1. PY3P05 Lecture 13: Diatomic orbitals oHydrogen molecule ion (H 2 + ) oOverlap and exchange integrals oBonding/Anti-bonding orbitals oMolecular orbitals

2. PY3P05 Schrödinger equation for hydrogen molecule ion oSimplest example of a chemical bond is the hydrogen molecule ion (H 2 + ). oConsists of two protons and a single electron. oIf nuclei are distant, electron is localised on one nucleus. Wavefunctions are then those of atomic hydrogen. r ab rbrb rara + + - a b o  a is the hydrogen atom wavefunction of electron belonging to nucleus a. Must therefore satisfy and correspondingly for other wavefunction,  b. The energies are therefore E a 0 = E b 0 = E 0 (1) HaHa

3. PY3P05 Schrödinger equation for hydrogen molecule ion oIf atoms are brought into close proximity, electron localised on b will now experience an attractive Coulomb force of nucleus a. oMust therefore modify Schrödinger equation to include Coulomb potentials of both nuclei: where o To find the coefficients c a and c b, substitute into Eqn. 2: (2) HaHa HbHb

4. PY3P05 Solving the Schrödinger equation oCan simplify last equation using Eqn. 1 and the corresponding equation for H a and H b. oBy writing E a 0  a in place of H a  a gives oRearranging, oAs E a 0 = E b 0 by symmetry, we can set E a 0 - E = E b 0 - E = -  E, (3)

5. PY3P05 Overlap and exchange integrals o  a and  b depend on position, while c a and c b do not. Now we know that for orthogonal wavefunctions but  a and  b are not orthoganal, so oIf we now multiply Eqn. 3 by  a and integrate the results, we obtain oAs -e|  a | 2 is the charge density of the electron => Eqn. 4 is the Coulomb interaction energy between electron charge density and nuclear charge e of nucleus b. oThe -e  a  b in Eqn. 5 means that electron is partly in state a and partly in state b => an exchange between states occurs. Eqn. 5 therefore called an exchange integral. Normalisation integral Overlap integral (4) (5)

6. PY3P05 Overlap and exchange integrals oEqn. 4 can be visualised via figure at right. oRepresents the Coulomb interaction energy of an electron density cloud in the Coulomb field on the nucleus. oEqn. 5 can be visualised via figure at right. oNon-vanishing contributions are only possible when the wavefunctions overlap. oSee Chapter 24 of Haken & Wolf for further details.

7. PY3P05 Orbital energies oIf we mutiply Eqn 3 by  a and integrate we get oCollecting terms gives, oSimilarly, oEqns. 6 and 7 can be solved for c 1 and c 2 via the matrix equation: oNon-trivial solutions exist when determinant vanishes: (6) (7)

8. PY3P05 Orbital energies oTherefore (-  E + C) 2 - (-  E·S + D) 2 = 0 => C -  E = ± (D -  E ·S) oThis gives two values for  E:and oAs C and D are negative,  E b <  E a o  E b correspond to bonding molecular orbital energies.  E a to anti-bonding MOs. oH 2 + atomic and molecular orbital energies are shown at right. Energy (eV) -13.6

9. PY3P05 Bonding and anti-bonding orbitals oSubstituting for  E a into Eqn. 6 => c b = -c a = - c.The total wavefunction is thus oSimilarly for  E b => c a = c b = c, which gives oFor symmetric case (top right), occupation probability for  is positive. Not the case for a asymmetric wavefunctions (bottom right). oEnergy splits depending on whether dealing with a bonding or an anti-bonding wavefunction. Anti-bonding orbital Bonding orbital

10. PY3P05 Bonding and anti-bonding orbitals oThe energy E of the hydrogen molecular ion can finally be written o‘+’ correspond to anti-bonding orbital energies, ‘-’ to bonding. oEnergy curves below are plotted to show their dependence on r ab

11. PY3P05 Molecular orbitals

12. PY3P05 Molecular orbitals

13. PY3P05 Molecular orbitals oEnergies of bonding and anti-bonding molecular orbitals for first row diatomic molecules. oTwo electrons in H 2 occupy bonding molecular orbital, with anti-parallel spins. If irradiated by UV light, molecule may absorb energy and promote one electron into its anti-bonding orbital.

14. PY3P05 Molecular orbitals