17.4 Hydrogen atom and atomic orbitals

The hydrogen atom is the only atom for which the Schrödinger equation can be solved exactly. That makes it the foundation for understanding every other atom and molecule in chemistry. The solutions give you quantized energy levels and wavefunctions that describe where electrons are likely to be found, and those wavefunctions are what we call atomic orbitals.

Hydrogen Atom Orbitals

Solving the Schrödinger Equation

The hydrogen atom consists of one electron bound to one proton by the Coulomb potential. The time-independent Schrödinger equation for this system can be separated into radial and angular parts by working in spherical coordinates $(r, \theta, \phi)$. Separation of variables splits the problem into a radial equation (depending on $r$) and angular equations (depending on $\theta$ and $\phi$), each of which must be solved independently.

The solutions yield two things:

• Energy eigenvalues that depend only on the principal quantum number $n$:

$E_n = -\frac{13.6 \text{ eV}}{n^2}$

This means all orbitals with the same $n$ are degenerate in hydrogen (they share the same energy). That degeneracy breaks in multi-electron atoms, but for hydrogen it holds exactly.

• Wavefunctions $\psi_{n,l,m_l}(r, \theta, \phi)$ that factor into a radial part $R_{n,l}(r)$ and an angular part $Y_l^{m_l}(\theta, \phi)$:

$\psi_{n,l,m_l}(r, \theta, \phi) = R_{n,l}(r) \cdot Y_l^{m_l}(\theta, \phi)$

The angular parts $Y_l^{m_l}$ are the spherical harmonics, which you may have already encountered. Three quantum numbers $(n, l, m_l)$ emerge naturally from the boundary conditions of the differential equations, not from any ad hoc assumption.

Probability and Wavefunctions

The wavefunction itself doesn't have a direct physical meaning, but its square does. The quantity $|\psi|^2$ gives the probability density, the probability per unit volume of finding the electron at a given point in space.

• The total probability over all space must equal 1 (normalization condition): $\int |\psi|^2 \, dV = 1$
• Regions where $|\psi|^2$ is large correspond to places you're most likely to find the electron.
• For hydrogen, probability density is generally highest near the nucleus for s orbitals and distributed further out as $n$ increases.

Atomic Orbital Visualization

s Orbitals

s orbitals have $l = 0$, so the angular part is a constant. That makes them spherically symmetric: the probability density depends only on $r$, not on direction.

• The 1s orbital is the simplest, with a single exponential decay away from the nucleus and no nodes.
• Higher s orbitals (2s, 3s, ...) grow larger and develop radial nodes, which are spherical shells where $\psi = 0$.
• The number of radial nodes in any s orbital is $n - 1$. So the 1s has 0 nodes, the 2s has 1, the 3s has 2, and so on.

p Orbitals

p orbitals have $l = 1$, giving three possible values of $m_l$: $-1, 0, +1$. Each corresponds to a different spatial orientation.

• The real-valued combinations are labeled $p_x$, $p_y$, and $p_z$. Each has a dumbbell shape with two lobes of opposite sign separated by a nodal plane through the nucleus.
• That nodal plane is an angular node. Every p orbital has exactly one angular node.
• p orbitals first appear at $n = 2$. As $n$ increases, the lobes extend further from the nucleus and additional radial nodes appear.
• The number of radial nodes is $n - 2$. So a 2p orbital has 0 radial nodes, a 3p has 1, and so on.

d Orbitals

d orbitals have $l = 2$, giving five possible values of $m_l$: $-2, -1, 0, +1, +2$. Their shapes are more complex.

• Four of the five ($d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{x^2-y^2}$) have a cloverleaf shape with four lobes. The fifth, $d_{z^2}$, has two lobes along the z-axis plus a torus (donut) in the xy-plane.
• Each d orbital has two angular nodes. These can be nodal planes or nodal cones, depending on the specific orbital.
• d orbitals first appear at $n = 3$. The number of radial nodes is $n - 3$.

A useful general pattern: the total number of nodes (radial + angular) for any orbital is always $n - 1$, and the number of angular nodes equals $l$.

Electron Density and Probability

Electron Density

Electron density at a point is proportional to $|\psi|^2$ at that point. This is the Born interpretation of the wavefunction.

• High electron density means a high probability of finding the electron there.
• For the hydrogen 1s orbital, the electron density is highest right at the nucleus. However, the most probable distance from the nucleus is not zero (more on this below in the radial distribution section).
• In multi-electron atoms, electron density maps become the basis for understanding chemical bonding and molecular shape.

Visualization

Several types of plots are used to represent orbitals:

• Probability density plots show $|\psi|^2$ as a function of position, often as a color map or dot-density diagram.
• Isosurface plots draw a 3D surface enclosing a region where there's, say, a 90% probability of finding the electron. These are the "orbital shapes" you typically see in textbooks (spheres for s, dumbbells for p, cloverleaves for d).
• Cross-sectional contour plots slice through the orbital to show how $|\psi|^2$ varies in a plane.

Each visualization has trade-offs. Isosurfaces show shape clearly but hide internal structure like radial nodes. Contour plots reveal nodes but only show one slice at a time.

Quantum Numbers and Orbital Properties

Principal Quantum Number ($n$)

• $n$ determines the energy (in hydrogen) and the overall size of the orbital.
• Allowed values: $n = 1, 2, 3, \ldots$
• Larger $n$ means higher energy (less negative $E_n$) and a more spatially extended orbital.
• The total number of orbitals at a given $n$ is $n^2$.

Angular Momentum Quantum Number ($l$)

• $l$ determines the shape of the orbital and the magnitude of the orbital angular momentum: $L = \hbar\sqrt{l(l+1)}$.
• Allowed values: $l = 0, 1, 2, \ldots, n-1$
• Each value of $l$ corresponds to a letter designation: $l = 0$ (s), $l = 1$ (p), $l = 2$ (d), $l = 3$ (f).
• For a given $n$, there are $n$ possible values of $l$.

Magnetic Quantum Number ($m_l$)

• $m_l$ determines the orientation of the orbital in space and the z-component of angular momentum: $L_z = m_l \hbar$.
• Allowed values: $m_l = -l, -l+1, \ldots, 0, \ldots, l-1, l$
• For a given $l$, there are $2l + 1$ possible values of $m_l$.
• In the absence of an external field, orbitals with different $m_l$ but the same $n$ and $l$ are degenerate.

Radial and Angular Probability Distributions

Radial Probability Distribution

The radial probability distribution $P(r)$ tells you the probability of finding the electron at a distance $r$ from the nucleus, summed over all directions. It's defined as:

$P(r) = r^2 [R_{n,l}(r)]^2$

The factor of $r^2$ comes from the volume element in spherical coordinates (the spherical shell at distance $r$ has area $4\pi r^2$). This is why, for the 1s orbital, even though $|\psi|^2$ is maximized at $r = 0$, the radial probability distribution peaks at $r = a_0$ (one Bohr radius). At $r = 0$ the shell has zero area, so the probability of finding the electron at the nucleus is zero.

Key features of radial distributions:

• The number of radial nodes (values of $r$ where $P(r) = 0$, excluding $r = 0$ and $r \to \infty$) is $n - l - 1$.
• As $n$ increases, the outermost peak moves further from the nucleus, consistent with higher-energy electrons being found at greater distances.
• Comparing 2s and 2p: the 2s has a small inner peak close to the nucleus (inside its radial node), while the 2p does not. This "penetration" of s electrons toward the nucleus becomes important in multi-electron atoms.

Angular Probability Distribution

The angular probability distribution comes from the spherical harmonics $|Y_l^{m_l}(\theta, \phi)|^2$ and describes how the probability varies with direction, independent of distance.

• For $l = 0$ (s orbitals), the angular distribution is uniform in all directions (a sphere).
• For $l = 1$ (p orbitals), the distribution has a dumbbell shape with a nodal plane.
• For $l = 2$ (d orbitals), the distribution shows cloverleaf or related patterns with two nodal surfaces.

The number of angular nodes always equals $l$. These nodes are planes or cones where the angular part of the wavefunction equals zero. Combined with the radial nodes, the total node count is $n - 1$ for every orbital.