Improper matrix representation

If a proper rotation is combined with a reflection with respect to the axis of rotation, it is called an improper rotation The matrix representation of silvan operation is found simply by replacing 1 by -1 in Eq. (104), The Scftfllfllies symbol for an improper rotation by y is S /tp- Hence, matrix the representation of an improper counter-clockwise rotation by y is of the form ... 

It should be noted that the trace of a matrix that represents a given geo] operation is equal to 2 cos y 1, the choice of signs is appropriate to or improper operations. Furthermore, it should be noted that the aim direction of rotation has no effect on the value of the trace, as a inverse sense corresponds only to a change in sign of the element sin y. TE se operations and their matrix representations will be employed in the following chapter, where the theory of groups is applied to the analysis of molecular symmetry. 

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... 

Let us set up a 2D unitary matrix representation for the transformation of the spin functions a and (1 in Civ. So far, we have established only a relation between 0(3)+ and SU(2). The matrix representations of reflections or improper rotations do not belong to 0(3)+ because their determinants have a value of -1. To find out how a and p behave under reflections, we notice that any reflection in a plane can be thought of as a rotation through n about an axis perpendicular to that plane followed by the inversion operation. For instance, 6XZ may be constructed as xz = Cz(y) i. Herein, it is not necessarily required... 

So far, we have been concerned only with proper rotations because in the spinor basis improper rotations are not defined. Since the electron spin is associated with an internal spinning of the electron around its axis, the electron spin is assigned an intrinsic positive parity. As we have seen before in Sect. 3.8, every improper rotation can be written as the product of a proper rotation and an inversion therefore, whenever an improper rotation acts on a spinor, we simply take the matrix representation for the proper factor in this improper rotation. 

S is a 3 X 3 real, orthogonal matrix representation of one of the proper or improper rotations of the point group of the space group, v(S) is a vector associated with S which is smaller than any primitive translation vector and t(m) = zm. a., where a. are the primitive translation vectors and m. ... 

Thereby the matrix 11(F) denotes a K-dimensional permutation matrix and F (F) a 3 by 3 orthogonal matrix. The form of this representation follows from the fact that each isometric transformation maps the NC Xk, Zk, Mk onto a NC which by definition has the same set of distances, i.e. is isometric to NC Xk, Zk, Mk. Expressed alternatively, the nuclear configurations NC Xk( ), Zk, Mk and NC Xk(F 1 ( )), Zk, Mk are properly or improperly congruent up to permutations of nuclei with equal charge and mass for any F G ( ). The set of matrices Eq. (2.12) forms a representation of J d) by linear transformations and will furtheron be denoted by... 

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