Atomic Structure

The periodic table is not arbitrary. Every trend on it — size, reactivity, color, conductivity — falls out of one idea: electrons live in quantized orbitals around a positively charged nucleus, and the rules for filling those orbitals determine everything else.

1. Why atomic structure matters

You already know the cartoon. An atom is a tiny nucleus with electrons buzzing around it. The nucleus has protons (positive) and neutrons (neutral); the electrons (negative) balance the protons to make the whole thing electrically neutral. Different elements have different numbers of protons, and chemistry is mostly about what the electrons do.

That cartoon is right, but flat. It doesn't tell you why carbon forms four bonds, why sodium and fluorine react violently while argon refuses, why copper is a conductor and diamond an insulator, why iron's d-electrons give it magnetism, or why every emission line in a sodium streetlamp is exactly where it is. All of that comes from one deeper fact: electrons inside atoms are restricted to a discrete set of quantum states, and the structure of those states is what makes chemistry predictable.

Why should you care, if you aren't a chemist?

• Materials and batteries. Why does lithium make good anodes? Because it has one loosely held valence electron and a small, light ion. Why is cobalt's role in a cathode hard to replace? Its d-orbital occupancy sets the voltage window. Every cell-chemistry tradeoff is an atomic-structure tradeoff.
• Pharmaceuticals. Drug-target binding is electron density docking with electron density. Medicinal chemists spend their days predicting how substituents change electronic structure and therefore binding affinity.
• Environmental and atmospheric. Ozone depletion, greenhouse warming, and photochemistry in the upper atmosphere are all driven by how specific photons interact with specific electronic transitions in specific molecules.
• Imaging and diagnostics. MRI sees nuclear spins. PET scans see positron emissions. X-ray crystallography reads electron density. All are atomic-structure tools.

2. Vocabulary cheat sheet

3. Atoms and isotopes

A neutral atom has three ingredients:

• Protons in the nucleus. Each carries charge $+e$. The count is $Z$, the atomic number, and it determines the element. $Z = 6$ is carbon, always. Change $Z$ and you change the element.
• Neutrons in the nucleus. Neutral, slightly more massive than protons. Their count is $N$, and it can vary without changing the element — atoms with the same $Z$ but different $N$ are called isotopes.
• Electrons outside the nucleus, each with charge $-e$. In a neutral atom there are $Z$ of them, one for every proton. Chemistry is mostly about these.

The total mass of the atom is dominated by the nucleus (protons and neutrons are about 1800 times heavier than an electron). We quote the mass in mass number:

When you look up an element's atomic weight on a periodic table (say, 12.011 for carbon), that decimal is the natural abundance-weighted average of the isotope masses. If your work cares about individual isotopes — in mass spectrometry, NMR, or nuclear chemistry — you will often need the specific value for a specific isotope, not the average.

Worked example: isotope arithmetic

Boron has two stable isotopes: $^{10}\text{B}$ (mass 10.013 u, abundance 19.9%) and $^{11}\text{B}$ (mass 11.009 u, abundance 80.1%). The weighted average is

4. The quantum atom

The first useful model of the atom was Niels Bohr's, in 1913. He assumed electrons orbit the nucleus in circular paths like planets, but with one non-classical twist: only certain orbits are allowed. The allowed orbits are those where the electron's angular momentum is an integer multiple of $\hbar = h/(2\pi)$ (Planck's constant over $2\pi$).

For a single-electron ion with nuclear charge $Ze$, the Bohr model predicts discrete energy levels

For hydrogen ($Z = 1$), $E_1 = -13.6$ eV, $E_2 = -3.4$ eV, $E_3 = -1.51$ eV, and so on, converging to 0 as $n \to \infty$. The energy you need to ionize a hydrogen atom from its ground state — pull its electron out to infinity — is $0 - (-13.6) = 13.6$ eV. That number is the first ionization energy of hydrogen, and it's the first prediction you can compute and check against experiment.

The Schrödinger picture in one paragraph

For a single electron in a potential $V(r)$ (for hydrogen, $V(r) = -Ze^2/(4\pi\epsilon_0 r)$), the time-independent Schrödinger equation is

5. Orbitals and quantum numbers

Every electron in an atom is described by four quantum numbers. Three come from the Schrödinger equation; the fourth is a purely quantum property called spin.

1. Principal quantum number $n = 1, 2, 3, \dots$ — the shell. Sets the main energy and rough distance from the nucleus.
2. Angular momentum quantum number $\ell = 0, 1, 2, \dots, n-1$ — the subshell. Sets the orbital shape. $\ell = 0$ is called $s$, $\ell = 1$ is $p$, $\ell = 2$ is $d$, $\ell = 3$ is $f$. (The letters are historical — they come from pre-quantum spectroscopy: sharp, principal, diffuse, fundamental.)
3. Magnetic quantum number $m_\ell = -\ell, \dots, 0, \dots, +\ell$ — labels the $2\ell + 1$ orbitals in the $\ell$-th subshell. For $\ell = 1$ (a $p$ subshell) you get $m_\ell \in \{-1, 0, +1\}$, the three $p$ orbitals usually called $p_x, p_y, p_z$.
4. Spin quantum number $m_s = \pm\tfrac{1}{2}$ — intrinsic angular momentum. Every electron is either spin-up or spin-down.

The Pauli exclusion principle says no two electrons in the same atom can share all four quantum numbers. In practice, that means each orbital (a specific $n, \ell, m_\ell$) holds at most two electrons — one spin-up and one spin-down.

Orbital shapes

The math of the Schrödinger equation, when separated into radial and angular parts, assigns every $(n, \ell, m_\ell)$ a specific three-dimensional probability density:

• $s$ orbitals ($\ell = 0$) are spherically symmetric. The $1s$ orbital is a ball of electron density centered on the nucleus, densest at $r = 0$ (well, actually at the Bohr radius $a_0 \approx 0.529$ Å if you look at the radial distribution function). The $2s$ has a spherical node — a thin shell where the density drops to zero — between two concentric regions of density.
• $p$ orbitals ($\ell = 1$) are dumbbell shaped, with a nodal plane through the nucleus. The three $p$ orbitals in a shell are oriented along mutually perpendicular axes: $p_x, p_y, p_z$.
• $d$ orbitals ($\ell = 2$) come in five shapes, four of them four-leaf-clovers in different orientations, plus the distinctive "dumbbell-with-a-donut" $d_{z^2}$.
• $f$ orbitals ($\ell = 3$) have seven even more baroque shapes. You mostly encounter them in the lanthanides and actinides.

These shapes are the reason molecular geometry is what it is. When atoms bond, their orbitals overlap, and the overlap geometry is set by the orbital shapes. See bonding for the consequences.

6. Interactive: orbital shapes

The figure below draws cross-sections of hydrogen-like orbitals. Pick an orbital by its $(n, \ell, m_\ell)$ quantum numbers and the display updates with a 2D slice of the probability density $|\psi|^2$ in the $xz$-plane. Brighter means more likely to find the electron.

Things to notice as you click through:

• The $1s$ is a single blob at the center. No nodes of any kind.
• The $2s$ has a radial node — a sphere of zero density nested inside the outer density. Very hard to see in 2D but visible as a thin dark ring.
• The $p$ orbitals have one angular node (a plane). The two lobes have opposite sign in the underlying wavefunction $\psi$, even though both show up bright in $|\psi|^2$.
• The $d$ orbitals have two angular nodes. Their shapes dictate the geometry of transition-metal complexes.

7. Electron configurations and the Aufbau principle

To build a multi-electron atom, you feed electrons into orbitals one at a time, in increasing order of energy, with two per orbital (Pauli). That procedure is called the Aufbau principle (German for "building up"), and it comes with two refinements:

• Hund's rule. When filling orbitals of equal energy (say, the three $p$ orbitals in a $p$ subshell), put one electron in each orbital, all with parallel spin, before pairing any up. Intuitively: electrons repel, so they spread out and keep their spins aligned to minimize Pauli repulsion within the same spatial orbital.
• Madelung rule (also called $n + \ell$). Orbitals fill in order of increasing $n + \ell$, with lower $n$ breaking ties. So $4s$ fills before $3d$ because $n + \ell = 4 + 0 = 4$ beats $3 + 2 = 5$. This is the origin of all the rows in the periodic table.

The standard filling order is: $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, \dots$

So carbon ($Z = 6$) fills as $1s^2\, 2s^2\, 2p^2$. Sodium ($Z = 11$) is $1s^2\, 2s^2\, 2p^6\, 3s^1$. Iron ($Z = 26$) is $1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6\, 4s^2\, 3d^6$. You can also write these using noble gas shorthand: Fe is $[\text{Ar}]\, 4s^2\, 3d^6$, because argon has $1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6$ and you don't need to rewrite it.

8. Periodic trends

The periodic table's rows correspond to shells filling up. Its columns correspond to atoms with matching outer-shell configurations — the valence electrons — and that's why columns share chemical behavior. All the alkali metals (Li, Na, K, Rb, Cs) have $ns^1$ valence configurations, and they all react the same way: lose one electron, become a $+1$ cation. All the halogens (F, Cl, Br, I) have $ns^2 np^5$ configurations, and they all want one more electron to close their shell.

Four trends fall out of this structure. Three of them you should commit to memory, because they come up constantly.

Atomic radius

Across a row (left to right), atoms shrink. The nuclear charge $Z$ goes up, but the additional electrons go into the same shell and don't shield each other well, so every electron feels a stronger effective nuclear charge and is pulled in tighter. Down a column, atoms grow. Each row adds a whole new shell, and the core electrons partially shield the outer ones from the nucleus, so the outer shell settles at a larger radius.

Ionization energy

The first ionization energy $\text{IE}_1$ is the energy required to remove one electron from a neutral gas-phase atom:

Across a row, $\text{IE}_1$ generally rises (the nucleus pulls harder, electrons are harder to remove). Down a column, it falls (the electron you're removing is in a higher, less tightly bound shell). There are two famous glitches: $\text{IE}_1$ drops slightly going from Be to B (because you start pulling a 2p electron instead of a 2s one, and 2p is higher in energy), and again from N to O (because the O you're pulling from has a paired 2p electron, which is less stable than the half-filled configuration of N).

Electron affinity and electronegativity

Electron affinity is the energy released when a neutral atom captures an electron to become a $-1$ anion. Electronegativity is a derived quantity (Pauling's scale) that captures how strongly an atom pulls shared electrons toward itself in a bond. Both grow across a row and shrink down a column, for the same reasons ionization energy does. Fluorine is the most electronegative element (Pauling scale 3.98). Francium is the least (0.7). Everything else falls in between, and the electronegativity difference between two bonded atoms controls whether the bond is purely covalent, polar, or ionic — see bonding.

9. Spectroscopy: how energy levels become light

Spectroscopy is the experimental handle on electronic structure. Shine light on an atom and it absorbs photons whose energies exactly match gaps between its electronic levels. Heat an atom and it emits photons at those same gaps as excited electrons fall back down. The pattern of absorbed or emitted wavelengths is a fingerprint of the atom's energy-level structure, and for simple atoms like hydrogen, you can read the level structure off the spectrum and confirm the theory.

For hydrogen, the transition from level $n_i$ to level $n_f$ emits a photon of energy

Connect to the photon: $E_{\text{photon}} = h\nu = hc/\lambda$, where $\nu$ is frequency and $\lambda$ is wavelength. If $\Delta E = 10.2$ eV (the first Lyman line, $n_i=2 \to n_f=1$), then $\lambda = hc/\Delta E = 121.6$ nm — in the vacuum ultraviolet. That line is one of the brightest in the spectrum of the early universe, and it's how astronomers find distant hydrogen.

Worked example: the Balmer alpha line

Compute the wavelength of the $n_i = 3 \to n_f = 2$ transition in hydrogen.

1. $\Delta E = R_H(1/4 - 1/9) = 13.6\,(0.25 - 0.1111) = 13.6 \times 0.1389 = 1.889$ eV.
2. Convert to joules: $\Delta E = 1.889 \times 1.602\times 10^{-19} = 3.026\times 10^{-19}$ J.
3. $\lambda = hc/\Delta E = (6.626\times 10^{-34})(3\times 10^8)/(3.026\times 10^{-19})$.
4. $= 6.566\times 10^{-7}$ m $= 656.6$ nm. Red light.

That's the H-alpha line, and it's exactly the red glow you see in pictures of nebulae. The prediction and the observation agree to parts per million.

10. Atomic structure in code

Two small computations. The first evaluates the radial part of a hydrogen orbital. The second computes a weighted isotope mass from abundances. Both are tiny; both are useful.

import numpy as np

# ---------- 1. Hydrogen 1s radial wavefunction ----------
# R_{10}(r) = 2 (1/a0)^{3/2} exp(-r/a0)
# a0 is the Bohr radius in meters.
a0 = 5.29177e-11
def R10(r):
    return 2.0 * a0**(-1.5) * np.exp(-r / a0)

# Radial probability density P(r) = 4 pi r^2 |R(r)|^2
# Most likely r is where dP/dr = 0, which happens at r = a0.
r_grid = np.linspace(0, 5 * a0, 200)
P = 4 * np.pi * r_grid**2 * R10(r_grid)**2
r_peak = r_grid[np.argmax(P)]
print(f"most likely r = {r_peak/a0:.3f} a0 (expected 1.000)")

# ---------- 2. Bohr energy levels ----------
RH = 13.6057  # eV
def E_bohr(n, Z=1):
    return -Z**2 * RH / n**2

# Balmer alpha: n=3 -> n=2
dE = E_bohr(3) - E_bohr(2)  # negative, emission
print(f"Balmer alpha photon = {-dE:.3f} eV")
# hc in eV*nm is a handy constant: 1240 eV*nm
lam_nm = 1240.0 / (-dE)
print(f"wavelength = {lam_nm:.1f} nm (expected ~656)")

# ---------- 3. Average atomic mass from isotopes ----------
def avg_mass(masses, abundances):
    a = np.asarray(abundances, dtype=float)
    a = a / a.sum()  # normalize in case percentages don't sum to 1
    return float(np.dot(masses, a))

# Boron: 10B (10.013 u, 19.9%) and 11B (11.009 u, 80.1%)
m_B = avg_mass([10.013, 11.009], [0.199, 0.801])
print(f"natural boron = {m_B:.3f} u (periodic table: 10.811)")

import math

# Same calculations without NumPy.

a0 = 5.29177e-11
RH = 13.6057

def R10(r):
    return 2.0 * a0**(-1.5) * math.exp(-r / a0)

def E_bohr(n, Z=1):
    return -Z * Z * RH / (n * n)

def photon_nm(n_hi, n_lo, Z=1):
    dE = E_bohr(n_hi, Z) - E_bohr(n_lo, Z)
    return 1240.0 / (-dE)

print(f"Lyman alpha  = {photon_nm(2,1):6.1f} nm")  # ~121.6
print(f"Balmer alpha = {photon_nm(3,2):6.1f} nm")  # ~656.5
print(f"Paschen alpha= {photon_nm(4,3):6.1f} nm")  # ~1875

# He+ (Z=2) Lyman alpha: factor of Z^2 = 4 in energy, 1/4 in wavelength
print(f"He+ Lyman alpha = {photon_nm(2,1,Z=2):.1f} nm")  # ~30.4

def avg_mass(masses, fractions):
    total = sum(fractions)
    return sum(m * (f / total) for m, f in zip(masses, fractions))

print(f"Cl avg = {avg_mass([34.969, 36.966], [0.7576, 0.2424]):.3f}")  # ~35.45

Three things worth noticing:

• $hc$ in electron-volt nanometers is 1239.84 eV·nm, usually rounded to 1240. Chemists carry it as a unit constant so they never have to convert back and forth through SI units.
• The Bohr energies scale like $Z^2$, so He$^+$ has four times hydrogen's ionization energy and quarter-sized wavelengths. Li$^{2+}$ has nine times, and so on. That's how you get from the Balmer lines to the Lyman lines of heavier one-electron ions.
• For multi-electron atoms, you can't just plug in $Z$ — electron-electron repulsion messes up the simple formula. You need an effective nuclear charge $Z_\text{eff}$ that each electron actually feels, which is smaller than $Z$ because of shielding. Slater's rules give empirical values of $Z_\text{eff}$, and Hartree-Fock computes them self-consistently; see quantum chemistry.

11. Cheat sheet

See also