Symmetry operation improper

If a molecule has no improper symmetry operations it will not be superposable on its mirror image, so it may be optically active. Note that this does not require the molecule to have no symmetry. 

Improper symmetry operation A symmetry operation that converts a right-handed object into a left-handed object. Such operations include mirror planes, centers of symmetry, and rotatory-inversion axes. 

Supramolecular architectures with T symmetry are chiral because they have no improper symmetry operations, but possess a combination of two- and three-fold rotation axes. They represent the most investigated metal-based chiral assemblies.There are three types of chiral tetrahedra with different metal-to-ligand stoichiometry, M4L6, M6L4 and M4L4 (see Figure 5.7). 

Dissymmetric. Lacking improper symmetry operations. A synonym for "chiral", but not the same as "asymmetric". 

Enantiotopic. The relationship between two regions of a molecule that are related only by an improper symmetry operation, typically a mirror plane. 

For a discussion of the transformation of the field operators under improper Lorentz transformations and discrete symmetry operations such as charge conjugation, see ... 

For each symmetry element of the second kind (planes of reflection and improper axes of rotation) one counts according to Eq. (1) the pairs of distinguishable ligands at ligand sites which are superimposable by symmetry operations of the second kind. 

If rotation about an axis by 360°ln followed by reflexion through a plane perpendicular to the axis produces an equivalent configuration of a molecule, then the molecule contains an improper axis of symmetry. Such an axis is denoted by Sn, the associated symmetry operation having been described in the previous sentence. The C3 axis of the PC15 molecule is also an S3 axis. The operation of S3 on PC15 causes the apical (i.e. out-of-plane) chlorine atoms to exchange places. 

Our convention is that a symmetry operation R changes the locations of points in space, while the coordinate axes remain fixed. In contrast in Section 1.2 we considered a change (proper or improper rotation) of coordinate axes, while, the points in space remained fixed Let x y z be a set of axes derived from the xyz axes by a proper or improper rotation. Consider a point fixed in space. We found that its coordinates in the x y z system are related to its coordinates in the xyz system by (1.120) or (2.29) ... 

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

All symmetry operations that we wish to consider can be regarded as either proper or improper rotations. 

The final requirement, that every element of the group have an inverse, is also satisfied. For a group composed of symmetry operations, we may define the inverse of a given operation as that second operation, which will exactly undo what the given operation does. In more sophisticated terms, the reciprocal S of an operation R must be such that RS = SR = E. Let us consider each type of symmetry operation. For <7, reflection in a plane, the inverse is clearly a itself a x a = cr = E. For proper rotation, C , the inverse is C" m, for C x C "m = C" = E. For improper rotation, S , the reciprocal depends on whether m and n are even or odd, but a reciprocal exists in each of the four possible cases. When n is even, the reciprocal of S% is S m whether m is even or odd. When n is odd and m is even, S% — C , the reciprocal of which is Q m. For S" with both n and m odd we may write 5 = C a. The reciprocal would be the product Q ma, which is equal to and which... 

The chief reason for pointing out these relationships is for systematization AO symmetry operations can be included in C. and S . Taken in the order in which they were introduced, c = S, i S2l E C,. Thus whoi we say that dural molecules are those without improper axes of rotation, the possibility of planes of symmetry and inversion centers has been included. 

It follows from Exercise 2.1-3(a) and Example 2.1-1 that the only necessary point symmetry operations are proper and improper rotations. Nevertheless, it is usually convenient to make use of reflections as well. However, if one can prove some result for R and IR, it will hold for all point symmetry operators. 

Then Q +IR Q is a point group P which is isomorphous with P and therefore has the same class structure as P. The isomorphism follows from the fact that I commutes with any proper or improper rotation and therefore with any other symmetry operator. Multiplication tables for P and P are shown in Table 2.7 we note that these have the same structure and that the two groups have corresponding classes, the only difference being that some products Xare replaced by IXin P. Examples are given below. 

This is a most useful result since we often need to calculate the inverse of a 3 x3 MR of a symmetry operator R. Equation (10) shows that when T(R) is real, I R)-1 is just the transpose of T(R). A matrix with this property is an orthogonal matrix. In configuration space the basis and the components of vectors are real, so that proper and improper rotations which leave all lengths and angles invariant are therefore represented by 3x3 real orthogonal matrices. Proper and improper rotations in configuration space may be distinguished by det T(R),... 

It will be more economical in the first two sections to label the coordinates of a point P by xi x2 x3. Symmetry operations transform points in space so that under a proper or improper rotation A, P(xi x2 x3) is transformed into P (xi x x3 ). The matrix representation of this... 

The space group G of a crystal is the set of all symmetry operators that leave the appearance of the crystal pattern unchanged from what it was before the operation. The most general kind of space-group operator (called a Seitz operator) consists of a point operator R (that is, a proper or improper rotation that leaves at least one point invariant) followed by a translation v. For historical reasons the Seitz operator is usually written R v. However, we shall write it as (R ) to simplify the notation for sets of space-group operators. When a space-group operator acts on a position vector r, the vector is transformed into... 

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