Answer to Question #109666 in Physical Chemistry for Shubham

Question

Draw the molecular orbitals formed by the s – s combination of atomic orbitals. Also illustrate the
electron densities for these orbitals and explain them

Answer 1

The drawing of molecular orbitals (MO) formed from s-s atomic orbitals (AO) is below. The axis shows increase of energy (E).

Electron densities are depicted as the clouds around the nuclei centered within them.

For the binding MO $\sigma$ at lower energy the electron density homogeneously surrounds both atoms with the maximum in the interatomic region. For the antibonding MO $\sigma$^* at higher energy electron density surrounds each atom, but has a minimum in the interatomic region.

Explanation: an orbital is a region with high enough probability of electron occurence (more than 90 %). The probability is a space integral of probability density, which is square of the absolute value of wavefunction $\psi$ . The s-wavefunction is spherical and the atomic s-orbital is spherical too. The wavefunction corresponding to bonding MO is a linear sum of atomic s-wavefunctions: $\Psi(\sigma)=\psi1+\psi2$ , that is why the probability (i.e. electron density) for bonding $\sigma$ MO is proportional to the absolute value of $\Psi(\sigma)$² = ($\psi1+\psi2$)² =

$\psi1$² + 2*$\psi1$*$\psi2$ + $\psi2$². The sum ψ1² + ψ2² is a sum of electron densities of atomic s-orbitals, and the additive 2*ψ1*ψ2 is positive in the region where s-orbitals intercept, i.e. between atoms.

The wavefunction corresponding to antibonding MO is еру linear difference of atomic s-wavefunctions: Ψ(σ^*)=ψ1 - ψ2 , that is why the electron density for antibonding σ^* MO is proportional to the absolute value of Ψ(σ^*)² = (ψ1 - ψ2)² = ψ1² - 2*ψ1*ψ2 + ψ2².

The sum ψ1² + ψ2² is a sum of electron densities of atomic s-orbitals, and the term (-2*ψ1*ψ2) is negative in the region where s-orbitals intercept, i.e. between atoms.

The green and violet colors show just the sign (+ or -) of atomic s-orbitals in the expressions for molecular orbitals.