Completed Multiplication Tables

The complete multiplication table of the point group, worked out using arguments similar to those leading to (equation A1.4.23) and (equation A1.4.24), is given in table A1.4.2. It is left as an exercise for the reader to use this table to show that the elements of satisfy the group axioms given in section Al.4.2.1. 

It is usual to refer to the compound operation produced by applying two operations in succession as the product of the two operations. In order to investigate further whether the products of the known covering operations of the water molecule produce any new ones, it is helpful to construct a complete multiplication table, shown in the box. Each entry in the table gives the operation resulting from applying hrst the operation at the head of the column containing the entry, and then the operation at the left of the row. 

There remain, now, the products of C3 and C3 with cr(1), a(2 and cr(3). Figure A5-5(b) shows a type of geometric construction that yields these products. For example, we can see that the reflection a(1) followed by the rotation C3 carries point 0 to point 2, which could have been reached directly by the operation [Pg.1315]

If s and g are known, it is usually found that (8) can be satisfied in only one way. When the complete multiplication table is known, the characters may be found as follows. The product of all the elements in class Qi by all those in class Qk is calculated. The elements of the resulting set will be found to give a unique arrangement of classes. A given product occurs several times or never. Let hik,j be the number of times the yth class appears. Then if each element of the product is taken as many times as it occurs (contrary to the rule for multiplication of complexes) ... 

Table 2.6 A completed multiplication table for the NH symmetry operator ...

Table 4.4 The complete multiplication table for (a) the Cj, group operations and (b) the B, representation.

Assignment of the signals is completed in Table 30.2. The criteria for assignment are the shift values (resonance effects on the electron density on C and N), multiplicities and coupling constants. Because the difference between them is so small, the assignment of N-8 and N-9 is interchangeable. 

Exercise 2.2-2 Find the products Cgae and multiplication table for this set of operators G E Cf C3 rrd ae at is shown in Table 2.3. The complete multiplication... 

For another simple, but more general, example of a symmetry group, let us recall our earlier examination of the ammonia molecule. We were able to discover six and only six symmetry operations that could be performed on this molecule. If this is indeed a complete list, they should constitute a group. The easiest way to see if they do is to attempt to write a multiplication table. This will contain 36 products, some of which we already know how to write. Thus we know the result of all multiplications involving E, and we know that... 

Complete the table, and then analyze the data to determine if Compounds I and II are the same compound. If the compounds are different, use the law of multiple proportions to show the relationship between them. 

Solution. The structure of H2O is shown below, where the labels on the H atoms are imaginary. VSEPR theory predicts tetrahedral electron geometry with a bent molecular geometry and bond angles less than 109.5°. The complete set of symmetry operations is , C2, f7y, and (7 2- The multiplication table is shown below. In this symmetry group, each operation is its own inverse. 

In cell Dl, enter the label °F . In cell D2, type the formula = (9/5) B2+32 . This is equivalent to writing °F = (9/5) °C + 32. The slash (/) is a division sign and the asterisk ( ) is a multiplication sign. Parentheses are used to make the computer do what we intend. Operations inside parentheses are carried out before operations outside the parentheses. The computer responds to this formula by writing —328 in cell D2. This is the Fahrenheit equivalent to —200°C. Select cells D2 through D6. In the Home ribbon, go to Editing and select Fill and then Down to complete the table shown in Figure 3-3c. 

It is clear that any two of these numbers multiplied together will equal any one of the numbers, so this representation satisfies the group multiplication table for the 2 group. The complete set of representations of transformation numbers illustrated in Fig. 3.7 is given in Table 3.5. 

If a transformation can be found which will put all the matrices of a given representation into this general form, the representation is said to be reducible. If no transformation can further diagonalize all submatrices such as c,/, and i in Eq. (3.16) then the set of matrices of a given representation is said to be completely reduced and the sets of submatrices are called the irreducible representations. The fact that the submatrices are also representations can be clarified by referring to Eq. (3.16). If the group multiplication table requires that AB = C then, following the rules for matrix multiplication, it is clear that ad = g be = h, and cf = i so that the set of submatrices a, d, and g, for example, form part of a representation which as described above cannot be further reduced. 

To identify all the possible products systematically, a multiplication table of symmetry operations can be drawn up in which each row and column of the table has one of the symmetry operations as a heading the body of the table then contains the operation resulting from the product of that row and column. The starting point for the case of HjO is given in Table 2.1 and it is left to the reader to complete this following the instructions in Problem 2.1. 

Before completing the multiplication table we will write down some observations on the types of answer to expect for these products this will help narrow down the possibilities for this exercise and for more complex cases. 

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