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Ionization energy

Ionization energy is the minimum energy required to remove an electron from an isolated gaseous atom or ion in its ground state, typically measured in electron volts (eV) or kilojoules per mole (kJ/mol). It quantifies the strength of the attraction between the nucleus and the outermost electron, serving as a key atomic property that influences chemical reactivity and bonding.^[1] Successive ionization energies refer to the energy needed to remove additional electrons from a positively charged ion, with each subsequent removal requiring progressively more energy due to increased effective nuclear charge on the remaining electrons.^[2] For example, the first ionization energy (IE₁) removes the outermost valence electron, while higher orders like IE₂ and IE₃ target core electrons, often showing sharp increases that reveal electron shell configurations. In the periodic table, ionization energy exhibits clear trends: it generally increases from left to right across a period due to rising nuclear charge and decreasing atomic radius, which strengthens the pull on electrons, and decreases down a group because of increasing atomic radius and shielding effects from inner electrons.^[3] Notable exceptions occur at half-filled or fully filled subshells, where stability leads to higher-than-expected values, such as nitrogen's higher IE than oxygen.^[4] Factors affecting ionization energy include atomic radius (larger radii lower IE by distancing electrons from the nucleus), effective nuclear charge (higher charge increases IE), and electron configuration (stable configurations resist removal).^[5] These elements contribute to broader periodic properties, like metallic character, which inversely correlates with IE—low IE favors electron donation in metals.^[1] Ionization energy plays a crucial role in understanding chemical behavior, such as predicting ion formation, electronegativity, and reaction tendencies; for instance, alkali metals with low IE are highly reactive, while noble gases with high IE are inert.^[2] Experimental values are precisely measured using techniques like photoelectron spectroscopy, with comprehensive data available from sources like NIST for atomic species.^[6]

Fundamentals

Definition and Importance

Ionization energy (IE), also known as ionization potential, is defined as the minimum energy required to remove an electron from a neutral atom in the gaseous phase in its ground state, resulting in the formation of a positively charged ion.^[7] This process typically refers to the adiabatic ionization energy, which corresponds to the energy difference between the ground state of the neutral species and the ground state of the ion./Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Ionization_Energy) Ionization energies are specified for successive electrons: the first IE removes the most loosely bound electron, the second IE removes another from the resulting cation, and so on, with each subsequent value generally increasing due to stronger electrostatic attraction in the more positively charged species.^[8] Mathematically, the ionization energy is expressed as

IE = E(\text{cation}) + E(e^-) - E(\text{neutral}),

where E denotes the total energy of the respective species, and the energy of the free electron E(e^-) is taken as zero at infinite separation./Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Ionization_Energy) Ionization energy is fundamental in chemistry and physics, as it quantifies an atom's tendency to lose electrons and thus predicts chemical reactivity, electronegativity, and metallic character across the periodic table.^[8] Elements with low first IE, such as alkali metals, readily form cations by removing their outer s-electron, facilitating ionic bonding and high reactivity; for instance, sodium's first IE of approximately 496 kJ/mol reflects its ease of electron donation./Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Ionization_Energy) In physics, IE governs ionization processes in plasmas, where it determines the energy threshold for creating ionized gases essential in fusion research and astrophysical phenomena.^[9]

Units and Notation

In atomic physics, ionization energies are typically expressed in electronvolts (eV), a unit representing the energy acquired by a single electron when accelerated through an electric potential difference of one volt.^[10] This unit is particularly suited for describing processes involving individual atoms or ions, such as electron removal in gaseous species. In contrast, thermochemical contexts often employ kilojoules per mole (kJ/mol) to quantify the energy required to ionize a mole of atoms, facilitating comparisons with reaction enthalpies; the precise conversion factor is 1 eV ≈ 96.485 kJ/mol.^[10] Standard notation distinguishes successive ionization energies as IE₁ for the first (removal of the most loosely bound electron from a neutral atom), IE₂ for the second (from the resulting singly charged ion), and generally IEₙ for the nth ionization energy./Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Ionization_Energy) In spectroscopic literature, the term ionization potential (IP) serves as a synonym, with analogous subscripting (e.g., IP₁), emphasizing the potential energy threshold for electron ejection.^[11] Historically, early determinations of ionization energies in spectroscopy relied on Rydberg series analysis, expressing values in wavenumbers (cm⁻¹) or Rydberg units (1 Ry = 13.605693122994 eV), as pioneered in the late 19th and early 20th centuries for spectral line fitting. The transition to modern SI-derived units like eV and kJ/mol occurred with the standardization of atomic data in the mid-20th century, aligning with the International System of Units (SI) for broader scientific consistency.^[12] Current precision standards are maintained by the National Institute of Standards and Technology (NIST) Atomic Spectra Database, which provides critically evaluated values with uncertainties as low as 10⁻⁶ eV, with its data content last comprehensively updated in November 2024.^[6] For neutral atoms, ionization energies span a wide range, from approximately 3.9 eV for cesium (the lowest, reflecting its large atomic radius and low effective nuclear charge) to about 24.6 eV for helium (the highest, due to its stable closed-shell configuration).^[13]^[14] Successive ionization energies increase progressively (IEₙ > IE_{n-1}), often dramatically after removing valence electrons, and are notated accordingly to track these escalations in multi-electron systems.^[10]

Measurement and Determination

Experimental Methods

Photoelectron spectroscopy (PES) serves as the primary experimental technique for measuring ionization energies in atoms and molecules, utilizing ultraviolet (UV) or X-ray photon sources to eject electrons and analyze their kinetic energies. In this method, a sample is irradiated with monochromatic photons of known energy h\nu, where h is Planck's constant and \nu is the photon frequency, causing photoionization; the ionization energy (IE) is then determined from the difference between the photon energy and the measured kinetic energy of the emitted photoelectron, given by the equation:

\text{IE} = h\nu - E_{\text{kinetic}}

where E_{\text{kinetic}} is the kinetic energy of the ejected electron.^[15] This approach provides direct empirical values for vertical ionization energies, reflecting electronic transitions without nuclear rearrangement. Other established methods include mass spectrometry for threshold ionization, where the onset energy for ion production is detected as the electron or photon energy is varied until ions appear, allowing determination of the ionization threshold.^[16] Electron impact ionization, involving bombardment of the sample with electrons of controlled energy followed by analysis of energy loss or ion appearance potentials, offers complementary measurements, particularly for gas-phase species.^[17] A key historical milestone is the Franck-Hertz experiment of 1914, which demonstrated quantized energy levels in mercury vapor through electron collisions, revealing discrete excitation and ionization thresholds that supported early quantum theory.^[18] Post-1980s advancements in laser-based PES have significantly enhanced precision, enabling sub-meV energy resolutions through tunable laser sources and improved electron analyzers.^[19] By the 2020s, resolutions as fine as 1.5 meV have been achieved, facilitating detailed studies of fine structure in ionization spectra.^[19] PES and related techniques are particularly valuable for probing transient species, such as radicals or excited states generated in situ, by rapidly detecting short-lived photoelectrons before decay.^[20] Typical error margins in these measurements range from 0.01 to 0.1 eV, depending on instrumental resolution and sample conditions, ensuring reliable values for most atomic and molecular systems. Comprehensive experimental data are compiled in databases such as the NIST Chemistry WebBook.^[21]

Theoretical Approaches

Theoretical approaches to predicting ionization energies rely on computational methods that solve the Schrödinger equation or its approximations, enabling estimates without experimental measurement. Ab initio wavefunction-based methods, such as Hartree-Fock (HF), provide a starting point by assuming a single Slater determinant for the molecular wavefunction and minimizing the energy with respect to orbital coefficients.^[22] Post-Hartree-Fock techniques, including second-order Møller-Plesset perturbation theory (MP2) and coupled-cluster methods like CCSD(T), incorporate electron correlation effects to improve accuracy, with CCSD(T) often achieving chemical accuracy (errors <1 kcal/mol or ~0.04 eV) for small molecules when extrapolated to the complete basis set limit.^[23]^[24] Density functional theory (DFT) offers a computationally efficient alternative by using the electron density rather than the wavefunction, with hybrid functionals like B3LYP combining exact HF exchange with Becke's gradient-corrected exchange and Lee-Yang-Parr correlation, yielding mean absolute errors of approximately 0.22 eV for vertical ionization energies of small molecules containing first-row atoms.^[25] A key approximation in these methods is Koopmans' theorem, which posits that the ionization energy (IE) is approximately the negative of the highest occupied molecular orbital (HOMO) energy from HF calculations, assuming no orbital relaxation or electron correlation upon ionization.

IE \approx - \epsilon_{\mathrm{HOMO}}

This theorem simplifies predictions but has limitations, as it neglects nuclear relaxation, electron correlation, and reorganization effects, leading to errors of 1-5 eV for typical systems; extensions like ΔSCF methods, which compute the energy difference between optimized neutral and cation states, mitigate these issues in both HF and DFT frameworks.^[25] Semi-empirical models reduce computational cost by parameterizing integrals based on experimental data, with extended Hückel theory (EHT) providing quick estimates of orbital energies and thus ionization potentials through a tight-binding approximation that includes overlap integrals and uses atomic ionization potentials to set diagonal elements.^[26] EHT is particularly useful for large hydrocarbons, offering qualitative insights into molecular orbital ordering, though quantitative accuracy is lower, with errors often exceeding 1 eV due to neglect of explicit electron-electron interactions. Recent advancements incorporate machine learning, such as neural networks trained on high-quality datasets like those from the NIST Atomic Spectra Database, enabling high-throughput predictions of ionization energies for thousands of organic molecules with root-mean-square errors below 0.3 eV, surpassing traditional semi-empirical methods for diverse chemical spaces.^[27]

Atomic Ionization Energies

Periodic Trends

Ionization energy exhibits distinct patterns across the periodic table for neutral atoms, primarily influenced by atomic structure and electron configuration. Across a period from left to right, the first ionization energy generally increases due to the progressive increase in nuclear charge, which enhances the effective nuclear charge experienced by valence electrons without a corresponding increase in shielding from inner shells. This pulls valence electrons closer, reducing atomic radius and making electron removal more difficult.^[1] Down a group, the first ionization energy typically decreases as the principal quantum number n of the valence electrons increases, leading to larger atomic radii and greater shielding by additional core electrons. These factors diminish the effective nuclear charge on valence electrons, facilitating easier ionization. The shell structure associated with higher n further contributes to this trend by placing valence electrons in higher-energy orbitals farther from the nucleus.^[1] These patterns are evident in quantitative data for the first ionization energies. Noble gases exhibit peak values, such as 24.59 eV for helium and 21.56 eV for neon, reflecting their stable, filled-shell configurations. Alkali metals show minima, with potassium at 4.34 eV, highlighting the ease of removing their single valence electron. In the second period, ionization energy rises steadily from lithium (5.39 eV) to neon (21.56 eV), while in the third period, sodium (5.14 eV) has a lower value than magnesium (7.65 eV), but both are below their period-2 counterparts due to the group trend.^[14]^[28]^[29]^[30]^[31] In the f-block elements, these general trends are modified by contraction effects. The lanthanide contraction results in a gradual decrease in atomic radius across the series, causing ionization energies to remain relatively constant or increase slightly rather than decrease as expected for a normal group trend. A similar actinide contraction, amplified by relativistic effects, leads to an even more pronounced stability in radii, resulting in ionization energies that increase slightly down the actinide series due to enhanced nuclear attraction on valence electrons.^[32]^[33]

Exceptions and Anomalies

While ionization energies generally increase across a period and decrease down a group, notable deviations arise due to the stability of specific electron configurations, particularly those involving half-filled or fully filled subshells.^[1] In the second period, beryllium (Be) exhibits a higher first ionization energy of 9.32 eV compared to boron (B) at 8.30 eV, contrary to the expected increase from group 2 to group 13.^[6] This anomaly occurs because Be has a stable [He] 2s² configuration, where removing an electron disrupts the fully filled s subshell; in contrast, B's [He] 2s² 2p¹ configuration allows easier removal of the p electron, as the 2p orbital is higher in energy than the 2s. A similar drop is observed between magnesium (Mg, 7.65 eV) and aluminum (Al, 5.99 eV) in the third period, attributed to the same shift from a stable ns² to ns² np¹ configuration.^[6] Another key exception appears between nitrogen (N) and oxygen (O) in group 15 and 16, where N has a higher first ionization energy of 14.53 eV than O's 13.62 eV.^[6] Nitrogen's [He] 2s² 2p³ configuration features a half-filled 2p subshell, which provides extra stability due to minimized electron-electron repulsion and maximized exchange energy; removing an electron from this configuration requires more energy. In oxygen's [He] 2s² 2p⁴ configuration, the paired electrons in one p orbital experience increased repulsion, facilitating easier removal of a p electron. This pattern repeats in the third period between phosphorus (P, 10.49 eV) and sulfur (S, 10.36 eV).^[6] In the d-block transition metals, ionization energies show irregularities due to Aufbau principle exceptions that favor half-filled or filled d subshells. For instance, chromium (Cr) has a first ionization energy of 6.77 eV, lower than manganese (Mn) at 7.43 eV, despite the general increasing trend across the period.^[6] Chromium's ground-state configuration is [Ar] 4s¹ 3d⁵, allowing removal of the single 4s electron to yield a stable half-filled 3d⁵ ion; manganese's [Ar] 4s² 3d⁵ configuration requires removing an electron from the paired 4s orbital, leading to a less stable [Ar] 4s¹ 3d⁵ ion. These deviations contribute to erratic trends throughout the first transition series, where similar stabilities affect elements like copper (Cu) with its [Ar] 4s¹ 3d¹⁰ configuration. The f-block lanthanides display further anomalies influenced by lanthanide contraction, where poor shielding by 4f electrons causes a gradual decrease in atomic and ionic radii across the series, leading to relatively constant but erratic first ionization energies, with fluctuations due to subshell effects. For example, values range from 5.58 eV for La to a peak of about 6.25 eV for Yb, then 5.43 eV for Lu, as the contraction counteracts the expected decrease down the series.^[6] This contraction also impacts post-lanthanide elements, contributing to higher-than-expected ionization energies in the 5d series compared to 4d analogs. For superheavy elements beyond the known periodic table trends, predictions reveal additional anomalies. Oganesson (Og, element 118), with a predicted first ionization energy of approximately 8.89 eV, deviates from the expected noble gas behavior due to relativistic effects destabilizing its 7s and 7p electrons, making it more reactive than lighter noble gases like radon (10.75 eV).^[34] These calculations, based on relativistic coupled cluster methods, highlight how strong spin-orbit coupling in superheavies alters subshell stabilities, leading to irregular ionization properties.

Theoretical Models

Bohr Model Application

The Bohr model, proposed by Niels Bohr in 1913, describes the hydrogen atom as consisting of a positively charged proton at the center with a negatively charged electron moving in discrete circular orbits around it. These orbits are characterized by a principal quantum number n = 1, 2, 3, \dots, and the model incorporates two key postulates: the electron occupies stationary states without radiating energy, and transitions between states occur by absorbing or emitting photons with energy equal to the difference between the states.^[35] To derive the energy levels, the model balances the classical centripetal force on the electron with the electrostatic attraction from the proton:

\frac{m v^2}{r} = \frac{k e^2}{r^2},

where m is the electron mass, v is the orbital velocity, r is the orbital radius, k = 1/(4\pi\epsilon_0) is Coulomb's constant, and e is the elementary charge. Bohr's second postulate quantizes the angular momentum as m v r = n \hbar, with \hbar = h/(2\pi) and h Planck's constant. Solving these equations yields the radius of the nth orbit:

r_n = n^2 a_0,

where a_0 \approx 0.529 Å is the Bohr radius, defined as a_0 = 4\pi\epsilon_0 \hbar^2 / (m e^2).^[36] The total energy of the electron in the nth state is then

E_n = -\frac{k e^2}{2 r_n} = -\frac{13.6}{n^2} \, \text{[eV](/page/EV)}.

This negative value indicates a bound state, with the zero of energy at infinite separation.^[35] The ionization energy of the hydrogen atom is the energy required to remove the electron from the ground state (n=1) to the ionized state (n \to \infty), where E_\infty = 0. Thus, it equals -E_1 = 13.6 eV, or more generally from the energy difference formula,

\text{IE}_H = 13.6 \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) \, \text{[eV](/page/EV)} = 13.6 \, \text{[eV](/page/EV)}.

This predicted value closely matches the experimental ionization energy of 13.59844 eV.^[37] For hydrogen-like ions with nuclear charge Z e, the model scales the energy as E_n = -13.6 Z^2 / n^2 eV, yielding an ionization energy of $13.6 Z^2 eV from the ground state.^[35] Bohr's model successfully explained the discrete spectral lines of the Balmer series in hydrogen, attributing them to transitions ending at n=2, and provided the first theoretical basis for the atom's ionization energy. The energy scale also connects to the Rydberg constant R_\infty, the limiting wavenumber for transitions to n=\infty, via \text{IE}_H = h c R_\infty, where c is the speed of light.^[38] While exact for hydrogen and hydrogen-like atoms, the Bohr model fails for multi-electron atoms because it neglects the shielding effects of inner electrons on the nuclear attraction experienced by outer electrons.^[38]

Quantum Mechanical Explanation

In quantum mechanics, the electronic structure of multi-electron atoms is described by wavefunctions composed of atomic orbitals (AOs), which are solutions to the Schrödinger equation accounting for electron-electron interactions through approximations. The first ionization energy (IE) corresponds to the energy needed to remove an electron from the highest occupied atomic orbital, and within the independent-electron approximation, it is directly linked to the negative of that orbital's energy eigenvalue. For molecules, this extends to molecular orbitals (MOs), where the highest occupied molecular orbital (HOMO) energy approximates the IE. Koopmans' theorem formalizes this relation in the Hartree-Fock framework, stating that the IE is equal to -\epsilon_i, the negative of the orbital energy for the i-th electron, under the assumptions of frozen orbitals and no electron correlation beyond mean-field effects.^[39] The effective nuclear charge Z_\text{eff} experienced by a valence electron governs the contraction of its radial wavefunction \psi(r), increasing the binding strength and thus the IE. In multi-electron atoms, inner electrons shield the nucleus, reducing the full nuclear charge [Z](/page/Z) to Z_\text{eff} = [Z](/page/Z) - \sigma, where \sigma is the shielding constant. Slater's empirical rules provide a practical method to estimate \sigma by grouping electrons into shells and assigning shielding contributions based on their principal quantum number n and orbital type, with core electrons contributing nearly fully to shielding while valence electrons contribute less. As Z_\text{eff} rises across a period, the valence orbitals contract, elevating the IE. This mean-field treatment is captured in the Hartree-Fock self-consistent field method, where each electron evolves in the average potential from the nucleus and other electrons, yielding orbital energies that approximate IEs via Koopmans' theorem; the Hartree equations iteratively solve for this potential, later refined by Fock to include exchange effects via antisymmetrized wavefunctions.^[40]^[41] Penetration and shielding effects further modulate IEs based on the angular momentum quantum number l. Orbitals with lower l (e.g., s orbitals, l=0) have radial probability distributions that extend closer to the nucleus due to fewer centrifugal barriers, allowing greater penetration and reduced shielding from inner electrons, resulting in a higher Z_\text{eff} and thus higher IE compared to p (l=1), d (l=2), or f (l=3) orbitals in the same shell. This penetration leads to more negative orbital energies for s electrons, making them more tightly bound. For higher l, the wavefunction is pushed outward by the centrifugal term in the radial equation, increasing shielding and lowering the IE. For greater accuracy beyond Hartree-Fock, configuration interaction (CI) methods incorporate electron correlation by expanding the wavefunction as a linear combination of multiple Slater determinants, capturing dynamic correlation that adjusts orbital relaxations upon ionization. Seminal CI applications have achieved near-exact non-relativistic IEs for first-row transition metals, with errors below 0.01 eV when using large basis sets and initiator full CI quantum Monte Carlo variants. In the 2020s, time-dependent density functional theory (TDDFT) has advanced the treatment of dynamic IEs in excited states, enabling calculations of ionization from non-ground-state configurations via linear-response formulations that include core-valence separations for X-ray spectroscopy and charge-transfer excitations.^[42]^[43]

Molecular Ionization Energies

Vertical Ionization Energy

In molecular systems, the vertical ionization energy is defined as the energy required to remove an electron from a neutral molecule to form a molecular cation while maintaining the same nuclear geometry, without allowing for vibrational or structural relaxation.^[10] This concept arises from the Franck-Condon principle, which posits that electronic transitions, such as ionization, occur on a timescale much faster than nuclear motion, resulting in the ion inheriting the equilibrium geometry of the neutral species and often leading to vibrational excitation in the cation.^[44] Consequently, vertical ionization energies correspond to the peak positions observed in photoelectron spectra, providing a direct measure of the energy difference between the initial and final electronic states at fixed geometry.^[10] Within molecular orbital (MO) theory, the vertical ionization energy can be approximated using Koopmans' theorem, which states that the energy needed to remove an electron from an occupied orbital equals the negative of that orbital's energy in the Hartree-Fock approximation, neglecting electron correlation and relaxation effects.^[39] Specifically, for the highest occupied molecular orbital (HOMO), IE_v \approx -\epsilon_{\ce{HOMO}}, where \epsilon_{\ce{HOMO}} is the orbital eigenvalue.^[45] This approximation highlights differences from atomic ionization energies due to the formation of bonding and antibonding MOs; for instance, in molecules, delocalization or hybridization can stabilize or destabilize orbitals relative to isolated atoms, altering ionization thresholds.^[46] A representative example is the water molecule (H₂O), where the vertical ionization energy from the HOMO (1b₁ orbital, primarily an oxygen 2p lone pair in the molecular plane) is approximately 12.62 eV, lower than the first ionization energy of the atomic oxygen (13.618 eV) due to the stabilizing influence of the molecular environment on the lone-pair orbital.^[47]^[48] This reduction reflects the partial antibonding character and electrostatic effects in the molecule compared to the free atom. Vertical ionization energies are typically greater than adiabatic values, as the latter account for geometry relaxation in the ion, lowering its energy.^[45] In ultraviolet photoelectron spectroscopy (UV-PES), vertical ionization energies enable the assignment of spectral bands to specific molecular orbitals by correlating peak positions with computed orbital energies, facilitating the mapping of electronic structures in organic and inorganic molecules.^[49] Recent X-ray photoelectron spectroscopy (XPS) studies on biomolecules, such as those examining solvent interactions with peptides, have utilized vertical ionization data to quantify valence orbital binding energies (e.g., HOMO at ~8-10 eV in aqueous environments), revealing specific versus nonspecific solvation effects on electronic properties.^[50]

Adiabatic Ionization Energy

The adiabatic ionization energy of a molecule is defined as the minimum energy required to remove an electron from the neutral molecule in its ground electronic, vibrational, and rotational state, thereby forming the molecular cation in its corresponding ground state while allowing the nuclei to relax to the ion's equilibrium geometry. This process corresponds to the enthalpy change, ΔH, at 0 K for the ionization reaction M → M⁺ + e⁻. Unlike instantaneous processes, it accounts for the full structural reorganization following electron ejection, making it the thermodynamically precise value for the ground-state transition.^[10]^[7] The adiabatic ionization energy is related to the vertical ionization energy by the reorganization energy, which quantifies the energy gained from nuclear relaxation in the ion; consequently, the adiabatic value is typically lower than the vertical one. This relationship can be expressed mathematically as

\text{IE}_{\text{adiab}} = E(\ce{M^+_{\min}}) - E(\ce{M_{\min}})

where E(\ce{M_{\min}}) and E(\ce{M^+_{\min}}) are the total energies at the equilibrium geometries of the neutral molecule and cation, respectively (with the free electron's energy taken as zero at infinite separation). For example, in nitric oxide (NO), the adiabatic ionization energy is 9.26 eV, illustrating the contribution from reorganization to the vertical value. Adiabatic ionization energies play a key role in thermochemical cycles, enabling accurate calculations of cationic bond dissociation energies and related reaction enthalpies.^[51]^[52] Experimentally, adiabatic ionization energies are determined using high-resolution techniques such as threshold photoelectron spectroscopy (PES) or zero kinetic energy (ZEKE) photoelectron spectroscopy, which detect electrons near zero kinetic energy at the ionization threshold and resolve fine vibrational progressions. These methods are essential for capturing the subtle onset of ionization, particularly in cases involving complex potential energy surfaces. In molecules where the cation ground state is electronically degenerate, such as in tetrahedral systems like CH₄⁺, Jahn-Teller distortions arise, splitting the degenerate electronic states and lowering the energy of the lowest adiabatic surface; this manifests as characteristic vibrational patterns in the spectrum, influencing the precise location of the ionization threshold and providing insights into the ion's dynamic stability.^[53]^[54]^[55]

Extensions to Other Systems

Electron Binding Energy

Electron binding energy is defined as the minimum energy required to remove an electron from its atomic or molecular orbital to a position at infinite distance from the ion, where it is at rest and with zero potential energy. This concept is synonymous with ionization energy in the gas phase for valence electrons but is more broadly applied to core electrons in techniques such as X-ray photoelectron spectroscopy (XPS).^[56] In XPS, the binding energy (BE) of a core electron is determined experimentally from the photoelectric effect, using the relation

\text{BE} = h\nu - E_{\text{kinetic}}

where h\nu is the energy of the incident X-ray photon (typically from an Al Kα source at 1486.6 eV) and E_{\text{kinetic}} is the measured kinetic energy of the emitted photoelectron, neglecting instrumental work function corrections for simplicity. This equation enables precise measurement of core-level binding energies, which are characteristic of the element and its chemical environment.^[57] For atomic systems, core-electron binding energies are significantly higher than valence ionization energies due to the proximity of core orbitals to the nucleus. For example, the carbon 1s binding energy in graphite is 284.2 eV.^[58] Similarly, the gold 4f_{7/2} binding energy serves as a standard reference in XPS calibration at 84 eV for metallic gold, providing a benchmark for energy scale alignment across samples.^[59] In molecular contexts, electron binding energies exhibit chemical shifts arising from the local electronic environment, which alter the effective nuclear charge experienced by the core electron. These shifts, typically on the order of 1–5 eV, arise primarily from initial-state effects (changes in orbital energies due to electronegativity differences) and final-state relaxation (screening of the core hole by surrounding electrons). For instance, in organic molecules, the C 1s binding energy for a carbonyl carbon (C=O) is shifted to higher values by about 3.7 eV compared to an alkane carbon (C-C or C-H at ~284.8 eV), due to the electron-withdrawing oxygen atom increasing the effective positive charge on carbon. Recent studies on functionalized organic surfaces confirm these shifts, with carbonyl groups in polyesters showing C 1s peaks at 288–289 eV, enabling identification of functional groups in polymers and biomolecules. Such shifts are valuable for probing molecular structure and bonding.^[60]^[61]^[62] These binding energy measurements find extensive applications in surface analysis, where XPS probes the top 5–10 nm of materials to reveal elemental composition and chemical states. In catalysis, core-level shifts help elucidate active site environments, such as oxidation states of metal centers or adsorbate interactions on catalyst surfaces, guiding the design of more efficient heterogeneous catalysts. For example, monitoring C 1s shifts in carbon-supported metal catalysts reveals carbon deposition or functionalization effects on reactivity.^[63]^[64]

Work Function in Solids

In solid-state physics, the work function φ represents the minimum energy required to remove an electron from the Fermi level of a solid to the vacuum level just outside its surface, mathematically expressed as φ = E_vac - E_F, where E_vac is the vacuum energy level and E_F is the Fermi energy.^[65] This parameter is crucial for understanding electron emission processes in materials like metals and semiconductors, as it quantifies the energy barrier at the surface.^[66] For metals, the work function typically falls in the range of 2 to 6 eV, which is notably lower than the atomic ionization energies of the constituent elements (often exceeding 7-10 eV) due to the delocalization of electrons within the metallic band structure, allowing electrons near the Fermi level to be more readily extracted.^[67]^[68] One common method to determine φ is through the photoelectric effect, where the threshold photon energy hν_threshold equals the work function, as electrons with kinetic energy gain zero at this frequency. For instance, cesium metal exhibits a low work function of approximately 2.1 eV, making it ideal for applications in photocathodes where efficient electron emission is required under illumination.^[67] The work function plays a key role in thermionic emission, where thermal energy enables electron escape from heated surfaces; this process is governed by the Richardson-Dushman equation, which relates emission current density to temperature and φ. Additionally, φ influences surface reactivity, such as in adsorption and catalytic processes on metal surfaces, by modulating the electron transfer at interfaces.^[69] In two-dimensional materials, such as graphene, the work function is around 4.6 eV for undoped sheets and can be tuned over a range of several electron volts through doping strategies, enabling tailored electronic properties for devices like transistors and sensors, as demonstrated in studies on amine-rich macromolecule adsorption.^[70] This tunability arises from shifts in the Fermi level induced by charge transfer, highlighting the extension of ionization energy concepts to low-dimensional solids.

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Ionization energy 198310.669 cm-1 (24.587387 eV) Ref. M02 He II Ground State 1s 2S1/2. Ionization energy 438908.8789 cm-1 (54.417760 eV) Ref. MK00b.Missing: cesium | Show results with:cesium
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[PDF] Basic Concepts of X-Ray Photoelectron Spectroscopy
The basic components necessary for performing an XPS experiment consist of a radiation source for excitation, the specimen to be studied, an electron energy ...<|control11|><|separator|>
[18]
Ionization energy measurements and electronic spectra for ThO
The ionization energy (IE) for ThO has been determined using photoionization efficiency and mass-analyzed threshold ionization measurements.
[19]
Electron Impact Ionization - an overview | ScienceDirect Topics
Electron impact ionization is defined as an ionization technique where a gaseous analyte is bombarded by energetic electrons, resulting in the formation of ...
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[PDF] The Franck-Hertz Experiment - Purdue Physics
Objective: To measure the first excitation potential and the ionization potential of mercury atoms and to show that the energies of bound electrons are ...
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Laser-based angle-resolved photoemission spectroscopy with ... - NIH
The setup achieves an energy resolution of 1.5 (5.5) meV without (with) the spin detection mode, compatible with a spatial resolution better than 10 µm. This ...Laser μ-Sarpes System · Specifications · Energy Resolution
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High-sensitivity photoelectron spectrometer for studying reactive ...
In this paper we describe the construction and perfor- mance of a high-sensitivity photoelectron spectrometer de- signed for studying reactive transient species ...
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Quantitative ionization energies and work functions of aqueous ...
Oct 5, 2016 · Ionization energies of aqueous solutions are measured by photoelectron spectroscopy (PES) using a liquid jet; however, published values for ...
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A mathematical and computational review of Hartree–Fock SCF ...
We present a review of the fundamental topics of Hartree–Fock theory in quantum chemistry. From the molecular Hamiltonian, using and discussing the Born– ...<|separator|>
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Accurate ab initio predictions of ionization energies and heats of ...
Jan 30, 2006 · Furthermore, the present study provides support for the conclusion that the CCSD(T)/CBS approach with high-level energy corrections can be used ...
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On the accuracy of density functional theory and wave function ...
May 21, 2015 · In general, ionization energies calculated directly from the orbital energies of the neutral species are less accurate and more sensitive to an ...
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Extended Hückel calculations of the ionization potentials of some ...
The prediction of ionization potentials of conjugated hydrocarbons using extended Hückel theory is reevaluated. Consequently, two major modifications.
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Machine Learning for Ionization Potentials and Photoionization ...
Apr 6, 2023 · ... typically be measured within a margin of error of <20%. (70−74)Error can arise from several factors, including (1) measurement uncertainty ...
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Atomic Data for Neon (Ne) - Physical Measurement Laboratory
Atomic Data for Neon (Ne) Atomic Number = 10 Atomic Weight = 20.1797 Reference E95 Ne I Ground State 1s 2 2s 2 2p 6 1 S 0 Ionization energy 173929.75 cm
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Atomic Data for Lithium (Li) - Physical Measurement Laboratory
Lithium's atomic number is 3, atomic weight is 6.941. Isotopes include 6Li (7.5% abundance) and 7Li (92.5% abundance). Ionization energy for Li I is 5.391719 ...Missing: first | Show results with:first
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Strong Lines of Sodium ( Na ) - Physical Measurement Laboratory
Strong Lines of Sodium ( Na ) ; 1, 10572.28, Na I, R56 ; 2, 10746.44, Na I, R56.Missing: first ionization
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Atomic Data for Magnesium (Mg) - Physical Measurement Laboratory
Magnesium's atomic number is 12, atomic weight is 24.3050. Its first ionization energy is 7.646235 eV, and second is 15.03527 eV.Missing: first | Show results with:first
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[PDF] Ground Levels and Ionization Potentials for Lanthanide and Actinide ...
Oct 29, 2009 · Values of the first four ionization potentials of the lanthanides (Z =57 -71) and of Hf have been compiled. All except the value for neutral ...
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Bohr radius - CODATA Value
Concise form, 5.291 772 105 44(82) x 10-11 m ; Click here for correlation coefficient of this constant with other constants ; Source: 2022 CODATA
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6.4 Bohr's Model of the Hydrogen Atom - University Physics Volume 3
Sep 29, 2016 · The constant R H = 1.09737 × 10 7 m −1 R H = 1.09737 × 10 7 m −1 is called the Rydberg constant for hydrogen. In Equation 6.31, the positive ...
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Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den ...
Es wird angegeben, wie man in eindeutiger Weise den einzelnen Elektronen bestimmte Wellenfunktionen und Eigenwerte zuordnen kann. Diese Eigenfunktionen genügen ...
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Atomic Shielding Constants | Phys. Rev.
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Näherungsmethode zur Lösung des quantenmechanischen ...
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Accurate Ab Initio Calculation of Ionization Potentials of the First ...
Jan 21, 2015 · Accurate ionization potentials of the first-row transition-metal atoms are obtained via the initiator full configuration quantum Monte Carlo ...
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Visualizing and characterizing excited states from time-dependent ...
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On Koopmans' theorem in density functional theory - AIP Publishing
Nov 1, 2010 · This paper clarifies why long-range corrected (LC) density functional theory gives orbital energies quantitatively.
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Water - the NIST WebBook
Ionization energy determinations ; 12.614 ± 0.005, PI, Brehm, 1966, RDSH ; 12.597 ± 0.010, PI, Nicholson, 1965, RDSH.Gas phase ion energetics data · References
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Oxygen, atomic - the NIST WebBook
Ionization energy determinations ; 14.0 ± 0.5, EI, Hildenbrand, 1975 ; 13.618, S · Cermak, 1975 ; 13.61806, S · Moore, 1970 ; 13.62, PE, Jonathan, Morris, et al., ...Gas phase ion energetics data · References
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UV Photoelectron Spectroscopy of Aqueous Solutions
Nov 28, 2022 · L-Tryptophan has the lowest vertical ionization energy, 7.3 eV, followed by tyrosine (7.8 eV) and phenylalanine (∼8.7 eV). Essentially, no ...
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Specific versus Nonspecific Solvent Interactions of a Biomolecule in ...
Nov 16, 2023 · The lowest-binding-energy valence photoelectron peaks correspond to the ionization of the HOMO and HOMO–1 electrons with binding energies of ...
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Experimental data for NO (Nitric oxide) - CCCBDB
Energy (cm-1), Degeneracy, reference, description. 0 ; Ionization Energy, I.E. unc. vertical I.E., v.I.E. unc. reference. 9.264 ; Electron Affinity, unc.
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On the accuracy of density functional theory and wave function ...
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Ionization energies and cationic bond dissociation ... - AIP Publishing
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On the adiabatic ionization energy of the propargyl radical
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Photoion Mass-Selected Threshold Photoelectron Spectroscopy to ...
Aug 21, 2023 · We measure the adiabatic ionisation energy of the benzyl radical to be 7.252(5) eV and observe a well-resolved vibrational progression in ...
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Jahn-Teller Effect in the Methane Cation: Rovibronic Structure and ...
Oct 27, 2006 · The ground state of CH 4 + is determined to be of F 2 nuclear spin symmetry and the adiabatic ionization energy of CH 4 amounts to 101753.0 ( 15 ) ...
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Photoelectron Spectroscopy - Chemistry LibreTexts
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X-ray photoelectron spectroscopy: Towards reliable binding energy ...
As electrons absorb over the entire photon energy, the conservation of energy requires that their kinetic energy is equal to h ν – W, where W is the work ...
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[PDF] Table 1-1. Electron binding energies, in electron volts, for the ...
Table 1-1 lists electron binding energies in electron volts for elements in their natural forms. For example, Hydrogen is 13.6 eV and Helium is 24.6 eV.
[58]
https://xdb.lbl.gov/Section1/Table_1-1.pdf
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XPS provides chemical bond information - EAG Laboratories
For a C-O single bond a chemical shift of ~1.5 eV is observed relative to C-C. For a C=O double bond a slightly larger positive charge exists on the carbon atom ...
[60]
Carbon | XPS Periodic Table | Thermo Fisher Scientific - US
For high concentrations of sp3-bonded carbon, the C1s peak will have a more symmetric shape and will also be slightly shifted to higher binding energy. C1s ...
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Origins of sp3C peaks in C1s X-ray Photoelectron Spectra of Carbon ...
Jun 4, 2016 · We calculated the carbon and oxygen 1s core level binding energies for oxygen and hydrogen functionalities such as graphane-like ...
[62]
Review on surface-characterization applications of X-ray ...
XPS has been used in the surface analysis in various fields such as corrosion, catalysis, electronics, nanomaterials, biomedicine, mineral processing, ...
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https://www.sciencedirect.com/science/article/pii/S2666523922001222
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[PDF] Understanding Low Work Function Perovskite Thermionic Emission ...
precise definition of work function was therefore developed: Φ = Evac − EF. Eq 2.2. With Evac and EF denoted to vacuum level and Fermi level, respectively.Missing: E_Fermi | Show results with:E_Fermi
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[PDF] 351-2: Introductory Physics of Materials
Apr 6, 2025 · In general, the work function Φ of a material is defined to be the energy difference between the vacuum level for a free electron and the ...
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Work Functions for Photoelectric Effect - HyperPhysics
Element, Work Function(eV). Aluminum. 4.08. Beryllium. 5.0. Cadmium. 4.07. Calcium. 2.9. Carbon. 4.81. Cesium. 2.1. Cobalt. 5.0. Copper. 4.7. Gold ...Missing: range 2-6
[67]
On the relationship between the ionization potential and the work ...
Aug 7, 2025 · A relationship is analyzed between the ionization energy of atoms I and the work function φ of a metal composed of these atoms.
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Application of work function measurements in the study of surface ...
The present article aims to show how work function measurements (WF) can be applied in the study of elementary surface reaction steps on metallic single ...
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[PDF] Highly Efficient n-Type Doping of Graphene by Vacuum Annealed ...
May 8, 2020 · The work function of HOPG and undoped graphene measured by Kelvin probe force microscopy (KPFM) were 4.61 and 4.39 eV, respectively (Figure 2C).