Hybrid Atomic Orbitals Lecture Notes

Standard valence bond theory is effective for explaining bond formation in diatomic molecules through the overlap of atomic orbitals, but it fails to accurately describe molecules with more than two atoms.
Case Study: The Water Molecule ( $H_2O$ )
- Oxygen has the ground-state electron configuration $1s^2 2s^2 2p^4$ .
- In its valence shell, oxygen contains two unpaired electrons, with one electron in each of two separate $2p$ orbitals. /
- Initial Prediction: Valence bond theory would suggest that the two $O-H$ bonds form via the overlap of these two oxygen $2p$ orbitals with the $1s$ orbitals of two hydrogen atoms.
- The Potential Geometric Error: Because $p$ orbitals are oriented perpendicularly to one another, this model predicts a bond angle of exactly $90^{\circ}$ (as illustrated in Figure 8.6).
- Experimental Contradiction: Scientific observation confirms the actual bond angle of a water molecule is $104.5^{\circ}$ , not $90^{\circ}$ . This discrepancy necessitates a more refined model.

Wave Functions ( $\psi$ ): Quantum mechanics uses mathematical expressions called wave functions to describe the wavelike properties of electrons and the specific information pertaining to each orbital in an isolated atom.
LCAO (Linear Combination of Atomic Orbitals): When atoms form a molecule, their individual wave functions are mathematically combined through a process called hybridization to create new mathematical descriptions of electron distribution.
Definition of Hybrid Orbitals: The result of the hybridization process is a set of new orbitals known as hybrid orbitals, which possess different shapes and orientations than the original atomic orbitals.
Water Molecule Hybridization:
- In isolated oxygen, the valence orbitals are one $2s$ orbital and three $2p$ orbitals.
- In a bonded water molecule, these combine into four equivalent hybrid orbitals.
- These orbitals point toward the corners of a tetrahedron (Figure 8.7).
- The Predicted vs. Observed Angle: Hybridization predicts a tetrahedral bond angle of $109.5^{\circ}$ . While different from the unhybridized $90^{\circ}$ , it is much closer to the observed $104.5^{\circ}$ . The variation between $109.5^{\circ}$ and $104.5^{\circ}$ is accounted for by oth er factors in valence bond theory, making hybridization a necessary component for accurate structural predictions.

1. Bonded Status Requirement: Hybrid orbitals are not found in isolated atoms; they are generated only when atoms are covalently bonded.
2. Unique Geometry: The shapes and orientations of hybrid orbitals are significantly different from the atomic orbitals found in isolated atoms.
3. Conservation of Orbitals: The total number of hybrid orbitals in a resulting set must exactly equal the number of individual atomic orbitals that were combined to create that set.
4. Internal Equivalence: Every orbital within a specific set of hybrid orbitals is equivalent to the others in terms of both shape and energy level.
5. VSEPR Integration: The specific type of hybrid orbitals formed on a bonded atom is determined by its electron-pair geometry, which is calculated using Valence Shell Electron-Pair Repulsion (VSEPR) theory.
6. Bond Classification:
- $\sigma$ (Sigma) Bonds: Formed by the overlap of hybrid orbitals.
- $\pi$ (Pi) Bonds: Formed by the overlap of unhybridized orbitals.

Applicability: $sp$ hybridization occurs in central atoms that have exactly two regions of valence electron density (electron domains) a nd no lone pairs, resulting in a linear arrangement.
Case Study: Gaseous Beryllium Chloride ( $BeCl_2$ )
- Beryllium acts as the central atom bonded to two chlorine atoms.
- To accommodate two electron domains (the two covalent $Be-Cl$ bonds), two of Beryllium's four valence orbitals must mix.
The Hybridization Process:
- One valence $s$ orbital is mixed with one valence $p$ orbital.
- This results in two equivalent $sp$ hybrid orbitals.
- Geometric Orientation: The two $sp$ orbitals are oriented at an angle of $180^{\circ}$ , producing a linear geometry (Figure 8.8).
Structural Characteristics of $sp$ Orbitals:
- While they may appear similar to original $p$ orbitals, each individual $sp$ orbital contains one lobe that is significantly larger than the other.
- Each orbital is oriented primarily in a single direction.
Electron Redistribution:
- The two electrons originally residing in the beryllium $s$ orbital are distributed across the two new $sp$ orbitals.
- This makes both hybrid orbitals half-filled and ready for bonding.
- These half-filled hybrid orbitals overlap with orbitals from chlorine atoms to form two identical $\sigma$ bonds.

Orbital Mapping (Figure 8.9): Energy diagrams represent orbitals as horizontal lines and electrons as arrows. Energy increases toward the top of the diagram.
Isolated Beryllium Atom:
- Electrons are paired in the $2s$ orbital.
- All three $2p$ orbitals remain empty and at a higher energy level.
Bonded/Hybridized Beryllium in $BeCl_2$ :
- Hybridized Set: The two $sp$ hybrid orbitals exist at an energy level between the original $2s$ and $2p$ levels.
- Unhybridized Orbitals: Two $2p$ orbitals remain unhybridized and empty, retaining their higher energy.
- Bond Formation: Each of the two valence electrons of Be occupies one half-filled $sp$ orbital. These electrons then pair with an unpaired electron from a chlorine atom's orbital (specifically a $Cl$ $3p$ orbital) during bond formation.

General Rule: Any central atom with just two areas of valence electron density will exhibit $sp$ hybridization.
Example 1: Mercury ( $Hg$ ) in the linear molecule $HgCl_2$ .
Example 2: Zinc ( $Zn$ ) in the compound $Zn(CH_3)_2$ , which features a linear $C-Zn-C$ atomic arrangement.
Example 3: Carbon ( $C$ ) in acetylene ( $HCCH$ ).n
Example 4: Carbon ( $C$ ) in carbon dioxide ( $CO_2$ ).

Octet rule fill valence shell with 8 electrons

Covalent is sharing electrons