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Step: 145
Summary
The knowledge of the symmetry has great importance for many subsections of chemistry. It is important aid to clarifying the geometrical structure of the molecules and crystals and the electronic structure of chemical compounds.
We differentiate between symmetry elements and symmetry operations.
Symmetry elements are geometrical objects within a molecule: Even one, levels and points. At them symmetry operations can be executed.
Symmetry operations are movements, those concerning the straight lines, levels and points at the molecule to be executed can and the molecule into an equivalent or identical arrangement indistinguishable in relation to the output arrangement transfer.
On the close relationship between symmetry element and symmetry operation is particularly referred to. Both cause themselves mutually. A symmetry operation can be defined and executed only regarding a symmetry element, while the presence of a symmetry element can be only shown, if in addition appropriate symmetry operations exist.
Symmetry elements and symmetry operations are named the same symbols.
Symmetry operations, which to the identical (output -) arrangement lead, are called identity operation E. In the following outline the symmetry elements and the symmetry operations produced by them are summarized:
| Symmetry element | Symbol | Symmetrioperation | Symbols |
| axis of proper rotation with stepsize n | C n | One or more turns over den angel (2 p / n) around this axis | C n ,C n 2 ..., C n n = E |
| Symmetry plane | s | n-multiple reflection to the symmetry plane | s n = s (n oddly) s n = E (n even) |
| Symmetry center | i | Inversion of all atoms at the symmetry center | i n = i (n oddly) i n = E (n even) |
| axis of improper rotation nth order | S n | improper rotation, i.e. turn around the angle 2 p / n and reflection on one level perpendicularly to the axis of rotation | n even: S n , C n/2 , S n 3 ... s n n/2 = i... s n n = E n oddly: S n , C n 2 ..., S n n = s S n 2n = E |
The product of symmetry operations is the stepsize made of these symmetry operations. Ci means that first the inversion is executed, then the rotation operation.
The product of two symmetry operations corresponds again to a symmetry operation.
The complete set of all at a given molecule executable symmetry operations defines one point group of symmetry. The designation point group follows from two reasons:
Always at least one point of the space remains in its position unchanged.
The group of points fulfills the criteria of a mathematical group. For molecules theoretically infinitely many point groups are conceivable, from which however in practice only a certain number occurs. The determination of the point group of symmetry of a molecule takes place with the help of a regulation algorithm, which is based on the fact that one the point groups.
1. according to the type of the available axes of rotation into different types to divide can. The groups of points of linear and cubic symmetry are characterised by its high symmetry and are recognizable as such.
| Type of the available axis of rotation | Group of points | Example |
one  (linear molecules) |
D  h |
CO 2 |
| v |
HCN | |
| several C n with n > 3 (cubic molecules) | I h | |
| O h | [ CrCl 6 ] 3 ,SF 6 | |
| T d | ||
| a C n with n > 2 | S 2n | C 6 H 6 , BF 3 |
| D nh | p-disubst. benzene | |
| D lp | C 2 H 4 staggered, Ferrocen | |
| D n | ||
| C nh | trans-dichlorethylen | |
| C nv | H 2 O, NH 3 , [ CrCl 5 H 2 O ], o and m disubst. benzene | |
| C n | ||
| no C n with n > 2 | C i | |
| C s | ||
| C 1 | FClSO |
2. by consideration of further symmetry elements, which with existence of several axis to the main or reference axis refer, finally determines.
(the two questions: 1. Which molecules possess a permanent dipole moment? 2. Which molecules are optically active? Can you answer now briefly in such a way:
1. All molecules, which belong to the groups of points of C s, C n or C nv, possess a dipole moment.
2. All molecules, which belong to the groups of points of C n or D n, are optically active.)
Literature recommendations
Group theory:
Franc A. Cotton, Chemical Applications OF Group Theory, Wiley, New York
Application oscillation spectroscopy:
Molecularly vibrations; E. Bright Wilson, Jr., J. C Decius and Paul C. CROSS, Dover Publications, New York
Thank saying (on-line version):
Florian Dufey for the valuable notes and literature quotations