Jones symbol

The set of components of the vector r1 in eq. (13) is the Jones symbol or Jones faithful representation of the symmetry operator R, and is usually written as (x / /) or x / z. For example, from eq. (15) the Jones symbol of the operator R (n/2 z) is (yxz) or yxz. In order to save space, particularly in tables, we will usually present Jones symbols without parentheses. A faithful representation is one which obeys the same multiplication table as the group elements (symmetry operators). 

Exercise 3.2-1 Write down the Jones symbol for the improper rotation SAz. 

Write down the Jones symbols for 7 6 C.4V and then the Jones symbols for 7 1. [Hints You have enough information from Problems 3.4 and 3.5 to do this very easily. Remember that the MRs of 7 are orthogonal matrices.] Write down the angular factor... 

Find the MR of R(—2n/3 [1 1 1]) for the basis (ei e2 e3. Hence write down the Jones representations of R and of i 1. Find the transformed d orbitals Rd, when d is dxy, dyz, or dzx. [Hint Remember that the unit vectors ei e2 e3 are oriented initially along OX, OY, OZ, but are transformed under symmetry operations. Observe the comparative simplicity with which the transformed functions are obtained from the Jones symbol for R 1 instead of trying to visualize the transformation of the contours of these functions under the configuration space operator R.]... 

Explain why the point group D2 = E C2z C2x C2y is an Abelian group. How many IRs are there in D2 Find the matrix representation based on (e e2 e31 for each of the four symmetry operators R e D2. The Jones symbols for R 1 were determined in Problem 3.8. Use this information to write down the characters of the IRs and their bases from the set of functions z xy. Because there are three equivalent C2 axes, the IRs are designated A, B1 B2, B3. Assign the bases Rx, Ry, Rz to these IRs. Using the result given in Problem 4.1 for the characters of a DP representation, find the IRs based on the quadratic functions x2, y2, z2, xy, yz, zx. 

Table 6.4. Jones symbols and character systems for AOs and MOs in the octahedral ML6 complex ion. 

Table 17.4. Jones symbols for the transformation offunctions (that is, R x y zj) for the twenty-four operations R C 0.

Exercise 17.4-5 AtX, P(k) is C2v the character table for which is shown in Table 17.7. The basis functions shown are those for the IRs of C2v when the principal axis is along a. Table 17.8 contains the character table for C3V with basis functions for a choice of principal axis along [1 1 1], The easiest way to transform functions is to perform the substitutions shown by the Jones symbols in these two tables. The states X2 and S4 are antisymmetric with respect to vertical planes at A and, as Table 17.8 shows, A2 is antisymmetric with respect to dihedral planes in C4v and from Table 17.2 we see that the bases for T j and T2 are antisymmetric with respect to fold axis at T is parallel to kz, and carrying out the permutation y —> z, z —> x, x y on the bases for A[ and A2 gives xy(x2 y2) and x2 y2 for the bases of Tj and T2, which are antisymmetric with respect to ab.)... 

Jones symbols for the set IR are obtained by changing the sign of the symbols for R). The principal axis has been chosen along v for the choices z orx, use cyclic permutations of xyz, or derive afresh, using the appropriate projection diagram. 

These same Jones symbols are used in the derivation of the ligand orbitals of an MLg molecule. 

To reduce writing we give only the transformed coordinates (Jones symbols) which are to be substituted for xy z) in eq. (52), rather than the actual functions, using parentheses to separate the classes of Oh. For j = rb... 

Table 17.14. Jones symbols and character tables for InSb (space group 216) at A and X.

Table 17.15. Character table and Jones symbols for the rotational part R of the space group operator (7f v) Fd3m (227 or 0 ) at A.

The additional symmetry elements in the silicon structure are given in Table 17.12. At A, P(k) C4V, which is isomorphous with Gg. Jones symbols and the character table are in Table 17.15. 

The Jones symbols which provide the substitutions for 1 i](A) are in the third line of Table 17.14. On making use of these substitutions, a second basis for the Ai representation is provided by... 

Table 18.2. Character table of the factor group G(q)/T at A [0 q 0], together with the corresponding classes of G(q)/T at T and the Jones symbols R(xyz), where (xyz) is an abbreviation for (ex ey ez).

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