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

In summary: A1gx3A2g=3A2g, ...In summary, the configuration (e1g)2(e2u)2 in D6h symmetry generates the following terms: 31A1g, 3A1g, 5A1g, 1A2g, 23A2g, 31E2g, and 23E2g. The incorrect answer is due to a typo and incorrect reduction of terms during the direct product calculation.
  • #1
magnesium12
19
0

Homework Statement


What terms are generated by the configuration (e1g)2(e2u)2 in D6h symmetry?

Homework Equations


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

(e1g)x(e1g) = A1g + [A2g] + E2g
(e2u)x(e2u) = A1g + [A2g] + E2g
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 seperately
(e1g)2x(e1g)2 = 1A1g + 3A2g + 1E2g
(e2u)2x(e2u)2 = 1A1g + 3A2g + 1E2g

Then find direct product of [(e1g)x(e1g)]x[(e2u)x(e2u)]
1A1g 1A1g
3A2g x 3A2g =
1E2g 1E2g

1A1gx1A1g + 1A1gx3A2g + 1A1gx 1E2g
3A2gx1A1g + 3A2gx3A2g + 3A2gx 1E2g =
1E2gx1A1g + 1E2gx3A2g + 1E2gx 1E2g

1A1g + 3A1g + 2A2g + 4A2g + 2E2g
1A1g + 3A1g + 5A1g + 2A2g + 4A2g + 3E2g
1A1g + 1A2g + 1E2g + 2E2g + 3E2g

= 31A1g + 23A1g + 5A1g +1A2g +22A2g +24A2g + 1E2g + 22E2g + 23E2g

The correct answer is
31A1g + 3A1g + 5A1g +1A2g +23A2g + 31E2g + 23E2g

I assume I'm messing up in when doing this direct product: [(e1g)x(e1g)]x[(e2u)x(e2u)].
but I'm not sure exactly what I'm doing wrong. Any help is really appreciated!
 
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  • #2
You messed some things up: This is probably a typo:
"So first I found the terms generated by e1gxe1g and e2uxe2u seperately
(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 seperately
(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, ...
 

1. What is the meaning of "Electronic Spec - Terms generated by config of (e1g)2(e2u)2"?

The term "Electronic Spec" refers to the electronic configuration of a molecule or atom, which describes the arrangement of electrons in the energy levels or orbitals. In this case, the configuration is (e1g)2(e2u)2, meaning that there are two electrons in the e1g orbital and two electrons in the e2u orbital.

2. How is the electronic configuration of a molecule or atom determined?

The electronic configuration of a molecule or atom is determined by the number of electrons present and the rules of electronic structure. These rules include the Aufbau principle, Hund's rule, and the Pauli exclusion principle, which dictate the order in which electrons fill the available energy levels or orbitals.

3. What is the significance of the (e1g)2(e2u)2 configuration?

The (e1g)2(e2u)2 configuration is significant because it describes the electronic structure of a molecule or atom, which is directly related to its chemical and physical properties. This configuration can also be used to predict the molecule's reactivity, stability, and bonding characteristics.

4. How does the electronic configuration affect the behavior of a molecule?

The electronic configuration of a molecule or atom plays a crucial role in determining its behavior. The arrangement of electrons in the energy levels or orbitals directly influences the molecule's chemical reactions, bonding, and reactivity. It also affects the molecule's physical properties, such as its shape, polarity, and magnetism.

5. Can the electronic configuration of a molecule change?

Yes, the electronic configuration of a molecule can change through chemical reactions or interactions with other molecules. When a molecule undergoes a chemical reaction, its electrons may be redistributed among different energy levels or orbitals, resulting in a new electronic configuration. This change in electronic structure can significantly alter the molecule's properties and behavior.

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