are said to exhibit atropisomerism. A chiral compound can contain no improper axis of rotation (Sn), which includes planes of symmetry and inversion center. Chiral molecules are always dissymmetric (lacking Sn) but not always asymmetric (lacking all symmetry elements except the trivial identity). Asymmetric molecules are always chiral. The following table shows some examples of chiral and achiral molecules, with the Schoenflies notation of the point group of the molecule. In the achiral molecules, X and Y (with no subscript) represent achiral groups, whereas X and X or Y and Y represent enantiomers. Note that there is no meaning to the orientation of an S axis, which is just an inversion. Any orientation will do, so long as it passes through the center of inversion. Also note that higher symmetries of chiral and achiral molecules also exist, and symmetries that do not include those in the table, such as the chiral C or the achiral S. Stereogenic centers Many chiral molecules have point chirality, namely a single chiral stereogenic center that coincides with an atom. This stereogenic center usually has four or more bonds to different groups, and may be carbon (as in many biological molecules), phosphorus (as in many organophosphates), silicon, or a metal (as in many chiral coordination compounds). However, a stereogenic center can also be a trivalent atom whose bonds are not in the same plane, such as phosphorus in P-chiral phosphines (PRR′R″) and sulfur in S-chiral sulfoxides (OSRR′), because a lone-pair of electrons is present instead of a fourth bond. Chirality can also arise from isotopic differences between atoms, such as in the deuterated benzyl alcohol PhCHDOH; which is chiral and optically active ([α]D = 0.715°), even though the non-deuterated compound PhCH2OH is not. If two enantiomers easily interconvert, the pure enantiomers may be practically impossible to separate, and only the racemic mixture is observable. This is the case, for example, of most amines with three different substituents (NRR′R″), because of the low energy barrier for nitrogen inversion. While the presence of a stereogenic center describes the great majority of chiral molecules, many variations and exceptions exist. For instance it is not necessary for the chiral substance to have a stereogenic center. Examples include 1-bromo-3-chloro-5-fluoroadamantane, methylethylphenyltetrahedrane, certain calixarenes and fullerenes, which have inherent chirality. The C2-symmetric species 1,1′-bi-2-naphthol (BINOL), 1,3-dichloroallene have axial chirality. (E)-cyclooctene and many ferrocenes have planar chirality. When the optical rotation for an