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Enantiomers

Chiral molecules that are mirror images of one another and non-superimposable.
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The statement of the theorem

Let MM be a molecular structure represented by a set of coordinates P={(xi,yi,zi)}i=1N⊂R3P = \{(x_i, y_i, z_i)\}_{i=1}^N \subset \mathbb{R}^3. Define the reflection operator ρ:R3→R3\rho: \mathbb{R}^3 \to \mathbb{R}^3 across a plane Π\Pi (e.g., the yzyz-plane, Π:x=0\Pi: x=0) such that ρ(x,y,z)=(−x,y,z)\rho(x, y, z) = (-x, y, z). The mirror image of MM is M′=ρ(P)={(ρ(xi),ρ(yi),ρ(zi))}i=1NM' = \rho(P) = \{(\rho(x_i), \rho(y_i), \rho(z_i))\}_{i=1}^N. The pair (M,M′)(M, M') constitutes a pair of enantiomers if MM is chiral, which means that MM cannot be mapped onto M′M' by any rigid body transformation TT belonging to the group of rotations and translations SE(3)SE(3) (i.e., M≇T(M′)M \not\cong T(M') for all T∈SE(3)T \in SE(3)).