Linear combination of atomic orbitals

The LCAO approximation builds a molecular orbital as a linear combination of single-centre atomic orbitals. For two equivalent normalized real $1s$ functions ${\varphi }_A$ and ${\varphi }_B$, choose their signs positive and define their orbital overlap $S$. At nonzero separation, $0<S<1$, and the normalized combinations areHere ${\sigma }_{+}={\sigma }_g$ and ${\sigma }_{-}={\sigma }_u^{\ast }$. Equal coefficients use the equivalence of the two centres; a general LCAO need not have equal coefficients. See Field, MIT 5.61, lecture 24, pp. 3–6: LCAO coefficients and overlap integrals.

For the Gaussian function toy's $a=1$, the two denominators are $\sqrt{2(1+S)}\approx 1.654013$ and $\sqrt{2(1-S)}\approx 1.124385$. To check wavefunction normalization, expand the square: $\int ({\varphi }_A\pm {\varphi }_B{)}^2=1+1\pm 2S$. Dividing by $2(1\pm S)$ gives one. The local probability density isThe 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.

A linear combination of atomic orbitals chosen to transform under an irreducible representation of the molecular point group. Relative orbital signs must be transformed as well as the positions of their centres. Such combinations organize a Hamiltonian operator into symmetry blocks; combinations of the same symmetry generally still mix and need not themselves be molecular orbitals.

Four equivalent $s$ atomic orbitals at the vertices of a regular tetrahedron carry the tetrahedral symmetry representation $A_1\oplus T_2$. The normalized equal sum is $A_1$; three coordinate-sign combinations span $T_2$. Every normalized coefficient vector has entries $\pm 1/2$ in a suitable orthonormal real basis. These functions simplify a Hamiltonian invariant under the tetrahedral operations, but equivalent symmetry copies on other sites may still mix.

The orbital overlap of normalized real atomic orbitals is their inner product $S=\int {\varphi }_A{\varphi }_B$. It is a dimensionless measure of their common amplitude, not an electron count. For the Gaussian function toy, increasing $a$ from $1$ to $2$ reduces $S$ from $e^{-1}\approx 0.367879$ to $e^{-4}\approx 0.018316$. As $S$ tends to zero, both LCAO normalization denominators tend to $\sqrt{2}$.