Lithium forms a weakly bound diatomic with a single bond between the two valence…

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

Lithium forms a weakly bound diatomic

\mathrm{Li_2}

with a single

\sigma_g

bond between the two

2s

valence electrons (the two filled

1s^2

cores on each atom are inert). Its measured dissociation energy is

D_e = 1.05\,\mathrm{eV}

R_e = 2.67\,\mathrm{\mathring{A}}

, with the Born-Oppenheimer electronic energy

E(R)\to 0

R\to\infty

(two separated ground-state lithium atoms) and

E(R_e) = -D_e

. ((a)) Using the molecular virial theorem

2\langle\hat T\rangle+\langle V\rangle = -R\,\mathrm{d}E/\mathrm{d}R

evaluated at

R_e

, find the changes in the total electronic kinetic energy and in the total potential energy (nuclear repulsion included),

\Delta\langle\hat T\rangle

and

\Delta\langle V\rangle

, between the separated atoms and equilibrium

\mathrm{Li_2}

. ((b)) The Heitler-London treatment of

\mathrm{H_2}

finds that the two-centre exchange integral

X

between the

1s

orbitals is negative at bonding distances, binding the singlet. Lithium's

2s

valence electron is bound by

5.39\,\mathrm{eV}

, against

13.6\,\mathrm{eV}

for hydrogen's

1s

. Using the central-field picture of atomic structure (Chapter 5) for why a

2s

valence electron sits, on average, well outside the filled

1s^2

core of the same atom, explain why

\mathrm{Li_2}

is bound by the same singlet-favouring exchange yet far more weakly than

\mathrm{H_2}

(

D_e=4.75\,\mathrm{eV}

), and at a much larger separation.