Quantum Chemistry
Atomic Orbitals
Quantum probability clouds where electrons live — s, p, d, f shapes
An atomic orbital is a region of space around a nucleus where there's a high probability of finding an electron. Orbitals are quantized — defined by 4 quantum numbers (n, l, m_l, m_s). Shapes determined by angular momentum quantum number l: s (l=0, spherical), p (l=1, dumbbell), d (l=2, four lobes), f (l=3, complex). Each orbital holds at most 2 electrons (Pauli exclusion). Orbital energies determine chemistry — bonding, reactivity, spectra. Replaced Bohr's planetary model around 1926 (Schrödinger).
- Quantum numbersn (size), l (shape), m_l (orientation), m_s (spin)
- s orbitall=0; spherical; 1 per shell
- p orbitall=1; dumbbell; 3 per shell (px, py, pz)
- d orbitall=2; 5 per shell; complex shapes
- f orbitall=3; 7 per shell; very complex
- Max electrons per orbital2 (Pauli exclusion principle)
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Why orbitals matter
- Chemical bonding. Foundation of covalent bonds.
- Molecular geometry. Orbital shapes determine angles.
- Spectroscopy. Electronic transitions between orbitals.
- Periodic trends. Patterns across the periodic table.
- Catalysis. d-orbital interactions in transition metals.
- Materials science. Electronic structure of solids.
- Quantum mechanics. First non-trivial QM application.
Common misconceptions
- Electrons orbit like planets. Obsolete Bohr model; electrons are wave-like.
- Orbital shape shows electron path. Shows probability, not trajectory.
- Orbitals are physical objects. Mathematical descriptions.
- Higher n always higher energy. 4s fills before 3d (sometimes).
- Orbitals are empty space. Probability distributions of electrons.
- p orbitals are 6 lobes. Each p orbital has 2 lobes; 3 p orbitals total.
Frequently asked questions
What is an atomic orbital?
A mathematical function describing the wave-like behavior of an electron in an atom. Solving Schrödinger's equation for the hydrogen atom gives a discrete set of solutions — orbitals. Each orbital has a specific energy, shape, and orientation. The square of the wavefunction at any point gives probability density for finding the electron there.
How do quantum numbers describe orbitals?
Four quantum numbers fully specify an electron's state. n (principal, 1, 2, 3...): size and energy. l (angular momentum, 0 to n-1): shape — s, p, d, f. m_l (magnetic, -l to +l): orientation in space. m_s (spin, ±½): electron spin. Pauli exclusion: no two electrons share all four numbers.
Why are orbitals shaped that way?
Wavefunction shapes come from angular momentum solutions of Schrödinger's equation. s (spherical): no angular nodes. p (dumbbell): one angular node. d (cloverleaf or donut): two nodes. f: three nodes. The shapes reflect probability distributions — where you're likely to find the electron, not its trajectory.
How are orbitals filled?
Aufbau principle: lowest energy first. Hund's rule: fill degenerate orbitals singly before pairing. Pauli exclusion: paired electrons have opposite spins. Order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d... Energies don't match shell number perfectly due to electron-electron interactions.
What's the difference between orbital and shell?
Shell = principal quantum number n (e.g., n=2 is the second shell). Subshell = angular momentum l within a shell (e.g., 2p subshell). Orbital = specific m_l within subshell (e.g., 2p_x is one specific orbital). Shell n contains n² orbitals total and 2n² electrons max.
Why can orbitals only hold 2 electrons?
Pauli exclusion principle. No two fermions (electrons) can occupy identical quantum states. Two electrons in same orbital must differ in spin (↑ and ↓). A third electron would need to share spin with one of the first two — forbidden. Forces electrons into higher orbitals once lower ones are full.
How do orbitals relate to chemistry?
Valence electrons (outermost) determine bonding. Orbital overlap creates molecular orbitals — covalent bonds. Hybridization mixes orbitals (sp, sp², sp³) to explain molecular geometry. Periodic trends (electronegativity, ionization energy) all reflect orbital structure. d orbitals enable transition metal catalysis; f orbitals give lanthanide chemistry.