Electron Configuration and Orbitals

To precisely describe the state of an electron in an atom, we use four quantum numbers:

• Principal Quantum Number (n): This number determines the main energy level or shell an electron occupies. It can be any positive integer (1, 2, 3, ...). Higher values of 'n' indicate higher energy levels and larger orbitals. For example, n=1 is the first shell, n=2 is the second shell, and so on.

• Azimuthal/Angular Momentum Quantum Number (l): Also known as the subshell quantum number, 'l' defines the shape of the orbital and the subshell within a given energy level. Its values range from 0 to (n-1).

  • l=0 corresponds to an s orbital (spherical shape).
  • l=1 corresponds to a p orbital (dumbbell shape).
  • l=2 corresponds to a d orbital (more complex shapes).
  • l=3 corresponds to an f orbital (even more complex shapes).

• Magnetic Quantum Number (ml): This number describes the orientation of the orbital in space. Its values range from -l to +l, including 0.

  • For l=0 (s orbital), ml=0 (1 orientation).
  • For l=1 (p orbital), ml=-1, 0, +1 (3 orientations: px, py, pz).
  • For l=2 (d orbital), ml=-2, -1, 0, +1, +2 (5 orientations).

• Spin Quantum Number (ms): This describes the intrinsic angular momentum (spin) of an electron. It can only take two values: +1/2 (spin up) or -1/2 (spin down). Each orbital can hold a maximum of two electrons, and they must have opposite spins (Pauli Exclusion Principle).

Understanding the shapes and relative energies of atomic orbitals is crucial for predicting electron configurations and chemical reactivity.

• s orbitals (l=0): These are spherical in shape. The 1s orbital is the smallest and lowest energy, followed by 2s, 3s, etc., which are progressively larger and higher in energy. All s orbitals are non-directional.

• p orbitals (l=1): There are three p orbitals (px, py, pz) in each p subshell, oriented along the x, y, and z axes respectively. Each has a dumbbell shape with two lobes separated by a nodal plane at the nucleus. The 2p orbitals are lower in energy than 3p orbitals.

• d orbitals (l=2): There are five d orbitals in each d subshell. Four of these have a cloverleaf shape (dxy, dxz, dyz, dx²-y²), while the fifth (dz²) has a unique shape resembling a dumbbell with a donut around its middle. These orbitals are more complex and higher in energy than s and p orbitals of the same principal quantum number.

• Energy Levels: In a multi-electron atom, the energy of orbitals generally increases with increasing 'n'. However, within a given shell, the order of energy is typically s < p < d < f. There are also overlaps in energy levels, for example, the 4s orbital is often lower in energy than the 3d orbitals, which is important for transition metals.

Electrons fill atomic orbitals according to specific rules, which help us determine the ground state electron configuration of an atom:

• Aufbau Principle (Building-Up Principle): This principle states that electrons will occupy the lowest energy orbitals available first before moving to higher energy orbitals. This sequential filling ensures the most stable electron configuration. The order of filling can be remembered using an energy level diagram or the diagonal rule (1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, etc.).

• Pauli Exclusion Principle: This fundamental principle states that no two electrons in an atom can have the exact same set of four quantum numbers. This means that an atomic orbital can hold a maximum of two electrons, and these two electrons must have opposite spins (one +1/2 and one -1/2). This is often represented by arrows pointing in opposite directions (↑↓) within an orbital box.

• Hund's Rule of Maximum Multiplicity: When electrons are filling a set of degenerate orbitals (orbitals of the same energy, e.g., the three p orbitals or five d orbitals), they will occupy each orbital singly with parallel spins before any orbital is doubly occupied. This maximizes the total spin and minimizes electron-electron repulsion, leading to a more stable configuration. For example, for a p subshell with three electrons, they would be placed as ↑ ↑ ↑ rather than ↑↓ ↑ .