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Atomic QM to Molecular QM (16.4-16.6)
• Solution of SE for molecules is more complicated due to much larger
number of electrons and multiple nuclei
– SE is still not exactly solvable since more than one electron is involved
– Atomic orbitals are not appropriate since multiple nuclei are involved
• Just as atoms combine to form molecules, atomic orbitals (AO) should
combine to form molecular orbitals (MO)
– Linear combination of atomic orbitals (LCAO) is an approximation used to solve the
molecular SE
– When creating MOs from AOs, there is a one-to-one correspondence
– Atomic orbital overlap is the driving force in whether an appropriate MO is generated
(this included orbital phases)
• MOs have similar properties to AOs (and other wavefunctions)
– Two electrons can reside in each MO
– MOs are orthogonal to one another
– Energy order is related to nodal character
Molecular Orbital Energy Diagrams (16.7)
• MO energy diagrams are useful in that they show how atomic orbitals
from different atoms may combine to molecular orbitals
– Sigma bonds form from s- and p-orbitals, pi bonds from p-orbitals
– Orbital energies and shapes dictate which AOs are used to generate MOs
• Valence atomic orbitals are used to form chemical bonds
– Two AOs combine to form a bonding orbital and and anti-bonding orbital
– Bonding orbitals have considerable electron density between atoms, anti-bonding
orbitals have a node in between the two atoms
– AOs of similar energy combine more readily to form MOs
• Electrons from each atom are used to fill the MO energy diagram
– Lowest energy orbitals are filled first
– Up to two electrons can be placed in a single MO
– For degenerate MOs, one electron is placed in each MO before the electron is paired up
MO Diagrams of Diatomic Molecules (16.7 and 16.11)
• For homonuclear diatomic molecules, the orbitals on each atom are
paired with their counterparts on the other atom since they are
degenerate
– s-orbitals combine to form σ- and σ*-orbitals, as do p-orbitals along bond axis (usually
designated as pz-orbitals)
– px- and py-orbitals combine to form a pair of degenerate π- and π*-bonds
• Oxygen is an interesting case, and only MO theory describes its behavior
correctly
– MO diagram differs from Lewis structure (how?)
• Heteronuclear diatomic molecules have similar MO diagrams, but AOs are
no longer degenerate
– More electronegative the atom, the lower in energy the atomic orbitals
– AOs of similar energy overlap more readily
Bond Orders, Energies, and Lengths (16.10)
• Bond orders of diatomic molecules can be determined by summing up
bonds and anti-bonds
– Each bond (σ or π) adds to the bond order
– Each anti-bond reduces the bond order
– Partially filled bonds and anti-bonds also contribute to bond order (e.g., oxygen)
• Bond order and bond energy can be determined from bond order
information
– More bonds means stronger bonds
– More bonds means shorter bonds
• One can also predict the stability of ions based on MO diagrams
– Adding electrons to bonding orbitals strengthen bonds, removing electrons from
bonding orbitals weakens them
– Adding electrons to anti-bonding orbitals weakens bonds, removing electrons from antibonding orbitals strengthens bonds
Bonding and Anti-bonding Orbitals of H2
Electron Density for H2 orbitals
MO Energy Diagram
MO Energy Diagrams for H2 and He2
MO Energy Diagram for F2
MO Energy Diagram for N2
MO Energy Diagram for Homonuclear Diatomics
MO Energy Diagram for HF