Electronic Structure of Polyatomic Molecules

Chem 3240 · Lecture 8.4

Davit Potoyan

Two Pictures of Bonding

  • Molecular orbitals (MO): electrons delocalized over the whole molecule.

  • Valence bond (VB): bonds built from overlapping atomic orbitals.

  • VB naturally gives hybrid orbitals that explain molecular shape.

  • Both are approximations to the same exact wavefunction.

Filling the Diatomic Series

  • Fill MOs in order of energy: each pair is bonding plus antibonding.

\[\text{BO} = \tfrac{1}{2}\left(N_\text{bonding} - N_\text{antibonding}\right)\]

  • Higher bond order means shorter \(R_e\) and larger \(D_e\).
  • \(\pi\) orbitals hold 4 electrons; \(\sigma\) hold 2.

The Diatomic Trend

Molecule Config BO \(R_e\) (Å) \(D_e\) (eV)
\(H_2\) \((1\sigma_g)^2\) 1.0 0.741 4.78
\(C_2\) \(Be_2(1\pi_u)^4\) 2.0 1.242 6.36
\(N_2\) \((3\sigma_g)^2\) 3.0 1.094 9.90
\(O_2\) \((1\pi_g)^2\) 2.0 1.207 5.21
  • \(N_2\) has the strongest bond: triple bond, shortest, deepest well.

The Non-Crossing Rule

  • States of the same symmetry never cross.

  • As \(R\) changes, same-symmetry levels avoid each other.

  • Sorts orbitals into clean bonding / antibonding ladders.

Valence Bond and Overlap

  • A bond forms where atomic orbitals have non-zero overlap.

  • Orbitals must share the same symmetry to overlap.

  • Hybrids are linear combinations of orbitals on a single atom.

  • They only form as other atoms approach, not in free atoms.

sp Hybrids: \(BeH_2\)

\[\psi_{sp}^1 = \tfrac{1}{\sqrt{2}}(2s + 2p_z)\]

\[\psi_{sp}^2 = \tfrac{1}{\sqrt{2}}(2s - 2p_z)\]

  • One \(s\) plus one \(p\) give two hybrids.
  • Geometry is linear: H-Be-H.

sp2 Hybrids: \(BH_3\)

  • Mix \(2s\), \(2p_z\), \(2p_x\) into three hybrids.

\[\psi^1_{sp^2} = \tfrac{1}{\sqrt{3}}\,2s + \sqrt{\tfrac{2}{3}}\,2p_z\]

  • Each forms a \(\sigma\) bond with an H.
  • Geometry is trigonal planar, angles \(120^\circ\).

sp3 Hybrids: \(CH_4\)

\[\psi^1_{sp^3} = \tfrac{1}{2}(2s + 2p_x + 2p_y + 2p_z)\]

  • One \(s\) plus three \(p\) give four hybrids.

  • Geometry is tetrahedral.

  • Count is conserved: orbitals in = hybrids out.

Lone Pairs: \(H_2O\)

  • Oxygen is \(sp^3\) hybridized.

  • Two hybrids bond to H; two hold lone pairs.

  • Predicted angle \(109^\circ\), observed \(104^\circ\).

  • Lone pairs squeeze the bond angle.

Reading an MO Diagram

  • Atomic orbitals on the sides, molecular orbitals in the middle.

  • Bonding orbitals drop below the atomic levels.

  • Antibonding rise above.

From Orbitals to Numbers

  • Real calculations replace atomic orbitals with Gaussian basis sets.

  • Gaussians make the required integrals analytic.

  • Used in Hartree-Fock (SCF) and beyond.

  • Handy rule: a product of Gaussians is a Gaussian.

Takeaway

The MO picture predicts bond order, length, and strength across the diatomic series, while the valence bond picture builds bonds from overlapping atomic orbitals and gives \(sp\), \(sp^2\), \(sp^3\) hybrids that fix molecular geometry.