A VCD robust mode analysis of induced chirality: The case of pulegone in chloroform

Authors

  • Valentin Paul Nicu,

    Corresponding author
    1. Theoretical Chemistry, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands
    • Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
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  • Elke Debie,

    1. BioTools, Inc., Jupiter, Florida 33458
    2. Department of Chemistry, Syracuse University, Syracuse, New York 13244
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  • Wouter Herrebout,

    1. Department of Chemistry, University of Antwerp, B-2020, Antwerp, Belgium
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  • Benjamin Van der Veken,

    1. Department of Chemistry, University of Antwerp, B-2020, Antwerp, Belgium
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  • Patrick Bultinck,

    Corresponding author
    1. Department of Inorganic and Physical Chemistry, Ghent University, B-9000, Ghent, Belgium
    • Department of Inorganic and Physical Chemistry, Ghent University, Krijgslaan 281-S3, B-9000, Ghent, Belgium
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  • Evert Jan Baerends

    Corresponding author
    1. Theoretical Chemistry, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands
    2. Department of Chemistry, Center for Superfunctional Materials, Pohang University of Science and Technology, Namgu Pohang 790-784, South-Korea
    • Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
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  • Contribution to the Special Thematic Project “Advances in Chiroptical Methods”

Abstract

Vibrational modes in an achiral molecule may acquire rotational strength by complexation to a chiral molecule, as happens for achiral solvent molecules complexed to a chiral solute. We investigate this transfer of chirality in vibrational circular dichroism for the pulegone molecule in CDCl3 solvent from the point of view of the robustness concept introduced recently. It turns out that the transfer of chirality yields nonrobust modes, which means that, although they are observed in vibrational circular dichroism (VCD) experiments, the sign of these modes cannot be predicted reliably with standard (Density Functional Theory) VCD calculations. This limits the usefulness of the induced chirality phenomenon for obtaining information on the intermolecular interactions that give rise to it. Chirality 21:E287–E297, 2009. © 2010 Wiley-Liss, Inc.

INTRODUCTION

Although most often associated to a chiral carbon, chirality is a more general geometrical property that is not limited to a chiral center. Indeed, there are many molecules that are chiral without having a chiral center. An example in this regard is the transfer of chirality where an achiral molecule—obviously lacking a chiral center—becomes optically active upon interacting (e.g., by intermolecular hydrogen bonding) with a chiral molecule. The phenomenon is most often encountered when considering (chiral) solute– (achiral) solvent interactions, although it is not limited to this particular case. Experimentally, such cases of induced chirality have been observed using both electronic (ECD)1 and vibrational circular dichroism (VCD)2–4 spectroscopy.

As suggested in Ref.1, the presence of signals associated to the achiral moiety in the ECD spectra might provide information about the relative orientation of the interacting chiral and achiral molecules. As VCD spectroscopy deals with vibrational transitions rather than electronic transitions, one can expect that under normal circumstances VCD spectroscopy should provide more detailed structural information on the relative orientation of the chiral and achiral moieties than ECD spectroscopy. However, compared to ECD spectroscopy, much less attention has been paid in VCD spectroscopy to the phenomenon of induced chirality in the solvent. For example, only six papers have been published on this subject in the period between 2007 and 2009, i.e., the combined experimental and computational studies of Xu et al.3, 5, 6 and Debie et al.4, 7 and the theoretical study of Nicu et al.8

The studies of Xu et al.3, 5, 6 and Debie et al.4, 7 very clearly show that (1) the presence of a chiral solute in an achiral solvent, such as for example H2O and CDCl3, may cause the modes of the achiral solvent to become VCD active, and (2) it is possible not only to find the VCD signals of the achiral solvent experimentally, but also to relate these VCD signals through calculations to interactions between the solute and solvent molecules.

The theoretical investigation of Nicu et al.8 has provided a very easy and intuitive explanation of the mechanism responsible for the chirality transfer in VCD. Due to symmetry constraints, the electric and magnetic transition dipole moments (ETDM and MTDM) associated to the fundamental transition of a given normal mode are perpendicular in isolated achiral molecules. As a result the rotational strength, i.e., the inner product between the ETDM and MTDM vectors of a given mode, is zero for all vibrational modes of isolated achiral molecules. As the VCD intensities are proportional to the rotational strengths, it is clear that all modes of achiral molecules will have zero VCD intensities. When an achiral molecule is involved in a molecular complex (MC) however, its ETDMs and MTDMs are perturbed and no longer perpendicular. Consequently, the modes of the achiral molecule can exhibit nonzero VCD signals. However, because in weak complexation of a solvent molecule to the solute, the perturbation is weak, it is expected that the angle ξ between the ETDM and MTDM will deviate only slightly from 90° and as a result the rotational strength will be small.

The findings in the above-mentioned papers, i.e., the possibility to measure and compute chirality transfer effects on VCD spectra, and the identification of the mechanism responsible for the chirality transfer, suggest that VCD spectra can indeed provide information about the complexation between solute and solvent molecules involved in a MC, including information on the relative orientation of the molecules. This would indeed be welcome, because current state-of-the-art calculations still have difficulty describing weak complexation accurately, in particular for large solute molecules. In that case one has to take recourse to Density Functional Theory (DFT) calculations, which describe (with a suitable functional) hydrogen bonding reasonably well, but not dispersion interactions. Moreover, the DFT calculations are not sufficiently accurate to compute reliable Boltzmann populations for the various conformations of the MC.

The usefulness of the induced chirality phenomenon is, however, not beyond doubt. As it is to be expected that the modes of the achiral moiety have angles ξ between the ETDM and MTDM close to 90°, they should be classified as nonrobust modes according to Refs.8, 9. In general nonrobust modes are defined as having angles ξ close to 90°. They can easily change the angle through 90° (hence change sign of the rotational strength) by small computational perturbations (other choice of functional or basis set) or experimental circumstances (e.g., different solvent).

The aim of the present work is to investigate whether induced chirality in VCD spectra yields indeed nonrobust signals. It might be hoped that, even though the perturbation due to the complexation is small, it would always lead to a deviation of the ξ angle in the same direction compared to 90°. In that case, we would have weak but still robust induced VCD bands. Using as an example the VCD spectrum of pulegone measured in CDCl3 solvent, we will investigate the robustness of the C–D stretching mode of the achiral CDCl3 solvent which as shown recently by Debie et al.7 exhibits a nonzero VCD signal in the 1:1 pulegone–CDCl3 MC.

The work is organized as follows: after some experimental, computational, and methodological details, the concept of robustness is introduced and exemplified with a discussion of the robust and nonrobust normal modes in the VCD spectrum of pulegone between 850 and 1700 cm−1. Then the robustness of the C–D stretch mode of the achiral solvent molecule that exhibits induced chirality is investigated in detail. We find that this mode, like other modes with ξ close to 90°, is nonrobust. The sign and magnitude of the rotational strength therefore cannot be reliably reproduced by standard DFT calculations. This limits the usefulness of the induced VCD signals for obtaining information about the structure and strength of the intermolecular interaction giving rise to the complexation and the induced chirality.

EXPERIMENTAL AND COMPUTATIONAL DETAILS

The samples of R-(+)-pulegone (98%) and S-(−)-pulegone (98%) were obtained from Sigma Aldrich. CDCl3 (99.8%) and CS2 (99.9%) were used as solvent, and were obtained from Sigma Aldrich and Riedel-de Haën, respectively. All samples and solvents were used without further purification. The VCD spectra were recorded on a Bruker IFS 66V FTIR spectrometer, coupled to a PMA37 VCD module. The IR absorbance spectra of pulegone dissolved in CDCl3 and CS2 were recorded at a resolution of 4 cm−1; the corresponding VCD spectra were recorded at a resolution of 6 cm−1. Depending on the spectral range to be studied and the solvent used, different experimental parameters were chosen. For the measurements below 1800 cm−1, a demountable KBr cell with a path length of 100 μm was used in combination with a 1850 cm−1 long wave pass filter. The concentration of pulegone for the solutions was set to 0.25 M. The CD stretching region was studied using a demountable CaF2 cell with a path length of 100 μm and a 3000 cm−1 long wave pass filter. The concentration of pulegone used in these studies was 1.5 M. The collection time for the VCD spectra in the CD stretching region was set to 240 min. For all VCD experiments, baseline corrections were introduced by recording the VCD spectra of a racemic mixture, using exactly the same conditions as those used for the samples.

All calculations (geometry optimization, and IR/VCD calculations) are performed using the Amsterdam Density Functional (ADF) program package.10–12 The vibrational rotational strengths are calculated using the implementation of Stephens' equations for VCD13 in the ADF program package.14 Analytical derivative techniques15 are employed for the calculation of the atomic polar tensors (APT) and atomic axial tensors (AAT), and harmonic force field, within the framework of DFT. For the calculation of AATs, London atomic orbitals16 based on Slater-type orbitals, and the common origin gauge are used.

The analysis of ETDM and MTDM, of the normal modes, and of the angles ξ was done using the ToolsVCD program.8

The calculated VCD spectra were simulated by representing the peaks as Lorentzian bands17 with a half-width of 4 cm−1. The harmonic frequencies obtained from the B3LYP/6-311+G(d,p) calculations were scaled by 0.967. No scaling factor was applied to the frequencies obtained from the OLYP/TZP and BP86/TZP calculations.

METHODOLOGY

Normal Modes Analysis

When comparing modes of a certain molecule in different calculations (e.g., different functional, different basis set) a one-to-one mapping between vibrational modes in the different calculations must be established first. We have done this by calculating overlaps between normal modes.

Using the mass-weighted Cartesian displacement vectors, i.e., the eigenvectors of the mass-weighted Hessian, a normalized 3N-dimensional vector (N being the number of atoms of the molecule) is constructed for each normal mode:

equation image(1)

where vmath image(i) is the kth Cartesian component of the mass-weighted displacement vector of atom λ in the normal mode i, and equation image is the 3 N-dimensional vector associated with the mode i.

The overlap of two modes i and j, Ω(i,j), obtained from two different calculations can be expressed as an inner product:

equation image(2)

where equation image labels the 3N-dimensional vector associated to the ith mode of one calculation, while equation image is the 3N-dimensional vector associated to the jth mode of the other calculation.

The value of the scalar product Ω(i,j), gives an indication of how well the nuclear displacement vectors of the modes i and j overlap. Two identical modes yield an overlap of one. Thus, the closer Ω is to one, the more similar are the modes i and j.

Finally, as the set of normal mode vectors of a particular calculation are an orthonormal basis (they are the eigenvectors of the mass-weighted Hessian), the normal modes of one calculation, { equation image}, can be expressed as linear combinations of the normal modes of another calculation, {equation image}:

equation image(3)

The expansion coefficients, Ω(i,j), are the normal mode overlaps defined in eq. (2). They satisfy:

equation image(4)

Thus, an overlap of 0.90 between two modes means that the two modes differ by 19%, i.e., 1.0 − 0.92 = 0.19, whereas an overlap of 0.70 means that the two modes differ by 51%, i.e., 1.0 − 0.72 = 0.51.

The Concept of Robustness in VCD

The VCD intensity of the fundamental transition (|0〉→|1〉) of the ith vibrational mode is given by the rotational strength (R)13:

equation image(5)

where equation image and equation image are the ETDM and MTDM of the fundamental vibrational transition of the ith normal mode. (Note that equation image is purely imaginary.)

The sign of the R(i) is determined by the angle ξ(i) between the vectors equation image and equation image:

equation image(6)

where equation image and equation image are the lengths of the vectors equation image and equation image, respectively. Thus, if ξ < 90° then R(i) > 0, if ξ > 90° then R(i) < 0.

Computational studies of Nicu and Baerends8, 9 have shown that when the angle ξ of a normal mode is close to 90° even a slight perturbation can change it through 90°. This observation led to the introduction of the concept of robustness of the normal modes in a VCD spectrum. Robust modes are characterized by angles ξ that are far from 90° (see below for quantification of “far”). The VCD sign of the robust modes is characteristic and can be accurately predicted by calculation as it is not sensitive to small perturbations. The nonrobust modes, on the other hand, have angles ξ that are close to 90°. Because of this, even the smallest perturbation, e.g., the use of slightly different computational parameters, can affect the VCD sign of the nonrobust modes. As a result, the VCD sign of nonrobust modes can not be accurately predicted from calculation and therefore should not be trusted. (When a mode has ξ angles that are close to 90°, also experimental “perturbations”, e.g., the change from one inert solvent to another, can change the angle ξ across 90°.)

We can thus formulate as criterion for robustness of a normal mode in both experiment and calculation that it should, in a particular situation (given computational settings, given experimental conditions of solvent, temperature, etc.) have a ξ that deviates more than a given threshold from 90°, i.e., ξ < 90° − τ or ξ > 90° + τ. A sensible value for τ has been determined in Ref.9 by investigating the changes induced in the angles ξ upon changing the exchange-correlation functional, i.e., BP86 vs. OLYP, using a test group of 27 molecules. We will summarize the main results of that investigation.

To determine a safe value of τ we need to obtain an estimate of typical changes Δξ of the angles ξ upon changing the functional, e.g., Δξ = ξ (BP86) − ξ (OLYP). For the relevant modes (see below) the differences Δξ for the modes in a molecule turn out to have a Gaussian distribution. The Δξ distributions are centered on the 0° value, indicating that one functional does not have a systematic bias to larger or smaller ξ for all modes with respect to the other functional. The Δξ distributions have, depending on the molecule, standard deviations between 6° and 10°. The maximum Δξ, i.e., Δξmax, was smaller than 30° (slightly more than three standard deviations) for all 27 molecules considered in Ref.9. Because 99.7% of the members of a distribution fall within the range of three standard deviations, it was concluded in Ref.9 that it is a conservative criterium that for a mode to be classified as robust, it should have an angle ξ that differs from 90° by at least 30° (τ is 30°).

A few cautioning remarks need to be made:

  • 1Only BP86–OLYP mode pairs with normal mode overlaps of at least 0.90 and significant VCD intensities were considered for the statistics of Δξ (the relevant modes for the analysis mentioned above). Actually, this covers a large fraction of all modes: 90% of the modes exhibit a normal mode overlap of at least 0.90 with the corresponding mode in the calculation with the other functional. The rest of the modes, i.e., 10%, showed overlaps with their corresponding modes of at least 0.70. As discussed in “Normal Modes Analysis”, the two modes of a pair with overlap 0.7 are very different, i.e., the modes differ by 51%. This is a consequence of mode mixing caused by the change of functional. Very large variation of the ξ angles (thus also of the VCD intensities) can be encountered upon mode mixing, i.e., variations of 60° −70° for ξ are not unusual.
  • 2Not only modes that mix cannot be used for analysis of the VCD spectrum, but also normal modes with small VCD intensities, i.e., small electric and magnetic dipole transition moments (ETDM and MTDM) cannot be used. Large deviations of the angles ξ have been observed for such modes. Although both functionals have predicted small magnitudes for the ETDMs and MTDMs of all these modes, often the relative orientation of these two vectors was very different in the BP86 and OLYP calculations. This suggests that the ETDMs and MTDMs of small magnitude are also very sensitive to small perturbations and therefore cannot be calculated very accurately. We note that an absolute measure, valid for all molecules, to determine what should be considered small in this context cannot be given as the magnitudes of the total ETDM and MTDM of a normal mode depend on the number of atoms involved in the normal mode motion, i.e. on the molecule. However, an unstable sign has been found in Ref.9 if for a given molecule, the ETDM (or MTDM) is <10% of the mean value of the ETDMs (MTDMs) of all the modes of the respective molecule. We take as a conservative threshold a magnitude of 10–20% of the mean value of the ETDM (or MTDM) values of the molecule.

Applying the Concept of Robustness

In practice, checking on the robustness of a mode is an additional test that improves the reliability of the VCD predictions. This should be performed in what can be called the “standard procedure” for determining the AC of a molecule from VCD calculation.

The standard approach is to perform first a single VCD calculation using computational parameters (basis set, numerical integration accuracy, and density functional) that are known to yield both good agreement with experiment and also converged VCD quantities, i.e., nuclear displacement vectors, and atomic axial and polar tensors. Then, using the criteria described above the robustness of the modes in the calculated VCD spectrum should be identified. We advocate to present the robustness of all modes in the plot of the computational VCD spectrum by a dot indicating the value of the angle ξ (90° at the baseline); (see Refs.8, 9 and Fig. 4 of this paper).

As the VCD sign of the robust modes can be accurately computed, one should expect to have a good agreement between calculation and experiment for the modes identified as robust (in the calculated VCD spectrum). If this is the case, one can simply ignore the discrepancies observed between calculation and experiment for the nonrobust modes.

This is a very quick and accurate way of using VCD for determining the absolute configuration of chiral compounds. However, it is important to realize that occasionally one can encounter sign differences also for modes identified according to the above criteria as robust. In case different signs are encountered for one or a few of the robust modes, the strategy to be followed is to examine whether the cause could be mixing of modes due to a perturbation. From the displacement vectors of the calculated mode (e.g., do they involve atoms that may participate in hydrogen bonding with the solvent?) and taking into account the type of solvent used, one may detect if there is the possibility of mixing of modes by solvent effects. In the calculations, one can check on proximity of other modes as a cause of mode mixing. Suspicion of mode mixing can then be confirmed by performing a second calculation where the interaction is explicitly specified (e.g., calculation for the hydrogen-bonded complex) or a computational perturbation (different functional) is applied.

ROBUSTNESS OF THE NORMAL MODES OF PULEGONE

Calculated and Experimental VCD Spectra of Pulegone

Figure 1 shows a comparison between the experimental VCD spectra measured in CDCl3 (continuous line) and CS2 (dotted line) solvents, and calculated VCD spectra obtained from vacuum calculations performed for the pulegone molecule using the BP86, OLYP, and B3LYP functionals.

Figure 1.

Comparison between experimental VCD spectrum (Exp.) and the calculated VCD spectra obtained from OLYP, BP86 and B3LYP vacuum calculations. The experimental VCD spectrum obtained using CDCl3 as a solvent is given as a solid line, while the spectrum recorded for the solution in CS2 is given as a dashed line.

To point out the similarities/differences between the experimental and the various calculated spectra, and to simplify their discussion, we have divided the frequency interval between 1000 and 1800 cm−1 into five regions labeled from A to E (see Fig. 1).

As can be seen in Figure 1, there is a fair agreement between calculations and experiment. The calculated spectra reproduce most of the features that stand out in the experimental spectrum. As a result the OLYP, BP86, and B3LYP VCD spectra in Figure 1 have a similar appearance. However, it should be noted that when compared in detail the calculated spectra are different. Thus, a close examination of the experimental and calculated VCD patterns in the regions A to E reveals the following hierarchies for the three functionals: OLYP ≃ BP86 ≃ B3LYP for region A (none of the calculations is able to reproduce the −/+ patterns in the experimental spectrum); B3LYP ≥ BP86 ≥ OLYP in region B; OLYP ≃ BP86 ≃ B3LYP in region C; OLYP ≃ BP86 ≃ B3LYP in region D; OLYP ≥ BP86 > B3LYP in region E. Clearly, none of the three functionals can be considered as being overall superior.

Regarding the discrepancies between calculations and experiment observed in region A, we note that, as Debie et al. have shown,7 they can be remedied by using continuum solvation models in calculations for simulating the solvent effects, or by performing calculations for the 1:1 pulegone–CDCl3 MC. In Ref.7, Debie et al. have found that when using the B3LYP functional, the best agreement between calculation and experiment is obtained when the 1:1 pulegone–CDCl3 MC is embedded in a continuum solvation model.

These conclusions are true also for the calculations performed here with the OLYP and BP86 exchange-correlation functionals. However, because here we investigate the significance of the changes induced in the VCD spectra by the use of various computational parameters (e.g., the use of solvation models and/or different functionals), we have chosen the vacuum calculations as the reference calculations for the discussion of the normal modes robustness in the next section. Hence, the comparison between vacuum calculations and experiment in Figure 1.

Identifying the Robust and Nonrobust Modes of Pulegone

We illustrate the usefulness of the robust modes concept using the pulegone molecule in CDCl3. We begin by assigning the robustness of the modes in the VCD spectrum of calculation 1 (the OLYP spectrum in Fig. 1). To do this, the values of the ξ angles of all modes in Figure 1 have been plotted as dots on top of the calculated VCD spectrum (see Fig. 2). In Figure 2, the baseline of the calculated VCD spectrum is the 90° line, the Y-coordinate of each dot gives the magnitude of each ξ angle (see the vertical right axis), and the X-coordinate gives the frequency of the mode. For comparison, the experimental spectrum is also shown in Figure 2.

Figure 2.

Comparison between experimental VCD spectrum (Exp.) and the VCD spectrum predicted by calculation 1 (Calc. 1) in Table 1. The dots on top of the calculated spectrum indicate the magnitude of the ξ angles of the modes of the calculated spectrum. The robust modes are indicated by arrows. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

In the calculated spectrum, there are 38 modes with frequencies between 800 and 1800 cm−1. As can be seen in Figure 2, only 13 of these 38 modes have angles ξ that differ from 90° by more than 30°. Furthermore, out of these 13 modes, only nine have significant VCD intensities and therefore can be classified as robust, according to the criteria introduced in “Robustness of the Normal Modes of Pulegone”. In Figure 2, the robust modes have been indicated by arrows, and also labeled with numbers. The modes that were not identified as robust, are nonrobust modes. Only the nonrobust modes that will be discussed have been labeled (with numbers).

Robust Modes of Pulegone

Having identified the robust and nonrobust modes in the VCD spectrum of pulegone obtained from calculation 1, we continue by investigating how the robust modes and (in the next section) the nonrobust modes are affected by the use of various computational parameters. To this end, we have investigated the differences between calculations performed (1) with different functionals (OLYP18, 19 vs. BP8620, 21), (2) with the COSMO solvation models vs. calculations performed for the isolated molecule, (3) with different criteria for geometry optimization, i.e., very tight criterium (10−6 Hartree for the energy and 10−4 Hartree/Ångstrom for the gradients) vs. the ADF default criterium (10−4 Hartree for the energy and 10−3 Hartree/Ångstrom for the gradients), and (4) for the free pulegone molecule vs. the 1:1 pulegone–CDCl3 MC. By combining the computational parameters mentioned above we have performed 16 different calculations for pulegone (labeled from 1 to 16). A list with the computational parameters used in each of the 16 calculations is given in Table 1.

Table 1. Computational parameters used in the 16 calculations performed for pulegone
CalculationsComputational details
1TZP, OLYP, Vacuum, tight geometry, free molecule
2TZP, OLYP, Vacuum, tight geometry, molecular complex
3TZP, OLYP, Vacuum, default geometry, free molecule
4TZP, OLYP, Vacuum, default geometry, molecular complex
5TZP, OLYP, COSMO, tight geometry, free molecule
6TZP, OLYP, COSMO, tight geometry, molecular complex
7TZP, OLYP, COSMO, default geometry, free molecule
8TZP, OLYP, COSMO, default geometry, molecular complex
9TZP, BP86, Vacuum, tight geometry, free molecule
10TZP, BP86, Vacuum, tight geometry, molecular complex
11TZP, BP86, Vacuum, default geometry, free molecule
12TZP, BP86, Vacuum, default geometry, molecular complex
13TZP, BP86, COSMO, tight geometry, free molecule
14TZP, BP86, COSMO, tight geometry, molecular complex
15TZP, BP86, COSMO, default geometry, free molecule
16TZP, BP86, COSMO, default geometry, molecular complex

Finally, for the 1:1 pulegone–CDCl3 MC, we have also investigated how the relative orientation of the pulegone and CDCl3 molecules influence the VCD spectra. The relative orientation of the solvent molecule towards pulegone was described using the dihedral angles τ1 and τ2 (see Fig. 3). A systematic conformational analysis performed using the MMFF force field yielded nine stable conformations (labeled A, B, … I) for the 1:1 pulegone–CDCl3 complex (see Ref.7 for a detailed description).

Figure 3.

The dihedral angles τ1 and τ2 varied during the conformational analysis of the 1:1 pulegone–CDCl3 molecular complex.

The differences between the 16 calculations have been judged with respect to our reference calculation, i.e., the results of the calculations 216 have been compared against the results of calculation 1. Thus, after establishing a one-to-one correspondence between the modes of all calculations (as described in “Methodology”), we have monitored the differences between the frequencies, rotational strengths, and ξ angles of the modes obtained from calculation 1 and their homologues obtained from the calculations 2–16. For the modes 28–65 of calculation 1 (i.e., the modes with frequencies between 850 and 1675 cm−1) all this information is given in the Electronic Supporting Information (ESI).

In Figure 4, the values of the angles ξ predicted by the 16 calculations have been plotted on top of the VCD spectrum of calculation 1. As before, the base line of the VCD spectrum represents the 90° line. The large vertical spreading of the dots in Figure 4 clearly shows that the angles ξ are very sensitive to the computational parameters used in the calculations. Because the various calculations predict slightly different frequencies for the normal modes, it is not immediately evident from the Figure 4, without further information like the mode overlaps we discussed earlier, which dots belong to the “same” mode in the different calculations. Thus, we have encircled the dots associated with one mode; this has been done for the modes that will be discussed in what follows (see Fig. 4).

Figure 4.

Comparison of the values predicted by the calculations 116 for the angles ξ of the modes 28–65. The VCD spectrum of calculation 1 is also shown. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

For the calculations performed with the same functional the differences in the normal modes frequencies caused by the various computational variables used are most of the time smaller than 5 cm−1. When comparing different functionals (OLYP and BP86) the differences in frequencies are larger but still small, i.e., smaller than 15 cm−1 for modes with frequency below 1300 cm−1, and smaller than 25 cm−1 for modes with frequency above 1300 cm−1. There are some exceptional cases (modes 64 and 65), whose frequencies are much more sensitive (variation up to 61.5 cm−1).

The ξ angles associated to robust modes exhibit variations that are always smaller than 30°, i.e., Δξ < 30° (see encircled dots associated with the modes 29, 31, 38, 45, 46, 48, 50, and 56 in Fig. 4). As a result, no changes across 90° have been encountered for these modes (robust modes by definition have ξ angles that differ from 90° by more than 30°). It is therefore clear that all 16 calculations predict the same signs for all robust modes, corroborating the definition of robustness. This is in complete agreement with the conclusions obtained in Ref.9, i.e., robust modes have a characteristic sign that is not affected by small perturbations such as the use of slightly different computational parameters.

We should again9 caution that sometimes modes strongly mix when going from one calculation to another one. A look at the normal mode overlaps (see ESI), reveals that approximately 75% of studied modes, i.e., the modes with frequencies between 800 and 1800 cm−1, are practically identical in all 16 calculations, i.e., the normal mode overlaps are very close to one (>0.95). The rest of the modes (25%) mix when changing computational parameters, i.e., these modes do not have an equivalent in calculation 1 and can be obtained only as linear combinations of the modes of calculation 1. As discussed in “Methodology”, two modes that exhibit an overlap of 0.7 differ by 51%. Thus, as shown previously,9 it is not unusual for modes with overlaps of 0.7 to have ξ angles that differ by 60° or more. This can be seen indeed in Figure 4 when looking at the mode 33 which mixes with mode 32 upon changing computational parameters and exhibits angle variation up to 66.5°.

Finally, we would like to draw attention to the modes 28 and 58 which exhibit very large variation of their angles ξ, i.e., 88.1° and 56.1°, respectively. These modes do not mix upon changing computational parameters. However, they have very small VCD intensities. As discussed in “The concept of Robustness in VCD”, it is not unusual for modes with weak VCD signals to exhibit large variation in ξ.

Nonrobust Modes of Pulegone: the Case of the C[DOUBLE BOND]O Stretch Vibration

In the case of the nonrobust modes, we have two distinct situations: modes that exhibit changes across 90° (e.g., mode 65), and modes that do not (e.g., modes 37, 44, etc.). As in Ref.7 a lot of computational effort was invested to reproduce correctly the sign of the C[DOUBLE BOND]O stretching mode of pulegone, i.e., mode 65 (a nonrobust mode), in the following we will investigate this mode in detail.

Figure 5 shows a comparison of the values predicted by the 16 calculations in Table 1 for the angle ξ of the C[DOUBLE BOND]O stretching mode of pulegone. In Figure 5, the Y-coordinate gives the magnitude of the ξ angles, whereas the X-coordinate indicates the calculation (from 1 to 16). The rotational strengths of the 16 calculations are also shown in Figure 5. As can be seen, all 16 calculations predict values for the angle ξ that are very close to 90°, i.e., ξ differs from 90° by <5° for 11 calculations, and by <9° for the other five calculations. Furthermore, the predicted values for ξ are both larger and smaller than 90°, i.e., in calculations 1, 3, 4, 9, and 11 ξ > 90° (thus, R < 0), whereas in the rest of the calculation ξ < 90° (thus, R > 0). This clearly shows that even the use of slightly different computational parameters, e.g., different convergence criteria for the geometry optimization, can affect the prediction of the sign for the rotational strength of the C[DOUBLE BOND]O stretching mode of pulegone. It should also be mentioned here that all COSMO calculations, i.e., calculations 5, 6, 7, 8, 13, 14, 15, and 16, have predicted the correct sign for the C[DOUBLE BOND]O stretching mode. However, in our opinion this is purely fortuitous. Firstly, the use of solvation models in calculations is often unsuccessful.3, 22 Secondly, as discussed above the changes induced in the ξ angles upon using COSMO in the calculations are too small (compared to the vacuum calculation) to be trusted.

Figure 5.

Comparison of the values predicted by the calculations 116 for the angle ξ of the C[DOUBLE BOND]O stretching mode of pulegone (mode 65 in Figs. 2 and 4). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Further, we investigate how the relative orientation of the pulegone and CDCl3 molecules in the 1:1 pulegone–CDCl3 MC affects the values of the angles ξ of the C[DOUBLE BOND]O stretching mode of pulegone. Figure 6 shows a comparison of the calculated values of the ξ angles for the nine conformations of the MC (A–I). The computational parameters used to obtain the ξ angles in Figure 6 are identical to the ones used in calculation 2 (see Table 1 for details). As can be seen, the situation in Figure 6 is very similar to the one in Figure 5. That is, the ξ angles have values that are both larger and smaller than 90° (89° < ξ < 94°).

Figure 6.

Comparison of the computed values for the angle ξ of the C[DOUBLE BOND]O stretching mode (mode 65 in Figs. 2 and 4) in the 9 conformations (AI) of the 1:1 pulegone–CDCl3 molecular complex. The computational parameters used are identical to the ones used in calculation 2 (see Table 1). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Thus, based on the comparisons done in Figures 5 and 6, and keeping in mind the large spread in the values of angle ξ in Figure 4, it is clear that sign differences between calculated and experimental VCD signals can easily occur for nonrobust modes. This clearly shows that sign discrepancies between experiment and calculation should be of no concern, if they are associated to non- robust modes.

TRANSFER OF CHIRALITY AND ROBUSTNESS

In Ref.7, Debie et al. have measured a weak but nonzero VCD signal for the C–D stretching mode of CDCl3. This is a typical case of chirality transfer.1–4 In this section, we investigate whether this mode is robust or nonrobust.

We start our analysis of the C–D stretching mode by comparing its frequency, rotational strength, and angle ξ in the free CDCl3 to those in the MC (see Table 2). As can be seen, in the free CDCl3 (FM) the ETDM and MTDM associated to the C–D stretching mode are perpendicular, i.e., ξ = 90.00°, whereas in the MC the angle ξ is 91.87°. As the ξ differs from 90° by <2°, it is clear that the C–D stretching mode (in the MC) would normally be classified as a nonrobust mode. To substantiate this classification, we investigate how the angle ξ associated to the C–D stretching mode of CDCl3 is affected by (1) the use of various computational parameters, and (2) the relative orientation of the pulegone and CDCl3 molecules. Figure 7 compares the calculated ξ angles of the C–D stretching mode obtained from the 8 calculations (in Table 1) performed for the MC, i.e., calculations 2, 4, 6, 8, 10, 12, 14, and 16. Figure 8 shows a comparison of the calculated ξ angles of the C–D stretching mode for the 9 conformations of the MC (A–I). (Figures 7 and 8 show also the calculated rotational strengths for the C–D stretching mode.)

Figure 7.

Comparison of the values predicted by the MC calculations 2, 4, 6, 8, 10, 12, 14, and 16 for the angle ξ of the C–D stretching mode of CDCl3 (that exhibited transfer of chirality) of the 1:1 pulegone–CDCl3 molecular complex. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 8.

Comparison of the computed values for the angle ξ of the C–D stretching mode of CDCl3 (that exhibited transfer of chirality) in the 9 conformations (AI) of the 1:1 pulegone–CDCl3 molecular complex. The computational parameters used are identical to the ones used in calculation 2 (see Table 1). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Table 2. Comparison of the frequencies, rotational strengths (R), and angles ξ associated to the C–D stretching mode in the 1:1 pulegone–CDCl3 molecular complex (MC) and in the free CDCl3
C–D stretchingFrequencyRξ
MC2308.15−1.1891.87
Free CDCl32306.980.0090.00

As can be seen, all calculations predict positive and negative rotational strengths of very small magnitude, and ξ angles that are very close to 90°, i.e., 88° < ξ < 93° in Figure 7, and 88° < ξ < 95° in Figure 8. The situation is completely analogous to the one studied in the previous section for the C[DOUBLE BOND]O stretching mode of pulegone. This clearly shows that the angle ξ close to 90° for the induced chiral mode behaves precisely as in other cases: it can change through 90° and therefore change sign of the rotational strength by a small perturbation. Induced chirality does not lead to robust modes.

It is interesting to point out that upon complexation the magnitudes of both the ETDM and the MTDM of the C–D stretching mode have increased with a factor of approximately 60. This can be seen in Table 3 where the magnitudes and the Cartesian components of both transition moments predicted by calculation 2, i.e., a MC calculation, are compared to those computed for the free CDCl3 using identical computational parameters. In fact, all 17 calculations performed here for the MC i.e., 2, 4, 6, 8, 10, 12, 14, and 16, and A–I, have predicted an enhancement of the magnitudes of the ETDM and MTDM.

Table 3. Comparison of the Cartesian components of the total electric and magnetic transition dipole moments (ETDM and MTDM) of the C–D stretching mode in the free CDCl3 (FM) and in the pulegone –CDCl3 molecular complex (MC)
TDMLengthXYZ
ETDMs (10−22 esu cm)
 ETDM in FM1.07−1.050.190.00
 ETDM in MC59.7759.48−4.63−3.53
MTDMs (10−24 esu cm)
 MTDM in FM1.23−0.08−0.43−1.15
 MTDM in MC60.342.8418.0957.49

The enhancement of the ETDM results in a significant increase of the IR intensity of the C–D stretching mode. This is observed in both calculation (see Table 3) and experiment (see Figure S5 in the ESI of Ref.4).

The VCD intensity of the C–D stretching mode on the other hand, was found to be very weak in experiment and also in most of the calculations (see Figs. 7 and 8). This apparent contradiction can be explained easily. Firstly, all calculations have predicted ξ angles that are very close to 90°, i.e. 88° < ξ < 95° (see Figs. 7 and 8). Thus, the large magnitudes of the two transition dipole moments are counteracted by the very small values taken by cos ξ. Secondly, as shown in Figure 8, the C–D stretching mode has different VCD signs in different conformations of the 1:1 CDCl3–pulegone complex. Thus, in experiment one can expect that the VCD signal will be even more reduced due to cancellation between various conformations.

We can conclude that there is good qualitative agreement between calculation and experiment for the frequency of the C–D stretching mode and for the enhancement of the ETDM (visible in the IR experiment). However, it should be clear that the VCD sign of the C–D stretching mode studied here cannot be computed accurately. Not only is this mode nonrobust and therefore its sign cannot be computed accurately but also the Boltzmann weights of the various conformations cannot be computed accurately. That is because the relative energies of the various conformations of the 1:1 pulegone–CDCl3 complex are within 1 kcal/mol which is below the accuracy of DFT.

The enhancement of the ETDMs and MTDMs (thus of the IR and VCD intensities) of the stretching modes is a very general phenomenon that is often encountered in complexation phenomena involving intra- and inter-molecular hydrogen-bonding interactions. It is a consequence of the electronic charge that flows into the stretched bond affecting both the magnitude of the electronic components of the ETDM and MTDM and also the frequency of the mode. A detailed theoretical description of the mechanism responsible for the enhancement on the IR and VCD intensities can be found in references8, 23; experimentally the phenomenon is well documented in Ref.24, 25. In the case of the 1:1 pulegone−CDCl3 MC, the enhancement of the magnitude of the ETDM and MTDM of the C–D stretching mode is caused by the donor–acceptor interaction between the σ* molecular orbital (MO) localized on the C–D bond as the acceptor, and the occupied MO 42 of pulegone as the donor (see Fig. 9). We note that this phenomenon is very often encountered for O[BOND]H and N[BOND]H bonds and that it is somewhat remarkable that it occurs also for the C–D stretching mode since the σ* MO of the C–D bond is much higher in energy (thus less likely to mix with occupied MOs) than the σ* MOs of the O[BOND]H and N[BOND]H bonds. The fact that the frequency of the C–D mode shifts with <2 cm−1 in the MC compared to the FM supports this last affirmation. (Upon complexation, calculations predict shifts up to a few hundred cm−1 for the frequencies of the O[BOND]H and N[BOND]H stretching modes8, 23) It is probable that the abundance of CDCl3, as it is the solvent, leads in this case to an observable effect. The CDCl3 do not have to compete with any other complexing agent, indeed the pulegone molecules cannot avoid association with CDCl3 molecules.

Figure 9.

Representation of the MOs that play an important role in the enhancement of the ETDM and MTDM of the C–D stretching mode, i.e., the σ* MO of CDCl3 and the occupied MO 42 of pulegone. The last occupied MO of the 1:1 pulegone –CDCl3 complex is a linear combination of the MO 42 of pulegone (99%), and the virtual MO 11 A1 of CDCl3 (<1%). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

CONCLUSIONS

In this article, we have investigated the robustness of the C–D stretch mode of an achiral molecule (CDCl3) that exhibits induced chirality upon complexation to the chiral pulegone molecule. This case of chirality transfer was recently studied by Debie et al.7 In the first part of this work we have defined the robustness concept and illustrated on the example of the pulegone molecule its usefulness for the interpretation of the differences between calculated and experimental VCD spectra. To highlight the difference between robust and nonrobust modes, we have performed 25 different calculations for pulegone, i.e., the 16 calculations in Table 1 (performed for the free pulegone and for the 1:1 pulegone−CDCl3 MC using eight different computational parameters), plus the nine calculations for the different conformers (labeled A–I) of the MC. The predictions of these 25 calculations have been monitored with respect to the experimental VCD spectrum of pulegone measured in CDCl3. The results of this detailed analysis have clearly confirmed the general usefulness of the concept of robustness.

Firstly, the analysis of the robust modes of pulegone (with frequencies between 800 and 1700 cm−1) has shown that indeed the sign of these modes is not affected by the use of different computational parameters (e.g., functional, solvation model, and different convergence criteria for the geometry optimization). We note that the signs of all signals associated to robust modes in the calculated VCD spectra of pulegone are in agreement with the experiment.

Secondly, the analysis of the C[DOUBLE BOND]O stretching mode of pulegone (a nonrobust mode) has clearly shown that the VCD sign of a nonrobust mode can be affected even by the use of slightly different computational parameters. That is, the changes of ξ across 90° can be observed even for normal modes (obtained from calculations performed with slightly different computational parameters) that have overlaps of 0.9 or higher. It is therefore clear that sign discrepancies between experiment and calculations for modes with ξ angles that are close to 90°—which are nonrobust by definition—bear little relevance as the VCD sign of these modes can not be computed accurately (see Figs. 5 and 6).

As already mentioned, Debie et al. have invested a lot of computational effort to reproduce correctly the signs of these VCD signals. However, although agreement in sign between theory and experiment was eventually obtained, the fact remains that the C[DOUBLE BOND]O stretching mode is a nonrobust mode and the agreement may be (entirely) fortuitous. As shown here, from the moment a single sufficiently good quality calculation, i.e., a calculation performed with a trusted functional and a good basis set, reveals an angle too close to 90°, the reliability of the predicted sign of the VCD band is heavily jeopardized.

In the second part of this work, the C–D stretching mode of the CDCl3 molecule in the 1:1 pulegone−CDCl3 MC (a typical case of chirality transfer) has been investigated. The complexation changes the angle between ETDM and MTDM of this mode, which in the free achiral molecule is 90°. As the complexation perturbation is weak, the angle will not differ much from 90°. This does not rule out that in the transfer of chirality phenomenon the deviation would consistently be in the same direction, and therefore the sign of the VCD signal still be robust. Our study revealed, however, that this particular case of an angle ξ close to 90° is no exception to the rule: the mode is nonrobust. As shown in Figure 8, the VCD sign of this mode is conformation dependent. Moreover, the use of slightly different computational parameters yields different signs for the rotational strength of a given conformation (see Fig. 7). In view of previous experience, we would also expect that if the MC would be in a different solvent (not the neat CDCl3 of the actual experiment) the sign of the VCD signal could also change. It is therefore clear that in cases like the one studied here, i.e., a MC formed via a hydrogen bond between a chiral solute molecule and an achiral solvent molecule, one cannot use the agreement between the calculated and experimental VCD signals of the modes of the achiral moiety to extract information about the molecular interaction, e.g., the relative orientation of the two molecules involved in the MC. This limits the usefulness of the induced chirality phenomenon for the interpretation of the intermolecular interactions that give rise to it. In future works, whenever induced chirality is observed experimentally, first the robustness of the associated modes should be checked carefully. If such a mode is not robust, which will usually be the case if the transfer of chirality is by a relatively weak intermolecular interaction, we feel that no conclusion on, e.g., the structure of the solvent solute complex can be drawn from an agreement in sign between theory and experiment.

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