Valence Bond Theory and Molecular Orbital Theory Notes

Valence Bond Theory and Hybridization

• Valence Bond Theory (VBT) states that a chemical bond forms when two atoms bring their atomic orbitals into the same space and overlap them, increasing the probability of finding electrons in that region. This overlap increases electron density in the bonding region, while nuclei remain positively charged and attract the shared electrons, stabilizing the system.
• Bonds correspond to overlapping orbitals with increased electron density between nuclei; the overlap lowers energy and stabilizes the molecule.
• Limitation of VBT: it excels at explaining molecular shapes and three-dimensional geometries (via hybridization) but is less predictive for electronic energy changes, excited states, and magnetic properties.
• Molecular Orbital Theory (MOT) provides a complementary view by treating electrons as occupying molecular orbitals (MOs) that belong to the molecule as a whole, formed by linear combinations of atomic orbitals.

Hybridization Theory (Hyb.)

• Purpose: reconcile observed molecular geometries with orbital overlaps by allowing the central atom to mix its s and p (and sometimes d) orbitals to form a new set of directional orbitals.
• Core idea: the central atom uses hybrid orbitals to maximize electron density spread and direct bonding in the directions needed for the observed geometry.
• Key rule: the number of hybrid orbitals around the central atom equals the number of electron groups (bonding pairs + lone pairs) around that atom.
• Different atoms can exhibit different hybridizations depending on their bonding context (similar to how a person may show different personalities in different settings).
• Common hybridizations:
  • 2 electron groups: sp
  • 3 electron groups: sp^2
  • 4 electron groups: sp^3
  • 5 electron groups: sp^3d (trigonal bipyramidal)
  • 6 electron groups: sp^3d^2 (octahedral)
  • Note: notation may appear as sp^3d^2, often written as sp^3d^2; some texts describe as d-type contributions (not essential for basic shapes yet).
• Examples:
  • CH4 (carbon with four groups): central C uses sp^3 hybridization → tetrahedral geometry.
  • BF3 (boron with three groups): central B uses sp^2 hybridization → trigonal planar geometry; boron often has an incomplete octet.
  • PCl5 (phosphorus with five groups): central P uses sp^3d hybridization → trigonal bipyramidal geometry.
  • ClF4− (chlorine with six groups): central Cl uses sp^3d^2 hybridization → square planar or related octahedral geometry depending on context; in this example six electron domains lead to sp^3d^2.
• Shortcut used in class (Lewis-structure-based): count central atom valence electrons and electrons donated by outer atoms to assess if a perfect electron match occurs (simplified method).
  • Example: phosphorus in PCl5 has 5 valence electrons; each outer halogen donates 1 electron to form 5 bonds; central atom has 5 electron groups → sp^3d hybridization.
  • Example: boron in BF3 has 3 valence electrons; three F atoms donate one electron each; central atom has 3 electron groups → sp^2 hybridization (boron often lacks a full octet, which is an octet exception justified by formal charges in practice).
  • Example: ClF4− has 6 electron groups around Cl (4 bonding pairs + 2 lone pairs) → sp^3d^2 hybridization.
• Practical note on ALEKS vs textbook practice: some platforms donate all atomic orbitals, while the common chemistry teaching often emphasizes valence orbitals for bonding; the core electrons typically do not participate directly in bonding, but some computational tools may include them depending on context.

Molecular Orbital Theory (MOT)

• Core idea: when atoms bond, their atomic orbitals donate electrons to form a new set of molecular orbitals that belong to the molecule. Electrons occupy these MOs according to energy: lower-energy (more stabilizing) orbitals fill first.
• Construction rules (conceptual, not full equations):
  • Atomic orbitals combine by addition or subtraction of their wavefunctions to form molecular orbitals.
  • Adding orbitals increases electron density in the resulting MO (lowering energy when bonding), while subtracting reduces electron density in the region (raising energy; related to antibonding behavior).
  • Only valence orbitals typically participate in forming the MOs most relevant to bonding; core orbitals are less involved in bonding in simple molecules (ALEKS may treat differently in some contexts).
  • Hybridization is a VB concept; MOT relies on combining orbitals to form bonding (lower energy) and antibonding (higher energy) MOs.
• Key concepts:
  • Bonding orbitals: lower energy, increase electron density between nuclei, stabilize the molecule.
  • Antibonding orbitals: higher energy, decrease electron density between nuclei, oppose bonding.
  • Orbital mixing: forming MOs from orbitals of the same type (s with s, p with p; same energy) yields better overlap; mixing across different types (s with p) is avoided in simple cases.
  • Aufbau for MOs: electrons fill from lowest-energy MO to higher-energy MO, with each MO holding at most 2 electrons with opposite spins (Hund’s rule applies when degenerate orbitals are available).
  • Bond order in MOT: $ext{Bond order} = \frac{N<em>b - N</em>a}{2}$ where $N<em>b$ is the number of electrons in bonding MOs and $N</em>a$ is the number of electrons in antibonding MOs.
• Practical workflow for diatomic molecules:
  1) Count valence electrons of the molecule (from both atoms).
  2) Choose the appropriate MO ordering pattern depending on the atoms involved (two common patterns for 2p-based diatomics).
  3) Draw the MO ladder and place electrons from lowest to highest energy, obeying the 2-electron limit per MO and Hund’s rule for degenerate orbitals.
  4) Compute bond order with the MO counts.
  5) Infer magnetic behavior: paramagnetic if there are unpaired electrons in the MO diagram; diamagnetic if all electrons are paired.
• Two common MO ordering patterns for 2p diatomics:
  • Pattern 1 (applies to B2, C2, N2 and related ions):
  • Order: ext{(lowest) } \sigma{2s}, \sigma^{}{2s},\ \ B
    ightarrow ext{pi}{2p} ext{ (degenerate, two orbitals)}, \ \ 6 loor{Note: the scissor-like ordering bears the bD shift: } \ \ ext{sigma}{2p}, \ \ ext{pi}^{2p}, \ \ ext{sigma}^*{2p}
    
  • In words: sigma 2s, sigma* 2s, pi 2p (two degenerate orbitals), sigma 2p, pi* 2p (degenerate), sigma* 2p.
  • Important caveat: the sigma 2p often sits above the pi 2p in this pattern, due to mixing and energy differences among orbitals.
  • Pattern 2 (applies to O2, F2, Ne and their ions):
  • Order: ext{(lowest) } \sigma{2s}, \ \  loor{sigma^{}{2s}}, \ \ 6 loor{sigma{2p}}, \ \ 6 loor{pi{2p}}, \ \ 6 loor{pi^{2p}}, \ \ 6 loor{sigma^*{2p}}
    
  • In words: sigma 2s, sigma* 2s, sigma 2p, pi 2p (degenerate set), pi* 2p (degenerate), sigma* 2p.
• Example applications:
  • H2 (diatomic hydrogen): both atoms contribute 1s orbitals; MOs formed are: sigma 1s (bonding) and sigma* 1s (antibonding).
  • After formation, both electrons occupy the bonding MO (sigma 1s) with paired spins; bond order $BO = \frac{2 - 0}{2} = 1.$
  • He2: both He atoms have 1s^2; after MO formation, both the sigma 1s and sigma* 1s are filled, giving $BO = \frac{4 - 2}{2} = 1$? Actually for He2 the occupied configuration leads to no net bonding because the energetic stabilization is offset by antibonding occupancy; the standard result is that He2 is not stable (bond order ~0). The important idea is that if antibonding orbitals are occupied as much as or more than bonding orbitals, bonding is not favored.
  • Heteronuclear and other diatomics follow the same MO construction rules; Li2 and Be2 illustrate s-s MO interactions with the same basic pattern:
  • For Li2 and Be2, the 2s MO set forms a bonding sigma 2s and antibonding sigma* 2s; electron counts lead to small or zero bond orders depending on occupancy.
  • O2 and beyond: p-orbital MOs become important; energy ordering differs between Pattern 1 and Pattern 2, affecting bond order and magnetic behavior.
• Magnetic properties in MOT:
  • Paramagnetic: presence of unpaired electrons in MO occupancy (e.g., O2 has two unpaired electrons in the π*2p set).
  • Diamagnetic: all electrons are paired (e.g., H2, He, N2 in its ground state is diamagnetic due to paired electrons in occupied MOs).

Examples Worked Through in MO Theory (Highlights)

• Hydrogen molecule (H2):
  • Before: two H atoms each with 1s^1.
  • After: formation creates a bonding MO (sigma 1s) and an antibonding MO (sigma* 1s).
  • Electron filling: 2 electrons fill sigma 1s; sigma* 1s remains empty.
  • Bond order: $BO = \frac{N<em>b - N</em>a}{2} = \frac{2 - 0}{2} = 1.$
• Helium molecule (He2) and He2+:
  • He2: 1s^2 per He, total 4 electrons in 1s set; fill sigma 1s (2 electrons) and sigma* 1s (2 electrons) → net bond order 0 (no stable bond).
  • He2+ (three electrons): fill sigma 1s with 2 electrons, sigma* 1s with 1 electron; bond order $BO = \frac{2 - 1}{2} = 0.5.$
• Lithium and Beryllium diatomics (Li2, Be2):
  • Use 2s-based MO set: sigma 2s and sigma* 2s; occupancy follows the same rules; patterns are analogous to H2 but with higher energy orbitals involved.
• Nitrogen and Oxygen diatomics (N2, O2):
  • Pattern choice depends on the diatomic; N2 follows Pattern 1 (pi 2p below sigma 2p with some mixing quirks); O2 follows Pattern 2 (sigma 2p below pi 2p).
  • N2: valence electrons = 10 (5 from each N); occupancy leads to a bond order of 3 (triple bond) and diamagnetic behavior (all electrons paired).
  • O2: valence electrons = 12; occupancy yields bond order 2 (double bond) and paramagnetic behavior due to two unpaired electrons in π* orbitals.
• Nitrogen ionization effects:
  • Removing electrons from N2 reduces bonding electron count more rapidly than antibonding electrons, generally lowering bond order and increasing bond length.
• Oxygen ionization effects:
  • In contrast to nitrogen, removing electrons from O2 (specifically removing antibonding electrons) can increase bond order, strengthening the bond and potentially shortening the bond length.

Connections to Foundational Principles and Real-World Relevance

• VSEPR vs Hybridization:
  • Hybridization helps explain observed molecular geometries by dictating directional bonding around the central atom, complementing the VSEPR framework.
• Magnetic properties and energy interactions:
  • MOT can predict magnetic properties (paramagnetism/diamagnetism) that VB theory cannot always explain.
• Energy and stability:
  • Bond formation generally lowers the system’s energy by placing electrons into bonding MOs; occupancy of antibonding MOs can counteract stabilization.
• Practical applications:
  • Understanding bond order and MO occupancy informs predictions about bond lengths, bond strengths, and reactivity in diatomic molecules and simple polyatomic species.
• Notable caveat:
  • For complex molecules, MO diagrams become more involved, and computational methods are often used to quantify orbital mixing and energy contributions; however, the simple diatomic MO framework captures essential trends.

Quick Reference: Key Formulas and Concepts

• Bond order (MOT):
  • $BO = \frac{N<em>b - N</em>a}{2}$ where $N<em>b$ is the number of electrons in bonding MOs and $N</em>a$ is the number of electrons in antibonding MOs.
• Two-electron occupancy rule:
  • Each MO can hold at most 2 electrons with opposite spins.
• Hund’s rule (degenerate orbitals):
  • Electrons singly occupy degenerate orbitals with parallel spins before pairing up.
• Sigma vs Pi nomenclature (MOT):
  • Bonding sigma orbitals: $ext{σ}$, formed by end-to-end overlap (e.g., $ext{σ}<em>{2s}, ext{σ}</em>{2p}$).
  • Bonding pi orbitals: $ext{π}$, formed by side-by-side overlap (e.g., $ext{π}_{2p}$).
  • Antibonding counterparts: $ext{σ}^<em>, ext{π}^</em>$.
• Magnetic behavior terminology:
  • Paramagnetic: species with unpaired electrons in its MO configuration.
  • Diamagnetic: species with all electrons paired.

// Summary takeaway

• Valence Bond Theory provides intuitive, geometry-focused pictures via hybridization (s/p/d mixing) to match observed shapes.
• Molecular Orbital Theory provides a global, energy-based framework to predict bond strengths, bond lengths, and magnetic properties by filling bonding and antibonding MOs according to energy, with the bond order giving a quantitative gauge of bond strength.
• In practice, chemists use a combination of these viewpoints to analyze real molecules and relate structure to properties and reactivity.