Electronic Structure and the Periodic Table

Light has a wave-particle nature: photons are energy packets $E = h\nu = hc/\lambda$, where $h = 6.626\times10^{-34}\,\mathrm{J\cdot s}$ is Planck's constant. This means light behaves both as a wave (interference, diffraction) and as a stream of particles (photoelectric effect).

The Bohr model (1913) describes the hydrogen atom by postulating that electrons can only occupy quantized orbits (stationary states), without emitting energy as long as they stay in the same orbit. Each orbit corresponds to a defined energy level: $$E_n = -\dfrac{13.6}{n^2}\,\mathrm{eV}$$ where $n=1$ is the ground state (lowest energy, -13.6 eV) and $n>1$ are excited states.

When an electron jumps from $n_i$ to $n_f$ (transition), it absorbs or emits a photon with energy equal to the difference between the two levels. The wavelength of the emitted radiation follows the Rydberg formula: $$\dfrac{1}{\lambda} = R_H\!\left(\dfrac{1}{n_f^2}-\dfrac{1}{n_i^2}\right)$$ where $R_H = 1.097\times10^7\,\mathrm{m^{-1}}$ is the Rydberg constant.

The spectral series of hydrogen are groups of lines corresponding to transitions ending at the same level $n_f$:

• Lyman series ($n_f=1$): ultraviolet
• Balmer series ($n_f=2$): visible (the red $H_\alpha$ line at 656 nm is the $3\to2$ transition)
• Paschen series ($n_f=3$): infrared

Real-world applications: atomic spectroscopy identifies elements in stars (each element has a unique spectrum, like a "fingerprint"), analyzes material composition, and enables gas lasers (e.g. He-Ne).

The Bohr model fails for atoms with more than one electron. The correct description comes from quantum mechanics. Heisenberg's uncertainty principle states that we cannot simultaneously know position and momentum with arbitrary precision: $$\Delta x\,\Delta p \geq \dfrac{\hbar}{2}, \qquad \hbar = \dfrac{h}{2\pi}$$ Instead of definite orbits, the electron is described by a wavefunction $\psi$ that satisfies the Schrödinger equation: $$\hat{H}\psi = E\psi$$ The square $|\psi|^2$ represents the probability density of finding the electron at a given position. The regions of space where the probability is high are called atomic orbitals.

Quantum numbers: each orbital is identified by three numbers, plus a fourth for the electron:

• $n$ (principal quantum number): determines the energy level and orbital size. Values: $1,2,3,\ldots$
• $\ell$ (azimuthal quantum number): determines orbital shape. Values: $0,1,\ldots,n-1$, conventionally denoted by letters: $\ell=0$ → $s$ (spherical), $\ell=1$ → $p$ (dumbbell), $\ell=2$ → $d$ (clover), $\ell=3$ → $f$ (complex shapes)
• $m_\ell$ (magnetic quantum number): determines the spatial orientation of the orbital. Values: $-\ell,\ldots,+\ell$ (e.g. for $\ell=1$ → $p_x, p_y, p_z$)
• $m_s$ (spin quantum number): $\pm1/2$, describes the electron's intrinsic rotation

Real-world applications: the scanning tunneling microscope (STM) uses quantum tunneling probability to "see" individual atoms; quantum dots use quantum confinement to emit different colors based on size (QLED displays).

The electron configuration of the ground state describes how electrons are distributed among orbitals, and is built using three fundamental rules:

1. Aufbau Principle (minimum energy): electrons fill the lowest-energy orbitals first. The filling order follows the Aufbau diagram: $$1s \to 2s \to 2p \to 3s \to 3p \to 4s \to 3d \to 4p \to 5s \to 4d \to 5p \to 6s \to 4f \to 5d \to 6p \ldots$$
Important note: the energy order does not simply follow increasing $n$ — for example, $4s$ fills before $3d$ because it has slightly lower energy.

2. Pauli Exclusion Principle: no two electrons in the same atom can have all four quantum numbers identical. Consequence: each orbital can hold at most 2 electrons, with opposite spins ($m_s = +1/2$ and $m_s = -1/2$).

3. Hund's Rule (maximum multiplicity): when filling degenerate orbitals (same energy, like the three $2p$ orbitals), electrons occupy them singly with parallel spins first, before pairing up. This minimizes electrostatic repulsion. Example: nitrogen ($Z=7$) has configuration $1s^2\,2s^2\,2p^3$, with the three $2p$ electrons in different orbitals, all with parallel spins.

Compact notation uses the preceding noble gas as a core: e.g. $\mathrm{Fe}$ ($Z=26$) → $[\mathrm{Ar}]\,4s^2\,3d^6$, where $[\mathrm{Ar}]$ represents the configuration of argon ($Z=18$: $1s^2\,2s^2\,2p^6\,3s^2\,3p^6$).

Anomalous configurations: some elements (like Cr and Cu) have different configurations than expected. Cr ($Z=24$) has $[\mathrm{Ar}]\,4s^1\,3d^5$ (not $4s^2\,3d^4$) because a half-filled or fully filled $d$ subshell provides extra stability.

Real-world applications: electron configuration explains why elements in the same group have similar chemical properties (e.g. alkali metals in group 1 all have $ns^1$ and are extremely reactive), and why noble gases (group 18, filled shells) are inert.

The modern periodic table orders elements by increasing atomic number $Z$, organized in periods (rows) and groups (columns). Elements in the same group have the same valence electron configuration and therefore similar chemical properties.

The main periodic properties vary regularly across the table:

• Atomic radius: decreases across a period (left to right) because the effective nuclear charge $Z_{eff}$ increases, pulling electrons more strongly toward the nucleus. Increases down a group because new electron shells are added. Example: $\mathrm{Li} > \mathrm{F}$ (same period), $\mathrm{Li} < \mathrm{Cs}$ (same group).
• Ionization energy ($E_i$): the minimum energy needed to remove an electron from a gaseous atom. Increases across a period (greater nuclear attraction) and decreases down a group (the electron is farther away and more shielded). Notable exception: $E_i(\mathrm{O}) < E_i(\mathrm{N})$ because in oxygen the second electron in a $2p$ orbital, being paired, is easier to remove (e-e repulsion).
• Electron affinity ($A_e$): energy released when a gaseous atom captures an electron. More negative values (greater energy release) indicate a stronger tendency to form anions. Generally increases (becomes more negative) across a period. Noble gases have $A_e > 0$ (they do not form anions).
• Electronegativity ($\chi$): a measure of an atom's tendency to attract electrons in a bond. The Pauling scale ranges from 0.7 (Fr) to 4.0 (F). Increases across a period and decreases down a group. Fluorine is the most electronegative element.
• Metallic character: decreases from left to right (metals are on the left, nonmetals on the right), increases from top to bottom.

Real-world applications: the periodic table is the fundamental tool for predicting chemical behavior. Alkali metals ($ns^1$) are used in batteries (Li, Na), noble gases ($ns^2np^6$) provide inert atmospheres for welding, and periodic properties guide the design of new materials (e.g. doped semiconductors in electronics).