21.9 Groups of Symmetry Operators 897
Exercise 21.16
Find the coordinates of the point to which the point (1, 2, 3) is moved by thêS 3 operator whose
symmetry element is thexaxis.A symmetry operator that does not change the potential energy when applied to the elec-
trons of a molecule commutes with the electronic Hamiltonian in the Born–
Oppenheimer approximation and is said to belong to the molecule. To test whether
a symmetry operator belongs to a molecule in its equilibrium nuclear conformation,
we can apply the symmetry operator to all of the nuclei. If a symmetry operator belongs
to a molecule, it either leaves each nucleus in its original location or moves it to the
original location of a nucleus of the same isotope of the same element. We can also test
the symmetry operator by applying it to the electrons without moving the nuclei and
determine whether this changes the potential energy, but this is usually less convenient.EXAMPLE21.14
Find the improper rotations that belong to the ethane molecule, C 2 H 6 , in the staggered
conformation.
Solution
We place the molecule so that a hydrogen atom bonded to the upper carbon atom is in the
xzplane with a positive value ofxand a hydrogen atom bonded to the lower carbon atom
is in thexzplane with a negative value ofx. There is anS 3 axis coinciding with thezaxis.
A rotation of 180◦about theyaxis exchanges these two hydrogens and also exchanges two
other pairs of hydrogens. The reflection through thexzplane exchanges the four hydrogens
not in thexzplane, so thêS 2 yoperator with theyaxis as its element belongs to the molecule.
There are two additional̂S 2 operators with axes perpendicular to planes containing hydrogen
nuclei, so there are threêS 2 operators that belong to the molecule as well as thêS 3 operator.We can think of a symmetry operator as a mathematical operator that can operate
both on points and functions, or we can think of asymmetry operationas a means of
moving a point in three-dimensional space. The effect of the symmetry operator on a
point is the same as the effect of the symmetry operation. A symmetry operation will
be denoted by the same symbol as a symmetry operator without the caret (∧). When
operating on the nuclei of a molecule we will sometimes use the terms “symmetry
operators” and “symmetry operations” interchangeably.EXAMPLE21.15
List the symmetry operations that belong to the H 2 O molecule in its equilibrium nuclear
conformation.
Solution
There is aC 2 axis that bisects the bond angle, and we orient the molecule so that this
symmetry axis coincides with thezaxis. There is aσvreflection plane in the plane of the
molecule, which we place in theyzplane, and there is anotherσvplane perpendicular to
the plane of the molecule, in thexzplane. As with any molecule there is also the identity
operation,E.