Symmetry-adapted linear combination

Symmetry- adapted linear combination ( SALK ) of atomic orbitals (AOs) is used to construct molecular orbitals (MOs) according to the LCAO approximation ( linear combination of atomic orbitals ).

To construct two MOs from two AOs, the following theorems are useful:

• If the overlap integral of the AOs is zero, then they are unsuitable
• The more the AOs differ energetically, the smaller the interaction
• All possible MOs must form bases for irreducible representations of the point group of the molecule.

The MOs of a molecule appear as irreducible representations in the character table of the molecule.

example

Combination of two 1s orbitals

There are two possible combinations: + - (odd) and + + (even)

Such a molecule belongs to the point group whose character table looks like this: ${\displaystyle D_{\infty h}}$

${\displaystyle D_{\infty h}}$	${\displaystyle E}$	${\displaystyle 2C_{\infty }}$	${\displaystyle \infty \sigma _{v}}$	${\displaystyle i}$	${\displaystyle 2S_{\infty }}$	${\displaystyle \infty C_{2}}$
${\displaystyle \Gamma _{1s}}$	2	2	2	0	0	0

The reducible representations are here 2,2,2,0,0,0. By Ausreduzieren gives the irreducible representations: . The names come from the fact that they are - bonds , because the electron density is particularly strong between the atomic nuclei . g stands for even and u for odd, see above. ${\displaystyle \Gamma _{1s}=\sigma _{g}^{+}+\sigma _{u}^{+}}$ ${\displaystyle \sigma }$

In the first column of the character board there are always only ones. In order to get to the reducible representations above by addition, 1 + 1 = 2 and 1 + (- 1) = 0, the irreducible representations and must look like this: ${\displaystyle \Gamma _{+}}$ ${\displaystyle \Gamma _{-}}$

${\displaystyle D_{\infty h}}$	${\displaystyle E}$	${\displaystyle 2C_{\infty }}$	${\displaystyle \infty \sigma _{v}}$	${\displaystyle i}$	${\displaystyle 2S_{\infty }}$	${\displaystyle \infty C_{2}}$
${\displaystyle \Gamma _{+}}$	1	1	1	1	1	1
${\displaystyle \Gamma _{-}}$	1	1	1	${\displaystyle -1}$	${\displaystyle -1}$	${\displaystyle -1}$

The irreducible representations can also be explained as follows:

• +1: nothing changes
• -1: the wave function is converted into its inverse

for example:

• In the even function , none of the operations change anything (+ + → + +) ${\displaystyle \sigma _{g}^{+}}$
• With the odd function , identity, rotation around an infinite axis or reflection around one of the infinitely many mirror planes change nothing. Inversion, rotation mirroring or rotation around one of the twofold axes invert the function (+ - → - +) ${\displaystyle \sigma _{u}^{+}}$

→ As a basis for an LCAO approximation with 1s orbitals one should use and . ${\displaystyle \Gamma _{+}}$ ${\displaystyle \Gamma _{-}}$