The Modern Quantum Mechanical Atomic Model

From Orbits to Orbitals: Why a New Model Was Needed

The Bohr–Sommerfeld model successfully explained some features of hydrogen-like atoms (one electron, like H, He⁺, Li²⁺) but failed for:

• Atoms with several electrons
• Fine details of spectra
• Chemical bonding and periodic trends in a consistent way

Experiments (especially atomic spectra and the behavior of electrons and light) showed:

• Light and electrons sometimes behave like particles, sometimes like waves
• Exact orbits with well-defined paths and energies cannot describe electrons correctly

This led to the quantum mechanical (wave-mechanical) model, in which the electron is described by a wave function and is no longer thought of as “orbiting” on a fixed path.

Core Ideas of the Quantum Mechanical Model

Wave–Particle Duality and the de Broglie Hypothesis

Electrons show both particle and wave properties. According to de Broglie, a particle of momentum $p$ has a wavelength:

$$
\lambda = \frac{h}{p}
$$

where $h$ is Planck’s constant.

For electrons in atoms, this wavelength is comparable to the dimensions of the atom. Therefore, electrons must be treated as matter waves, not just as tiny balls.

The Schrödinger Equation and the Wave Function

In the quantum mechanical model, the state of an electron in an atom is described by a wave function $\psi(\mathbf{r})$ (often just written $\psi$), which is a mathematical function of position (and, more generally, time).

The wave function is obtained by solving the Schrödinger equation (time-independent form for stationary states):

$$
\hat{H}\,\psi = E\psi
$$

• $\hat{H}$ is the Hamiltonian operator (the mathematical representation of the total energy)
• $E$ is the energy of the state
• $\psi$ is the wave function associated with that state

For the hydrogen atom, solving this equation gives a set of allowed wave functions $\psi$ with specific allowed energies $E$. These wave functions correspond to the orbitals of the atom.

You do not need the detailed mathematics here; what matters is what $\psi$ means physically.

Probability Interpretation: Born’s Rule

The wave function itself is not directly observable. Its physical meaning is given by the Born interpretation:

• The probability of finding the electron near a point in space is proportional to:

$$
|\psi(\mathbf{r})|^2
$$

Thus:

• $|\psi(\mathbf{r})|^2$ is the probability density of the electron
• The electron is not “on a track” but is spread out like a cloud of probability around the nucleus

This leads to the idea of electron clouds and orbitals as regions of space where the electron is likely to be found.

The Heisenberg Uncertainty Principle

In classical physics, you can in principle know both the exact position and exact momentum (velocity) of a particle. Quantum mechanics imposes a fundamental limit:

$$
\Delta x \,\Delta p_x \geq \frac{\hbar}{2}
$$

• $\Delta x$ = uncertainty in position
• $\Delta p_x$ = uncertainty in momentum in the $x$-direction
• $\hbar = \dfrac{h}{2\pi}$

Consequences for the atomic model:

• You cannot picture electrons as tiny planets with both precisely known positions and momenta in well-defined orbits
• Instead, you describe probability distributions (orbitals)
• The “orbit” concept is replaced by the orbital concept

Atomic Orbitals: Quantum Numbers and Shapes

Solving the Schrödinger equation for the hydrogen atom yields wave functions $\psi$ that can be labeled by quantum numbers. Each set of quantum numbers corresponds to an atomic orbital with a certain energy and shape. These orbitals are the building blocks for understanding atomic structure and later, the periodic table and bonding.

Principal Quantum Number $n$

The principal quantum number $n$ takes positive integer values:

$$
n = 1, 2, 3, \dots
$$

It is related to:

• The overall size of the orbital (larger $n$ → electron, on average, farther from the nucleus)
• The energy of the orbital (for hydrogen and other one-electron ions, energy depends only on $n$)

Orbitals with the same $n$ form a shell (e.g. $n=2$ is the second shell).

Angular Momentum Quantum Number $l$

The angular momentum quantum number $l$ describes the shape of the orbital. For a given $n$:

$$
l = 0, 1, 2, \dots, n-1
$$

By convention, we use letters:

• $l=0$ → s
• $l=1$ → p
• $l=2$ → d
• $l=3$ → f
• $l=4$ → g (rarely needed in basic chemistry)

Thus:

• $n=1$ allows only $l=0$ → 1s
• $n=2$ allows $l=0,1$ → 2s, 2p
• $n=3$ allows $l=0,1,2$ → 3s, 3p, 3d

In multi-electron atoms, for a given $n$, orbitals with different $l$ (e.g. 2s vs 2p) have different energies.

Magnetic Quantum Number $m_l$

The magnetic quantum number $m_l$ describes the orientation of the orbital in space. For a given $l$:

$$
m_l = -l, -l+1, \dots, 0, \dots, l-1, l
$$

So there are $2l+1$ possible $m_l$ values for each $l$. Examples:

• For $l=0$ (s): $m_l = 0$ → 1 orientation → 1 s-orbital per shell
• For $l=1$ (p): $m_l = -1, 0, +1$ → 3 orientations → 3 p-orbitals per shell ($p_x$, $p_y$, $p_z$)
• For $l=2$ (d): $m_l = -2, -1, 0, +1, +2$ → 5 d-orbitals per shell
• For $l=3$ (f): $m_l = -3, -2, -1, 0, +1, +2, +3$ → 7 f-orbitals per shell

In the presence of magnetic fields, different $m_l$ values can correspond to slightly different energies; this helps explain spectral line splittings.

Spin Quantum Number $m_s$

Experimentally, electrons show an intrinsic angular momentum called spin. This property cannot be explained as literal spinning of a charged ball; it is a purely quantum mechanical property.

The spin quantum number $m_s$ can take two values:

$$
m_s = +\frac{1}{2} \quad \text{or} \quad m_s = -\frac{1}{2}
$$

These are often called “spin-up” and “spin-down”.

Spin has important consequences for how electrons can be arranged in orbitals and leads to the Pauli exclusion principle.

The Pauli Exclusion Principle

The Pauli exclusion principle states:

No two electrons in the same atom can have the same set of all four quantum numbers $(n, l, m_l, m_s)$.

Consequences:

• Each orbital (fixed $n, l, m_l$) can hold at most two electrons
• When two electrons share an orbital, they must have opposite spins ($m_s = +1/2$ and $m_s = -1/2$)

This principle underlies:

• The structure of the periodic table
• The arrangement of electrons into shells and subshells
• Many chemical and physical properties of elements

Shapes and Spatial Distribution of Orbitals

In the quantum mechanical model, orbitals are visualized as regions in space where the electron is likely to be found. One convenient way to visualize them is via surface diagrams enclosing a certain percentage (e.g. 90–95%) of the electron probability.

Key features:

• s-orbitals ($l=0$)
• Spherically symmetric (no angular dependence)
• Probability density depends only on distance from nucleus
• Higher $n$ s-orbitals (2s, 3s, …) have radial nodes (regions at certain distances where $|\psi|^2 = 0$)
• p-orbitals ($l=1$)
• Dumbbell-shaped, with two lobes of opposite sign of $\psi$
• Three orientations: $p_x$, $p_y$, $p_z$ (correspond to $m_l = -1, 0, +1$ in some orientation basis)
• Nodal plane passing through the nucleus (where $\psi = 0$)
• d- and f-orbitals ($l=2,3$)
• More complex shapes (e.g. cloverleaf shapes for many d-orbitals)
• Essential for understanding transition metals and lanthanides/actinides but not detailed here

The sign of the wave function (positive or negative) is important in bonding (because wave functions can add or cancel), but for simple probability pictures we often focus only on $|\psi|^2$.

Nodes

A node is a region where the probability of finding the electron is zero ($\psi = 0$).

• Radial nodes: spherical surfaces at certain radii
• Angular nodes: planes or cones (from the angular part of $\psi$)

The number and type of nodes are determined by $n$ and $l$ and affect the energy and shape of orbitals.

Energies of Orbitals in One- and Many-Electron Atoms

Hydrogen-Like (One-Electron) Atoms

For hydrogen and hydrogen-like ions (He⁺, Li²⁺, …):

• The energy depends only on $n$:

$$
E_n = -\frac{R_H}{n^2}
$$

where $R_H$ is the Rydberg constant (in energy units).

• All orbitals with the same $n$ (e.g. 2s and 2p) are degenerate (have the same energy).

Multi-Electron Atoms: Lifting of Degeneracy

In atoms with more than one electron, electron–electron repulsions and shielding modify orbital energies:

• For a given $n$, orbitals with different $l$ have different energies
• Typically: $E(\text{s}) < E(\text{p}) < E(\text{d}) < E(\text{f})$ within the same shell
• As $n$ increases, some subshells can have energies close to or even cross those of lower $n$ subshells
• Example: 4s is lower in energy than 3d in isolated atoms

The resulting energy ordering of subshells (1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < …) is crucial for understanding electron configurations and the periodic table, which are treated separately.

Conceptual Differences from the Bohr–Sommerfeld Model

In moving from the older model to the quantum mechanical model:

• Orbits → Orbitals
• No fixed circular or elliptical paths
• Orbitals are stationary probability distributions, not trajectories
• Definite position and velocity → Probability distributions
• Heisenberg’s uncertainty principle forbids specifying both exactly
• You work with $|\psi|^2$ instead of classical paths
• Ad hoc quantization → Natural quantization
• In Bohr’s model, quantized orbits were postulated
• In the quantum model, quantization of energy and angular momentum emerges from solving the Schrödinger equation with appropriate boundary conditions
• Classical particles → Quantum objects with wave character
• Electron interference, diffraction, and tunneling are naturally described
• Chemical bonding (overlap of wave functions), shapes of molecules, and modern spectroscopy are all based on these quantum states

Why the Quantum Mechanical Model Matters for Chemistry

The modern quantum mechanical atomic model provides:

• A framework for quantum numbers, orbitals, and electron configurations
• The basis for understanding:
• Periodic trends (atomic radii, ionization energies, electron affinities)
• Types and strengths of chemical bonds (covalent, ionic, metallic, etc.)
• Molecular shape and hybridization
• Spectroscopic behavior (absorption, emission, selection rules)

Later chapters build on this model to explain how electrons are arranged (electron configurations) and how that arrangement governs the chemical properties summarized in the periodic table.