1.2 Atomic Structure: Orbitals

The quantum mechanical model describes electrons not as tiny particles orbiting the nucleus, but as waves spread across regions of space. Their positions aren't fixed; instead, we talk about the probability of finding an electron in a given region. The mathematical functions that map out these probability regions are called atomic orbitals, and they come in different shapes, sizes, and energy levels (s, p, and d for organic chemistry purposes). Understanding orbitals is the foundation for everything that follows in this course: electron configurations, bonding, and molecular shape all depend on it.

Quantum Mechanical Model and Atomic Orbitals

Wave Functions and Orbitals

A wave function ($\Psi$) is a mathematical solution to the Schrödinger equation for an atom. It describes the probability of finding an electron in a given region of space. You don't need to solve the equation yourself, but you do need to understand what the solutions tell us.

• Each solution corresponds to an atomic orbital with a specific energy level and shape (e.g., 1s, 2p, 3d).
• The square of the wave function ($\Psi^2$) at any point gives the probability density at that location. High $\Psi^2$ means you're likely to find the electron there; low $\Psi^2$ means you're not.
• Electrons fill orbitals in order of increasing energy, following the Aufbau principle: 1s < 2s < 2p < 3s < 3p, and so on.
• Each orbital is specified by three quantum numbers:
  • Principal quantum number ($n$): energy level and size ($n$ = 1, 2, 3…)
  • Angular momentum quantum number ($l$): shape (0 = s, 1 = p, 2 = d)
  • Magnetic quantum number ($m_l$): orientation in space

Shapes of s, p, and d Orbitals

• s orbitals ($l = 0$)
  • Spherically symmetric around the nucleus (1s, 2s, 3s, etc.)
  • One orbital per energy level, holding a maximum of 2 electrons
  • Present at every principal energy level ($n$ = 1, 2, 3…)
  • Higher $n$ means a larger sphere with more nodes (the 2s orbital is bigger than the 1s)
• p orbitals ($l = 1$)
  • Dumbbell-shaped, with two lobes on opposite sides of the nucleus
  • Three orbitals per energy level ($p_x$, $p_y$, $p_z$), oriented along the x, y, and z axes
  • Each orbital holds 2 electrons, so a full set of p orbitals holds 6 electrons total
  • First appear at $n = 2$
• d orbitals ($l = 2$)
  • More complex shapes, including cloverleaf patterns ($d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{x^2-y^2}$) and a unique dumbbell-with-ring shape ($d_{z^2}$)
  • Five orbitals per energy level, each holding 2 electrons, for a maximum of 10 electrons total
  • First appear at $n = 3$
  • Less central to introductory organic chemistry, but relevant for transition metal chemistry

Spatial Orientation of p Orbitals

The three p orbitals ($p_x$, $p_y$, $p_z$) are oriented perpendicular to each other along the three coordinate axes. This mutual perpendicularity matters a lot when you get to molecular geometry and bonding.

• Each p orbital has two lobes with opposite algebraic signs (+ and −). These signs represent the phase of the wave function, not electrical charge. Think of it like the crest and trough of a wave.
• A node is a region where the probability of finding an electron is zero. Every p orbital has a nodal plane that passes through the nucleus, separating the two lobes. Nodes are a direct consequence of the wave-like behavior of electrons.
• Phase matters for bonding: when orbitals on adjacent atoms overlap, lobes of the same phase produce constructive interference (a bonding interaction), while lobes of opposite phase produce destructive interference (an antibonding interaction). You'll revisit this concept heavily when you study molecular orbital theory.

Electron Configuration and Orbital Filling

Electron configuration is the specific arrangement of electrons across an atom's orbitals. Three rules govern how electrons fill orbitals:

1. Aufbau principle: Fill orbitals starting from the lowest energy and work up (1s → 2s → 2p → 3s → 3p…).
2. Pauli exclusion principle: No two electrons in an atom can share the same set of four quantum numbers. In practice, this means each orbital holds at most 2 electrons, and those two must have opposite spins (+1/2 and −1/2).
3. Hund's rule: When filling degenerate orbitals (orbitals at the same energy, like $2p_x$, $2p_y$, and $2p_z$), place one electron in each orbital with parallel spins before pairing any up. Think of it like passengers on a bus: everyone takes their own seat before anyone doubles up.

For example, carbon has 6 electrons. Its electron configuration is 1s² 2s² 2p². By Hund's rule, those two 2p electrons go into separate p orbitals with parallel spins rather than pairing up in the same one. This detail directly affects how carbon forms bonds.