Atomic and Molecular Orbitals: Quantum Mechanics and Bonding in General Chemistry

Quantum Mechanics, Atomic Orbitals (AO's), and Molecular Orbitals (MO's)

Introduction to Quantum Mechanics in Chemistry

Quantum mechanics provides a mathematical framework for understanding the behavior of electrons in atoms and molecules. Unlike classical (Newtonian) mechanics, quantum mechanics describes electrons as occupying regions of space called orbitals, which are solutions to the wave equation (ψ).

• Bond formation lowers the energy of a molecule, making it more stable.

• Atoms/lone pairs prefer to be as far apart as possible to minimize repulsion.

• Spreading out charge reduces energy, especially through resonance.

Key Terms:

• Atomic Orbital (AO): A region of space around an atom where an electron is likely to be found.

• Molecular Orbital (MO): A region of space in a molecule where electrons are likely to be found, formed by the combination of atomic orbitals.

• Wave Equation (ψ): Mathematical function describing the probability distribution of an electron.

Example: The total energy of a system can be described by a mathematical function called the wave equation (ψ or "psi").

Additional info: The (+) and (–) signs in the wave equation do not have physical meaning; they are not related to charge.

Shapes of Atomic Orbitals (AO's)

Atomic orbitals have distinct shapes that influence how atoms bond and interact.

• s-orbitals: Spherical in shape (e.g., 1s, 2s).

• p-orbitals: "Dumbbell" shaped, oriented along x, y, or z axes (e.g., 2px, 2py, 2pz).

• Node: A region where the probability of finding an electron is zero. Electrons are never found at a node.

Example: The 2s orbital has a spherical shape with a node, while the 2p orbital has a dumbbell shape with a node at the nucleus.

Combining Atomic Orbitals: Molecular Orbitals (MO's)

Rules for Combining Orbitals

When atoms bond, their atomic orbitals combine to form molecular orbitals. The combination depends on the signs and overlap of the wave functions.

• Sign of Wave Equation: The wave equation (ψ) has a sign (+ or –). When orbitals of the same sign overlap, a new orbital is created.

• Same Sign Overlap: Orbitals reinforce or add together, increasing electron density between nuclei (bonding).

• Opposite Sign Overlap: Orbitals destructively interfere, reducing electron density between nuclei (antibonding).

• Number of Orbitals: The number of molecular orbitals formed equals the number of atomic orbitals combined.

• Orbitals May Be Empty: Orbitals exist whether or not they contain electrons.

Example: In H2 formation, the bonding MO (ψMO) leads to increased electron density between nuclei, stabilizing the molecule. The antibonding MO (ψMO*) has no electron density between nuclei, so no bond forms.

Additional info: The additive combination of orbitals puts electron density between positively charged nuclei, acting as the "glue" that holds the molecule together.

Hybridization of Atomic Orbitals

Hybrid Atomic Orbitals (hAO's)

Atoms often rearrange or hybridize their atomic orbitals to maximize bonding. Hybridization allows atoms to form more bonds and achieve lower total energy.

• Hybridization: The process of mixing atomic orbitals to form new, equivalent hybrid orbitals suitable for bonding.

• Types of Hybrid Orbitals:

  • sp: Combination of one s and one p orbital.

  • sp2: Combination of one s and two p orbitals.

  • sp3: Combination of one s and three p orbitals.

• Valence Electrons: Only the electrons in the valence shell participate in bonding and hybridization.

Example: Non-hybridized carbon (2s, 2px, 2py, 2pz) can only form two bonds, not achieving an octet. Hybridization (e.g., sp3) allows carbon to form four bonds, achieving an octet.

Additional info: The resulting shapes of molecules constructed using hybrid atomic orbitals (hAO's) resemble those predicted by VSEPR theory.

Summary Table: Types of Atomic and Hybrid Orbitals

AO's Combined	Hybrid AO's (hAO's)	Example
s + p	sp	Linear geometry (e.g., BeCl2)
s + p + p	sp2	Trigonal planar geometry (e.g., BF3)
s + p + p + p	sp3	Tetrahedral geometry (e.g., CH4)

Additional info: Hybridization always starts with an s orbital and includes as many p orbitals as needed for the geometry.

Key Equations

• Wave Equation for Orbitals: $\psi(x, y, z)$

• Bonding Molecular Orbital: $\psi_{MO} = \psi_{AO1} + \psi_{AO2}$

• Antibonding Molecular Orbital: $\psi_{MO^*} = \psi_{AO1} - \psi_{AO2}$

Conclusion

Understanding atomic and molecular orbitals, as well as hybridization, is essential for predicting molecular shapes, bond formation, and the stability of chemical compounds. Quantum mechanics provides the foundation for these concepts, which are central to General Chemistry.