Symmetry centre, reflection through

Centre of symmetry. A point through which there is reflection to an identical point in the pattern. 

If a molecule has a centre of inversion (or centre of symmetry), i, reflection of each nucleus through the centre of the molecule to an equal distance on the opposite side of the centre produces a configuration indistinguishable from the initial one. Figure 4.4 shows s-trans-buta-1,3-diene (the x refers to trans about a nominally single bond) and sulphur hexafluoride, both of which have inversion centres. 

The associated symmetry operations are reflection through the plane of symmetry, o, rotation by 1 In of a complete revolution about the axis of symmetry, Cn, and inversion through the centre of symmetry, i. 

Reflection through a centre of symmetry (inversion centre)... 

Simple mirror planes are not the only symmetry elements that use reflection. If a molecule possesses an improper rotation axis, then the reflection through the mirror plane used in the symmetry operation will also link the mirror images. So any molecule containing an improper rotation axis as a symmetry element cannot be chiral. Also, since 82 = i, the inversion centre also precludes chirality. 

Of particular importance in the physical sciences is the fact that the symmetry operations of any symmetrical system constitute a group under the operators that effect symmetry transformations, such as rotations or reflections. A symmetry transformation is an operation that leaves a physical system invariant. Thus any rotation of a circle about the perpendicular axis through its centre is a symmetry transformation for the circle. The permutation of any two identical atoms in a molecule is a symmetry transformation... 

In classical systems spatial inversion symmetry can be considered completely independent of time. In three dimensions it may refer to inversion through a plane (mirror reflection), a line, or a point (centre), represented by diagonal transformation matrices such as... 

The simplest symmetry elements are the centre, plane, and axes of symmetry. A cube, for example, is symmetrical about its body-centre, that is, every point (xyz) on its surface is matched by a point (xyz). It is said to possess a centre of symmetry or to be centrosymmetrical a tetrahedron does not possess this type of symmetry. Reflection of one-half of an object across a plane of symmetry (regarded as a mirror, hence the alternative name mirror plane) reproduces the other half. It can easily be checked that a cube has no fewer than nine planes of symmetry. The presence of an -fold axis of symmetry implies that the appearance of an object is the same after rotation through 3607 l a cube has six 2-fold, four 3-fold, and three 4-fold axes of symmetry. We postpone further discussion of the symmetry of finite solid bodies because we shall adopt a more general approach to the symmetry of repeating patterns which will eventually bring us back to a consideration of the symmetry of finite groups of points. 

If reflection of all parts of a molecule through the centre of the molecule produces an indistinguishable configuration, the centre is a centre of symmetry, also called a centre of inversion (see also Box 1.9) it is designated by the symbol i. Each of the molecules CO2 (3.3), trans-N2F2 (see worked example 3.1), SFg (3.4) and benzene (3.5) possesses a centre of symmetry, but H2S (3.6), CW-N2F2 (3.7) and SiH4... 

Atoms have complete spherical symmetry, and the angular momentum states can be considered as different symmetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower symmetry. Symmetry operations for the molecule are transformations such as rotations about an axis, reflection in a plane, or inversion through a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule form a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A1A on molecular symmetry. 

Table C.l. Symmetry operations of the ammonia molecule (the reflections pertain to the mirror planes perpendicular to the triangle, Fig. C.2, and go through the centre of the triangle)...

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