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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

queen_honey_blossom answered 2 years ago
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