Linear combination of atomic orbitals

Linear Combination of Atomic Orbitals (LCAO) is a method used in quantum chemistry and solid state physics to describe the molecular orbitals (MOs) of a molecule in terms of the atomic orbitals (AOs) of the individual atoms that make up the molecule. The fundamental idea is that the molecular orbitals can be approximated as linear combinations of the atomic orbitals.

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1 s

ϕ_{A}

and

ϕ_{B}

, choose their signs positive and define their orbital overlap

S

. At nonzero separation,

0 < S < 1

, and the normalized combinations are

σ_{\pm} (r) = \frac{ϕ _{A} ( r ) \pm ϕ _{B} ( r )}{2 ( 1 \pm S )}, S = \int_{R^{3}} ϕ_{A} (r) ϕ_{B} (r) d^{3} r .

Here

σ_{+} = σ_{g}

and

σ_{-} = σ_{u}^{*}

a = 1

, the two denominators are

2 (1 + S) \approx 1.654013

and

2 (1 - S) \approx 1.124385

\int (ϕ_{A} \pm ϕ_{B})^{2} = 1 + 1 \pm 2 S

. Dividing by

2 (1 \pm S)

gives one. The local probability density is

∣ σ_{\pm} ∣^{2} = \frac{ϕ _{A}^{2} + ϕ _{B}^{2} \pm 2 ϕ _{A} ϕ _{B}}{2 ( 1 \pm S )} .

The cross term is the interference; adding the separate densities first would lose it. These formulas normalize the chosen linear combination exactly, while the restricted choice of atomic orbitals remains an approximation to an actual molecular orbital.