Molecular Orbital Description of the Hydrogen Molecule

Chem 3240 · Lecture 8.3

Davit Potoyan

From One Electron to Two

• \(H_2^+\) had a single electron and an exact MO.

• Neutral \(H_2\) adds a second electron.

• New term: electron-electron repulsion \(1/r_{12}\).

• This term couples the electrons together.

• No analytic solution exists for the \(H_2\) electronic equation.

• We proceed approximately with LCAO-MO.

The Molecular Hamiltonian

\[\begin{aligned} H = &-\frac{\hbar^2}{2m_e}(\Delta_1 + \Delta_2) \\ &+ \frac{e^2}{4\pi\epsilon_0}\Big(\tfrac{1}{R} + \tfrac{1}{r_{12}} \\ &- \tfrac{1}{r_{A1}} - \tfrac{1}{r_{A2}} - \tfrac{1}{r_{B1}} - \tfrac{1}{r_{B2}}\Big) \end{aligned}\]

• The troublesome term is \(1/r_{12}\).
• A simple product wavefunction is not enough.

Filling the Bonding Orbital

• Pauli: two opposite-spin electrons share one spatial orbital.

• Assume the \(H_2\) orbitals match those of \(H_2^+\).

• Both electrons occupy \(1\sigma_g\): configuration \((1\sigma_g)^2\).

\[1\sigma_g(1) = \frac{1}{\sqrt{2(1+S)}}\big(1s_A(1) + 1s_B(1)\big)\]

The Total Wavefunction Must Be Antisymmetric

• Exchange of electrons must flip the sign.
• Build it as a Slater determinant.

\[\psi_{MO}^{(1\sigma_g)^2} = \frac{1}{\sqrt{2}}\begin{vmatrix} 1\sigma_g(1)\alpha(1) & 1\sigma_g(1)\beta(1)\\ 1\sigma_g(2)\alpha(2) & 1\sigma_g(2)\beta(2) \end{vmatrix}\]

• \(\alpha, \beta\) are the two spin states.

Space and Spin Factorize

• Expanding the determinant separates the parts.

\[\psi_{MO}^{(1\sigma_g)^2} = \frac{(1s_A(1)+1s_B(1))(1s_A(2)+1s_B(2))}{2\sqrt{2}(1+S_{AB})}\big(\alpha(1)\beta(2) - \alpha(2)\beta(1)\big)\]

• Symmetric spatial part times an antisymmetric spin singlet.
• Not an exact eigenfunction: energy comes from \(\langle \psi | H | \psi\rangle\).

The Energy Expression

\[E(R) = 2E_{1s} + \frac{e^2}{4\pi\epsilon_0 R} - \textnormal{integrals}\]

• \(2E_{1s}\): two separated H atoms.
• \(e^2/4\pi\epsilon_0 R\): nuclear repulsion.
• “integrals”: Coulomb, exchange, overlap of charge clouds.

How Good Is the Simple MO?

• The model predicts a bound molecule. The method works.

Quantity	Simple MO	Experiment
\(R_e\)	84 pm	74.1 pm
\(D_e\)	255 kJ/mol	458 kJ/mol

• Bond too long, binding too weak.
• Time to improve the wavefunction.

Ionic and Covalent Pieces

• The MO product hides distinct physical terms.

• Covalent: one electron on each atom (H + H).

• Ionic: both electrons on one atom (\(H^- + H^+\)).

• Simple MO weights them equally, overcounting ionic.

Weighting Them Separately

\[\psi = c_1\,\psi_{\textnormal{covalent}} + c_2\,\psi_{\textnormal{ionic}}\]

\[\psi_{\textnormal{covalent}} = 1s_A(1)1s_B(2) + 1s_A(2)1s_B(1)\] \[\psi_{\textnormal{ionic}} = 1s_A(1)1s_A(2) + 1s_B(1)1s_B(2)\]

• \(c_1, c_2\) are variational and depend on \(R\).
• Result: \(R_e\) = 74.9 pm, \(D_e\) = 386 kJ/mol. Much closer.

Toward Exact Agreement

• Add higher atomic orbitals to the basis.

• Hartree-Fock solves this efficiently but ignores correlation.

• Full configuration interaction captures correlation.

• \(D_e\) = 36117.8 cm\(^{-1}\) (CI) vs \(36117.3 \pm 1.0\) cm\(^{-1}\) (expt).

• \(R_e\) = 74.140 pm vs 74.139 pm. Essentially exact.

Takeaway

Neutral \(H_2\) has no analytic solution because of \(1/r_{12}\), so we place two opposite-spin electrons in \(1\sigma_g\) as a Slater determinant. This simple LCAO-MO already binds the molecule, and separating ionic from covalent terms, then adding configuration interaction, drives the bond length and binding energy to essentially exact agreement.