Difference Matters Energies of Enantiomers

The small, weak, and pseudo-scalar potential V PNC contributes to the attracting forces and the binding energy of molecules in a way, that a tiny amount of energy is added to one enantiomer whereas the same amount is subtracted from the energy of the mirror-image enantiomer giving rise to definite AEPNC values. Usually - and as we have outlined in Chap. 2 - it has been assumed that all scalar values of enan-tiomers, with the exception of its capacity to rotate the plane of linearly-polarized light, are strictly identical. The enthalpy, the entropy, and hence the standard Gibbs energy of an isomerization reaction between the R and S-enantiomers of a chiral molecule were assumed furthermore to be exactly zero at all temperatures, including the temperature 0 K. The equilibrium constant K for this isomerization was assumed to be exactly 1. But now we have to bear in mind, that chiral molecules are little more (or little less) stable than their mirror-image enantiomers!

Based on this knowledge on the physical basics of chemistry, symmetry, and chi-rality, we will now go a step further and ask the question whether the atoms of a left-handed molecule are more strongly (or less strongly) bound compared to the atoms of a right-handed molecule. We may try to measure the equilibrium constant K to very high precision. If K were found to be systematically different from 1 within high experimental confidence, the symmetry would be proven by experiment to be absent and thus the absolute handedness would become observable (Quack 2002). The literature is full of exiting examples of chiral molecules such as chiral hydrogen peroxide H2O2 (Mason and Tranter 1983b; Bakasov et al. 1996; Berger and Quack 2000; Gottselig et al. 2001), chiral disulfane H2S2 including its deuterium and tritium isotopomers (Gottselig et al. 2001), chiral dichlorodisulfane ClSSCl (Berger et al. 2001), the chiral archetypes bromochlorofluoromethane CHBrClF, and the deuterated CDBrClF (Quack and Stohner 2001), or chiral Buckminsterfullerenes (MacDermott 1993) for which parity non-conserving energy differences have been calculated. Even for transition-states involved in the first induction of chirality in amino acid-forming reactions, non equivalent electronic binding energies were calculated for the enantiomers of a-amino-propionitrile (Tranter 1985b). We will not discuss amino acid or ribose molecules as first examples since the applied ab initio methods calculating the spatial electron distribution are complex. Our first introducing case will concern energy differences of the trouble-free ethylene molecule H2C = CH2.

Ethylene is composed of two carbon atoms and four hydrogen atoms. It has 16 electrons and a plane of symmetry. In order to make the ethylene molecule chiral, we will twist it by 10° around the carbon-carbon axis (Rein et al. 1979; Rein 1992). A torsion angle q = 10° twist in one direction will give the P-helical ethylene (right-helix), a — 10° twist in the other direction will result in M-ethylene. The twisted ethylene molecule is now chiral, composed of two enantiomers. At this instant, a suitable software programme for ab initio calculations can be fed with bond distances and angles of the chiral ethylene in order to calculate (and minimize) the total energy. The programme will provide a precise picture of the electron distribution. Such calculations provide the advantage that we can calculate total energies for various molecular conformations. We are not limited to conformations that are preferably realized in nature, but we can study conformations not used naturally. For example we can change the ethylene conformation by twisting the ethylene molecule by the torsion angle p. This was done and indeed for the two twisted ethylene enantiomers a parity non-conserving energy difference of AEPNC = 4 • 10-20 a.u. = 10-18 eV was found (Rein 1992). The two twisted ethylene enantiomers are of different energy. The P-helix of twisted ethylene was favoured by 10-18 eV to M-ethylene and was thus calculated to be more stable. This is exciting and encouraging, even if such energy differences are too small to be directly measured for example with available spectroscopic methods.

With a certain approximation, we can generally use an energy difference AEPNC for enantiomers that is specified by Eq. 5.1. The energy difference can be given in atomic units (a.u.), defined as the doubled ionization energy of hydrogen (27.2 eV). Alternatively, AEPNC can of course be given in eV.

AEPNC « 10-20 • Z5a.u. = 3 • 10-19 • Z5eV (5.1)

Equation 5.1 includes a large exponent of five on the atomic number of protons Z in the nucleus, which results in the case of carbon (Z = 6) in approximately four orders of magnitude. Particularly for higher elements this exponent was assumed to help in making the energy difference experimentally accessible. Later, we will see that chiral cobalt- (Z = 27) and iridium- (Z = 77) complexes have been studied to measure AEPNC values experimentally.

In spite of the extremely small AEPNC values, we have to face the consequence that R- and S-enantiomers of the same molecule are now by definition really diastereoisomers, not enantiomers. Remember that diastereoisomers show distinguishable scalar properties! The strict enantiomer of an L-amino acid is the D-amino acid made of antimatter, but the 'classical' D-amino acid is not energetically equivalent (Mason 1984). Based on the weak interaction in atoms' nuclei also atoms themselves become optically active! The 'strict' enantiomer of an atom with its electrons is an antimatter atom including positrons. This gives rise to a small electroweak optical rotation, which was measured experimentally for the heavy metal atoms Tl, Pb, Bi, and Cs in the gas phase (Emmons et al. 1983) in accordance with quantum mechanical calculations.

Hence the two natural enantiomers of a chiral molecule differ slightly in many ways, including energetically. According to the Yamagata-Rein hypothesis such differences in the electronic energy might have been important for the deterministic generation of biomolecular asymmetry in prebiotic evolution on the Early Earth. The condition is that L-amino acids must have been favoured compared to D-amino acids and that D-sugar molecules are favoured to L-sugars.

Values of AEPNC do influence rate constants whenever chiral molecules (a) are formed, (b) racemize, (c) decompose, or otherwise react to give achiral products. The activation energies will in general be slightly different for the d- vs. L-enantiomers (Hegstrom 1984). A tiny difference in the energy of a chiral molecules compared to its mirror image enantiomer also influences the stability of chiral molecules in chemical reactions. L-Amino acids might be little more stable than D-amino acids or vice versa. They are not equally stable if we consider the contribution of the weak interaction to the electromagnetic interaction carefully. And this is what we have to do, independently on the size of the effect.

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