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Symmetry Adopted Linear Combinations (SALCs)
Symmetry Adapted Linear Combinations(SALCs), also known as group orbitals, are the linear combinations of atomic orbitals of ligands or outer atoms that possess the equivalent symmetry to interact with the central atom.

Deriving SALCs
Construction of SALCs involves assigning directions(x,y, and z) to a molecule and use the valence atomic orbitals of interest on the outer atoms as a basis set to generate a reducible representation. Each orbital in the same group that is unchanged or unmoved after a transformation takes place, it contributes +1 to the character. Any orbital that is transformed into the opposite of itself will contribute -1, and 0 if it is moved.

For a water molecule, which belongs to the C2v point group, only the 1s orbitals of the hydrogen atoms will be considered because each hydrogen only has 1s orbital that can interact with the valence orbitals of matching symmetry on the central atom.

SALCs are determined by reducing the reducible representations which represent the basis set of valence orbitals of the ligands. The reduction formula to reduce reducible representations to irreducible representations is:

$$n_i = \frac{1}{h} \sum N X_{RR} X_{IRR}$$

Where:

ni is the number of times the irreducible representation i occurs in the reducible representation

h is the order of the group

N is the coefficient in each class of symmetry

XRR is the character for reducible representation

XIRR is the character for irreducible representation

By applying the above formula, the reducible representation for 1s orbitals of the hydrogen atoms can be reduced:

$$n_{A1} = \frac{1}{4}[1(2)(1)+1(0)(1)+1(2)(1)+1(0)(1)] = 1 A_1$$

$$n_{A2} = \frac{1}{4}[1(2)(1)+1(0)(1)-1(2)(1)-1(0)(1)] = 0$$

$$n_{B1} = \frac{1}{4}[1(2)(1)-1(0)(1)+1(2)(1)-1(0)(1)] = 1 B_1$$

$$n_{B2} = \frac{1}{4}[1(2)(1)-1(0)(1)-1(2)(1)+1(0)(1)] = 0$$

Summing the values gives the SALCs that have the same symmetry as the valence orbitals on the central atom:

$$\Gamma_s = A_1 + B_1$$

Next step is to find atomic orbitals on the central atom with matching symmetry as SALCs, then take the linear combination of atomic orbitals(LCAOs) by combining the symmetrically related atomic orbitals on the central atom with SALCs. These combinations give rise to molecular orbitals(MO).

Projection Operator Technique - A Systematic Way to Derive SALCs
A systematic method may be required in order to properly select the correct linear combination of atomic orbitals to form the right SALCs. Such a method is called the Projection Operator Technique.

In the Projection Operator Technique, one atomic orbital can be selected to represent all the others in each set of identical atomic orbitals to determine how they are transformed under each class of symmetry operation belonging to the point group. The projections are then used to derive the wave functions of the SALCs or group orbitals by multiplying the projections by the characters of each irreducible representation and summing them.

It is important that the SALCs belonging to the same point group are orthogonal to each other. This can be quickly verified by the great orthogonality theorem in which the scalar product between the two SALCs must equal to zero.

Here is a step by step approach to derive SALCs using the projection operator technique for the water molecule:



Normalization of SALCs
In addition to deriving SALCs, normalization must also be considered for the relative proportions of individual atomic orbitals. A normalizing factor is assigned to each group orbital and represents the relative size of the atomic orbitals in the group compared to that of the central atom. It is given by the formula:

$$N = \frac{1}{\surd\sum c_i ^2}$$

Where:

N represents the normalizing factor

c is the coefficient or scaler of each atomic orbital in the same group