Construction of the molecular orbital diagram for a species M4

I need to construct the molecular orbital diagram for the hypothetical species $${Li_4}$$, 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): $$C_{4}$$(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.

• Could you explain why you think "It is clearly seen that the only symmetry element present is an axis of rotation of order 4"? Unless I am misunderstanding your diagram I would have thought there are more than just that. What about reflections for starters? Jun 15 '21 at 16:34
• You are right. On closer inspection, you can see that there are more elements of symmetry, such as a mirror plane. I have commented that there is only one axis of rotation of order 4, because that is how it appears in the study slides. Perhaps this assumption is motivated by the fact that we want to study cyclic molecules ($C_{n}$ symmetry), and this exercise is presented as an introductory example. @IanBush Jun 15 '21 at 16:38
• Try the book "Molecular Symmetry and Group Theory: A Programmed Introduction to Chemical Applications" by Alan Vincent (link). This book helped me a lot when I first started learning group theory for chemistry. Also, when using projection operator method, check whether you are using all of the symmetry operations. For example a $\ce{C_2}$ operation in the graph actually represents two symmetry operations, $\ce{C_2}$ and $\ce{C_2^{-1}}$, similarly $\ce{C_4}$ actually has 4 operations which you need to consider. Jun 15 '21 at 19:06
• Thank you, but I checked and it is not available at my university library. @ShoubhikRMaiti Jun 15 '21 at 19:15
• Haha yeah it does sound a bit confusing now that I read it again. If I have time, I will try to write a more detailed answer tomorrow. Jun 15 '21 at 19:32