[jestquim]

I need to construct the molecular orbital diagram for the hypothetical species Li4, which has the following geometrical arrangement:



The first step is to identify the point symmetry group. In this particular case, we consider that there is only one axis of rotation of order four (actually, other symmetry elements can be observed, but this is a previous consideration of the exercise): C4 (Schöenflies notation).

Once the point group has been identified, we consult the literature for the table of characteristics. For this symmetry:



The associated reducible representation is then constructed. In this step, I have a doubt, because I do not understand the concept of "reducible representation" and its usefulness in this theory. According to what I have given in class, it is the number of atomic orbitals that remain unchanged when a symmetry element is applied on the solid. If we go by this definition, such representations would be:



Thus, the irreducible representation of this molecule would be: A + B + E

And, now, according to what has been taught in the course, it would be necessary to use the projection operator to determine the linear combinations adapted to the symmetry. But, here the truth is that I'm starting to make a mess.

Once here, how could I continue? Or, perhaps, they know of a simpler way of constructing orbital diagrams. Sorry, but I don't know why the images are not placed where they should be.

[sjb]

Compare and contrast https://chemistry.stackexchange.com/questions/152969/construction-of-the-molecular-orbital-diagram-for-a-species-m4

[Orcio_87]

C4 (Schöenflies notation). Once the point group has been identified

Are you blind !? Li₄ square has D_4h molecular symmetry !

About writing of orbitals - maybe this will help you:

http://www.huntresearchgroup.org.uk/teaching/teaching_MOs_year2/L2_Notes_web.pdf

After all - you should get 6 MO, first 4 are 1s Li atomic orbitals.

5 MO is bonding for all four Li atoms.

6 MO is antibonding to diagonal Li-Li.

I think that this orbital (6 MO) is degenerated (both Li-Li diagonals are the same) and (in normal circumstances) will led to fragmentation: Li₄ ---> 2 Li₂

[Corribus]

@OP

The significance of the reducible representation can be better understood by considering what the LCAO-MO approach is: molecular electrons exist in molecular orbitals that are approximated as sums (linear combinations) of atomic orbitals centered on each nucleus. The reason symmetry treatments are useful is that implicit in this model is the idea that molecular orbitals must have symmetry similar to the underlying nuclear spatial arrangement, which limits the different combinations of atomic orbitals that would be allowed. Group theory helps you determine which of the large number of possible combinations of atomic orbitals have the appropriate symmetry to serve as molecular orbitals (and, with some approximations and semi-empirical treatments, their approximate energies). The process is rather simple in concept, although application of it can be confusing when you first get started: First, you identify what symmetry class the molecule fits in. Second, the associated character table is sort of a decoder tool that allows you to perform symmetry operations on the atomic orbitals and generate the group of molecular orbitals that satisfy the symmetry requirements. When you do this, you get a "reducible representation", which is an aggregate representation of all the symmetry-allowed molecular orbitals (one molecular orbital output for each atomic orbital input) lumped together. Finally, the projection operator splits up the grouped molecular orbitals (reducible representation) based on their respective symmetry representations (irreducible representations) - i.e., it allows us to conclude that the group of allowed molarular orbitals includes one of this type, two of this type, and one of this type. The "type" (irreducible representations) that is assigned to each molecular orbital provides further information on the nature of each molecular orbital and the way that the atomic orbitals are combined to construct it - which can further tell us a lot about the molecular wavefunction, where electrons and charge are likely to be concentrated, and certain spectroscopic predictions.

Hope that helps you understand the process involved. If you still have trouble with the specific problem, don't hesitate to come back and ask.