Conformational Effects on Specific Rotation - American Chemical

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Conformational Effects on Specific Rotation: A Theoretical Study Based on the Sk̃ Method Marco Caricato* Department of Chemistry, University of Kansas, 1251 Wescoe Hall Drive, Lawrence, Kansas 66045, United States ABSTRACT: In this work, we study the difference in specific rotation of the stable conformers of two test chiral molecules: (S)-(+)-2-carene and (R)-3-methylcyclopentanone. We perform the analysis of the specific rotation in terms of rotational strength in configuration space, Sk̃ , which provides information about the contribution of occupied-virtual molecular orbital pairs to this property. We show that, although a considerable number of excited configurations contribute to the total value of the specific rotation, only a limited number of configurations are necessary to explain the different sign and magnitude of the rotation between different conformers. The results in this work thus offer a promising picture for our ability to better understand and possibly predict the value of specific rotation of chiral molecules.

1. INTRODUCTION The optical activity of chiral molecules is one of the primary tools to discern different enantiomers, which otherwise present the same physicochemical characteristics, e.g., same energy, UV/vis spectra, NMR spectra, melting point, and so forth. This separation is essential since Nature has selected only one specific enantiomer for the amino acids (L-) and sugars (D-) that constitute the building blocks of any life form. This phenomenon, known as “homochirality”, is still unexplained,1,2 but has profound consequences in disciplines like biochemistry and drug design since, for instance, only one enantiomer of a new synthesized drug will have a constructive interaction with our body while the other enantiomer will be inactive or even damaging. After the discovery about two centuries ago of the strong optical activity of chiral compounds,3−5 a number of chiroptical spectroscopies have been developed to help characterize these molecules, e.g., optical rotatory dispersion, circular dichroism in electronic and vibrational form, and so forth.1 These techniques are fast and noninvasive; thus, they have become routine in biochemical research. Experiments, however, do not provide a direct structure/property relationship. Theoretical simulations, on the other hand, are able to provide this final link that allows a complete molecular characterization. Thanks to the advent of response theory and its efficient implementation in the framework of density functional theory (DFT), simulations of the spectra measured with the techniques mentioned above are also routine.6−23 A complete recollection of the efforts of the many researchers involved in this field is beyond the scope of this work. We limit ourselves to mention that many shortcomings in the calculations have been addressed. For instance, origin dependence is removed with gauge invariant atomic orbitals (GIAOs)24−26 in variational methods or through the modified velocity gauge approach of Pedersen et al. for coupled cluster (CC) methods.27 Still, poor accuracy of © 2015 American Chemical Society

approximate functionals can lead to incorrect assignment of the sign of the specific rotation for species with small rotation angles.28,29 Autschbach and co-workers reported benchmarking of numerous functionals against experiment.30 The same investigators also introduced an approach to correct some deficiencies in certain functionals due to overdelocalization of the electron density.31 Crawford also reported benchmarking data computed at the CC level,32,33 and investigated the role of basis set effects.34 Vibrational effects have also been shown to provide considerable contributions, sometimes close in magnitude to the electronic contribution, and they have been addressed including temperature dependence.35,36 An outstanding problem is that of solvation, which has been shown to be able to even change the sign of the specific rotation.37 Thanks to the effort of Vaccaro and co-workers38,39 in developing the first efficient polarimeter to perform measurement of specific rotation in gas phase, Lahiri et al.40 recently reported the dramatic solvation effect on the specific rotation of (1R,4R)-norbornenone. For this relatively simple organic molecule, the difference in specific rotation between a low polarity solvent like cyclohexane and a polar solvent like acetonitrile is smaller than the difference between gas phase and solution. Classical solvation models are not able to explain such behavior, even if in some cases these models are able to provide the correct sign of the specific rotation contrary to gas phase calculations.41 Classical models of solvation have also been used to suggest that a chiral solvation shell surrounding a chiral solute can provide a significant contribution to the total value of the measured specific rotation.42 However, more work is needed for a satisfactory description of solvation. Overall, even with such unresolved issues, experiment and theory (mostly) Received: May 28, 2015 Revised: July 10, 2015 Published: July 13, 2015 8303

DOI: 10.1021/acs.jpca.5b05103 J. Phys. Chem. A 2015, 119, 8303−8310

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The Journal of Physical Chemistry A

2. THEORY AND COMPUTATIONAL DETAILS A detailed description of the Sk̃ method can be found in ref 51. Here we only report a short review. The Sk̃ method can be viewed as a SOS in configuration space in the sense that the specific rotation can be written as

successfully complement each other and have become inseparable in research that involves chiral species. Despite the remarkable progress in experimental and theoretical techniques, a deep understanding of the origin of the specific rotation is still missing. An a priori prediction of the sign of the rotation based on the 3D structure of a compound has been elusive. Empirical methods like the octant rule43 have only met limited success and have been largely abandoned. More recently, a variety of approaches based on quantum mechanical calculations have been suggested to dissect the specific rotation in terms of fundamental contributions. To name a few, Kondru et al.44 proposed a Mulliken-type decomposition of the specific rotation to provide contributions in terms of functional groups in their relative orientation. Wiberg and co-workers have extensively used a sum-over-states (SOS) approach45−47 where the Kramer-Kronig relations48 between specific rotation and circular dichroism are used to decompose the specific rotation in terms of electronic excited state contributions. The Autschbach group proposed a molecular orbital (MO) decomposition of the linear response tensor to study the MOs contribution to the tensor individual diagonal components.49 Murphy and Kahr50 extended the analysis of optical activity to include nonchiral molecules resorting to a simple and effective Hückel theory decomposition. We have also proposed an approach that decomposes the specific rotation in terms of MO contributions.51 Our approach is similar to that of Autschbach,49 but focuses the attention on the trace of the linear response tensor rather than on its individual diagonal elements. All of these approaches are complementary as they are trying to tackle the same problem from different points of view. In this work, we apply our MO decomposition of the specific rotation, called S̃k method, to study how conformational changes in the structure of chiral molecules affect specific rotation. In fact, for molecules that present several stable conformers that can be easily accessed at room temperature, the observed specific rotation is the Boltzmann weighted sum of the specific rotation of each conformer. Each conformer, however, may present a widely different value of specific rotation including a change in sign. We want to gain some insights into the origin of such differences. We focus on two relatively simple systems, (S)-(+)-2-carene and (R)-3-methylcyclopentanone, recently studied experimentally and computationally by Lahiri, Wiberg, and Vaccaro.52,53 For simplicity, in the following we will refer to these systems as 2-carene and 3MCP, respectively. Both systems present a flexible ring and one functional group: a 6-membered ring and a CC group in 2-carene, and a 5-membered ring and a CO group in 3MCP. Both molecules have two conformers close in energy deriving from different puckering of the ring, which present different sign of the specific rotation, and in the case of 2-carene, also different magnitude. It has been suggested that the difference in sign is related to the different helicity created by the puckered rings.53 However, here we are interested in determining explicitly the origin of the sign as well as magnitude change. This work is organized as follows. Section 2 summarizes the main points of the S̃k method, and describes our computational protocol. Sections 3 and 4 present the results of our calculations, and section 5 reports final discussion and concluding remarks.

Nex

[α]ω = C ∑ Sk̃

(1)

k=1

where Nex is the number of excited configurations, and C is a proportionality factor:16 C=−

6 2 2 (72 × 10 )ℏ NAω 3 c 2me2M

(2)

In eqs 1−2, S̃k and ω are given in atomic units, NA is Avogadro’s number, c is the speed of light (m/s), me is the electron rest mass (kg), and M is the molecular mass (amu). The initial factor of 2/3 is due to the fact that we will consider only the α-spin part of S̃k since our molecules are closed shell (numerator), and we need to consider the orientation averaging of the molecules in space (denominator). With choice of units, [α]ω is expressed in deg dm−1 (g/mL)−1. Sk̃ is defined as a rotational strength in configuration space: ⎫ ⎧ 3 ̃ Sk = Im⎨∑ ⟨0|μi |ϕk ⟩⟨ϕk |X m+i|0⟩⎬ ⎩ i=1 ⎭ ⎪

⎪

⎪

⎪

(3)

where k refers to a singly excited determinant, ϕk ≡ ϕai , and i indicates an occupied MO, and a a virtual MO. μ is the electric dipole operator, and X+mi is the perturbed density due to the magnetic field perturbation evaluated through a linear response (LR) calculation:16 ⟨ϕk |X m±i|0⟩ =

Nex

∑ ⟨ϕk|(H − E0 ∓ ω)|ϕl⟩−1⟨ϕl|mi|0⟩ l=1

(4)

where m is the magnetic dipole operator, H is the Hamiltonian operator, and E0 the ground state energy. The sum in eq 3 runs over the Cartesian components of the electric and magnetic terms so that Sk̃ can be seen as the inner product between two vectors. In other words, each S̃k gives the contribution of the configuration k to the trace of the Rosenfeld G′ tensor. The sum of all possible values of Sk̃ gives the trace of the LR tensor, which is proportional to the specific rotation as shown in eq 1. Therefore, the Sk̃ values can be used to analyze the specific rotation of a molecule in terms of singly excited configurations, isolating the major MO contributions that can be visualized as electric and magnetic vectors. The magnitude and sign of each S̃k will depend on the length of these vectors and their relative orientation. The Sk̃ expression in eq 3 uses the 2n+1 rule of perturbation theory,25,26 and is applicable to the case of nonresonant, isotropic specific rotation. The calculations presented here are performed at B3LYP/ aug-cc-pVDZ level of theory54−56 with a development version of the GAUSSIAN program57 using the length gauge and GIAOs.24 The geometries are taken from refs 52 and 53. We stress that it is not our goal to investigate the accuracy of this level of theory, nor to provide a comparison with experiment, both of which have been done in the works of Lahiri et al. Here we consider a level of theory that is accurate enough (and computationally efficient) for a qualitative understanding of the contributions to the specific rotation of these compounds. For 8304


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The Journal of Physical Chemistry A the same reason, we used default values for integrals and grid thresholds.

3. (S)-(+)-2-CARENE The structure of the two conformers of 2-carene without the H atoms is shown in Figure 1: (a) for the A and (b) for the B

Figure 1. Structure of the two conformers of 2-carene. The H centers are not displayed.

conformer, respectively. In conformer A, the C5 is tilted below the C1C2C3C10 plane (the C1C2C3C5 dihedral angle is +10°), while it is above the plane in conformer B (the C1C2C3C5 dihedral angle is −18°). At the level of theory used in this work, conformer A is about 8 cm−1 more stable than conformer B. More importantly, the specific rotation is [α]A355 = −243 deg dm−1 (g/mL)−1 and [α]B355 = 1052 deg dm−1 (g/mL)−1 for the two conformers, respectively. These values are in reasonable agreement with the more accurate data computed with the aug-cc-pVTZ basis set in ref 52: −248 deg dm−1 (g/mL)−1 and 1060 deg dm−1 (g/mL)−1, respectively. The configuration contributions to the specific rotation of the conformers can now be decomposed in Sk̃ values, which are shown in Figure 2(a) for the A and (b) for the B conformer, respectively. The S̃k values are reported according to the occupied-virtual orbital pairs that constitute the excited configuration k. The orbitals are sorted in energy order (a.u.): lower to higher energy from right to left for the occupied orbitals, and from left to right for the virtual orbitals. Although Figure 2 carries a lot of information, some trends are evident: most of the lower energy occupied orbitals, and most of the higher energy virtual orbitals provide negligible contribution to the specific rotation. Among the higher energy occupied orbitals, only a few seem to provide large enough S̃k values to be responsible for the change of sign in the specific rotation. This is shown in Figure 3, which reports the sum of the contributions of all virtual orbitals for each occupied orbital Sĩ =

∑ Sĩ a a

Figure 2. S̃k values (a.u.) for the two conformers of 2-carene. The orbitals are sorted in energy order (increasing from bottom to top for the occupied MOs, and from left to right for the virtual MOs).

(5)

Figure 3. S̃i values, eq 5, for the two conformers of 2-carene. Only the occupied orbitals with non-negligible contribution to the specific rotation are shown. The plot also reports the change in S̃i between the two conformers. S indicates the sum of all contributions.

where each excited configuration k is represented by the corresponding occupied-virtual orbital pair i-a. The Sĩ values for the two conformers and their difference are shown in Figure 3 for a selection of occupied orbitals with nonnegligible contribution. Taking the difference is justified because the orbitals vary little between the two conformers. Although several Sĩ values contribute significantly to the final sum, Figure 3 clearly shows that it is the drastic change in the HOMO contribution (i = 38) that is mainly responsible for the different sign of specific rotation between the two conformers. Figure 4 reports the S̃k values for the HOMO and the lower

energy virtual orbitals. As before, the major cause of sign change passing from one conformer to another can be attributed to a single virtual orbital, the LUMO+2 (a = 3). Certainly there are effects from other orbitals that mitigate the HOMO−LUMO+2 effect, like the LUMO+4 (a = 5). 8305


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Figure 6. Occupied and virtual orbitals for S̃338 for the two conformers of 2-carene.

Table 1. Magnitude in a.u. for the Electric (μ → Ve) and magnetic (Xm → Vm) vector components of S̃338 for the two conformers of 2-carenea

Figure 4. HOMO S̃k values for the two conformers of 2-carene and their difference. Only the virtual orbitals with non-negligible contribution to the specific rotation are shown.

̃ may be considered the main player, and we are However, S338 going to focus the rest of our analysis on this term. The magnetic and electric components of S̃338 for the two conformers are reported in Figure 5 where the molecules are

A B a

Ve

Vm

θ

0.96 0.79

2.0 2.3

84 108

θ is the angle between the two vectors.

conformer A, the angle between the vectors is 90°. Since the Sk̃ values depend on the scalar product between the two vectors, which in turn depends on the cosine of the angle between them, this difference explains the sign change. The difference in magnitude of the specific rotation of the two conformers can also be correlated with the value of the angle θ. The angle in conformer A is closer to 90° than that in conformer B, which results in the much smaller magnitude of S̃338 for A than for B. The angle between the vectors can be correlated with the degree of planarity of the C1C2C3C10 moiety. The C1 C2C3C10 dihedral is −179.5° for A, and −177.1° for B. The better planarity of the double bond in A induces a better planarity and orthogonality of the electric and magnetic vectors, respectively, with respect to the CC group: The VeC1 C2C3 dihedral is −4.0° and the VmC1C2C3 dihedral is 90.8°. In B, the VeC1C2C3 dihedral is −14.4° and the VmC1C2C3 dihedral is 104.4°. Incidentally, for (S)(+)-3-carene, which was also studied in ref 52 and where the double bond is on C3C4, the S̃338 value (same π → π* configuration) is half of that of conformer A of 2-carene. The reason is that the ring structure of 3-carene is basically planar, and the angle between the electric and magnetic components is 85°, which is consistent with the results obtained for 2-carene A. We now quantitatively discuss the cumulative contributions of all S̃k values to the total specific rotation. At this level of theory, there are 38 alpha electrons and 336 unoccupied orbitals, for a total of 12 768 excited configurations. If we count the configurations that have S̃k values that are within a factor of 10−3 from the total sum, we obtain 5483 contributions for conformer A and 3518 for conformer B. The value of specific rotation computed only with these configurations yields a 3.1% error for A and 2.9% error for B compared to the full value of [α]ω. In other words, only 43% of configurations for A and 28% for B provide about 97% of the entire value of the specific rotation.

Figure 5. Three views of the electric (red) and magnetic (blue) components of S̃338 for the two conformers of 2-carene.

oriented such that the origin is in C1 and the magnetic (blue) vectors are aligned with the z axis. The figures were produced with the VMD software.58 This excited configuration is mainly of π → π* character as shown in Figure 6, where the orbitals involved look pretty much the same except for an inconsequential change of phase. A π → π* transition induces a slight charge transfer along the direction of the double bond. The directionality would be perfectly aligned for a symmetric alkene like ethylene. The magnetic component is, on the other hand, on a direction almost orthogonal to the C1C2C3 C10 plane since the transition conserves this as a nodal plane. From a visual inspection of the orbitals and of the vectors in Figure 5, it would seem that there should be little difference ̃ of the two conformers. between the S338 However, the data in Table 1 for the vectors’ magnitude and their relative angle shows the reason for the difference: for 8306


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4. 3MCP The two conformers of 3MCP can be distinguished by the orientation of the methyl group with respect to the ring as axial (A) and equatorial (E). The two structures53 are shown in Figure 7: (a) for the axial and (b) for the equatorial conformer,

Figure 7. Structure of the two conformers of 3MCP.

respectively. The specific rotation is similar in magnitude but opposite in sign for the two conformers: [α]A355 = −2212 deg dm−1 (g/mL)−1 and [α]E355 = 2107 deg dm−1 (g/mL)−1, respectively. These values are in reasonable agreement with those computed with the aug-cc-pVTZ basis set:53 [α]A355 = −2155 deg dm−1 (g/mL)−1 and [α]E355 = 2052 deg dm−1 (g/ mL)−1, respectively. At this level of theory, E is more stable by about 395 cm−1. However, as in the previous case, here we are only interested in the difference in specific rotation. The collection of all S̃k values for this case is shown in Figure 8(a) for the axial and (b) for the equatorial conformer, respectively. Also in this case the number of significant S̃k values is much smaller than the total. In order to identify the main players, a plot of S̃i (see eq 5) is shown in Figure 9, and the sign change is driven by the HOMO contribution (i = 27). The Sk̃ values that involve this orbital are shown in Figure 10. The axial conformer is dominated by the HOMO−LUMO configuration (a = 1). The equatorial conformer shows two competing effects: a negative one for the HOMO−LUMO configuration (of opposite sign with respect to A), and a positive one for the HOMO−LUMO+1 configuration (a = 2). The latter has negligible contribution in A. A pictorial representation of the magnetic and electric ̃ and S227 ̃ is shown in Figure 11. Both components of S127 molecules are oriented with the origin on the carbonyl C and the CO aligned with the z axis. Since the length of the vectors would make the visualization difficult, we scaled them: ̃ ; Vm: ×2 and Ve: ×10 for S227 ̃ . The Vm: ×0.5 and Ve: ×20 for S127 orbitals involved in the excited configurations are shown in Figure 12. The actual length of the vectors together with the angle between them and the corresponding Sk̃ values are reported in Table 2. Starting from S̃127 (HOMO−LUMO), the magnetic vector Vm is large in magnitude and oriented along the carbonyl group, which is typical for carbonyl compounds and corresponds to a n/σ → π* transition. This is confirmed by inspection of the HOMO and LUMO in Figure 12, and it is the same for both conformers. The electric vectors, on the other hand, are much smaller in magnitude, and point in opposite directions for the two conformers. Ve and Vm are almost perfectly aligned in both molecules, about 20° away from collinearity, therefore maximizing the inner product. The large magnitude of Vm compensates for the small magnitude of Ve to provide a significant value of S̃k. The opposite direction of Ve for A and E is mainly due to the orientation of the methyl

Figure 8. Sk̃ values (a.u.) for the two conformers of 3MCP. The orbitals are sorted in energy order (increasing from bottom to top for the occupied MOs, and from left to right for the virtual MOs).

Figure 9. S̃i values (eq 5) for the two conformers of 3MCP. Only the occupied orbitals with non-negligible contributions to the specific rotation are shown.

group. In a n/σ → π* transition in a carbonyl, there is partial charge transfer from O to C, which would result in a red arrow pointing toward the oxygen (in the convention of the Gaussian 8307


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Table 2. Magnitude in a.u. for Sk̃ , Its Electric (μ → Ve) and Magnetic (Xm → Vm) Vector Components, And Their Relative angle (θ in degrees) for the HOMO−LUMO (H−L) and the HOMO−LUMO+1 (H−L+1) Excited Configurations of the Two Conformers of 3MCP A E

H−L H−L+1 H−L H−L+1

S ̃k

Ve

Vm

θ

0.58 0.04 −0.96 0.43

0.04 0.19 0.07 0.29

14.23 1.47 13.65 2.17

20 81 162 46

software, the dipole arrow points from the negative to the positive charge). This is what happens in the axial conformer since the methyl group is out-of-plane and does not contribute to the ring electronic cloud. In E, however, the methyl is almost in-plane and it contributes to the transition, moving electronic charge toward the carbonyl group. This offsets the O → C charge transfer and Ve now points in the opposite direction. In fact, this contribution is rather large and the magnitude of Ve in E is almost twice that of A so that S̃127 is also almost twice as large for E than for A. In the second excited conformation, HOMO−LUMO+1, the virtual orbital still has partial π* character, so that the magnitude of Vm is still substantial, although an order of magnitude smaller than for the HOMO− LUMO case. The direction of Vm is still mostly aligned with the carbonyl group, but opposite due to a sign change in the LUMO+1 orbital lobes, as shown in Figure 12. The Ve vectors are 1 order of magnitude larger than in the HOMO−LUMO case, and indicate a partial charge transfer away from the ring. In A, the charge transfer is toward one side of the ring, which therefore brings Ve and Vm in an almost orthogonal orientation that produces a small S̃227 value. On the other hand, in E the position of the methyl group distorts the charge transfer toward the methyl itself, providing an acute angle between Ve and Vm ̃ . The latter partially and a significant and positive value of S227 compensates the value S̃127 in this conformer, yielding an overall similar contribution to the total specific rotation for the two conformers. For a quantitative analysis of the contribution of the configurations to the specific rotation, consider that, at this level of theory, this molecule has 27 alpha electrons and 224 unoccupied alpha orbitals, for a total of 6048 excited configurations. Counting the configurations with values of Sk̃ within a factor of 10−3 from the total sum, we obtain 1673 contributions for A and 1647 for E. If we compute [α]ω only with these configurations, we obtain an error of 2.3% for A and 1.2% for E. In other words, the largest part of the final value of specific rotation (98% for A and 99% for E) is obtained considering only 27−28% of all possible configurations.

Figure 10. HOMO Sk̃ values for the two conformers of 3MCP. Only the virtual orbitals with non-negligible contributions to the specific rotation are shown.

Figure 11. Electric (red) and magnetic (blue) components of S1̃27 (H− L) and S̃227 (H−L+1) for the two conformers of 3MCP. For clarity’s sake, the length of the vectors has been scaled as follows: for H−L, blue = ×0.5, red = ×20; for H−L+1, blue = ×2, red = ×10.

5. DISCUSSION AND CONCLUSIONS In the previous sections we have compared two stable conformers of 2-carene and 3MCP through a Sk̃ decomposition of their specific rotation. For both molecules, the two conformers have different sign of the rotation due to the opposite puckering on the central ring. Although a significant number of excited-state configurations contribute to the total value of [α], we showed that the major contribution to the sign change comes form a handful of MOs. For 2-carene, we can focus on the HOMO and LUMO+2. The change of sign is related to the angle between the electric and magnetic components of S̃k, which are slightly below and above 90° for

Figure 12. Occupied and virtual orbitals for S̃127 and S̃227 for the two conformers of 3MCP.

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conformer A and B, respectively. The magnitude of [α] can be correlated to the degree of planarity of the ring structure around the CC group. Conformer A is considerably more planar than B, thus closer to a typical olefin where the electric and magnetic components are orthogonal. For 3MCP, the specific rotation is influenced by the different orientation of the methyl group. For this molecule, we can focus on the HOMO, LUMO, and LUMO+1 orbitals. The HOMO−LUMO excited configuration produces S̃k values of opposite sign because the position of the methyl groups changes the direction of the small charge transfer since in the equatorial conformation this group participates in the ring electronic cloud. This group also influences the HOMO−LUMO+1 excited configuration changing the direction of the charge transfer in the E conformer. This produces a significant value of Sk̃ , contrary to the A conformer where the electric and magnetic vectors are basically orthogonal (even if their magnitude is comparable in both conformers). The S̃k values compensate, producing a net effect of [α] very similar in magnitude between the two conformers. From a computational point of view, the S̃k analysis shows that only 27−43% of all the possible configurations are necessary to account for 97−98% of the total specific rotation of the systems considered here. This is consistent with previous results,51 and may be used to preventively select a window of active orbitals for LR calculations and significantly reduce the computational cost of these calculations on large compounds. This strategy may then be used to investigate solvation effects on specific rotation, which were shown to be substantial and difficult to interpret,40 by considering multiple explicit solvent molecules quantum mechanically. The results in this work indicate that differences in specific rotation between stable conformers of chiral molecules can be reduced to the contribution of a limited number of excited configurations, i.e., occupied-virtual MO pairs. Further research will explore whether this behavior is consistent across different levels of theory, for instance, different density functionals and basis sets. Additionally, we will extend the S̃k method to highlevel wave function methods such as those belonging to coupled cluster theory, where higher levels of excited configurations are considered. The Sk̃ method may be used to further improve our understanding of the optical activity of chiral species, and may provide some clue on how to predict sign and magnitude of their specific rotation based solely on their 3D structure.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1 785 864 6509. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This investigation was supported by the University of Kansas General Research Fund allocation #2302049, and by the University of Kansas startup fund. The author thanks Tal Aharon for suggestions on the manuscript.

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REFERENCES

(1) Berova, N.; Polavarapu, P. L.; Nakanishi, K.; Woody, R. W. Comprehensive Chiroptical Spectroscopy; John Wiley & Sons, Inc.: Hoboken, NJ, 2012. 8309


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