Electronic Spec - Terms generated by config of (e1g)2(e2u)2

[magnesium12]

Homework Statement

What terms are generated by the configuration (e_1g)²(e_2u)² in D_6h symmetry?

Homework Equations

configuration (χ_d)² gives terms ¹(symmetric product) + ³(antisymmetric product) where χ=symmetry of an orbital and d = degenerate

(e_1g)x(e_1g) = A_1g + [A_2g] + E_2g
(e_2u)x(e_2u) = A_1g + [A_2g] + E_2g
where brackets [ ] indicate antisymmetric product

s=singlet, d=doublet, t=triplet, q=quartet, qu=quintet
sxs = s
sxd = d
sxt = t
sxq = q
dxd = s + t
dxt = d+q
dxq = t+qu
txt = s+t+qu
txq = d+q+sextet
qxq = s+t+qu +septet

The Attempt at a Solution

So first I found the terms generated by e1gxe1g and e2uxe2u separately
(e_1g)²x(e_1g)² = ¹A_1g + ³A_2g + ¹E_2g
(e_2u)²x(e_2u)² = ¹A_1g + ³A_2g + ¹E_2g

Then find direct product of [(e_1g)x(e_1g)]x[(e_2u)x(e_2u)]
¹A_1g ¹A_1g
³A_2g x ³A_2g =
¹E_2g ¹E_2g

¹A_1gx¹A_1g + ¹A_1gx³A_2g + ¹A_1gx ¹E_2g
³A_2gx¹A_1g + ³A_2gx³A_2g + ³A_2gx ¹E_2g =
¹E_2gx¹A_1g + ¹E_2gx³A_2g + ¹E_2gx ¹E_2g

¹A_1g + ³A_1g + ²A_2g + ⁴A_2g + ²E_2g
¹A_1g + ³A_1g + ⁵A_1g + ²A_2g + ⁴A_2g + ³E_2g
¹A_1g + ¹A_2g + ¹E_2g + ²E_2g + ³E_2g

= 3¹A_1g + 2³A_1g + ⁵A_1g +¹A_2g +2²A_2g +2⁴A_2g + ¹E_2g + 2²E_2g + 2³E_2g

The correct answer is
3¹A_1g + ³A_1g + ⁵A_1g +¹A_2g +2³A_2g + 3¹E_2g + 2³E_2g

I assume I'm messing up in when doing this direct product: [(e_1g)x(e_1g)]x[(e_2u)x(e_2u)].
but I'm not sure exactly what I'm doing wrong. Any help is really appreciated!

 

[DrDu]

You messed some things up: This is probably a typo:
"So first I found the terms generated by e1gxe1g and e2uxe2u separately
(e1g)2x(e1g)2 = 1A1g + 3A2g + 1E2g
(e2u)2x(e2u)2 = 1A1g + 3A2g + 1E2g" and should read
"So first I found the terms generated by e1gxe1g and e2uxe2u separately
(e1g)x(e1g) = 1A1g + 3A2g + 1E2g
(e2u)x(e2u) = 1A1g + 3A2g + 1E2g"

Then a singlet times a triplet does not give a quartett:
1A1gx3A2g is not 4A2g but 3A2g.
Similarly 1A1gx 1E2g is not 2E2g but 1E2g. There are more of this kind.
It would be helpful if you could write down for each term in the sum how it is reduced out.
E.g. 1A1gx1A1g=1A1g, ...