Ionization Energy

Ionization energy (IE, also denoted I or E_i; historically termed ionization potential when expressed in volts) is the minimum energy required to detach the most loosely bound electron from an isolated gaseous atom, radical, molecule, or ion in its ground electronic, vibrational, and rotational state and carry that electron to rest at infinite distance (r = ∞, zero translational kinetic energy). In chemical thermodynamics, the first ionization energy (IE₁) corresponds to the standard internal energy change Δ*U*° at T = 0 K for the gas-phase reaction:

M(g) → M⁺(g) + e⁻(g) (IE₁ = Δ*U*°_{0 K} > 0)

In atomic physics and photoelectron spectroscopy, ionization energy is measured in electronvolts (eV) per atom, whereas in thermochemistry it is expressed in kilojoules per mole (kJ mol⁻¹), related by the Faraday conversion factor 1 eV = 96.4853 kJ mol⁻¹. Because bound electrons occupy negative potential energy wells generated by electrostatic attraction to the positively charged atomic nucleus, removing an electron from any neutral atom or positively charged cation is always endothermic (IE > 0).

Across the periodic table, first ionization energies span more than a sixfold range—from the lowest measured value in cesium (Cs, IE₁ = 3.894 eV = 375.7 kJ mol⁻¹) and francium (Fr, 4.073 eV = 393.0 kJ mol⁻¹) to the highest value in helium (He, IE₁ = 24.587 eV = 2,372.3 kJ mol⁻¹). Together with electron affinity, ionization energy governs whether an element forms cations in ionic bonding, shares electrons in a covalent bond, or delocalizes valence electrons in metallic bonding.

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The Hydrogenic Rydberg Limit and Effective Nuclear Charge (Z_eff)

For a one-electron hydrogenic atom or ion with nuclear charge Z and principal quantum number n, the exact non-relativistic energy required to promote the electron from orbital n to the ionization continuum (n = ∞) is given by the Rydberg formula:

IE = R_∞ hc (Z² / n²) = 13.5984 eV × (Z² / n²) = 1,312.0 kJ mol⁻¹ × (Z² / n²)

In a multi-electron atom, inner core electrons and fellow valence electrons partially shield the outer electron from the full nuclear charge Z via inter-electronic Coulombic repulsion. Introducing the Slater screening constant (σ) defines the effective nuclear charge (Z_eff = Z − σ) experienced by the valence electron, yielding the semi-empirical one-electron approximation:

IE₁ ≈ 13.598 eV × (Z_eff / n_eff)²

Consequently, ionization energy is controlled by three coupled quantum parameters: (1) effective nuclear charge (Z_eff), (2) principal shell distance (n and atomic radius), and (3) subshell penetration (s > p > d > f) and spin-pairing exchange energy.

Successive Ionization Energies (IE₁ < IE₂ < IE₃)

Removing second, third, and subsequent electrons from an ion defines the successive ionization energies (IE₂, IE₃, ..., IE_k):

M^(k−1)+(g) → M^k+(g) + e⁻(g) (IE_k)

For every element without exception, successive ionization energies increase monotonically:

IE₁ < IE₂ < IE₃ < ... < IE_Z

Each electron removal decreases the number of mutually repulsive electrons while leaving the nuclear charge Z unchanged; this lowers the screening constant σ, increases Z_eff, and contracts the remaining orbitals closer to the nucleus. Superimposed on this steady electrostatic rise is a dramatic discontinuous jump (typically 4-fold to 10-fold) whenever ionization crosses from the outermost valence shell into an underlying closed noble gas core shell of lower principal quantum number (n − 1):

These core-break discontinuities explain why sodium forms Na⁺ but never Na²⁺ in chemical compounds, why magnesium forms Mg²⁺ but never Mg³⁺, and why aluminum forms Al³⁺.

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General Periodic Trends Across Periods and Down Groups

1. Increase Across a Period (Left to Right): Moving across a period (e.g., Li → Ne in Period 2, or Na → Ar in Period 3), each added proton increases Z by +1 while the added electron enters the same principal valence shell n. Because electrons within the same shell shield one another poorly (by Slater's rules, Δ*σ* ≈ 0.35 per fellow ns/np electron), the effective nuclear charge increases by Δ*Z*_eff ≈ +0.65 per step, contracting the atomic radius and raising IE₁ from alkali metal minima to noble gas maxima.
2. Decrease Down a Group (Top to Bottom): Moving down a main group (e.g., He → Ne → Ar → Kr → Xe → Rn), each step increments the principal quantum number n by +1, placing the outermost valence electron at a larger mean radial distance ⟨*r*⟩ from the nucleus behind an additional core shell, which lowers electrostatic attraction and decreases IE₁. (Exceptions occur in the 6th-period d-block and heavy p-block due to the lanthanide contraction and relativistic 6*s* stabilization.)

The Two Classic Periodic Inversions: Group 2 → 13 and Group 15 → 16

Against the general left-to-right rise across Periods 2 and 3, two reproducible saw-tooth inversions occur in every short period:

• Group 2 → Group 13 Inversion (ns² → ns²np¹: Be > B and Mg > Al): The first ionization energy of beryllium (Be, 1*s*² 2*s*², IE₁ = 899.5 kJ mol⁻¹) exceeds that of boron (B, 1*s*² 2*s*² 2*p*¹, IE₁ = 800.6 kJ mol⁻¹), and magnesium (Mg, 737.7 kJ mol⁻¹) exceeds aluminum (Al, 577.5 kJ mol⁻¹). In Group 13, the electron being removed occupies an np subshell rather than the ns subshell. Because np orbitals possess an angular nodal plane at the nucleus and lack the inner radial lobe of ns orbitals, the np¹ electron is shielded by the underlying ns² pair and lies higher in energy, requiring less energy to ionize.
• Group 15 → Group 16 Inversion (np³ → np⁴: N > O and P > S): The first ionization energy of nitrogen (N, 2*p*³, ⁴S_3/2, IE₁ = 1,402.3 kJ mol⁻¹) exceeds that of oxygen (O, 2*p*⁴, ³P₂, IE₁ = 1,313.9 kJ mol⁻¹), and phosphorus (P, 1,011.8 kJ mol⁻¹) exceeds sulfur (S, 999.6 kJ mol⁻¹). By Hund's rules, Group 15 atoms possess a half-filled np³ subshell with one spin-parallel electron in each of the three orthogonal p orbitals, maximizing quantum-mechanical exchange stabilization. In Group 16 (np⁴), the fourth p electron must pair with an opposite-spin electron in an already occupied p orbital, introducing localized intra-orbital Coulombic pairing repulsion (Π_c) that makes this paired electron easier to remove, leaving behind a stable half-filled np³ cation (O⁺, 2*p*³).

Photoelectron Spectroscopy (PES) and Koopmans' Theorem

Experimentally, orbital-specific ionization energies of both valence and inner core shells are measured by ultraviolet and X-ray photoelectron spectroscopy (UPS and XPS). Monochromatic photons of energy hν irradiate a sample, ejecting photoelectrons whose translational kinetic energy E_k = ½*m*_ev² is analyzed by an electrostatic hemispherical analyzer. Conservation of energy gives the electron binding (ionization) energy:

IE_i = hν − E_k

By Koopmans' theorem (formulated by Tjalling Koopmans in 1934), within the "frozen-orbital" Hartree–Fock approximation (assuming the remaining N − 1 electrons do not relax spatially upon ionization), the ionization energy required to eject an electron from canonical spin-orbital φ_i is equal to the negative of its computed Hartree–Fock orbital eigenvalue: IE_i ≈ −*ε*_i. Deviations from Koopmans' theorem reflect electronic orbital relaxation (which lowers the true IE) and electron correlation (which generally raises the true IE).

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Why do transition metals show only a very gradual change in first ionization energy across the d-block?

Across a d-block series such as Sc (IE₁ = 633 kJ mol⁻¹) to Zn (906 kJ mol⁻¹), each successive proton added to the nucleus is accompanied by an electron added to the inner (n − 1)*d* subshell rather than the outer ns valence shell. Because the inner (n − 1)*d* electrons radial distributions lie mostly inside the diffuse ns orbital, they shield the outer ns electrons with high efficiency (by Slater's rules, Δ*σ* = 0.85 per d electron), so the effective nuclear charge Z_eff felt by the ionizing ns electron increases by only ~0.15 per element across the row.

What is the difference between adiabatic and vertical ionization energy in molecules?

When a polyatomic molecule M is ionized to a molecular cation M^•+ (typically by removing a bonding, antibonding, or lone pair electron in molecular orbital theory), the equilibrium bond lengths and angles of M^•+ generally differ from those of neutral M. By the Franck–Condon principle, electronic photoionization occurs on an attosecond timescale before nuclei can move, producing M^•+ in a vibrationally excited state that preserves the neutral geometry (vertical ionization energy, the peak maximum in PES), whereas the adiabatic ionization energy is the slightly lower 0–0 transition energy between the vibrational ground states of M and M^•+.

Why is the first ionization energy of gold (Au) much higher than that of silver (Ag) or copper (Cu)?

Moving down Group 11 from copper (Z = 29, IE₁ = 745.5 kJ mol⁻¹) to silver (Z = 47, IE₁ = 731.0 kJ mol⁻¹), IE₁ decreases as expected for a larger principal shell. However, between silver and gold (Z = 79, IE₁ = 890.1 kJ mol⁻¹), 14 poorly shielding 4*f* electrons are inserted (lanthanide contraction) and the penetrating 6*s*¹ valence electron experiences strong Dirac relativistic contraction, pulling the 6*s* electron closer to the Z = 79 nucleus and raising gold's ionization energy by 159 kJ mol⁻¹, which accounts for gold's resistance to atmospheric oxidation.

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2. Slater, J. C. (1930). "Atomic shielding constants." Physical Review, 36(1), 57–64. https://doi.org/10.1103/PhysRev.36.57
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