Operations and symmetry elements
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص405-406
2025-12-02
57
Operations and symmetry elements
The classification of objects according to symmetry elements corresponding to operations that leave at least one common point unchanged gives rise to the point groups. There are five kinds of symmetry operation (and five kinds of symmetry element) of this kind. When we consider crystals (Chapter 20), we shall meet symmetries arising from translation through space. These more extensive groups are called space groups. The identity, E, consists of doing nothing; the corresponding symmetry element is the entire object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity element. One reason for including the identity is that some molecules have only this symmetry element (1); another reason is technical and connected with the detailed formulation of group theory. Ann-fold rotation (the operation) about an n-fold axis of symmetry, Cn (the corresponding element) is a rotation through 360°/n. The operation C1 is a rotation through 360°, and is equivalent to the identity operation E. An H2O molecule has one twofold axis, C2. An NH3 molecule has one threefold axis, C3, with which is associated two symmetry operations, one being 120° rotation in a clockwise sense and the other 120° rotation in a counter-clockwise sense. A pentagon has a C5 axis, with two (clock wise and counterclockwise) rotations through 72° associated with it. It also has an axis denoted C5 2, corresponding to two successive C5 rotations; there are two such operations, one through 144° in a clockwise sense and the other through 144° in a counter clockwise sense. A cube has three C4 axes, four C3 axes, and six C2 axes. However, even this high symmetry is exceeded by a sphere, which possesses an infinite number of symmetry axes (along any diameter) of all possible integral values of n. If a molecule possesses several rotation axes, then the one (or more) with the greatest value of n is called the principal axis. The principal axis of a benzene molecule is the sixfold axis perpendicular to the hexagonal ring (2).
Fig. 12.3 An H2O molecule has two mirror planes. They are both vertical (i.e. contain the principal axis), so are denoted σv and σv ′.
Fig. 12.4 Dihedral mirror planes (σd) bisect the C2 axes perpendicular to the principal axis.
Fig. 12.5 A regular octahedron has a centre of inversion (i).
A reflection (the operation) in a mirror plane, σ (the element), may contain the principal axis of a molecule or be perpendicular to it. If the plane is parallel to the principal axis, it is called ‘vertical’ and denoted σv. An H2O molecule has two vertical planes of symmetry (Fig. 12.3) and an NH3 molecule has three. A vertical mirror plane that bisects the angle between two C2 axes is called a ‘dihedral plane’ and is denoted σd (Fig. 12.4). When the plane of symmetry is perpendicular to the principal axis it is called ‘horizontal’ and denoted σ h. A C6H6 molecule has a C6 principal axis and a horizontal mirror plane (as well as several other symmetry elements). In an inversion (the operation) through a centre of symmetry, i (the element), we imagine taking each point in a molecule, moving it to the centre of the molecule, and then moving it out the same distance on the other side; that is, the point (x, y, z) is taken into the point (−x, −y, −z). Neither an H2O molecule nor an NH3 molecule has a centre of inversion, but a sphere and a cube do have one. A C6H6 molecule does have a centre of inversion, as does a regular octahedron (Fig. 12.5); a regular tetrahedron and a CH4 molecule do not. An n-fold improper rotation (the operation) about an n-fold axis of improper rotation or an n-fold improper rotation axis, Sn, (the symmetry element) is com posed of two successive transformations. The first component is a rotation through 360°/n, and the second is a reflection through a plane perpendicular to the axis of that rotation; neither operation alone needs to be a symmetry operation. A CH4 molecule has three S4 axes (Fig. 12.6).
Fig. 12.6 (a) A CH4 molecule has a fourfold improper rotation axis (S4): the molecule is indistinguishable after a 90° rotation followed by a reflection across the horizontal plane, but neither operation alone is a symmetry operation. (b) The staggered form of ethane has an S6 axis composed of a 60° rotation followed by a reflection.
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