Symmetry-adapted linear combinations

So far, we have only asserted the forms of the linear combinations (such as s1, etc.) that have a particular symmetry. Group theory also provides machinery that takes an arbitrary basis, or set of atomic orbitals (sA, etc.), as input and generates combinations of the specified symmetry. Because these combinations are adapted to the symmetry of the molecule, they are called symmetry-adapted linear combinations (SALC). Symmetry-adapted linear combinations are the building blocks of LCAO molecular orbitals, for they include combinations such as those used to construct molecular orbitals in benzene. The construction of SALCs is the first step in any molecular orbital treatment of molecules. The technique for building SALCs is derived by using the full power of group theory. We shall not show the derivation (see Further reading), which is very lengthy, but present the main conclusions as a set of rules: 1 Construct a table showing the effect of each operation on each orbital of the original basis.

2 To generate the combination of a specified symmetry species, take each column in turn and: (i) Multiply each member of the column by the character of the corresponding operation. (ii) Add together all the orbitals in each column with the factors as determined in (i). (iii) Divide the sum by the order of the group.

For example, from the (s_N, s_A, s_B, s_C) basis in NH₃ we form the table shown in the margin. To generate the A1 combination, we take the characters for A1 (1,1,1,1,1,1); then rules (i) and (ii) lead to

ψ∝ s_N+ s_N+· · ·=6s_N

The order of the group (the number of elements) is 6, so the combination of A1 symmetry that can be generated from sN is sN itself. Applying the same technique to the column under sA gives

ψ = (sA + sB + sC + sA +sB+ sC) =  (sA + sB + sC)

The same combination is built from the other two columns, so they give no further information. The combination we have just formed is the s1 combination we used before (apart from the numerical factor). We now form the overall molecular orbital by forming a linear combination of all the SALCs of the specified symmetry species. In this case, therefore, the a1 molecular orbital is

ψ=c_Ns_N+c₁s₁

This is as far as group theory can take us. The coefficients are found by solving the Schrödinger equation; they do not come directly from the symmetry of the system. We run into a problem when we try to generate an SALC of symmetry species E, because, for representations of dimension 2 or more, the rules generate sums of SALCs. This problem can be illustrated as follows. In C3v, the E characters are 2, −1, −1, 0, 0, 0, so the column under sN gives

ψ= (2s_N − sN − sN +0+0+0) =0

The other columns give

 (2s_A − s_B − sC)  (2s_B − s_A − s_C)  (2s_C − s_B − s_A)

However, any one of these three expressions can be expressed as a sum of the other two (they are not ‘linearly independent’). The difference of the second and third gives   (sB −sC), and this combination and the first,  (2s_A − s_B − s_C) are the two (now linearly independent) SALCs we have used in the discussion of e orbitals.

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