JEE Chemistry Chemical Bonding and Molecular Structure Notes Set 09

Molecular Orbital Theory: Basic Ideas

In molecular orbital (MO) theory the electrons of a molecule occupy molecular orbitals that extend over all the nuclei, just as atomic electrons occupy atomic orbitals around one nucleus. MOs are formed by the linear combination of atomic orbitals (LCAO), and the number of MOs equals the number of atomic orbitals combined. Two atomic orbitals give one bonding MO (waves added in phase, electron density piles up between the nuclei, lower energy) and one antibonding MO (waves subtracted, a node between the nuclei, higher energy).
MOs are filled by the Aufbau principle, the Pauli exclusion principle and Hund's rule; the energy order of the MOs is fixed by experiment, not by an (n + l) rule.

Conditions for Combination and Types of MOs

Atomic orbitals combine only if (1) they have nearly the same energy (1s with 1s, not 1s with 2s), (2) they have the same symmetry about the molecular axis (with z as axis, 2pz combines with 2pz but not with 2px or 2py), and (3) they overlap to a large extent.
A sigma MO is symmetric about the internuclear axis; it comes from s-s, s-pz and pz-pz overlap. A pi MO comes from side-on overlap of px-px or py-py, has a nodal plane containing the axis, and its electron density is zero on the axis. The bonding sigma MO is gerade (symmetric on inversion) while the bonding pi MO is ungerade; for antibonding orbitals it is the opposite. Delta MOs arise from d orbitals.

Energy Order of MOs

For O2, F2 and Ne2: sigma 1s < sigma* 1s < sigma 2s < sigma* 2s < sigma 2pz < (pi 2px = pi 2py) < (pi* 2px = pi* 2py) < sigma* 2pz.
For Li2 to N2 (and their ions, as well as CO, CN-, NO+): sigma 1s < sigma* 1s < sigma 2s < sigma* 2s < (pi 2px = pi 2py) < sigma 2pz < (pi* 2px = pi* 2py) < sigma* 2pz. Here the 2s and 2pz orbitals are close in energy and mix, pushing sigma 2pz above the pi 2p pair. The 2s-2p gap grows from B to F, so the mixing fades by O2, which returns to the normal order.

Bond Order, Stability, Bond Length and Magnetism

Bond order = (Nb - Na)/2, where Nb and Na are the numbers of electrons in bonding and antibonding MOs. A positive bond order means a stable species; zero means no stable bond. Higher bond order means shorter bond length and larger bond energy. If two species have the same bond order, the one with more antibonding electrons is less stable ({N2^-} is less stable than {N2^+}).
If all MOs are doubly filled the species is diamagnetic; unpaired electrons make it paramagnetic. Magnetic moment = \( \sqrt{n(n+2)} \) Bohr magnetons for n unpaired electrons.

Homonuclear Diatomic Molecules

\( \ce{H2} \): bond order 1, 74 pm, diamagnetic. \( \ce{He2} \): bond order 0, does not exist. \( \ce{Li2} \): bond order 1, diamagnetic, known in vapour. \( \ce{Be2} \): bond order 0, not stable.
\( \ce{B2} \): 10 electrons, the last two in separate pi 2p orbitals (because of s-p mixing), so bond order 1 and paramagnetic, a fact Lewis structures cannot explain. \( \ce{C2} \): bond order 2, diamagnetic, and both bonds are pi bonds (four electrons in the two pi orbitals). \( \ce{C2^2-} \) (acetylide) has bond order 3 like \( \ce{N2} \).
\( \ce{N2} \): bond order 3 (one sigma, two pi), 109.8 pm, 942 kJ per mole, diamagnetic. \( \ce{O2} \): bond order 2, two unpaired electrons in the pi* orbitals, so paramagnetic. \( \ce{F2} \): bond order 1, diamagnetic. \( \ce{Ne2} \): bond order 0.
Oxygen species: \( \ce{O2^+} \) 2.5 (112 pm, 1 unpaired), \( \ce{O2} \) 2 (120 pm, 2 unpaired), \( \ce{O2^-} \) 1.5 (128 pm, 1 unpaired), \( \ce{O2^2-} \) 1 (149 pm, none). Nitrogen species: \( \ce{N2} \) 3, \( \ce{N2^+} \) 2.5, \( \ce{N2^-} \) 2.5 (the extra electron goes into pi*, so it is less stable than \( \ce{N2^+} \)), \( \ce{N2^2-} \) 2.

Isoelectronic Species and Colour of Oxygen

\( \ce{CO} \), \( \ce{CN^-} \), \( \ce{NO^+} \) and \( \ce{C2^2-} \) have 14 electrons like \( \ce{N2} \), so each has bond order 3 and is diamagnetic. Species with 13 or 15 electrons such as \( \ce{NO} \) (bond order 2.5, one unpaired electron in pi*) are paramagnetic.
Although \( \ce{O2} \) is paramagnetic, it is colourless as a gas because the HOMO-LUMO gap is large and absorption lies in the ultraviolet, not the visible region. Liquid oxygen is pale blue.

Metallic Bonding

Most metals are close-packed with 8 to 12 neighbours. Metallic bonding is the attraction between the positive metal ions and the mobile, delocalised electrons of the crystal. Two models explain it. The electron-sea model pictures ions in a sea of freely moving valence electrons. The band model treats the crystal as one huge molecule whose orbitals merge into bands: a metal has a partly filled band, or a filled band overlapping an empty band, so electrons can move into the conduction band. Insulators have a filled band separated from the empty band by a large band gap.
Conductivity of a metal falls on heating because thermal vibration of the ions hinders electron flow. Metallic bond strength rises with the number of valence electrons available and with a smaller atomic radius, which explains the high melting points of d-block metals (d electrons also bond) and the low enthalpy of atomisation of group 12 metals.

Worked Examples

1. Order of bond energy of \( \ce{N2} \), \( \ce{N2^+} \), \( \ce{N2^-} \), \( \ce{N2^2-} \): bond orders are 3, 2.5, 2.5 and 2. \( \ce{N2^+} \) has fewer antibonding electrons than \( \ce{N2^-} \), so the order is \( \ce{N2} \) > \( \ce{N2^+} \) > \( \ce{N2^-} \) > \( \ce{N2^2-} \).
2. Diamagnetic among \( \ce{C2} \), \( \ce{O2^2-} \), \( \ce{Li2} \), \( \ce{N2^+} \): the first three have all electrons paired; \( \ce{N2^+} \) has one unpaired electron in sigma 2p.
3. Which have bond order 3 among CO, \( \ce{CN^-} \), \( \ce{NO^+} \), \( \ce{O2^+} \)? The first three (14 electrons); \( \ce{O2^+} \) has 2.5.