By Rosann Kozlowski – Updated Aug 30, 2022

Energy levels and orbitals define an atom’s electronic structure, revealing how electrons are arranged around the nucleus. These concepts arise from quantum theory, which describes the discrete energy states that electrons can occupy.

Quantum Theory in Brief

Quantum theory states that atoms can exist only in specific energy states. When an electron transitions between these states, it absorbs or emits a precise quantum of energy equal to the difference between the initial and final states. This quantization is expressed through a set of four quantum numbers.

Four Quantum Numbers Explained

Each electron is uniquely identified by:

• n – Principal quantum number (energy level)
• l – Azimuthal quantum number (subshell type)
• m_l – Magnetic quantum number (orbital orientation)
• m_s – Spin quantum number (+½ or –½)

Principal Quantum Number (n)

The value of n determines the size and energy of an orbital. It takes integer values starting at 1. Each level is also labeled by a letter: n = 1 (K), n = 2 (L), n = 3 (M), n = 4 (N), and so forth.

The number of orbitals in a given energy level is calculated by n²:

• n = 1 → 1² = 1 orbital (s)
• n = 2 → 2² = 4 orbitals (s + p)
• n = 3 → 3² = 9 orbitals (s + p + d)
• n = 4 → 4² = 16 orbitals (s + p + d + f)

The maximum number of electrons per energy level follows from the Pauli exclusion principle and is given by 2n²:

• n = 1 → 2 electrons
• n = 2 → 8 electrons
• n = 3 → 18 electrons
• n = 4 → 32 electrons

Azimuthal Quantum Number (l)

For a fixed n, l can range from 0 to n‑1. The integer values correspond to subshells: 0 = s, 1 = p, 2 = d, 3 = f. Each subshell’s capacity is:

• s – 2 electrons
• p – 6 electrons
• d – 10 electrons
• f – 14 electrons

Magnetic Quantum Number (m_l)

Given an l, m_l can take integer values from –l to +l, including zero. This determines the spatial orientation of each orbital:

• l = 0 (s): m_l = 0 → 1 orbital
• l = 1 (p): m_l = –1, 0, +1 → 3 orbitals (px, py, pz)
• l = 2 (d): m_l = –2 … +2 → 5 orbitals
• l = 3 (f): m_l = –3 … +3 → 7 orbitals

Spin Quantum Number (m_s)

Each orbital can hold two electrons with opposite spins: +½ or –½. This ensures compliance with the Pauli exclusion principle, which prohibits two electrons from sharing all four quantum numbers.

Putting It All Together

To verify the orbital count for a specific energy level, sum the orbitals contributed by each subshell. For example, for n = 3 (M shell):

• s (l = 0): 1 orbital
• p (l = 1): 3 orbitals
• d (l = 2): 5 orbitals

1 + 3 + 5 = 9 orbitals, matching the n² rule.

Understanding these relationships is essential for interpreting electron configurations, predicting chemical behavior, and mastering advanced topics in quantum chemistry.