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
A_1g	1	1	1	1	1	1	1	1	1	1
A_2g	1	1	1	-1	-1	1	1	1	-1	-1
B_1g	1	-1	1	1	-1	1	-1	1	1	-1
B_2g	1	-1	1	-1	1	1	-1	1	-1	1
E_g	2	0	-2	0	0	2	0	-2	0	0
A_1u	1	1	1	1	1	-1	-1	-1	-1	-1
A_2u	1	1	1	-1	-1	-1	-1	-1	1	1
B_1u	1	-1	1	1	-1	-1	1	-1	-1	1
B_2u	1	-1	1	-1	1	-1	1	-1	1	-1
E_u	2	0	-2	0	0	-2	0	2	0	0

Now we can evaluate the products:-

a) A_1g x E_g = [ 1x2 , 1x0 , 1x(-2) , 1x0 ,1x0 , 1x2 ,1x0 ,1x(-2) ,1x0 ,1x0 ]

= [ 2 , 0 , -2 , 0 , 0 ,2 ,0 ,-2, 0 , 0 ] =E_g

b) A_1u xB_2u = [ 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 ] = B_2g

c) E_g x E_u = [ 2x2 , 0x0 , (-2)x(-2) , 0x0 ,0x0 , (-2)x2 ,0x0 ,2x(-2) ,0x0 ,0x0 ]

= [ 4 , 0 , 4 , 0 , 0 , -4 , 0 , -4 , 0 , 0 ] = A_1u + A_2u +B_1u +B_2u

d) B_2g xA_1u xB_2u = [ 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] = A_1g

queen_honey_blossom answered 1 year ago

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