[heyhey]

The molecular orbitals of an allyl radical are shown in the attachment below.

How do I derive the symmetry label and multiplicity of the following states: HOMO-1^2, HOMO^1, LUMO^0 (refer to attachment)?

I did the following steps:

(1) Assigned the point group to C2V
(2) Noted that the label A corresponds to symmetry w/ respect to the principal axis (C2), the label 1 refers to symmetry with respect to a C2 axis perpindicular to the principal axis and ' refers to symmetry with respect to σh mirror plane
(3) with those in mind, I thought the symmetry labels would be:
HOMO-1: A1
HOMO-1^2: A1*A1 = A1
LUMO-0 ->(what is LUMO-0 supposed to even mean?!?!): A1

Can any one spot my error with regards to the symmetry labels for HOMO-1, HOMO-1^2 and LUMO-0? What is LUMO-0 even supposed to look like?

I have no idea to do the multiplicity part....any one know?

I would be SO grateful for a response!!!

[Corribus]

Could you please provide the exact language of the question you need to answer?

[heyhey]

Could you please provide the exact language of the question you need to answer?

The question (labeled 2.1a in the attachment) is "The ground state of the allyl radical arises from the electron configuration HOMO-1² , HOMO¹ and LUMO⁰ . Describe this state by its symmetry label and multiplicity."

[Corribus]

Ok, that's one state you need to worry about, not three - defined by the electronic configuration where there are two electrons in the HOMO-1 orbital, one electron in the HOMO orbital and zero electrons in the LUMO orbital (the superscript refers to the number of electrons in the orbital). Essentially, what is the symmetry of the state where one electron has been removed from the molecule - allyl cation instead of the allyl radical.

You're kind of on the right track. The important thing to do first is assign symmetry representations to each molecular orbital. Then you can do some basic multiplication to arrive at a symmetry representation for the entire state.

[heyhey]

Thank you for the tips. So for the symmetry I got: A1 x B2 x A1 = B2. Does that sound right?

How do you derive the muliplicity of the state?

[Corribus]

Thank you for the tips. So for the symmetry I got: A1 x B2 x A1 = B2. Does that sound right?

It depends on how you assign your axes. Assuming the xz plane is the molecular plane, then yes this would be right. (Another quick tip: for conjugated organic molecules, pi molecular orbitals always belong to symmetry groups that are antisymmetric (have -1 character) for symmetry operations involving reflections through the plane. In this case, the σ_xz operation. So, immediately you should know that none of the molecular orbitals can have A₁ or B₁ symmetry.)

How do you derive the muliplicity of the state?

Multiplicity of a state is determined by the equation 2S + 1, where S is the total spin angular momentum for the state. Effectively all you have to do is count how many unpaired electrons you have.