INORGANIC CHEMISTRY: BONDING THEORIES (VBT & MOT)
Valence Bond Theory (VBT)
VBT describes bonding as the overlap of atomic orbitals (AOs) from atoms that come together to form bonds between atoms.
Focuses on localized bonds created by direct orbital overlap (overlap of orbitals on adjacent atoms).
Type of bond in VBT is determined by the type of orbital overlap: sigma (σ), pi (π), and delta (δ).
Sigma (σ) bonds
Formed by head-on overlap of atomic orbitals along the internuclear axis (z-axis).
Exhibit cylindrical symmetry around the bond axis.
Example: H–H bond in H₂ is a σ bond formed by overlap of 1s(AO) from each H.
Pi (π) bonds
Formed by sideways overlap of atomic orbitals (p orbitals) with lobes above and below the internuclear axis.
Do not have cylindrical symmetry about the bond axis.
Example: The π bond in an alkene (C=C) from sideways overlap of p orbitals.
Delta (δ) bonds
Formed by sideways overlap involving dx2−y2 or dxy type orbitals with a four-lobe geometry.
Less common than σ and π bonds but can contribute in certain metal-ligand bonding situations.
Origin of σ, π, and δ bonds
σ bonds arise from end-to-end overlap that builds a single shared electron density along the internuclear axis.
π bonds arise from sideways overlap producing electron density above and below the bond axis.
δ bonds arise from more complex multi-lobe overlaps (e.g., involving dx2−y2 and dxy) leading to additional bonding interactions.
As the number and arrangement of overlapping AOs change, the bond type and bond strength change accordingly.
VB description of H₂ (vdw): example of a simple bond
In H₂, a σ bond forms with cylindrical symmetry around the internuclear axis.
The molecular energy curve describes how the molecule’s energy depends on internuclear distance (bond length).
Depth of the energy minimum reflects bond strength: deeper minimum → stronger bond.
Steepness of the curve reflects bond stiffness: steeper curve → larger vibrational frequency when stretched or compressed.
The energy required to dissociate H₂ (bond dissociation energy) is shown as a deep well; typical approximate value in many texts is around D_0 \,\approx\; 432\ \text{kJ mol}^{-1} for H–H in H₂.
The minimum corresponds to the equilibrium H–H bond length (internuclear distance at which energy is minimized).
Vibrational frequency of the molecule is governed by the curvature of this potential energy surface near the minimum.
VB description of molecules (homonuclear diatomic vs polyatomic)
Homonuclear diatomic molecules: bond formation can be described by combining AOs on each atom to maximize direct overlap (σ and π bonds).
Polyatomic molecules: central atoms often undergo hybridization to maximize overlap with multiple neighbors and to adopt geometries consistent with VSEPR theory.
Hybridization explains observed valencies and molecular geometries by forming new hybrid orbitals (combinations of s and p, and sometimes d orbitals).
When unhybridized p orbitals are present, they can participate in π-bond formation.
The arrangement often aligns with VSEPR predictions for minimizing electron-pair repulsion.
Hybridization (overview)
AOs of the same energy combine to form hybrid orbitals that participate in bond formation.
Common hybridization schemes (Atkins et al., 2010):
2-coordinate: sp
3-coordinate: sp^2
4-coordinate: sp^3
5-coordinate: sp^3d, spd^3
6-coordinate: sp^3d^2
Mixed/hybridized orbitals enable formation of σ bonds to several atoms and accommodate lone pairs.
Example: In H₂O, the oxygen atom forms four sp^3 hybrid orbitals (one 2s + three 2p from O).
Two of these sp^3 hybrids form σ bonds with the 1s orbitals of two H atoms.
The remaining two sp^3 hybrids contain lone pairs on O.
Overcoming VBT deficiencies
Excitation/higher-energy orbitals: to form stronger bonds, electrons can be promoted to higher-energy orbitals to enable additional bonding.
Hypervalent atoms (expanded octet): some elements (notably period-3 and below) can accommodate more than eight electrons around a central atom by engaging d-orbitals or through resonance/hypervalent bonding.
Expanded octet and sp-mixing: for elements like carbon, sp^3 hybridization explains tetravalency; for Li₂–N₂, sp-m mixing places degenerate π MOs at lower energy, giving s-character to π MOs (involves mixing of 2s and 2p AOs).
Quick examples of VB concepts
Alkene and alkyne bonding:
An alkene double bond contains 1 σ bond and 1 π bond.
An alkyne triple bond contains 1 σ bond and 2 π bonds.
Relationship to molecular geometry and properties
VBT helps rationalize valence (bonding capacity) and shapes for simple and some polyatomic molecules.
However, VBT is complemented by Molecular Orbital Theory (MOT) for a full description of delocalized bonding and magnetism in many molecules.
Molecular Orbital Theory (MOT)
MOT overview
Describes bonding as combinations of atomic orbitals (AOs) to form molecular orbitals (MOs) that extend over the entire molecule.
AOs from each atom combine to form MOs; electrons occupy MOs according to Aufbau, Hund, and Pauli principles.
MOs have definite energies and can be bonding (stabilizing), antibonding (destabilizing), or nonbonding.
The number of MOs equals the total number of AOs used in the construction.
The more similar the energy of AOs, the stronger the mixing and the more complex the MO picture becomes (e.g., sp-mixing).
Linear Combination of Atomic Orbitals (LCAO)
MOs are formed as linear combinations of AOs: a weighted sum of AOs from all atoms in the molecule.
Bonding MOs result from constructive interference of AO lobes; antibonding MOs from destructive interference.
Example: for H₂, the combination of two 1s AOs gives σ(1s) (bonding) and σ*(1s) (antibonding).
Energies: bonding MOs have lower energy than the contributing AOs; antibonding MOs have higher energy than the contributing AOs.
Orbitals in diatomic molecules: σ and π MOs
σ MOs: formed by end-on overlaps (s–s, s–pz, pz–p_z).
π MOs: formed by sideways overlaps of p orbitals (px–px, py–py).
For σ bonds: overlap is along the internuclear axis; for π bonds: overlap occurs above and below the axis.
For homonuclear diatomics (e.g., Li₂ to N₂): degenerate π MOs appear from px and py combinations; σ MOs arise from s and p_z combinations.
s–p mixing (interaction between 2s and 2p AOs)
In Li₂ to N₂ (and some other light diatomics), the degeneracy and energy closeness of 2s and 2p AOs allow sp-mixing.
Result: some s-character is transferred to π-type MOs, altering ordering of MOs and bond characters.
This effect helps explain deviations from simple MO orderings and contributes to bond strength and bond character in these species.
Orbitals and energy level diagrams (MO energy ordering)
General rules:
MOs fill following Aufbau, Hund, and Pauli principles.
Lower-energy MOs fill first; high-energy antibonding MOs are populated last (or not at all if electrons are insufficient).
For the H–H case (H₂): two AOs combine to give σ(1s) bonding MO and σ*(1s) antibonding MO; total electrons = 2, so both occupy the bonding MO, giving a bond order of 1.
Homonuclear diatomics (Li₂ to N₂):
Four AOs from each atom combine to form multiple MOs: σ(2s), σ*(2s), π(2px), π(2py), σ(2p_z), etc.
π orbitals form a doubly degenerate pair of bonding MOs and a doubly degenerate pair of antibonding MOs.
In Li₂–N₂, the degenerate π set can be lowered in energy by sp-mixing, which places the π MOs at lower energy than some σ MOs in this series.
For O₂ and F₂, MO energy ordering differs due to differences in orbital energies; multiple MOs (σ and π) exist with bonding/antibonding characters, influencing bond order and properties.
Bond properties predicted by MOT
Bond order (BO) from MO picture:
Formula: BO = \frac{n - n^}{2} where n = number of electrons in bonding MOs and n = number of electrons in antibonding MOs.
Higher BO generally correlates with stronger bond enthalpy and shorter bond length.
Nonbonding electrons do not contribute to the bond order calculation.
Bond enthalpy and bond length correlations
Higher bond order tends to correspond to higher bond enthalpy and shorter bond length.
Electronic structure descriptors used in MOT
HOMO: Highest Occupied Molecular Orbital
LUMO: Lowest Unoccupied Molecular Orbital
The HOMO/HOMO energy levels and the LUMO are important for predicting reactivity, spectroscopic transitions, and polarizability.
For simple molecules (e.g., H₂), HOMO is the bonding MO that contains electrons; LUMO is the corresponding antibonding or higher-energy nonbonding MO.
In heteronuclear diatomic molecules, MOs can be polarized toward the more electronegative atom, yielding partial ionic character and polar bonds.
Magnetic properties from MOT
Paramagnetic: molecules with at least one unpaired electron are attracted to a magnetic field.
Diamagnetic: molecules with all electrons paired are weakly repelled by a magnetic field.
Examples (illustrative, from MOT context):
O₂ is paramagnetic due to unpaired electrons in π* MOs.
Be₂ or Li₂ can be diamagnetic or paramagnetic depending on specific MO filling; some small diatomics exhibit diamagnetism.
Substances with unpaired electrons (e.g., O₂, certain transition-metal complexes) show paramagnetism; substances with all paired electrons (e.g., N₂, H₂O) show diamagnetism.
Orbitals and energy level diagrams: key concepts
HOMO and LUMO define the frontier orbitals governing reactivity and spectroscopy.
In MO diagrams, bonding MOs lie lower in energy than AOs, while antibonding MOs lie higher.
Relative energies of σ and π MOs depend on the molecule and on s–p mixing effects.
Summary of MOT vs VBT
MOT provides a delocalized description of bonding that can explain phenomena like paramagnetism and bond delocalization in conjugated systems.
VBT explains localized bond formation and valence-based geometries; it is intuitive for simple bond counting and VSEPR-type reasoning.
In many systems, both approaches are complementary; MOT is often better for describing conjugation, diradicals, and metal–ligand bonding in coordination chemistry.
Symmetry & Group Theory in Inorganic Chemistry
Purpose and scope
Symmetry and group theory provide a mathematical framework to classify molecular shapes and to predict spectral transitions, selection rules, and molecular vibrations.
Key ideas: symmetry operations, symmetry elements, and point groups.
Symmetry operations and elements
Identity (E): doing nothing; leaves the molecule unchanged.
Rotation (Cn): rotation by 360°/n about an axis; a molecule can look identical after rotation by this angle; Cn axes are designated; principal axis is the highest-order Cn axis.
Reflection (σ): reflection through a mirror plane; σ planes can be vertical (σv) or horizontal (σh) relative to the principal axis.
Inversion (i): inversion through the center of symmetry; each atom is moved to an equivalent position on the opposite side of the center.
Improper rotation (Sn): rotation by 360°/n about an axis followed by reflection through a plane perpendicular to that axis.
Polar vs dihedral notations: for molecules with multiple symmetry axes, the one with the highest n is the principal axis.
Notation conventions: rotations are typically taken counterclockwise about the principal axis; mirror planes can be parallel or perpendicular to the principal axis.
Point groups and symmetry operations
A point group is the complete set of symmetry operations that describe a molecule’s symmetry.
Common point groups include: C1, Ci, Cn, Cnv, Dn, Dnh, Dnd, Oh, Td, etc.
Example mappings:
H₂O is in the C₂v point group (E, C₂, σv, σv’).
NH₃ is in the C3v point group (E, 2C3, 3σv).
SF₄ is in C2v or related groups depending on geometry (see specific structures).
Linear groups: D∞h (idealized linear molecules) is a limiting case; many real molecules approximate C∞v or D∞h depending on symmetry features.
Cubic groups: Oh, Td, O, etc., used for highly symmetric species.
Practical use of point groups
Determine selection rules for transitions in spectroscopy (infrared, Raman).
Predict which vibrational modes are active in IR or Raman based on symmetry species.
Assign point groups to molecules from observed shapes and symmetry elements.
Correlate molecular geometry with expected electronic structure and reactivity.
Examples and applications from the slides
Polyatomic systems such as PF₄, SF₄, PCl₃, and others are discussed as examples of determining their point groups.
Ferrocene and related metallocenes are discussed in the context of D5h or related symmetry depending on conformation (eclipsed vs staggered; D5d vs D5h considerations).
For CH₄ (and similar molecules), typical analysis yields Td symmetry, while other molecules yield C2v, C3v, D3h, etc., depending on geometry.
Connections to bonding theories
Symmetry helps predict which orbitals (AOs, MOs) can combine to form bonding interactions in a manner consistent with the molecule’s geometry.
Group theory underpins selection rules for electronic transitions and vibrational activity, linking geometry to spectral properties.
Key Formulas and Definitions to Remember
Bond order from MO theory:
BO = \frac{n - n^*}{2}
n = number of electrons in bonding MOs; n^* = number of electrons in antibonding MOs.
Bond length and bond enthalpy correlations (qualitative):
Higher BO typically correlates with shorter bond length and higher bond enthalpy.
HOMO and LUMO definitions
HOMO: Highest Occupied Molecular Orbital
LUMO: Lowest Unoccupied Molecular Orbital
Symmetry operations (short list)
E, Cn, i, σ, Sn, etc. (Identity, n-fold rotation, inversion, mirror plane, improper rotation)
Real-World Significance and Takeaways
Bonding theories provide essential tools for predicting reactivity, stability, and spectra of inorganic and organometallic compounds.
VBT remains intuitive for local bonding and shapes; MOT explains delocalization, magnetism, and conjugation more generally.
Symmetry and group theory streamline the interpretation of experimental data (IR/Raman spectra, magnetic properties) and guide the design of molecules with desired properties.
Quick Practice Prompts (conceptual)
Describe how a σ bond forms in a diatomic molecule like H₂ and what features of the potential energy curve reflect bond strength and stiffness.
For a simple homonuclear diatomic (e.g., O₂, N₂), explain how sp-mixing could affect the ordering of MOs and the resulting bond order.
Given a molecule with a C₂v geometry, identify its most likely point group and list symmetry elements present.
Explain why paramagnetism arises in MO terms for a molecule with unpaired electrons in π* orbitals.
Provide the MO-based reasoning behind why a quadruple-bonded species would exhibit higher bond order and shorter bond length than a single-bonded species.
Note: The above notes consolidate the key ideas from the provided transcript, including Valence Bond Theory (VBT), Molecular Orbital Theory (MOT), and Symmetry & Group Theory, with explicit formulas where appropriate and examples to anchor understanding. Where numerical values appeared in the slides (e.g., H–H bond energy around ~432 kJ/mol), they are presented as illustrative typical values; consult your course materials for the exact figures used in your syllabus.