that continuum methods in some cases fail to reproduce observed trends. A promising, if computationally expensive, alternative is to combine molecular dynamics (MD) simulation with quantum chemical sampling of the solute and explicit solvent molecules.40–42 In an attempt to keep the computational cost of the MD-based approach at a minimum, Kundrat and Autschbach43, 44 have shown that a simple point charge representation of the solvent molecules (water) may be used to simulate explicit solvent effects in the quantum chemical sampling. As noted by these authors, the MD-based scheme may also be used as an alternative to Boltzmann averaging to account for conformational effects.
In our recent study20 of the gas phase optical rotation of fluoro-oxirane we observed that the main source of discrepancy between vibrationally averaged B3LYP and CCSD results is the difference in electronic contributions at the equilibrium geometry. The dependence of the optical rotation on vibrational degrees of freedom is reasonably well described by B3LYP. As this is also the case for methyloxirane30 and methylthiirane,45 we here assess the accuracy of the following procedure for seven molecules and two wavelengths for which experimental gas phase results are available:
1Determine the equilibrium geometry at the B3LYP level of theory.
2Calculate the CCSD electronic optical rotation at the B3LYP equilibrium geometry.
3Add the zero-point vibrational correction (ZPVC) calculated at the B3LYP level.
4For conformationally flexible molecules, perform points 1, 2, and 3 for each low-lying conformer and compute the final optical rotation by Boltzmann averaging.
Using the electronic optical rotation calculated with the approximate coupled cluster singles, doubles, and triples model CC346 by Kongsted et al.28 in point 2, essentially the same procedure has been successfully applied to the difficult case of methyloxirane by Ruud and Zanasi.29 Requiring only a single-point calculation of the optical rotation at the CCSD level of theory, this procedure is significantly less demanding in terms of computer time than a complete CCSD simulation. A somewhat less demanding alternative is obtained by replacing CCSD with the aproximate coupled cluster singles and doubles model CC2,47 and the accuracy of this approach is assessed here as well.
The specific optical rotation, in units of °(dm g/cm3)−1, in transparent spectral regions can be calculated according to the Rosenfeld expression11
where NA is Avogadro's number, a0 is the Bohr radius in cm, is the light frequency in cm−1, ω is the light frequency in atomic units, M is the molecular weight in g/mol, and Rn and ωn are the rotatory strength and excitation energy for the molecular transition 0 → n, respectively, in atomic units. The rotatory strength is given in terms of electric and magnetic dipole transition moments as
where |0〉 and |n〉 are ground and excited states of the molecular Hamiltonian, respectively, is the electric dipole operator, and is the magnetic dipole operator (in atomic units).
Invoking the Born-Oppenheimer approximation and neglecting couplings between nuclear degrees of freedom and the electromagnetic field, the specific optical rotation is calculated from purely electronic states for fixed nuclear positions, employing linear response theory48–50 to avoid explicit evaluation of the sum over states in eq. 1. The electronic contribution to the optical rotation thus becomes a function of the nuclear degrees of freedom. Zero-point vibrational effects can be taken into account by evaluating the expectation value in the vibrational ground state of the electronic ground state. The ZPVC thus is given by
where [α]eq is the electronic contribution at the equilibrium geometry. Ruud et al.21 included harmonic and anharmonic effects by a Taylor expansion of [α] to second order about an effective geometry,51, 52 while Mort and Autschbach22–25 used a minor modification of the general formula of Sauer and Packer.53 These studies clearly show that vibrational effects are non-negligible and that anharmonic contributions in some cases are more important than harmonic ones. Nevertheless, aiming at a reasonable compromise between accuracy and computational cost, we here subscribe to the harmonic approximation and thus Taylor expand [α] to second order in the normal coordinates Q about the equilibrium geometry to obtain
where ωi is the frequency of the ith vibrational mode. The second derivative is evaluated numerically according to the finite difference formula
where is the displacement from equilibrium of normal mode i. For an N-atom molecule, 6N – 11 calculations of the optical rotation are thus required to evaluate the ZPVC at each wavelength. For conformationally flexible molecules, the electronic optical rotation and ZPVC is calculated for each low-lying conformer and the total optical rotation is computed from the Boltzmann average
where Xi is the mole fraction of conformer i. The mole fractions are calculated using Gibbs free energies, including zero-point vibrational energies and entropy corrections (see26 for a more detailed discussion of Boltzmann averaging of optical rotations). The Boltzmann average is calculated at 298.15 K in all cases.
Equilibrium geometries, normal coordinates, and harmonic vibrational frequencies are calculated at the B3LYP level with the 6-31G* basis set.54 Subsequently, the ZPVC is calculated at the B3LYP level and [α]eq is calculated at the B3LYP, CC2, and CCSD levels using augmented correlation consistent basis sets.55, 56 To keep the number of atomic orbitals to a minimum, we employ a few nonstandard basis sets. A basis set denoted A/B specifies that basis set A is used on all nonhydrogen atoms and basis set B on all hydrogen atoms. The basis set denoted aug-cc-pVD[T]Z is obtained by using aug-cc-pVDZ on all atoms except those involved in double- or triple-bonds where the aug-cc-pVTZ basis set is used. To ensure origin invariance, gauge including atomic orbitals (GIAOs)57, 58 are used in conjunction with B3LYP, whereas the modified velocity gauge (MVG) formulation59 is used for the CC2 and CCSD models. In addition, despite its origin-dependence, the length gauge (LG) formulation is used for the CC2 and CCSD calculations; the center of mass is chosen as coordinate origin for these calculations. Core orbitals are frozen in the coupled cluster calculations.
The specific optical rotation is calculated at two wavelengths (355 and 633 nm) for five rigid molecules [(1S,4R)-fenchone, (1R,5R)-α-pinene, (1R,5R)-β-pinene, (1R,2S,5R)- cis-pinane, and (R)-2-chloropropionitrile] and two molecules with three low-lying conformers [(S)-3-chloro-1-butene and (S)-limonene], see Figure 1. For the latter, the three lowest-lying conformers were used for calculating the vibrationally corrected optical rotation via Boltzmann averaging. Specific optical rotations for these molecules have been measured in the gas phase at 355 and 633 nm by Vaccaro and coworkers using Cavity Ring-Down Polarimetry.60, 61 (For (1S,4R)-fenchone, the experimental rotation is only available at 355 nm.60)
The B3LYP and CC2 calculations were performed with a development version of the Dalton quantum chemistry program.62 The Cholesky decomposition-based implementation described in63 was used for the CC2 calculations with a decomposition threshold of 10−6 which is sufficient to give virtually exact results.63, 64 All CCSD calculations were performed with a development version of the PSI3 electronic structure package.65
RESULTS AND DISCUSSION
Electronic specific optical rotations at 355 and 633 nm calculated at the B3LYP/6-31G* equilibrium geometries using B3LYP, CC2, and CCSD are reported in Tables 1–3, respectively. We first note that including triple-zeta basis sets on multiple-bonded atoms overall has relatively little impact on the B3LYP rotations. The effect is somewhat larger for CC2, in particular for α-pinene. We also note that the impact is significantly greater at 355 nm than at 633 nm for both B3LYP and CC2, which can be explained by the fact that 355 nm is closer to the resonant spectral region (singularities in the Rosenfeld expression, eq. 1) where the total rotation is more sensitive to the relative location of the excitation energies and corresponding rotatory strengths. This is also reflected in the higher degree of agreement between B3LYP and the coupled cluster models at 633 nm than at 355 nm.
Table 1. Electronic specific optical rotation [°(dm g/cm3)−1] calculated at the B3LYP level of theory using GIAOs
Equilibrium structures optimized at the B3LYP/6-31G* level.
We label the three conformers of (S)-limonene by the dihedral angle connecting carbons 1 and 4 as denoted in Figure 1: conformer #1 has a dihedral angle of −110.3°, conformer #2 +110.4°, and conformer #3 +22.7°. Similarly for 3-chloro-1-butene, using the dihedral angle associated with the carbon backbone, conformer #1 has a dihedral angle of −122.3° conformer #2 −0.6°, and conformer #3 +120.4°. Calculating mole fractions from electronic energies only (i.e., excluding zero-point vibrations and entropy contributions), the following mole fractions are obtained for limonene at 298.15 K: 48.23% conformer #1, 32.41% conformer #2, and 19.37% conformer #3. For 3-chloro-1-butene we find: 80.49% conformer #1, 14.26% conformer #2, and 5.26% conformer #3. Using these mole fractions in eq. 6 and excluding ZPVCs, the following purely electronic optical rotations are obtained for limonene: B3LYP/aug-cc-pVDZ: −292.2° (dm g/cm3)−1 at 355 nm and −83.0° (dm g/cm3)−1 at 633 nm; CC2/aug-cc-pVD[T]Z: −321.3° (dm g/cm3)−1 at 355 nm and −87.3° (dm g/cm3)−1 at 633 nm; CCSD/aug-cc-pVDZ: −240.4° (dm g/cm3)−1 at 355 nm and −63.7° (dm g/cm3)−1 at 633 nm. For 3-chloro-1-butene we obtain: B3LYP/aug-cc-pVTZ: 539.7° (dm g/cm3)−1 at 355 nm and 96.8° (dm g/cm3)−1 at 633 nm; CC2/aug-cc-pVTZ: 293.2° (dm g/cm3)−1 at 355 nm and 55.6° (dm g/cm3)−1 at 633 nm; CCSD/aug-cc-pVTZ/aug-cc-pVDZ: 275.3° (dm g/cm3)−1 at 355 nm and 55.1° (dm g/cm3)−1 at 633 nm.
Harmonic ZPVCs calculated using B3LYP are given in Table 4. Again, we find a slightly more pronounced basis set dependence at 355 nm than at 633 nm, although it is relatively small in all cases. Comparing to the results of Table 1, we see that ZPVCs range from insignificant (e.g., 0.4% for β-pinene at 355 nm) to important (e.g., 21% for cis-pinane at 355 nm), in agreement with the findings of Mort and Autschbach.22 In absolute terms, the vibrational effects are larger at 355 nm than at 633 nm, except for β-pinene. In this case, the B3LYP equilibrium electronic contribution to the optical rotation is nearly halved by the ZPVC at 633 nm.
Table 4. Harmonic zero-point vibrational correction to the specific optical rotation [°(dm g/cm3)−1] calculated at the B3LYP level of theory using GIAOs
Equilibrium structures, vibrational frequencies, and normal coordinates computed at the B3LYP/6-31G* level.
As the coupled cluster models are not gauge invariant,50, 66 it is no surprise that the LG and MVG results differ for both CC2 (Table 2) and CCSD (Table 3). They do, however, agree on sign and order of magnitude and we will only use the origin independent MVG results for calculating vibrationally averaged specific optical rotations. Using the theoretically best (i.e., most flexible) basis set for both the equilibrium electronic rotation and the ZPVC, we obtain the vibrationally averaged specific optical rotations reported in Table 5 along with the experimental results. (In some cases, we have inverted the sign of the experimental result to match the absolute configuration used in the theoretical calculations.) Including zero-point vibrations and entropy via Gibbs free energies, the mole fractions used for Boltzmann averaging are (at 298.15 K): 46.13% conformer #1, 30.64% conformer #2, and 23.23% conformer #3 for limonene, and 84.99% conformer #1, 9.81% conformer #2, and 5.20% conformer #3 for 3-chloro-1-butene. For both limonene and 3-chloro-1-butene, none of the conformers can account for the observed optical rotation by itself, not even the lowest energy conformer of 3-chloro-1-butene which is by far the most populated. This confirms that Boltzmann averaging is mandatory for conformationally flexible molecules.
Table 5. Experimental and computed specific optical rotations [°(dm g/cm3)−1]
The methods B3LYP, CC2, and CCSD refer to the calculation of the electronic contribution with the theoretically best basis set of Tables 1, 2, and 3, respectively. The MVG is used for the coupled cluster methods. Similarly, the zero-point vibrational corrections are calculated at the B3LYP level with the largest basis set of Table 4. Results for limonene and 3-chloro-1-butene have been Boltzmann averaged over the three lowest-lying conformers.
At 355 nm, all three methods predict the correct sign of the rotation for all molecules, whereas only CCSD gives the correct sign in all cases at 633 nm. For β-pinene at 633 nm, the correct sign obtained with CCSD is due to the relatively large positive ZPVC which, however, is not large enough to make the vibrationally averaged rotation positive with B3LYP and CC2. For β-pinene at 355 nm, B3LYP and CC2 overestimate the magnitude by a factor of 3.7 and 2.9, respectively. In contrast, CCSD is 30% off. All three methods overestimate the optical rotation of fenchone at 355 nm by a factor of 2.3 or more, while the largest absolute error is observed for 3-chloro-1-butene at 355 nm where the B3LYP rotation is more than 400° (dm g/cm3)−1 larger than the experimental result. A likely explanation is that the theoretical excitation energies are too small in these cases, forcing the optical rotation dispersion curve upward due to the pole structure of the Rosenfeld expression.
Figure 2 compares the experimental and computed specific optical rotations with (Fig. 2b) and without (Fig. 2a) ZPVCs. Although the addition of B3LYP ZPVCs tends to shift the results in the direction of the experimental values, in particular for small-angle rotations, it evidently is more important to use CCSD for calculating the electronic optical rotation at the equilibrium geometry than to perform vibrational averaging. This can also be seen from the absolute errors defined as |(Computed-Experimental)/Experimental|: for B3LYP, the average absolute error is reduced from 1.15 to 1.11 by including ZPVCs (the standard deviation is reduced from 0.41 to 0.30); for CC2 and CCSD, the average goes from 0.89 to 0.76 (standard deviation from 0.43 to 0.31) and from 0.35 to 0.29 (standard deviation from 0.13 to 0.11), respectively. Purely electronic optical rotations using CCSD without ZPVCs thus compare more favorably to experiment than vibrationally averaged B3LYP and CC2 results. We also note that the performance of the computationally less demanding CC2 model is intermediate between B3LYP and CCSD.
Aiming at a reasonable balance between computational effort and accuracy, we have investigated an approach to the calculation of gas phase specific optical rotation in which different electronic structure methods are employed for computing purely electronic and vibrational contributions. Specifically, equilibrium geometries, normal coordinates, and harmonic vibrational frequencies are computed at the B3LYP level with a relatively small basis set. Subsequently, the ZPVC (i.e., second derivatives of the optical rotation with respect to the normal coordinates) is calculated at the B3LYP level with a diffuse basis set. The purely electronic rotation at the equilibrium geometry is calculated at the coupled cluster (or B3LYP) level with a diffuse basis set. Finally, Boltzmann averaging over low-lying conformers is employed for flexible molecules. This approach has been applied to seven molecules (five rigid molecules and two molecules with three low-lying conformers) and the results have been compared with recent experimental gas phase data. We find that this is a viable approach for calculating vibrationally averaged specific optical rotations, especially for small-angle rotations. The accuracy of the computed rotations improves with the level of electron correlation employed in the order B3LYP, CC2, and CCSD. The B3LYP and CC2 models predict the wrong sign of the optical rotation in one case: β-pinene at 633 nm. The experimentally determined61 rotation is 4.7° (dm g/cm3)−1 and the correct sign is obtained with CCSD only after addition of the ZPVC, confirming that vibrational corrections are particularly important for small-angle optical rotation.20, 30 It should be noted, however, that more diffuse basis sets as well as higher-order electron correlation effects could improve the description of the excited electronic states which, in turn, could have a significant impact on the computed electronic optical rotations. Overall, the results of the present investigation indicate that it may be more important to employ CCSD for the calculation of the optical rotation at the equilibrium geometry than to include vibrational effects. For flexible molecules, conformational averaging is indispensable.