# Is there a better coordinate framework and symmetry operator for constructing molecular orbitals of molecules with higher-order geometry?

The following excerpt is taken from [1] (with a few rewording, emphasizes are mine).

In an attempt to construct molecular orbital (MO) of any molecule, one needs to determine the symmetries of possible symmetry-adapted linear combinations (SALCs) that are formed from pendant-atom atomic orbitals. This prequisite can be proceeded with following steps:

1. Use directional properties (i.e. vectors) of orbitals on the outer atoms as a basis for a representation of the SALCs in the point group of the molecule.
2. Generate a reducible representation for all possible SALCs by noting whether vectors are shifted or not by each class of operations of the group.
3. Decompose the representation into its component (irreducible representations) to determine the symmetry species of the SALCs.

So far, I have practiced of constructing MOs of several molecules having simple geometry (low-order molecules) e.g. $$\ce{H2O}$$, $$\ce{NH3}$$, and others but I am worry with higher-order molecules, particularly adamantane, $$\ce{Re2Cl8}$$ cluster, arachno-$$\ce{B4H10}$$, and others.

I want to accomplish those tasks with hands and pens. I know there are several softwares where one can draw the molecules, assign the point group, and draw the molecular orbital but here, I want to know the process behind it and I want to figure it without using any software, so I am alright if the tasks will be very exhaustive and tedious. I want to know if there exists any method to simplify the process. At there, I want to ask several questions as below.

On the first step above from the excerpt, to generate group orbitals based on the representation of basic symmetry requirements, I am getting difficult with the natural way to place several atoms on Cartesian coordinate and I am wondering if there is any applicable projection or cartography that can ease up the process of generating the coordinates because most molecules are not planar. There are some books that present $$\ce{CH4}$$ molecule in a cubic framework which can ease the assignment of the coordinates for the atoms but I can not find any evidence of placing those higher-order molecules on polyhedral framework (perhaps like Bravais lattices that I can define the unit cells and so on). I got several papers of how these molecules had processed on the softwares but it turned out that the molecules were having coordinates that I worry I will get sloppy when I need to symmetrize it again. Is there any literature that can deal with this problem?

For the second step from the excerpt, is there a generator or a (matrix) operator that can ease up the process? Most matrices I found are already worked in backward process, that after visual inspection of symmetry operation, then the matrices were generated, which is totally very useless to me in this case. If this exists, it would be very relieving because so far I can not find any digestable mathematical method (not visual inspection) or algorithm that can perform such symmetry operations on those higher-order molecules.

Thank you.

P.S.: If possible, please illustrate your answers like a student trying to understand and answer the problems so I can work on those molecules later. Use molecules that are not as simple as water but not fullerene (small cluster molecules that are on par with molecules that I will work on) because most papers give examples of the latter one that I feel too strenuous to draw it with hands and pens.

References

1. Carter, R. L. Molecular Symmetry and Group Theory (Wiley, 1998).