The exact electronic dissociation energy of is at the equilibrium separation

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Molecular structure and spectra

Deriving the Spectrum, Ch The Born-Oppenheimer separation and the chemical bond

Problem

The exact electronic dissociation energy of H2\mathrm{H_2} is De=4.75 eVD_e = 4.75\,\mathrm{eV} at the equilibrium separation Re=0.741 A˚R_e = 0.741\,\mathrm{\mathring{A}}, with the Born-Oppenheimer electronic energy E(R)E(R) taken as zero at R→∞R\to\infty (two separated ground-state hydrogen atoms) and equal to −De-D_e at ReR_e, where E(R)E(R) is stationary. ((a)) Using the molecular virial theorem, 2⟨T^⟩+⟨V⟩=−R dE/dR2\langle\hat T\rangle + \langle V\rangle = -R\,\mathrm{d}E/\mathrm{d}R, find the changes Δ⟨T^⟩\Delta\langle\hat T\rangle and Δ⟨V⟩\Delta\langle V\rangle of the two-electron kinetic and potential energy between the separated atoms and the equilibrium molecule. ((b)) In helium's variational treatment, the two-electron exchange integral KK is positive and lowers the energy of the parallel-spin (triplet) state relative to the antiparallel (singlet) state, which is the origin of Hund's first rule for atomic terms. Explain, referring to the Heitler-London exchange integral XX between orbitals centred on the two protons, why the analogous exchange term has the opposite effect in H2\mathrm{H_2}: it is the singlet, not the triplet, that is bound.

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