Question
In: Chemistry
Write out the characters of the following direct products in D4h symmetry and determine the irreducible...
Write out the characters of the following direct products in D4h symmetry and determine the irreducible representations which comprise them. Show your work in detail. The D4h table can be found online. a. A1g x Eg b. A1u x B2u c. Eg x Eu d. B2g x A1u x B2u
Solutions
Expert Solution
| D4h | E | 2C4 | C2 | 2C'2 | 2C''2 | i | 2S4 |  |
 |
 |
| A1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| A2g | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 |
| B1g | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 |
| B2g | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 |
| Eg | 2 | 0 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 |
| A1u | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 |
| A2u | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 |
| B1u | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 |
| B2u | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 |
| Eu | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 |
Now we can evaluate the products:-
a) A1g x Eg = [ 1x2 , 1x0 , 1x(-2) , 1x0 ,1x0 , 1x2 ,1x0 ,1x(-2) ,1x0 ,1x0 ]
= [ 2 , 0 , -2 , 0 , 0 ,2 ,0 ,-2, 0 , 0 ] =Eg
b) A1u xB2u = [ 1x1 , 1x(-1) , 1x(1) , 1x(-1) ,1x1 , -1x(-1) ,-1x1 ,-1x(-1) ,-1x1 ,-1x(-1) ]
= [ 1 , -1 , 1 , -1 , 1 , 1 , -1 , 1 , -1 , 1 ] = B2g
c) Eg x Eu = [ 2x2 , 0x0 , (-2)x(-2) , 0x0 ,0x0 , (-2)x2 ,0x0 ,2x(-2) ,0x0 ,0x0 ]
= [ 4 , 0 , 4 , 0 , 0 , -4 , 0 , -4 , 0 , 0 ] = A1u + A2u +B1u +B2u
d) B2g xA1u xB2u = [ 1x1x1 , -1x1x(-1) , 1x(1)x1 , -1x1x(-1) ,1x1x1 , 1x(-1)x(-1) ,-1x(-1)x1 ,1x(-1)x(-1) ,-1x(-1)x1 ,1x(-1)x(-1) ]
= [ 1, 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1, 1] = A1g
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