Atomic Structure and Wavefunctions

Wave-Particle Duality

The concept of wave-particle duality describes how electrons exhibit both wave-like and particle-like properties. According to de Broglie, electrons can be described as waves with a wavelength inversely proportional to their momentum. This duality is essential for understanding electron behavior in atoms and molecules.

• de Broglie wavelength: , where is Planck's constant and is momentum.

• Wave properties are most useful for understanding electron distributions in organic molecules.

Wavefunctions

A wavefunction is a mathematical function that describes the shape and behavior of an electron's wave in an atom or molecule. The square of the wavefunction () gives the probability density of finding an electron at a particular location.

• Wavefunctions are solutions to the Schrödinger equation for electrons in atoms.

• Standing waves (as in vibrating strings) are analogous to electron wavefunctions, with only certain allowed frequencies and shapes.

• Nodes are points where the wavefunction passes through zero; more nodes mean higher energy.

Energy of the Wavefunction

The energy of an electron in an atom is related to its wavefunction:

• Wavefunctions with more nodes are higher in energy than those with fewer nodes.

• Wavefunctions that are farther from the nucleus (more delocalized) are higher in energy.

Atomic Wavefunctions (Orbitals)

Atomic orbitals are specific wavefunctions for electrons in atoms, characterized by quantum numbers (, , ):

• 1s, 2s, 2p, 3s, 3p, 3d, ... are common orbital labels.

• Each orbital has a distinct shape and energy.

• For hydrogen, orbitals are solutions to the Schrödinger equation.

Valence Bond Theory, Hybridization, and Lewis Structures

Valence Bond Theory (Heitler, London, Pauling)

Valence bond theory explains covalent bond formation as the overlap of atomic orbitals (wavefunctions) from two atoms:

• A covalent bond forms when two electrons are shared between two atoms.

• The bond is localized in the region where the wavefunctions overlap.

Orbital Directionality Problem

Valence bond theory requires that the overlapping orbitals point in the correct directions for bond formation:

• 2s orbitals are spherical and do not point in any direction.

• 2p orbitals are oriented along the x, y, and z axes.

• Hybridization is used to generate new orbitals with appropriate directionality for bonding.

Orbital Hybridization (Pauling)

Hybridization is the mixing of atomic orbitals to form new, equivalent orbitals suitable for bonding:

• sp3 hybridization: Mixing one 2s and three 2p orbitals forms four sp3 hybrid orbitals (tetrahedral geometry, 109.5° bond angles).

• sp2 hybridization: Mixing one 2s and two 2p orbitals forms three sp2 hybrid orbitals (trigonal planar geometry, 120° bond angles).

• sp hybridization: Mixing one 2s and one 2p orbital forms two sp hybrid orbitals (linear geometry, 180° bond angles).

Making Bonds and Drawing Lewis Structures

Lewis structures are diagrams that represent the arrangement of electrons in molecules:

• Each line represents a pair of shared electrons (a bond).

• Lone pairs are shown as pairs of dots.

• Formal charges are assigned to atoms based on electron count.

• Resonance structures show alternative arrangements of electrons; the true structure is a hybrid.

Steps for Drawing Lewis Structures

1. Count total valence electrons for all atoms.

2. Arrange atoms and connect with single bonds.

3. Distribute remaining electrons to complete octets (or duets for hydrogen).

4. Assign formal charges as needed.

5. Draw resonance structures if applicable.

Formal Charge Calculation

• Formal charge = (valence electrons in free atom) – (nonbonding electrons) – (1/2 × bonding electrons)

Resonance Structures

• Resonance structures are different possible Lewis structures for a molecule that differ only in the placement of electrons.

• The actual molecule is a resonance hybrid of all valid structures.

Examples

• Methane (CH4): Carbon is sp3 hybridized, forms four single bonds in a tetrahedral geometry.

• Ethylene (C2H4): Each carbon is sp2 hybridized, forms a double bond and three single bonds in a planar geometry.

• Acetylene (C2H2): Each carbon is sp hybridized, forms a triple bond and one single bond in a linear geometry.

Additional info: These notes expand on the original file by providing definitions, formulas, and structured tables for hybridization, as well as stepwise instructions for drawing Lewis structures and calculating formal charges. The examples illustrate key applications in organic molecules.