The Hydrogen Molecule Ion

Chem 3240 · Lecture 8.2

Davit Potoyan

Why \(H_2^+\)?

  • The simplest molecule: two nuclei, one electron.

  • Solvable exactly, but the math is ugly.

  • Instead we build an approximate wavefunction from atomic orbitals.

  • Goal: intuition for chemical bonding from a model we can do by hand.

Building MOs from AOs

  • Combine two hydrogen 1s orbitals.

\[\psi_{\pm} = c\,\big(1s_A \pm 1s_B\big)\]

  • This is the LCAO approximation.
  • Identical nuclei force \(c_1 = c_2 \equiv c\) by symmetry.

The Overlap Integral

  • Normalizing the bonding combination:

\[1 = c^2 \int (1s_A + 1s_B)^2\, d\tau = c^2(2 + 2S)\]

\[S = \int 1s_A(\vec{r})\,1s_B(\vec{r})\, d\tau\]

  • The overlap \(S\) measures how much the AOs share space.
  • Fixes the normalization constant \(c = 1/\sqrt{2(1+S)}\).

Bonding and Antibonding MOs

\[\psi_{+} \equiv \psi_g = \frac{1}{\sqrt{2(1+S)}}\left( 1s_A + 1s_B \right)\]

\[\psi_{-} \equiv \psi_u = \frac{1}{\sqrt{2(1-S)}}\left( 1s_A - 1s_B \right)\]

  • The antibonding orbital has a node between the nuclei: zero density there.

Where the Electron Goes

  • Bonding \((+)\): density piled up between the nuclei.

  • Antibonding \((-)\): density scooped out, a node at the midpoint.

  • This buildup of charge glues the nuclei together.

Symmetry Labels: \(g\) and \(u\)

  • Invert through the center of symmetry at the midpoint.

  • \(\psi(-\vec{r}) = +\psi(\vec{r})\): even parity, label \(g\).

  • \(\psi(-\vec{r}) = -\psi(\vec{r})\): odd parity, label \(u\).

  • For \(\sigma\) orbitals: bonding is \(g\), antibonding is \(u\).

  • The analytic overlap: \(S(R) = e^{-R}\left( 1 + R + \tfrac{R^3}{3}\right)\), with \(S(0)=1\).

Variational Energy

  • Plug the LCAO trial function into the variational energy:

\[E = \frac{c_1^2 H_{AA} + 2c_1 c_2 H_{AB} + c_2^2 H_{BB}}{c_1^2 + 2c_1 c_2 S + c_2^2}\]

  • \(H_{AA} = H_{BB}\): the Coulomb integral (attractive).
  • \(H_{AB} = H_{BA}\): the resonance integral.

The Secular Determinant

  • Minimizing over \(c_1, c_2\) gives a generalized eigenvalue problem:

\[\begin{pmatrix}H_{AA} - E & H_{AB} - SE\\ H_{AB} - SE & H_{BB} - E\end{pmatrix}\begin{pmatrix} c_1\\ c_2\end{pmatrix} = 0\]

  • Non-trivial \(c\)’s exist only if the determinant vanishes: \[\begin{vmatrix}H_{AA} - E & H_{AB} - SE\\ H_{AB} - SE & H_{BB} - E\end{vmatrix} = 0\]

Two Energy Roots

\[E_g(R) = E_{1s} + \frac{J(R) + K(R)}{1 + S(R)}\]

\[E_u(R) = E_{1s} + \frac{J(R) - K(R)}{1 - S(R)}\]

  • \(J(R)\): Coulomb term. \(K(R)\): resonance term, the source of bonding.
  • \(E_g\) lowers the energy, \(E_u\) raises it relative to separated atoms.

Energy vs Internuclear Distance

  • \(E_g\) has a minimum: a stable bond.

  • \(E_u\) is repulsive at all \(R\): a purely excited state.

  • Crude model: bond length 132 pm vs 106 pm experiment.

  • Improved by adding more terms to the LCAO.

The MO Diagram

  • Two AOs split into a lower \(\sigma_g\) and a higher \(\sigma_u^{*}\).

  • \(\sigma\): zero angular momentum about the axis, \(\lambda = 0\).

  • \(\pi\) orbitals from \(p_{x,y}\) are doubly degenerate.

  • Same recipe scales to all homonuclear diatomics.

Takeaway

Mixing two 1s orbitals gives a bonding \(\sigma_g\) that piles electron density between the nuclei and lowers the energy, plus an antibonding \(\sigma_u^{*}\) with a node that raises it. The variational secular determinant turns this LCAO picture into the two energy curves \(E_g\) and \(E_u\).