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For all microhelices on aromatic rings of inherently chiral calix[4]arene, an expression was derived from one approximation and one hypothesis on the basis of the electron-on-a-helix model of Tinoco and Woody as follows:

Inherently chiral calixarenes, whose chiralities result from the dissymmetric substitution of achiral residues on calixarene skeletons, are a type of attractive chiral molecule, because of their potential applications in chiral recognition and asymmetrical catalysis [

A few approaches have been used to assign the absolute configuration of inherently chiral calixarenes, including chemical interconversions [

By keeping the calixarene skeleton integrated, the common dissymmetrical substitutions to create inherent chirality are

As illustrated in ^{1} (^{1} (

Structural analysis of inherently chiral Calix[4]arene and representative inherently chiral Calix[4]arene (−)-

The dihedral angles and six pairs of exterior angles on phenyl ring ^{1} of (−)-

_{C1−Cf−Ca−Br} = 4.2828 |
_{Br−Ca−Cb−N} |
_{N−Cb−Cc−N} |

_{N−Cc−Cd−C4} = −6.3989 |
_{C4−Cd−Ce−O} = 6.7100 |
_{O−Ce−Cf−C1} = −7.0711 |

^{a}C^{b} |
^{b}C^{a} |
^{c}C^{b} |

^{a}C^{f} |
^{b}C^{c} |
^{c}C^{d} |

^{4}^{d}C^{c} |
^{e}C^{d} |
^{1}^{f}C^{e} |

^{4}^{d}C^{e} |
^{e}C^{f} |
^{1}^{f}C^{a} |

The nature of the sp^{2} hybrid orbital of phenyl carbon can theoretically impel each phenyl and its substituents to be almost located on one plane and their six pairs of exterior angles to be almost 120°, although their substituents can slightly destroy the tendency. For example, in a representative ^{1} from its crystal data are shown in ^{1}, these dihedral angles are indeed small and can be omitted, and these exterior angles are all approximately equal to 120°.

In a macrocyclic skeleton helix, four dummy atoms (D^{1}, D^{2}, D^{3} and D^{4}) on four phenyl centers are used to replace four phenyls, respectively. Then, one geometrical helix model (^{1}D^{3}. Since the substitution pattern of all bridging carbons are same, the distances between each dummy atom and bridging carbon should normally be equal. When the relation of the angles exists as ^{1}^{1}^{2} = ^{1}^{4}^{4}, ^{1}^{2}^{2} = ^{4}^{4}^{3} and ^{2}^{2}^{3} = ^{4}^{3}^{3}, the geometrical helices on the two parts will be canceled. For example, in (−)-

The distances between dummy atoms and bridging carbons and three pairs of angles in the geometrical helix model of (−)-

_{C1D1} = 2.9240 |
_{C4D1} = 2.9161 |
_{C1D2} = 2.9212 |
_{C4D4} = 2.9209 |
||

_{C2D2} = 2.9308 |
_{C3D4} = 2.9385 |
_{C2D3} = 2.9094 |
_{C3D3} = 2.9144 |
||

^{1}^{1}^{2} = 104.257 |
^{1}^{2}^{2} = 121.606 |
^{2}^{2}^{3} = 102.747 |
|||

^{1}^{4}^{4} = 108.995 |
^{4}^{4}^{3} = 121.172 |
^{4}^{3}^{3} = 102.767 |

Four substituents (one equatorial hydrogen (_{e}_{a}^{1} and ^{2}) exist on bridging carbon ^{1}. _{e}_{a}^{1}–_{e}^{1}–_{a}_{e}C^{1}^{1} and _{a}C^{1}^{1} and the angle difference between _{e}C^{1}^{2} and _{a}C^{1}^{2} are commonly slight and can almost be ignored. For example, in (−)-_{C1He} = 0.9904 Å, _{C1Ha} = 0.9909 Å, _{e}C^{1}^{f}_{a}C^{1}^{f}_{e}C^{1}^{g}_{a}C^{1}^{g}_{e}_{a}_{e}_{a}^{1} (^{1} can be omitted.

In the phenyl ring helix in ^{1} and its six substituents can be almost treated as coplanar. Then, the geometrical helix on phenyl ring ^{1} can be omitted, and only its physical helix needs be considered. Therefore, after the omission of the macrocyclic skeleton helix, bridging carbon helices and geometrical helices on four phenyl rings, only physical helices on four phenyl rings need be considered for the helical character analysis of inherently chiral Calix[4]arenes.

Since all aromatic rings are asymmetric and the architectures on two sides of each aromatic ring plane are different, then inherently chiral Calix[4]arenes can theoretically be regarded as a type of complex planar chiral molecule (^{1} in ^{4}, ^{4}^{1} and ^{1}

Helical character analysis of microhelix ^{1} (^{1} (_{X}_{Y}_{X}_{Y}_{X}_{Y}_{X}_{Y}

The Calix[4]arene cavity is comprised of four aromatic rings and has a high electronic density. Each aromatic ring has a tendency to move away from the cavity to weaken their electrostatic repulsion. As a result, each aromatic ring is essentially equal to being placed in a dissymmetrical electric field. Therefore, the electrons on bonds

Now, a Cartesian coordinate was introduced to analyze the electron movement of bond ^{1} is set as the

According to the polarizability difference of atom _{X}_{Y}_{X}_{Y}

If the directly connected substituents on all aromatic rings of inherently chiral Calix[4]arene are different, the length of all aromatic bonds and their directly connected bonds should be different. In order to facilitate the calculation of the microhelix radius, one approximation was tentatively made that the length of all aromatic bonds and their directly connected bonds is equal to the average length of all aromatic bonds.

In

The nature of the sp^{2} hybrid orbital of phenyl carbon can theoretically impel each phenyl and its substituents to be almost located on one plane and their six pairs of exterior angles to be almost 120°, which has been proven by a representative

Moreover, in order to facilitate the calculation of the height difference (_{Z}

Then, the height difference (

The length of segment

From Expressions (1), (2) and (5), the length of segment

From sine theorem, the radius (

From Expressions (2) and (6), the microhelical radius can be derived as:

The states and eigenvalues of an electron constrained to move on a helix were successfully solved on the basis of the electron-on-a-helix model by Tinoco and Woody [

Since microhelix

Then, the reciprocal of electronic energy (

Since the interior structure and exterior environment of all microhelices on aromatic rings of inherently chiral Calix[4]arene are similar, the variable _{Z}

Essentially, the electronic energy of the electron in a helix is the same whether the helix is right- or left-handed. However, similar to other energy concepts, such as potential energy, the sign of the electronic energy can be artificially stipulated based on a selected reference point. Therefore, in order to distinguish left-handed and right-handed microhelices, we stipulate that Expression (13) is only suitable for a right-handed microhelix and should be changed into Expression (14) when the microhelix is left-handed, as follows:

Actually, the above two expressions can be united into:

By the way, two concepts, “helical character” and “helical electronic energy”, should be tentatively discussed. Wang noted that “in the context of this helix theory, the terms, that is, the helical character and the local energy of electrons of a helix, are equivalent” [

There are two rules presented by Wang that “a molecular helix’s helical character (local electronic energy) increases as its length, which usually correlates to its ring size, decreases (Rule I); and, at a fixed helix length, increases as its radius, which correlates to the relevant groups’ polarizability distinctions that result in the bonds’ helical deformations, decreases (Rule II)” [

Since the above deduction is suitable for all microhelices on aromatic rings of inherently chiral Calix[4]arene, Expression (15) can be universally used to qualitatively calculate their microhelical electronic energy. Therefore, if the sum of the reciprocal of the helical electronic energy of inherently chiral Calix[4]arene is less than zero, it can be assigned as a right-handed helix and dextrorotatory, and

Although a variety of inherently chiral Calix[4]arenes were synthesized until now, there are only enumerable entities whose absolute configurations and optical rotation signs have been ascertained. Here, the scientificity of the qualitative calculation of helical electronic energy based on Expression (15) can be verified with inherently chiral Calix[4]arene (−)-_{e}_{a}_{e}_{a}

In (−)-_{Br}_{C1} ≈ _{C2} > _{N in NH2}_{H}_{Br}_{H}_{C1} (1.061) + α_{N in NH2} (0.964)) [^{1}–Br^{2}–H

Therefore, it should be a left-handed helix and levorotatory.

Inherently chiral Calix[4]arenes, whose absolute configuration and optical rotation sign have been ascertained.

In (+)-_{N in NO2} > _{C1} ≈ _{C2} > _{N in NH2} > _{H}_{N in NO2} (1.090) + _{H}_{C1} (1.061) + _{N in NH2} (0.964)) [^{1}–N^{1}^{1}–N^{2}^{2}–H^{2}–H

Therefore, it should be a right-handed helix and dextrorotatory.

In (−)-_{N in N=C}_{C1} ≈ _{C2} > _{H}^{1}–N^{2}–H

Therefore, it should be a left-handed helix and levorotatory.

In (+)-_{C in Ar} > _{C1} ≈ _{C2} ≥ _{C in CMe3} > _{H}_{C in Ar} (1.352) + _{H}_{C1} (1.061) + _{C in CMe3} (1.061)) [^{1}–C^{4}^{4}–C^{3}^{3}–H^{2}–H

Therefore, it should be a right-handed helix and dextrorotatory.

In (+)-_{3})_{2} > _{A4} > _{A3} >> _{A2} ≈ _{A1} and the bridging carbon polarizability should be _{C3} > _{C2} ≈ _{C4} >> _{C1} (because the polarizability is electron-rich groups > electron-poor analogues) [_{O1} ≈ _{O2} ≈ _{O4} > _{O3}. Then, besides those symmetrical microhelices, microhelices ^{2}–O^{2}^{4}–O^{1}^{1}–O^{2}^{1}–O^{1}^{3}–O^{3}^{2}–O^{3}^{3}–O^{4}^{4}–O^{4}

Therefore, it should be a right-handed helix and dextrorotatory.

In (−)-^{4} ^{2}_{2}^{3}^{1}_{A3} > _{A4} ≈ _{A2} >> _{A1} and the bridging carbon polarizability should be _{C3} ≈ _{C2} >> _{C4} ≈ _{C1}. Moreover, since the polarizability sequences are _{carbonyl carbon} > _{alkyl carbon} [_{O3} ≈ _{O1} > _{O2} > _{O4}. Then, besides those symmetrical microhelices, microhelices ^{1}–O^{1}^{4}–O^{1}^{2}–O^{3}^{3}–O^{3}^{2}–O^{2}^{1}–O^{2}^{4}–O^{4}^{3}–O^{4}

Therefore, it should be a left-handed helix and levorotatory.

In (+)-^{4}_{2}^{2}^{1}^{3}^{4}_{2}^{2}^{1}^{3}_{A3} > _{A2} ≈ _{A1} ≥ _{A4} and the bridging carbon polarizability should be _{C3} ≥ _{C2} > _{C1} ≥ _{C4}. Moreover, since the polarizability sequences are _{carbonyl carbon} > _{alkyl carbon} [_{O3} > _{O4} > _{O2} ≈ _{O1}. Then, besides those symmetrical microhelices,microhelices ^{1}–O^{2}^{1}–O^{1}^{2}–O^{3}^{2}–O^{2}^{3}–O^{4}^{3}–O^{3}^{4}–O^{1}^{4}–O^{4}

Since _{C3} ≥ _{C2} and _{O2} ≈ _{O1}, then:

Therefore, they should be right-handed helices and dextrorotatory.

In (+)-^{1}_{2}^{2}_{2}^{3}^{4}_{A3} ≈ _{A4} > _{A1} ≈ _{A2} and the bridging carbon polarizability should be _{C3} > _{C2} ≈ _{C4} > _{C1}. Moreover, since the group polarizabilities are _{O3} ≈ _{O4} > _{O1} ≈ _{O2}. Then, besides those symmetrical microhelices, microhelices ^{3}–O^{3}^{3}–O^{4}^{2}–O^{3}^{4}–O^{4}^{2}–H^{3}) and ^{4}–H^{4}) can also be canceled by each other. Without consideration of microhelices canceled by each other, the reciprocal of its helical electronic energy can be deduced from microhelices ^{2}–H^{2}–O^{2}^{1}–O^{2}^{1}–H^{2} and ^{1}–H^{1}–O^{1}^{4}–O^{1}^{4}–H^{1} as:

Therefore, it should be a left-handed helix and levorotatory.

It should be mentioned that geometrical helices, resulting from an intermolecular or intramolecular non-bonded interaction in a high concentration and polar solvents, are not taken into consideration in the above analysis and calculation from Expression (15). Therefore, the calculated results only can be compared with those measured in low concentration and non-polar solvents. It is very surprising and satisfying that all of optical rotations from the above analysis are consistent with the actual facts, except (+)-

Due to structural similarity, Expression (15) can be popularized to assign the absolute configurations of other inherently chiral calix[_{Z}, need to be quantified with theoretic deduction and experimental data. Moreover, the exception from (+)-

In summary, inherently chiral Calix[4]arenes can be theoretically regarded as a type of complex planar chiral molecule when bridging carbons are treated as achiral and each phenyl ring and its six substituents are treated as coplanar. Based on one approximation and one hypothesis, we derive Expression (15) to qualitatively analyze microhelical electronic energy. Its scientificity and effectivity in absolute configuration assignments of inherently chiral Calix[4]arenes were almost entirely confirmed with all of the entities, whose absolute configurations and optical rotation signs have been ascertained.

We thank the National Natural Science Foundation of China (Nos. 21272173 and 21272292) for financial support. Special thanks go to David Zhigang Wang at the Shenzhen Graduate School of Peking University, whose instructive advice made it possible to complete this work.

Theoretic deduction: Jing Zhou, Jing-Wei Fu, Qing-Wei Zhang, Shao-Yong Li, Wei Qiao and Jun-Min Liu; Wrote the paper: Shuang Zheng, Ming-Liang Chang and Shao-Yong Li.

The authors declare no conflict of interest.