©2004,2005,2006 Alan M. Schwartz, All Rights Reserved

Revised May 2006

CHI=1 Maximally Parity Divergent Molecules

Michel Petitjean published theory and software to calculate the quantitative parity divergence (gif) (chirality along all coordinate axes) of countable sets (arbitrary; molecules, crystal lattices) of points (atoms) possessing three finite moments of inertia. Parity divergence varies on a scale of CHI=0 (achiral) to CHI=1 (perfect theoretical parity divergence) inclusive. Presented herein are three classes of molecules - [m.n]chiralanes, [n]chirolanes, fullerenes - whose carbon skeletons calculate as CHI=1.

Review of quantitative parity (pdf)
Overview of theory
Software for calculating quantitative parity divergence
Petitjean, Michel, J. Math. Phys. 40(9) 4587 (1999)
J. Math. Phys. 43(8) 4147 (2002)
Compt. Rend. Acad. Sci. (Paris), serie IIc, 4(5) 331 (2001)
J. Math. Chem. 22(2-4) 185 (1997)
J. Math. Chem. 23 429 (1998)

(S)-[6.6]Chiralane optimized by Hartree-Fock 6-31G(d) has overall CHI=0.982423 and CHI=1 exactly for its carbon skeleton alone. Here is a different view of [6.6]chiralane.

[6.6]Chiralane is a condensed alkane, C27H28, built of fused homochiral twist-boat cyclohexane rings. Its skeletal graph is non-planar by Kuratowski's theorem. (A graph is planar if and only if it has no subgraph homeomorphic to K5 or K3,3). It has three C2-axes and four C3-axes of rotation present overall - point group T (not Td or Th) - to 15 decimal places. Chiralane synthesis is a significant challenge, and made no easier by substituting a central quaternized nitrogen cation (azonia) or quaternized boron anion (boranuida) for its uncharged central quaternary carbon. Homochiral azoniachiralane borachiralanuide might be the definitive chiral ionic crystal lattice.

Many other point group T C30-C50 skeletons give CHIs below 0.06. We believe large CHI generation requiries a convex mass distribution. Several ring size variations of the chiralane skeleton generate CHI=1. Flat graphs in point group T or I can also deliver CHI=1 (hollow chirolanes and fullerenes).

[6.6]Chiralane in 3-D manipulable VRML (e.g., Cortona viewer) here (wrl), and here (wrl) with more display options.


Graphics   MoleculeOverall
[5.5]Chiralane 1.000000 CHIME   C44 Fullerene
point group T
1.000000 CHIME
[5.7]Chiralane 0.985962 CHIME C52 Fullerene
point group T
1.000000 CHIME
[6.6]Chiralane 0.982423 CHIME
C92 Fullerene
point group T
1.000000 CHIME
[6.7]Chiralane 0.979747 CHIME C100 Fullerene
point group T
1.000000 CHIME
[7.7]Chiralane 0.984151 CHIME C140 Fullerene
point group I
1.000000 CHIME
[8.8]Chiralane 0.989588 CHIME C260 Fullerene
point group I
1.000000 CHIME
[7]Chirolane 0.986908 CHIME
[8]Chirolane 0.989588 CHIME
Browser "Back" button to return from graphics

Gravitation theory can be parity-even math: Galileo, Newton, Green's function; Einstein and metric gravitation overall; non-heterotic string theory. Gravitation theory can be parity-odd math: teleparallel Cartan, affine Weitzenböck. Parity-even gravitation postulates the Equivalence Principle (EP): all local centers of mass vacuum free fall identically regardless of chemical composition or mass distribution; inertial and gravitational masses are fundamentally indistinguishable. Parity-odd gravitation contains a chiral pseudoscalar vacuum background that diastereotopically interacts with opposite parity mass distributions. Only one class of theory can be correct. All other predictions are identical within testable limits.

The proper challenge of spacetime geometry is test mass geometry. Extremal parity divergence of otherwise chemically and macroscopically identical paired (sets of) test masses, a parity Eötvös experiment, is an important and untried challenge (pdf) to the Equivalence Principle. Practical considerations would run a parity Eötvös experiment with single crystal solid spheres of enantiomorphic space group P3121 (right-handed screw axes) and P3221 (left-handed screw axes) quartz sparse, quartz dense, Point scatter in these plots of log(1-CHI) vs. radius is not noise. It arises from variation of moments of inertia of increasing radius spherical samples of anisiotropic lattice. Placement of the sphere's center is arbitrary. Each unique origin will give a different CHI value versus a given radius. The overall graph will remain unchanged.

Dr. Penelope Smith at Lehigh University notes that CHI is a connection between eigenvalues, special functions, and their representation theory with solid angles and exponentials of fractions of pi. The graph intercept is the solid angle subtended by the smallest vertex angle of a polyhedron (the supplement of its dihedral angle) defined by the c-axis helix,

log(1-CHI)= -2[log(radius)] + [(180-alpha) (pi)/60] - pi
(alpha is the smallest vertex angle in the helix. The slightly distorted tetrahedral O-Si-O helix angle is 110.56° vs. 109.47° undistorted)

log(1-CHI) = -2[log(radius)] + 0.494277
3-cm quartz test mass has CHI = 1 - 1.387·10-16, theory
3-cm quartz test mass has CHI = 1 - 1.535·10-16, graph fit

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