1. MODULE 11 Diatomic Molecules and Beyond Our next task is to extend the procedures and ideas that we have developed for H 2 + to larger molecules. The track we follow is the customary one, that is, start with homonuclear diatomics, of which H 2 is the paradigm, then consider heteronuclear diatomics, and finally polyatomic molecules. The overall procedure is to construct sets of MOs and then put in the electrons according to the Pauli principle analogous to the aufbau method for atoms The result is similar because we end up with the electronic configurations of molecules.

2. MODULE 11 The Hydrogen Molecule H 2 has an additional electron over its molecule-ion and we can imagine a scheme such as shown in Figure. The electron configuration follows the same rule as for the He atom, where the added electron was placed into the lowest energy 1s orbital with its spin opposed (Pauli). Thus the added electron is placed into the bonding orbital defined for the 1s A + 1s B combination.

3. MODULE 11 The extra electron adds the distances r 12, r 2B, and r 2A. This adds complexity in that an electron-electron repulsion term appears, as well as two more electron-nucleon potential energy terms. The hamiltonian for the hydrogen molecule in the Born- Oppenheimer approximation and in atomic units is

4. MODULE 11 where  b is the bonding orbital (1  ). This can be rearranged to Since the hamiltonian is independent of the spin terms, we calculate the energy using only the spatial part of equation. Thus our molecular wavefunction   is given by Where the first term on the RHS is the normalization constant (it is the square of the normalization constant for   

5. MODULE 11 Thus our molecular wavefunction is a product of molecular orbitals both of which are linear combinations of atomic orbitals. The procedure for constructing molecular wavefunctions is known as the Linear Combination of Atomic Orbitals-Molecular Orbitals, or LCAO-MO, method. This is a commonly used procedure for a variety of molecules. The ground state energy of H 2 can then be calculated as before using Where the hamiltonian and the wavefunction are as above

6. MODULE 11 The results of the integration are shown in figure. As for the molecule ion the results lack accuracy, but the form is clearly shown and the results can be improved using a larger basis set. The ground state configuration of H 2 is classified as 1  g, which echoes the term symbols for atoms (not yet covered). The superscript 1 is the multiplicity of the state (here the 2 electrons are paired, S = 0). The  (analogous to atomic S) indicates that the total OAM around the inter-nuclear axis is zero because both electrons are in s-type AOs

7. MODULE 11 Whereas for atoms we used L = l 1 +l 2, etc to compute the total OAM, in molecules we use  = 1 + 2 to compute the total OAM In H 2,  = 0, hence . The subscript g indicates the overall parity of the state and we calculate this from the individual values for the electrons by the products g x g = g; g x u = u; u x u = g In H 2 both electrons occupy the  g orbital, hence the overall parity is g. Had one occupied a 2  u orbital the overall parity would have been u.

8. MODULE 11 Homonuclear Diatomic Molecules Beyond Hydrogen The MOs for homonuclear diatomics beyond H 2 are formed from pairs of AOs of many-electron atoms Therefore we must to use orbitals beyond 1s (higher energy). Nevertheless the 1s AOs are involved and we can start there and the first pair of MOs is given by In Module 10 we saw that the two MOs have cylindrical symmetry with respect to the inter-nuclear axis. For this reason they are termed  orbitals.

9. MODULE 11 Because they are constructed from a pair of 1s orbitals they are conventionally labeled  1s, indicating they are  orbitals formed from 1s AOs, and nothing else. The positive combination,    builds up electron density between the nucleons and is the bonding orbital, The negative combination,    excludes electron density from that region, and is the antibonding orbital, often written as  1s. As we saw in Module 10 the bonding orbital has gerade inversion symmetry, hence it is symbolized as  g 1s, and the antibonding orbital is u, therefore  u 1s. We do not include the asterisk in the symbol since the subscript gives the antibonding designation.

10. MODULE 11 Next consider combinations (sum and diff) of a pair of 2s AOs. Both AOs have the same energy and the same symmetry and these are factors important in the LCAO procedure. The two MOs resulting from 2s A +/-2s B are  g 2s and  u 2s, one bonding and the other antibonding. These MOs have similar shape to those shown in the figure above, except they are larger because the composite AOs are larger. By the same token the energy of the 2s-based orbitals are higher than those based on 1s. The energies can be calculated in an analogous way to that outlined above for the 1s combination.

11. MODULE 11 The relative energies are in the sequence In the hydrogenic ions the 2s and 2p orbitals are degenerate, but this degeneracy is lifted in many- electron atoms because of differences in the nuclear screening, thus E 2s < E 2p In Figure the way in which the atomic 2p orbitals combine is indicated. 2p z AOs are oriented along the inter-nuclear axis. Both u and g combinations of the 2p z orbitals are symmetric around the axis and are therefore of  -type.

12. MODULE 11 The 2p x and 2p y combinations overlap “sideways” and the result is that the four MOs generated do not have cylindrical symmetry. This change in symmetry leads to the designation of  -bonds. [An easy way of determining whether a MO is  or  is to “view” the orbital along the inter-nuclear axis. If you see a circle of electron density centered on the nucleus (like an s-AO), then that orbital is . If what you see looks like a p-AO (two circles separated by a nodal plane) then you are dealing with a  -orbital.] The negative combination 2p zA -2p zB is the one that yields the bonding MO; for the other combinations the positive ones are bonding. The bonding orbital resulting from the 2p z combination transforms as g on inversion, whereas the opposite is true for the bonding orbitals resulting from the 2p x and 2p y pair combinations.

13. MODULE 11 From figure we see that the energies depend on Z. Moreover the  g 2p z orbital changes so much with Z that between N 2 and O 2 it switches with the degenerate x and y-pairs. Also Figure shows the electron occupancy of the MOs built up by the LCAO- MO procedure, with aufbau. Not shown are H 2 and He 2. The former we have already considered, it has two electrons of opposed spins in the  g 1s MO and its configuration is (  g 1s) 2.

14. MODULE 11 He 2 has a configuration (  g 1s) 2 (  u 1s) 2 in which there are an equal number of bonding and antibonding electrons. The two sets cancel such that there is no net bond, according to the simple version of MO theory used here (He 2 has been detected spectrometrically in the gas phase at T~0.001 K). The molecular ion He 2 + is a stable, but reactive, species. Some properties of hydrogen and helium molecules are in the Table

15. MODULE 11 bond order = (n b -n a )/2 where n a and n b are the number of electrons in bonding and antibonding orbitals, respectively. Note that bond order can be a half-integer. Diatomic lithium has an electron configuration of (  g 1s) 2 (  u 1s) 2 (  g 2s) 2 and thus the bond order is one. Li 2 exists in Li vapor and its bond energy is 105 kJmol -1. Electron density contour maps, generated by computer solutions of the Schrödinger equation are presented in the figure over page

16. MODULE 11 The important thing to note about these diagrams is that the  g 1s and  u 1s electrons are found close to the nuclei and play virtually no part in the bonding interaction, which is mostly due to the electrons in the  g 2s MO. This leads to an alternative way of writing molecular electronic configuration for Li 2 as KK(  g 2s) 2 in which K represents the filled n = 1 shell of the Li atom.

17. MODULE 11 This becomes even more accurate as Z increases and the 1s electrons are held progressively more tightly than in the Li case. At this level of approximation only the valence electrons need to be considered for the homonuclear diatomics beyond He 2. Returning to the orbital energy diagram and focusing on B 2 and O 2, we see that each molecule has a degenerate pair of orbitals (  u 2p x,y in the case of B and  g 2p x.y in the case of O). Hund’s rule informs us that we place electrons into degenerate sets of orbitals one at a time  in order to maximize the spin multiplicity. Both molecular boron and molecular oxygen have been shown to have paramagnetic (triplet) ground states.

18. MODULE 11 In the Table are shown the ground state electron configurations of the 2 nd row homonuclear diatomics

19. MODULE 11 Q

20. Q

21. Q