Quantum Numbers, Orbitals & Their Shapes

Quantum Numbers

In the solution of the Schrödinger equation for the hydrogen atom, a set of quantum numbers arises that describe allowed states of an electron. Three quantum numbers (n, l, m_l) come from the spatial form of the wavefunction and a fourth quantum number (m_s) arises from the intrinsic spin of the electron. According to the Pauli exclusion principle, no two electrons in an atom can have the same set of all four quantum numbers; this restriction determines how electrons fill atomic orbitals and is fundamental to the structure of the periodic table.

Principal quantum number (n)

The principal quantum number, n, results from the radial solution of the Schrödinger equation for the hydrogen atom and takes integral values:

• Permissible values: n = 1, 2, 3, ... (all positive integers).
• It identifies the shell and largely determines the average distance of the electron from the nucleus (higher n → larger orbital size).
• For the hydrogen atom the bound-state energy is given by the expression E_n = -13.6 eV / n², so energy depends only on n in hydrogen.

• In many-electron atoms, energy depends on both n and l, so orbital ordering can change; for example, in many atoms the general ordering for s-orbitals is 4s > 3s > 2s > 1s in terms of size and average distance from the nucleus (but careful: the energy ordering among subshells depends on electron-electron interactions).
• The total number of orbitals in the nth shell equals n².
• The maximum number of electrons that can occupy the nth shell is 2n².
• The principal quantum number helps determine the radial probability distribution of an orbital.
• The principal quantum number also restricts allowed values of the azimuthal quantum number l (see next section).
• The orbital angular momentum magnitude is related to l by L = √(l(l+1)) × h/2π.  

Azimuthal (angular momentum) quantum number (l)

The azimuthal or angular momentum quantum number, l, arises from the angular part of the wavefunction and determines the shape of the orbital.

• Permissible values: l = 0, 1, 2, ..., (n - 1). Thus, the allowed values of l are limited by the principal quantum number n.
• The value of l identifies the subshell (s, p, d, f, ...) and the general shape and number of angular nodes of the orbital.
• The number of possible values of l for a given n equals n.
• The number of orbitals in a subshell with quantum number l is 2l + 1 (these correspond to the different m_l values).

Magnetic quantum number (m_l)

The magnetic quantum number, m_l, specifies the orientation of an orbital angular momentum in space relative to a chosen axis (usually the z-axis).

• Permissible values: m_l = -l, -l+1, ..., 0, ..., +l (integer steps).
• The z-component of angular momentum is quantized: L_z = m_l × h / 2π.
• There are 2l + 1 possible orientations for a given l, corresponding to the number of orbitals in that subshell.
• In the presence of an external magnetic field, these orientations have different energies and give rise to the Zeeman effect (splitting of spectral lines).
• The vector model of angular momentum provides a qualitative way to visualise how the angular momentum vector projects onto an axis; m_l determines the allowed projections.

Spin quantum number (m_s)

Electrons have an intrinsic angular momentum called spin. Spin is an intrinsic (non-classical) property of electrons and is not derived from spatial wavefunctions.

• The spin quantum number for an electron is s = 1/2 and its projection is given by m_s = +1/2 or m_s = -1/2.
• Because there are two possible m_s values, each orbital (specified by n, l, m_l) can hold up to two electrons with opposite spins (this is the origin of the factor 2 in the 2n² rule).
• The two spin orientations correspond to two possible z-components of spin angular momentum and are associated with two values of the electron's magnetic moment. Interaction of the spin magnetic moment with an external magnetic field leads to additional energy splitting (fine structure and spin-orbit effects).

Orbitals

An orbital is a region in space, described by a wavefunction ψ(n, l, m_l), where the probability of finding an electron is appreciable. Orbitals are designated by the set of three quantum numbers (n, l, m_l) that arise from the solution of the Schrödinger equation.

Different Atomic Orbitals

Shapes of atomic orbitals

The wavefunction ψ itself is a mathematical function; its square, |ψ|², gives the probability density for an electron at a point in space (Max Born interpretation). Plots of |ψ|² or constant |ψ|² boundary-surface diagrams show the shapes and sizes of orbitals.

• Boundary-surface diagrams are drawn at a constant value of |ψ|² that encloses a region where the probability of finding the electron is large (commonly chosen ≈ 90%).
• There is no finite boundary corresponding to 100% probability because |ψ|² tails extend to infinity, though with rapidly decreasing values.

s orbitals

• All s orbitals (l = 0) are spherically symmetric about the nucleus; the boundary surface is a sphere (appearing as a circle in two dimensions).
• The probability density for a 1s orbital is maximum at the nucleus and decreases monotonically with distance.
• Higher ns orbitals have radial nodes. An ns orbital has (n - 1) total nodes; for s orbitals these are all radial nodes. For example, 2s has one radial node, 3s has two radial nodes, and so on.
• The size of s orbitals increases with n: 4s > 3s > 2s > 1s (in terms of average radius and boundary size).

Spheres for s-Orbitals

Charge Cloud Diagram of s-Orbitals

p orbitals

• For l = 1 the orbitals are p orbitals. There are three p orbitals for a given n (except when n = 1): 2p has three orientations corresponding to m_l = -1, 0, +1.
• Each p orbital consists of two lobes separated by a nodal plane passing through the nucleus where |ψ|² = 0.
• Designations such as 2p_x, 2p_y, 2p_z indicate orientation along the x, y or z axis. The three p orbitals are mutually perpendicular and equivalent in energy in hydrogenic atoms.
• Radial nodes for np orbitals are given by n - l - 1 = n - 2; for example, 3p has one radial node, 4p has two radial nodes, etc.
• With increasing principal quantum number the size and energy of p orbitals increase: 4p > 3p > 2p (as regards size and average distance from nucleus).

2p Orbitals

d orbitals

• For l = 2 the orbitals are d orbitals. The minimum principal quantum number allowing d orbitals is n = 3.
• There are five possible m_l values for l = 2: -2, -1, 0, +1, +2, so five d orbitals exist in a d subshell.
• The five d orbitals are commonly written as d_xy, d_yz, d_xz, d_x²-y², d_z²; four of these have similar cloverleaf shapes while d_z² has a distinct doughnut-and-lobes shape.
• In hydrogen-like atoms the five d orbitals are degenerate (same energy). In many-electron atoms crystal field or ligand interactions and electron-electron repulsion can lift this degeneracy.

d Orbitals and Their Shapes

Nodes

• Radial nodes are spherical surfaces where the radial part of the wavefunction changes sign and |ψ|² = 0. Number of radial nodes = n - l - 1.
• Angular nodes are planes or cones where the angular part of the wavefunction is zero (for p orbitals the nodal plane is one; for d orbitals there are two angular nodes, etc.). Number of angular nodes = l.
• Total number of nodes (radial + angular) = n - 1.

Angular nodes for the p_z orbital correspond to the xy-plane being a nodal plane.
For d_xy, nodal planes lie along axes that cause the characteristic orientation of that orbital's lobes.