# Advanced Inorganic Chemistry/SALCs and the projection operator technique

SALCs refers to Symmetry Adapted Linear Combinations, which are generated via use of the projection operator technique. This technique is a mathematical method which outputs a function called a SALC that models the orbitals of the atoms of interest.<ref> These SALCs are mathematical representations and therefore bare no physical meaning. They are commonly used in the generation of molecular orbitals.

## Projection Operator Technique

The Projection Operator Technique utilizes the extended character table which includes each symmetry operation separately. The technique involves multiple steps, listed below in the form of the BF$_3$ example.
1. We begin by determining the reducible representations of the orbitals in question. For BF$_3$, which has the D$_{3h}$ point group, the D$_{3h}$ character table is used. For our example, we will consider the sigma and pi bonds of the fluorine atoms and determine their reducible representations. Using the character table, we identify the reducible representations. They are listed below.
Γ$_σ$ = Α$_1$'+ Ε'
Γ$_{πx}$ = Α$_2$' + Ε'
Γ$_{πy}$ = Α$_2$" + Ε"
Γ$_{πz}$ = Α$_1$' + Ε'
It is noted here that the π$_z$ orbitals transform as σ orbitals.
2. We now use the extended character table for the D$_{3h}$ Character table to generate the SALCs of the stated orbitals. To do this, we apply each symmetry operation to the given orbital and note which orbital it transforms into. We then add up each projector operator function in accordance to each irreducible representation.
3. Common techniques for dealing with the double degeneracy of the E' and E" representations<ref>:
i. If the principal axis is a C$_3$ axis, we subtract the functions corresponding to the σ$_2$ and σ$_3$ orbitals.
ii. If the principal axis is a C$_4$ axis, we apply the projection operator technique to the σ$_2$ and use the function that is derived.
iii. We must ensure that the representations are orthogonal to one another in order for them to be correct.
4. We can now apply the coefficients within the matrix to determine the coefficients of the function found via the projection operator technique.
We determine the SALCs for the example orbitals to be the following:
SALC$_σ$(A$_1$') = 3σ$_1$ + 3σ$_2$ + 3σ$_3$ = σ$_1$ + σ$_2$ + σ$_3$
SALC$_σ$(E') = 4σ$_1$ - 2σ$_2$ - 2σ$_3$ = 2σ$_1$ - σ$_2$ - σ$_3$
SALC$_σ$(E') = 2σ$_2$ - 2σ$_3$ = σ$_2$ - σ$_3$
SALC$_{πz}$(A$_1$') = 3π$_1$ + 3π$_2$ + 3π$_3$ = π$_1$ + π$_2$ + π$_3$
SALC$_{πz}$(E') = 4π$_1$ - 2π$_2$ - 2π$_3$ = 2π$_1$ - π$_2$ - π$_3$
SALC$_{πz}$(E') = 2π$_2$ - 2π$_3$ = π$_2$ - π$_3$
SALC$_{πx}$(A$_2$') = 3π$_1$ + 3π$_2$ + 3π$_3$ = π$_1$ + π$_2$ + π$_3$
SALC$_{πx}$(E') = 4π$_1$ - 2π$_2$ - 2π$_3$ = 2π$_1$ - π$_2$ - π$_3$
SALC$_{πx}$(E') = 2π$_2$ - 2π$_3$ = π$_2$ - π$_3$
SALC$_{πy}$(A$_2$") = 3π$_1$ + 3π$_2$ + 3π$_3$ = π$_1$ + π$_2$ + π$_3$
SALC$_{πy}$(E") = 4π$_1$ - 2π$_2$ - 2π$_3$ = 2π$_1$ - π$_2$ - π$_3$
SALC$_{πy}$(E") = 2π$_2$ - 2π$_3$ = π$_2$ - π$_3$

## Uses of SALCs

SALCs help us to understand which orbitals will be bonding, antibonding, and nonbonding. For example, by comparing the symmetry of the SALCs to that of the orbitals of the central atom, it is possible to generate the corresponding molecular orbitals. Although SALCs are mathematical representations that have no physical meaning, they are useful in providing a tool for deriving molecular orbitals.

## References

1.
http://www4.ncsu.edu/~franzen/public_html/CH437/lec8/pdf/projection_1.pdf
2.
http://troglerlab.ucsd.edu/GroupTheory224/Chap2B.pdf