Introduction: What Orbitals Really Are
An orbital is a wavefunction for an electron. The probability density \(|\psi|^2\) forms a "cloud" showing where the electron is likely to be found. These shapes determine chemistry.
Radial Probability
Probability of finding electron between \(r\) and \(r + dr\):
\[P(r)dr = |R(r)|^2 \cdot 4\pi r^2 dr = |u(r)|^2 dr\]The factor \(4\pi r^2\) accounts for the shell's volume.
Orbital Shapes
- s orbitals (\(l = 0\)): Spherical, non-zero at nucleus
- p orbitals (\(l = 1\)): Dumbbell shape, node at nucleus
- d orbitals (\(l = 2\)): Cloverleaf shapes, complex nodal patterns
- f orbitals (\(l = 3\)): Even more complex shapes
Node Counting
- Total nodes = \(n - 1\)
- Angular nodes = \(l\)
- Radial nodes = \(n - l - 1\)
Example: 3d orbital has \(n = 3, l = 2\): 2 angular nodes, 0 radial nodes.
The Quantum Connection
These probability clouds aren't just pictures—they determine chemical bonding. Overlapping orbitals form molecular bonds. The shapes of s, p, d orbitals explain molecular geometries (tetrahedral for sp³, linear for sp, etc.). Quantum mechanics isn't abstract here; it's the theory behind the entire chemical world.