Rotation-reflection group

The simplest molecules are atoms, which belong to point group %h (often called the full rotation-reflection group). The character table (which we omit) contains irreducible representations of dimensions 1,3,5,... these representations correspond to energy levels with electronic orbital angular-momentum quantum number /=0,1,2,... we have the (2/+1)-fold degeneracy associated with different values of the quantum number... 

The functions, here occurring in standard order, are our standard basis functions for the real irreducible representations of the full three-dimensional rotation-reflection group, Rg x I, and for its subgroups, Aoft. and Coo . All functions are normalized to 4nj(2l- -1), where / is the azimuthal quantum number. 

The Hamiltonian /lclcc(f f) has the same invariance under the rotation-reflection group 0(3) as does the full translationally invariant Hamiltonian (6), and it has a somewhat extended invariance under nuclear permutations, since it contains the nuclear masses only in symmetrical sums. Since it contains the translationally invariant nuclear coordinates as multiplicative operators, its domain is of... 

The three-dimensional rotation-reflection group (that is, the full group of V ) is obtained by adjoining an inversion operation to the operations of C(3). This generates an infinite number of reflection planes and the number of reps is just double that in C(3). Each character of C(3) is multiplied by +1, so instead of labeling the reps with an index I, an r " or l is used to indicate the parity under inversion. In the direct product (+) X (+) = (+) ( + ) X (-) = (-), and (-) X (-) = (+). For e.xample,... 

The axial vector and the polar vector representations are one and the same in the three-dimensional rotation group C(3), since the behavior of Rx and x cannot be distinguished unless the group contains other than pure rotations. In the rotation-reflection group each rep Z) of C(3) is... 

It can be shown that under the operations of the full rotation-reflection group in three dimensions, 0(3) ... 

The point groups T, and /j. consist of all rotation, reflection and rotation-reflection synnnetry operations of a regular tetrahedron, cube and icosahedron, respectively. 

The remaining fifty-eight magnetic point groups include the time reversal operator only in combination with rotation and rotation-reflection operators. The representations of these groups may be obtained from Eq. (12-27). 

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

Chemists use the Schoenflies Notation and rotation-reflection for improper rotation in assigning point groups. Crystallographers... 

Use italic type for the letters in symmetry operations and structural point groups. The symbols (Schoenflies) are as follows E, identity C, cyclic D, dihedral T, tetrahedral O, octahedral I, icosahedral S, rotation-reflection and a, mirror plane. Align subscripts and superscripts. 

Enantiotopic groups are those which can be interchanged one with the other, by a rotation-reflection operation. If two such groups, a and a" in Ca a"bc are separately substituted by an achiral group d, the products are the two enantiomers of Cabcd. The central carbon atom is described as prochiral. The two H atoms in... 

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