Editing Ionization energy

{{Short description|Energy needed to remove an electron}}
{{Hatnote|For the values of the ionization energies of the elements, see [[Molar ionization energies of the elements]] and [[Ionization energies of the elements (data page)]]}}
{{more citations needed|date=September 2020}}

[[File:First Ionization Energy blocks.svg|thumb|right|512px|Ionization energy trends plotted against the [[atomic number]], in units [[Electronvolt|eV]]. The ionization energy gradually increases from the [[alkali metals]] to the [[noble gas]]es. The maximum ionization energy also decreases from the first to the last row in a given column, due to the increasing distance of the valence electron shell from the nucleus. Predicted values are used for elements beyond 104.]]

In [[physics]] and [[chemistry]], '''ionization energy''' ('''IE''') is the minimum energy required to remove the most loosely bound [[electron]] of an isolated gaseous [[atom]], [[Ion|positive ion]], or [[molecule]].<ref>{{Cite web|date=2013-10-02|title=Periodic Trends|url=https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Modules_and_Websites_(Inorganic_Chemistry)/Descriptive_Chemistry/Periodic_Trends_of_Elemental_Properties/Periodic_Trends|access-date=2020-09-13|website=Chemistry LibreTexts|language=en}}</ref> The first ionization energy is quantitatively expressed as
:X(g) + energy ⟶ X<sup>+</sup>(g) + e<sup>−</sup>
where X is any atom or molecule, X<sup>+</sup> is the resultant ion when the original atom was stripped of a single electron, and e<sup>−</sup> is the removed electron.<ref name=Miessler>{{cite book |last1=Miessler |first1=Gary L. |last2=Tarr |first2=Donald A. |title=Inorganic Chemistry |date=1999 |publisher=Prentice Hall |isbn=0-13-841891-8 |page=41 |edition=2nd}}</ref> Ionization energy is positive for neutral atoms, meaning that the ionization is an [[endothermic process]]. Roughly speaking, the closer the outermost electrons are to the [[atomic nucleus|nucleus of the atom]], the higher the atom's ionization energy.

In physics, ionization energy (IE) is usually expressed in [[electronvolt]]s (eV) or [[joule]]s (J). In chemistry, it is expressed as the energy to ionize a [[Mole (unit)|mole]] of atoms or molecules, usually as [[Joule per mole|kilojoules per mole]] (kJ/mol) or [[Kilocalorie per mole|kilocalories per mole]] (kcal/mol).<ref>{{cite web|url=http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Descriptive_Chemistry/Periodic_Table_of_the_Elements/Ionization_Energy|title=Ionization Energy|work=ChemWiki|publisher=University of California, Davis|date=2013-10-02|access-date=2014-01-05|archive-date=2010-04-30|archive-url=https://web.archive.org/web/20100430215110/http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Descriptive_Chemistry/Periodic_Table_of_the_Elements/Ionization_Energy|url-status=dead}}</ref>

Comparison of ionization energies of atoms in the [[periodic table]] reveals two [[periodic trends]] which follow the rules of [[Coulombic attraction]]:<ref>{{cite web | date=January 15, 2018 | title=Chapter 9: Quantum Mechanics | url=http://faculty.chem.queensu.ca/people/faculty/mombourquette/FirstYrChem/Theory/ | access-date=October 31, 2020 | website=faculty.chem.queesu.ca | language=en | archive-date=July 24, 2020 | archive-url=https://web.archive.org/web/20200724131229/http://faculty.chem.queensu.ca/people/faculty/mombourquette/FirstYrChem/Theory/ | url-status=dead }}</ref>

# Ionization energy generally increases from left to right within a given [[Period (periodic table)|period]] (that is, row).
# Ionization energy generally decreases from top to bottom in a given [[Group (periodic table)|group]] (that is, column).

The latter trend results from the outer [[electron shell]] being progressively farther from the nucleus, with the addition of one inner shell per row as one moves down the column.

The ''n''th ionization energy refers to the amount of energy required to remove the most loosely bound electron from the species having a positive charge of (''n'' − 1). For example, the first three ionization energies are defined as follows:
:1st ionization energy is the energy that enables the reaction X ⟶ X<sup>+</sup> + e<sup>−</sup>

:2nd ionization energy is the energy that enables the reaction X<sup>+</sup> ⟶ X<sup>2+</sup> + e<sup>−</sup>

:3rd ionization energy is the energy that enables the reaction X<sup>2+</sup> ⟶ X<sup>3+</sup> + e<sup>−</sup>

The most notable influences that determine ionization energy include:
* Electron configuration: This accounts for most elements' IE, as all of their chemical and physical characteristics can be ascertained just by determining their respective electron configuration (EC).
* Nuclear charge: If the nuclear charge (atomic number) is greater, the electrons are held more tightly by the nucleus and hence the ionization energy will be greater (leading to the mentioned trend 1 within a given period).
* Number of [[electron shell]]s: If the size of the atom is greater due to the presence of more shells, the electrons are held less tightly by the nucleus and the ionization energy will be smaller.
* [[Effective nuclear charge]] (''Z''<sub>eff</sub>): If the magnitude of electron [[Shielding effect|shielding]] and penetration are greater, the electrons are held less tightly by the nucleus, the ''Z''<sub>eff</sub> of the electron and the ionization energy is smaller.<ref name="Lang & Smith 2003"/>
* Stability: An atom having a more stable [[Electron configuration|electronic configuration]] has a reduced tendency to lose electrons and consequently has a higher ionization energy.

Minor influences include:
* [[Relativistic quantum chemistry|Relativistic effects]]: Heavier elements (especially those whose [[atomic number]] is greater than about 70) are affected by these as their electrons are approaching the speed of light. They therefore have smaller atomic radii and higher ionization energies.
* [[Lanthanide contraction|Lanthanide and actinide contraction]] (and [[d-block contraction|scandide contraction]]): The shrinking of the elements affects the ionization energy, as the net charge of the nucleus is more strongly felt.
* [[Electron pair|Electron pairing energies]]: Half-filled [[Electron shell#subshells|subshells]] usually result in higher ionization energies.

The term ''ionization potential'' is an older and obsolete term<ref>{{GoldBookRef |title=ionization potential |file=I03208 }}</ref> for ionization energy,<ref>{{cite book |first1=F. Albert |last1=Cotton |author1-link=F. Albert Cotton |first2=Geoffrey |last2=Wilkinson |author2-link=Geoffrey Wilkinson |title=Advanced Inorganic Chemistry |edition=5th |publisher=John Wiley |date=1988 |page=1381 |isbn=0-471-84997-9}}</ref> because the oldest method of measuring ionization energy was based on ionizing a sample and accelerating the electron removed using an [[Particle accelerator#Electrostatic particle accelerators|electrostatic potential]].

== Determination of ionization energies ==
[[File:Measurement of ionization energy of atoms - schematic.svg|thumb|304x304px|Ionization energy measurement apparatus. |alt=]]
The ionization energy of atoms, denoted ''E''<sub>i</sub>, is measured<ref>{{Cite web|last=Mahan|first=Bruce H.|date=1962|title=Ionization Energy|url=https://archive.org/details/ionization_energy|access-date=2020-09-13|publisher=College of Chemistry, University of California Berkeley}}</ref> by finding the minimal energy of light quanta ([[photon]]s) or electrons accelerated to a known energy that will kick out the least bound atomic electrons. The measurement is performed in the gas phase on single atoms. While only noble gases occur as [[monatomic gas]]es, other gases can be split into single atoms.{{fact|date=February 2025}} Also, many solid elements can be heated and vaporized into single atoms. Monatomic vapor is contained in a previously evacuated tube that has two parallel electrodes connected to a voltage source. The ionizing excitation is introduced through the walls of the tube or produced within.

When ultraviolet light is used, the wavelength is swept down the ultraviolet range.  At a certain wavelength (λ) and frequency of light (ν=c/λ, where c is the speed of light), the light quanta, whose energy is proportional to the frequency, will have energy high enough to dislodge the least bound electrons. These electrons will be attracted to the positive electrode, and the positive ions remaining after the [[photoionization]] will get attracted to the negatively charged electrode. These electrons and ions will establish a current through the tube. The ionization energy will be the energy of photons ''hν''<sub>i</sub> (''h'' is the [[Planck constant]]) that caused a steep rise in the current: ''E''<sub>i</sub> = ''hν''<sub>i</sub>.

When high-velocity electrons are used to ionize the atoms, they are produced by an [[electron gun]] inside a similar evacuated tube. The energy of the electron beam can be controlled by the acceleration voltages. The energy of these electrons that gives rise to a sharp onset of the current of ions and freed electrons through the tube will match the ionization energy of the atoms.

== Atoms: values and trends ==

Generally, the (''N''+1)th ionization energy of a particular element is larger than the ''N''th ionization energy (it may also be noted that the ionization energy of an anion is generally less than that of cations and neutral atom for the same element). When the next ionization energy involves removing an electron from the same electron shell, the increase in ionization energy is primarily due to the increased net charge of the ion from which the electron is being removed. Electrons removed from more highly charged ions experience greater forces of electrostatic attraction; thus, their removal requires more energy. In addition, when the next ionization energy involves removing an electron from a lower electron shell, the greatly decreased distance between the nucleus and the electron also increases both the electrostatic force and the distance over which that force must be overcome to remove the electron. Both of these factors further increase the ionization energy.

Some values for elements of the third period are given in the following table:

{| class="sortable"
|+''Successive ionization energy values /'' [[joule|kJ]]&nbsp;[[mole (unit)|mol]]<sup>−1</sup> <br> (96.485 kJ&nbsp;mol<sup>−1</sup> ≡ 1&nbsp;[[electron volt|eV]])
|-
!   Element
!   First
!   Second
!   Third
!   Fourth
!   Fifth
!   Sixth
!   Seventh
|- align="right"
! align="center" | [[Sodium|Na]]
| 496
| 4,560 ||  ||  ||  ||  ||
|- align="right"
! align="center" | [[Magnesium|Mg]]
| 738
| 1,450
| 7,730 ||  ||  ||  ||
|- align="right"
! align="center" | [[Aluminium|Al]]
| 577
| 1,816
| 2,881
| 11,600 ||  ||  ||
|- align="right"
! align="center" | [[Silicon|Si]]
| 786
| 1,577
| 3,228
| 4,354
| 16,100 ||  ||
|- align="right"
! align="center" | [[Phosphorus|P]]
| 1,060
| 1,890
| 2,905
| 4,950
| 6,270
| 21,200 ||
|- align="right"
! align="center" | [[Sulfur|S]]
| 1,000
| 2,295
| 3,375
| 4,565
| 6,950
| 8,490
| 27,107
|- align="right"
! align="center" | [[Chlorine|Cl]]
| 1,256
| 2,260
| 3,850
| 5,160
| 6,560
| 9,360
| 11,000
|- align="right"
! align="center" | [[Argon|Ar]]
| 1,520
| 2,665
| 3,945
| 5,770
| 7,230
| 8,780
| 12,000
|}

Large jumps in the successive molar ionization energies occur when passing noble gas configurations. For example, as can be seen in the table above, the first two molar ionization energies of magnesium (stripping the two 3s electrons from a magnesium atom) are much smaller than the third, which requires stripping off a 2p electron from the [[neon]] configuration of Mg<sup>2+</sup>. That 2p electron is much closer to the nucleus than the 3s electrons removed previously.
[[File:Ionization energies of atoms - labeled - atomic orbital filling indicated.svg|thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill.]]
Ionization energy is also a [[periodic trends|periodic trend]] within the periodic table. Moving left to right within a [[Period (periodic table)|period]], or upward within a [[Group (periodic table)|group]], the first ionization energy generally increases,<ref name=":1">{{cite web |url=https://www.chem.tamu.edu/class/fyp/stone/tutorialnotefiles/fundamentals/trends.htm |title=Atomic Structure : Periodic Trends |last=Stone |first=E.G. |date=December 19, 2020v |department=Department of Chemistry |website=chem.tamu.edu |publisher=Texas A&M University |location=400 Bizzell St, College Station, TX 77843, Texas, United States |language=en |access-date=December 19, 2020 |archive-date=October 11, 2018 |archive-url=https://web.archive.org/web/20181011230430/http://www.chem.tamu.edu/class/fyp/stone/tutorialnotefiles/fundamentals/trends.htm |url-status=dead }}</ref> with exceptions such as aluminium and sulfur in the table above. As the nuclear charge of the nucleus increases across the period, the electrostatic attraction increases between electrons and protons, hence the [[atomic radius]] decreases, and the electron cloud comes closer to the nucleus<ref>{{Cite web|title=Anomalous trends in ionization energy|url=https://chemistry.stackexchange.com/questions/32363/anomalous-trends-in-ionization-energy|access-date=2020-09-20|website=Chemistry Stack Exchange}}</ref> because the electrons, especially the outermost one, are held more tightly by the higher effective nuclear charge.

On moving downward within a given group, the electrons are held in higher-energy shells with higher principal quantum number n, further from the nucleus and therefore are more loosely bound so that the ionization energy decreases. The [[effective nuclear charge]] increases only slowly so that its effect is outweighed by the increase in n.<ref>{{cite book |last1=Petrucci |first1=Ralph H. |last2=Harwood |first2=William S. |last3=Herring |first3=F. Geoffrey |title=General Chemistry |date=2002 |publisher=Prentice Hall |isbn=0-13-014329-4 |page=370 |edition=8th}}</ref>

===Exceptions in ionization energies===
{{original research|section|date=December 2022}}
There are exceptions to the general trend of rising ionization energies within a period. For example, the value decreases from [[beryllium]] ({{nuclide|Be|&nbsp;}}: 9.3 eV) to [[boron]] ({{nuclide|B|&nbsp;}}: 8.3 eV), and from [[nitrogen]] ({{nuclide|N|&nbsp;}}: 14.5 eV) to [[oxygen]] ({{nuclide|O|&nbsp;}}: 13.6 eV). These dips can be explained in terms of electron configurations.<ref name=":Grandinetti">{{Cite web|url=https://www.grandinetti.org/ionization-energy-trends|title=Ionization Energy Trends {{!}} Grandinetti Group |last=Grandinetti |first=Philip J. |date=September 8, 2019 |access-date=2020-09-13|website=www.grandinetti.org}}</ref>

[[File:BerylliumVsBoronElectronConfiguration.jpg|thumb|200px|The added electron in boron occupies a [[atomic orbital|p-orbital]].]]
Boron has its last electron in a 2p orbital, which has its [[electron density]] further away from the nucleus on average than the 2s electrons in the same shell. The 2s electrons then shield the 2p electron from the nucleus to some extent, and it is easier to remove the 2p electron from boron than to remove a 2s electron from beryllium, resulting in a lower ionization energy for B.<ref name=Miessler/>

{{Multiple images
| image1          = NitrogenVsOxygenElectronConfiguration.jpg
| alt1            = Nitrogen and oxygen's electron configuration
| caption1        = These electron configurations do not show the full and half-filled orbitals.
| image2          = NitrogenVsOxygenElectronConfigurationBoxAndArrows.jpg
| alt2            = Nitrogen and oxygen's electron configuration using box and arrows
| caption2        = Here the added electron has a spin opposed to the other 2p electrons. This decreases the ionization energy of oxygen}}
In oxygen, the last electron shares a doubly occupied p-orbital with an electron of opposing [[Spin (physics)|spin]]. The two electrons in the same orbital are closer together on average than two electrons in different orbitals, so that they [[Shielding effect|shield each other from the nucleus]] more effectively and it is easier to remove one electron, resulting in a lower ionization energy.<ref name=Miessler/><ref name=":0">{{cite web |url=https://www.kentchemistry.com/links/PT/PTIonE.htm |title=First Ionization Energy |last=Kent |first=Mr. |website=kentchemistry.com |publisher=KentChemistry |access-date=December 6, 2020 |quote=...The addition of the second electron into an already occupied orbital introduces repulsion between the electrons, thus it is easier to remove. that is why there is a dip in the ionization energy.}}</ref>

Furthermore, after every noble gas element, the ionization energy drastically drops. This occurs because the outer electron in the [[alkali metal]]s requires a much lower amount of energy to be removed from the atom than the inner shells. This also gives rise to low [[electronegativity]] values for the alkali metals.<ref>{{Cite web|title=Group IA|url=https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch9/alkali.php|access-date=2020-09-20|website=chemed.chem.purdue.edu}}</ref><ref>{{Cite web|title=Alkali Metals|url=http://hyperphysics.phy-astr.gsu.edu/hbase/pertab/alkmet.html|access-date=2020-09-13|website=hyperphysics.phy-astr.gsu.edu}}</ref><ref>{{Cite web|title=The Alkali Metals {{!}} Introduction to Chemistry|url=https://courses.lumenlearning.com/introchem/chapter/the-alkali-metals/|access-date=2020-09-13|website=courses.lumenlearning.com}}</ref>

{{multiple image
| image1            = ZincVsGalliumElectronConfiguration.jpg
| alt1              = Zinc and Gallium's respective electron configurations
| caption1          = Because of a single p-orbital electron in [[gallium]]'s configuration, makes the overall structure less stable, hence the dip in ionization energy values<ref name="Lang & Smith 2003"/>
| image2            = RadiumVsActiniumElectronConfiguration.jpg
| alt2              = Radium and Actinium's Electron Configuration (condensed)
| caption2          = [[Actinium]]'s electron configuration predetermines that it would require less energy to remove that single d-orbital electron, therefore even though it has a larger EC, [[radium]] still has the higher IE<ref>{{cite web |url=https://www.lenntech.com/periodic-chart-elements/ionization-energy.htm |title=Chemical elements listed by ionization energy |author=<!--Not stated--> |date=2018 |website=lenntech.com |publisher=Lenntech BV |access-date=December 6, 2020 |quote=The elements of the periodic table sorted by ionization energy click on any element's name for further information on chemical properties, environmental data or health effects. This list contains the 118 elements of chemistry.}}</ref>
}}

The trends and exceptions are summarized in the following subsections:

====Ionization energy decreases when====
* Transitioning to a new period: an alkali metal easily loses one electron to leave an [[octet rule|octet]] or pseudo-[[noble gas configuration]], so those elements have only small values for IE.
* Moving from the s-block to the p-block: a p-orbital loses an electron more easily. An example is beryllium to boron, with electron configuration 1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>1</sup>. The 2s electrons shield the higher-energy 2p electron from the nucleus, making it slightly easier to remove. This also happens from [[magnesium]] to [[aluminium]].<ref>{{cite web |url=https://www.angelo.edu/faculty/kboudrea/periodic/trends_ionization_energy.htm |title=The Parts of the Periodic Table |last=Boudreaux |first=K.A. |date=August 13, 2020 |orig-date=July 26, 2006 |department=Department of Chemistry and Biochemistry |website=angelo.edu/faculty/kboudrea/<!--this is the real website, pls dont change--> |publisher=Angelo State University |location=2601 W. Avenue N, San Angelo, TX 76909, Texas |language=en |access-date=December 19, 2020 |via=angelo.edu |archive-date=July 10, 2022 |archive-url=https://web.archive.org/web/20220710025232/https://www.angelo.edu/faculty/kboudrea/periodic/trends_ionization_energy.htm |url-status=dead }}</ref>
* Occupying a p-subshell with its '''first''' electron with spin opposed to the other electrons: such as in nitrogen ({{nuclide|N|&nbsp;}}: 14.5 eV) to oxygen ({{nuclide|O|&nbsp;}}: 13.6 eV), as well as [[phosphorus]] ({{nuclide|P|&nbsp;}}: 10.48 eV) to [[sulfur]] ({{nuclide|S|&nbsp;}}: 10.36 eV). The reason for this is because oxygen, sulfur and selenium all have dipping ionization energies because of shielding effects.<ref>{{Cite web|date=2014-07-02|title=18.10: The Group 6A Elements|url=https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_(Zumdahl_and_Decoste)/18%3A_The_Representative_Elements/18.10%3A_The_Group_6A_Elements|access-date=2020-09-20|website=Chemistry LibreTexts|language=en}}</ref> However, this discontinues starting from [[tellurium]] where the shielding is too small to produce a dip.
* Moving from the d-block to the p-block: as in the case of [[zinc]] ({{nuclide|Zn|&nbsp;}}: 9.4 eV) to [[gallium]] ({{nuclide|Ga|&nbsp;}}: 6.0 eV)
* Special case: decrease from [[lead]] ({{nuclide|Pb|&nbsp;}}: 7.42 eV) to [[bismuth]] ({{nuclide|Bi|&nbsp;}}: 7.29 eV). This cannot be attributed to size (the difference is minimal: lead has a covalent radius of 146 [[picometer|pm]] whereas [[bismuth]]'s is 148 pm<ref>{{Cite web|title=Covalent Radius for all the elements in the Periodic Table|url=https://periodictable.com/Properties/A/CovalentRadius.v.log.html|access-date=2020-09-13|website=periodictable.com}}</ref>). This is due to the spin-orbit splitting of the 6p shell (lead is removing an electron from the stabilised 6p<sub>1/2</sub> level, but bismuth is removing one from the destabilised 6p<sub>3/2</sub> level). Predicted ionization energies show a much greater decrease from [[flerovium]] to [[moscovium]], one row further down the periodic table and with much larger spin-orbit effects.
* Special case: decrease from radium ({{nuclide|Ra|&nbsp;}}: 5.27 eV) to [[actinium]] ({{nuclide|Ac|&nbsp;}}: 5.17 eV), which is a switch from an s to a d orbital. However the analogous switch from [[barium]] ({{nuclide|Ba|&nbsp;}}: 5.2 eV) to [[lanthanum]] ({{nuclide|La|&nbsp;}}: 5.6 eV) does not show a downward change.
* [[Lutetium]] ({{nuclide|Lu|&nbsp;}}) and [[lawrencium]] ({{nuclide|Lr|&nbsp;}}) both have ionization energies lower than the previous elements. In both cases the last electron added [[Electron configurations of the elements (data page)|starts a new subshell]]: 5d for Lu with electron configuration [Xe] 4f<sup>14</sup> 5d<sup>1</sup> 6s<sup>2</sup>, and 7p for Lr with configuration [Rn] 5f<sup>4</sup> 7s<sup>2</sup> 7p<sup>1</sup>. These dips in ionization energies for lutetium and especially lawrencium show that these elements belong in the d-block, and not lanthanum and actinium.<ref name="JensenLr">{{cite web|url=https://www.che.uc.edu/jensen/W.%20B.%20Jensen/Reprints/251.%20Lawrencium.pdf |title=Some Comments on the Position of Lawrencium in the Periodic Table |last1=Jensen |first1=W. B. |date=2015 |access-date=20 September 2015 |url-status=dead |archive-url=https://web.archive.org/web/20151223091325/https://www.che.uc.edu/jensen/W.%20B.%20Jensen/Reprints/251.%20Lawrencium.pdf |archive-date=23 December 2015 }}</ref>

====Ionization energy increases when====
* Reaching Group 18 [[noble gas]] elements: This is due to their complete electron subshells,<ref>{{cite book |last1=Singh |first1=Jasvinder |chapter=Inert Gases |page=122 |chapter-url=https://books.google.com/books?id=eKnrhryjqn0C&pg=PA122 |title=Sterling Dictionary of Physics |date=1999 |publisher=Sterling Publishers Pvt. Ltd |isbn=978-81-7359-124-2 }}</ref> so that these elements require large amounts of energy to remove one electron.
* Group 12: The elements here, zinc ({{nuclide|Zn|&nbsp;}}: 9.4 eV), [[cadmium]] ({{nuclide|Cd|&nbsp;}}: 9.0 eV) and [[mercury (element)|mercury]] ({{nuclide|Hg|&nbsp;}}: 10.4 eV) all record sudden rising IE values in contrast to their preceding elements: [[copper]] ({{nuclide|Cu|&nbsp;}}: 7.7 eV), [[silver]] ({{nuclide|Ag|&nbsp;}}: 7.6 eV) and [[gold]] ({{nuclide|Au|&nbsp;}}: 9.2 eV), respectively. For mercury, it can be extrapolated that the [[relativistic quantum chemistry|relativistic]] stabilization of the 6s electrons increases the ionization energy, in addition to poor shielding by 4f electrons that increases the effective nuclear charge on the outer valence electrons. In addition, the closed-subshells electron configurations: [Ar] 3d<sup>10</sup> 4s<sup>2</sup>, [Kr] 4d<sup>10</sup>5s<sup>2</sup> and [Xe] 4f<sup>14</sup> 5d<sup>10</sup> 6s<sup>2</sup> provide increased stability.
* Special case: shift from [[rhodium]] ({{nuclide|Rh|&nbsp;}}: 7.5 eV) to [[palladium]] ({{nuclide|Pd|&nbsp;}}: 8.3 eV). Unlike other Group 10 elements, palladium has a higher ionization energy than the preceding atom, due to its electron configuration. In contrast to [[nickel]]'s [Ar] 3d<sup>8</sup> 4s<sup>2</sup>, and [[platinum]]'s [Xe] 4f<sup>14</sup> 5d<sup>9</sup> 6s<sup>1</sup>, palladium's electron configuration is [Kr] 4d<sup>10</sup> 5s<sup>0</sup> (even though the [[Aufbau principle#Exceptions to the rule in the transition metal|Madelung rule]] predicts [Kr] 4d<sup>8</sup> 5s<sup>2</sup>). Finally, [[silver]]'s lower IE ({{nuclide|Ag|&nbsp;}}: 7.6 eV) further accentuates the high value for palladium; the single added s electron is removed with a lower ionization energy than palladium,<ref>{{cite book |doi=10.1016/B978-0-7506-3365-9.50028-6 |chapter=Vanadium, Niobium and Tantalum |title=Chemistry of the Elements |year=1997 |pages=976–1001 |isbn=978-0-7506-3365-9 }}</ref> which emphasizes palladium's high IE (as shown in the above linear table values for IE)
* The IE of [[gadolinium]] ({{nuclide|Gd|&nbsp;}}: 6.15 eV) is somewhat higher than both the preceding ({{nuclide|Sm|&nbsp;}}: 5.64 eV), ({{nuclide|Eu|&nbsp;}}: 5.67 eV) and following elements ({{nuclide|Tb|&nbsp;}}: 5.86 eV), ({{nuclide|Dy|&nbsp;}}: 5.94 eV). This anomaly is due to the fact that gadolinium valence d-subshell borrows 1 electron from the valence f-subshell. Now the valence subshell is the d-subshell, and due to the poor shielding of positive nuclear charge by electrons of the f-subshell, the electron of the valence d-subshell experiences a greater attraction to the nucleus, therefore increasing the energy required to remove the (outermost) valence electron.
* Moving into d-block elements: The elements Sc with a 3d<sup>1</sup> electronic configuration has a ''higher'' IP ({{nuclide|Sc|&nbsp;}}: 6.56 eV) than the preceding element ({{nuclide|Ca|&nbsp;}}: 6.11 eV), contrary to the decreases on moving into s-block and p-block elements. The 4s and 3d electrons have similar shielding ability: the 3d orbital forms part of the n=3 shell whose average position is closer to the nucleus than the 4s orbital and the n=4 shell, but electrons in s orbitals experience greater penetration into the nucleus than electrons in d orbitals. So the mutual shielding of 3d and 4s electrons is weak, and the effective nuclear charge acting on the ionized electron is relatively large. Yttrium ({{nuclide|Y|&nbsp;}}) similarly has a higher IP (6.22 eV) than {{nuclide|Sr|&nbsp;}}: 5.69 eV.
* Moving into f-block elements; The elements ({{nuclide|La|&nbsp;}}: 5.18 eV) and ({{nuclide|Ac|&nbsp;}}: 5.17 eV) have only very slightly lower IP's than their preceding elements ({{nuclide|Ba|&nbsp;}}: 5.21 eV) and ({{nuclide|Ra|&nbsp;}}: 5.18 eV), though their atoms are anomalies in that they add a d-electron rather than an f-electron. As can be seen in the above graph for ionization energies, the sharp rise in IE values from ({{nuclide|Cs|&nbsp;}}: 3.89 eV) to ({{nuclide|Ba|&nbsp;}}: 5.21 eV) is followed by a small increase (with some fluctuations) as the f-block proceeds from {{nuclide|Ba|&nbsp;}} to {{nuclide|Yb|&nbsp;}}. This is due to the [[lanthanide contraction]] (for lanthanides).<ref name=Housecroft>{{cite book |last1=Housecroft |first1=C.E. |last2=Sharpe |first2=A.G. |date=November 1, 1993 |title=Inorganic Chemistry |url=https://www.pearson.com/us/higher-education/program/Housecroft-Inorganic-Chemistry-5th-Edition/PGM2178749.html |type=eBook |language=en |volume=3 |edition=15th |location=Switzerland |publisher=Pearson Prentice-Hall |publication-date=November 1, 1993 |pages=536, 649, 743 |doi=10.1021/ed070pA304.1 |isbn=978-0-273-74275-3 |archive-url=https://web.archive.org/web/20210414235943/https://www.pearson.com/us/higher-education/program/Housecroft-Inorganic-Chemistry-5th-Edition/PGM2178749.html |archive-date=April 14, 2021 |access-date=December 14, 2020 |url-status=bot: unknown }}</ref><ref name=Cotton>{{Cotton&Wilkinson5th|pages=776, 955}}</ref><ref name=Jolly>{{cite journal |doi=10.1021/ed062pA137.1 |title=Modern Inorganic Chemistry (Jolly, William L.) |year=1985 |last1=Billo |first1=E. J. |journal=Journal of Chemical Education |volume=62 |issue=4 |pages=A137 |bibcode=1985JChEd..62..137B |doi-access=free }}</ref> This decrease in ionic radius is associated with an increase in ionization energy in turn increases, since the two properties correlate to each other.<ref name=":1" /> As for d-block elements, the electrons are added in an inner shell, so that no new shells are formed. The shape of the added orbitals prevents them from penetrating to the nucleus so that the electrons occupying them have less shielding capacity.

====Ionization energy anomalies in groups====

Ionization energy values tend to decrease on going to heavier elements within a group<ref name=":Grandinetti" /> as shielding is provided by more electrons and overall, the valence shells experience a weaker attraction from the nucleus, attributed to the larger covalent radius which increase on going down a group<ref>{{Cite web|title=Patterns and trends in the periodic table - Periodicity - Higher Chemistry Revision|url=https://www.bbc.co.uk/bitesize/guides/zxc99j6/revision/6|access-date=2020-09-20|website=BBC Bitesize|language=en-GB}}</ref> Nonetheless, this is not always the case. As one exception, in Group 10 palladium ({{nuclide|Pd|&nbsp;}}: 8.34 eV) has a higher ionization energy than nickel ({{nuclide|Ni|&nbsp;}}: 7.64 eV), contrary to the general decrease for the elements from technetium {{nuclide|Tc|&nbsp;}} to xenon {{nuclide|Xe|&nbsp;}}. Such anomalies are summarized below:
* Group 1:
** [[Hydrogen]]'s ionization energy is very high (at 13.59844 eV), compared to the alkali metals. This is due to its single electron (and hence, very small [[electron cloud]]), which is close to the nucleus. Likewise, since there are not any other electrons that may cause shielding, that single electron experiences the full net positive charge of the nucleus.<ref>{{Cite web|date=2013-10-03|title=Ionization Energies|url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Ionization_Energy/Ionization_Energies|access-date=2020-09-20|website=Chemistry LibreTexts|language=en}}</ref>
** [[Francium]]'s ionization energy is higher than the precedent [[alkali metal]], [[cesium]]. This is due to its (and radium's) small ionic radii owing to relativistic effects. Because of their large mass and size, this means that its electrons are traveling at extremely high speeds, which results in the electrons coming closer to the nucleus than expected, and they are consequently harder to remove (higher IE).<ref>{{Cite web|date=2019-11-06|title=IYPT 2019 Elements 087: Francium: Not the most reactive Group 1 element|url=https://www.compoundchem.com/2019/11/06/iypt087-francium/|access-date=2020-09-20|website=Compound Interest|language=en-GB}}</ref>
* Group 2: [[Radium]]'s ionization energy is higher than its antecedent [[alkaline earth metal]] [[barium]], like francium, which is also due to relativistic effects. The electrons, especially the 1s electrons, experience ''very high effective nuclear charges''. To avoid falling into the nucleus, the 1s electrons must move at very high speeds, which causes the special relativistic corrections to be substantially higher than the approximate classical momenta. By the [[uncertainty principle]], this causes a relativistic contraction of the 1s orbital (and other orbitals with electron density close to the nucleus, especially ns and np orbitals). Hence this causes a cascade of electron changes, which finally results in the outermost electron shells contracting and getting closer to the nucleus.
* Group 4:
** [[Hafnium]]'s near similarity in IE with [[zirconium]]. The effects of the lanthanide contraction can still be felt ''[[lanthanide contraction#Influence on the post-lanthanides|after the lanthanides]]''.<ref name=Cotton/> It can be seen through the former's smaller atomic radius (which contradicts the [https://www.chem.tamu.edu/class/fyp/stone/tutorialnotefiles/fundamentals/trends.htm#:~:text=WHY%3F%20%2D%20The%20number%20of%20energy,a%20period%2C%20atomic%20radius%20decreases. observed periodic trend] {{Webarchive|url=https://web.archive.org/web/20181011230430/http://www.chem.tamu.edu/class/fyp/stone/tutorialnotefiles/fundamentals/trends.htm#:~:text=WHY%3F%20%2D%20The%20number%20of%20energy,a%20period%2C%20atomic%20radius%20decreases. |date=2018-10-11 }}) at 159 pm<ref>{{cite web |url=https://www.gordonengland.co.uk/elements/hf.htm |title=Hafnium |author=<!--Not stated--> |date=2020 |website=gordonengland.co.uk |publisher=Gordon England |access-date=December 7, 2020 |quote=...Atomic Radius 159 pm...}}</ref> ([[atomic radius#Notes|empirical value]]), which differs from the latter's 155 pm.<ref>{{cite web |url=https://pubchem.ncbi.nlm.nih.gov/element/Zirconium#section=Atomic-Radius |title=Zirconium (Element) - Atomic Radius |author=<!--Not stated-->|website=pubchem.ncbi.nlm.nih.gov |publisher=PubChem |access-date=December 8, 2020 |quote=155 pm (Empirical)}}
</ref><ref>{{cite journal |last1=Slater |first1=J. C. |title=Atomic Radii in Crystals |journal=The Journal of Chemical Physics |date=15 November 1964 |volume=41 |issue=10 |pages=3199–3204 |doi=10.1063/1.1725697 |bibcode=1964JChPh..41.3199S }}</ref> This in turn makes its ionization energies increase by 18 kJ/mol<sup>−1</sup>.
** [[Titanium]]'s IE is smaller than that of both hafnium and zirconium. Hafnium's ionization energy is similar to zirconium's due to lanthanide contraction. However, why zirconium's ionization energy is higher than the preceding elements' remains unclear; we cannot attribute it to atomic radius as it is higher for zirconium and hafnium by 15 pm.<ref>{{Cite web|title=WebElements Periodic Table » Titanium » radii of atoms and ions|url=https://www.webelements.com/titanium/atom_sizes.html|access-date=2020-09-20|website=www.webelements.com}}</ref> We also cannot invoke the ''condensed'' ionization energy, as it is more or less the same ([Ar] 3d<sup>2</sup> 4s<sup>2</sup> for titanium, whereas [Kr] 4d<sup>2</sup> 5s<sup>2</sup> for zirconium). Additionally, there are no half-filled nor fully filled orbitals we might compare. Hence, we can only invoke zirconium's ''full'' electron configuration, which is 1s<sup>2</sup>2s<sup>2</sup>2p<sup>6</sup>3s<sup>2</sup>3p<sup>6</sup>'''3d<sup>10</sup>'''4s<sup>2</sup>4p<sup>6</sup>4d<sup>2</sup>5s<sup>2</sup>.<ref>{{Cite web|last=Straka |first=J. |title=Periodic Table of the Elements: Zirconium - Electronic configuration|url=https://www.tabulka.cz/english/elements/configuration.asp?id=40|access-date=2020-09-20|website=www.tabulka.cz}}</ref> The presence of a full 3d-block sublevel is tantamount to a higher shielding efficiency compared to the 4d-block elements (which are only two electrons).{{efn|Nonetheless, further research is still needed to corroborate this mere inference.}}
* Group 5: akin to Group 4, [[niobium]] and [[tantalum]] are analogous to each other, due to their electron configuration and to the lanthanide contraction affecting the latter element.<ref>{{Cite web|title=Tantalum {{!}} chemical element|url=https://www.britannica.com/science/tantalum|access-date=2020-09-20|website=Encyclopedia Britannica|language=en}}</ref> Ipso facto, their significant rise in IE compared to the foremost element in the group, [[vanadium]], can be attributed due to their full d-block electrons, in addition to their electron configuration. Another intriguing notion is niobium's half-filled 5s orbital; due to repulsion and exchange energy (in other words the ''"costs"'' of putting an electron in a low-energy sublevel to completely fill it instead of putting the electron in a high-energy one) overcoming the energy gap between s- and d-(or f) block electrons, the EC does not follow the Madelung rule.
* Group 6: like its forerunners groups 4 and 5, group 6 also record high values when moving downward. [[Tungsten]] is once again similar to [[molybdenum]] due to their electron configurations.<ref>{{cite book |doi=10.1002/0471435139.tox038 |chapter=Chromium, Molybdenum, and Tungsten |title=Patty's Toxicology |year=2015 |last1=Langård |first1=Sverre |isbn=978-0-471-12547-1 }}</ref> Likewise, it is also attributed to the full 3d-orbital in its electron configuration. Another reason is molybdenum's half filled 4d orbital due to electron pair energies violating the aufbau principle.
* Groups 7-12 6th period elements ([[rhenium]], [[osmium]], [[iridium]], [[platinum]], [[gold]] and [[mercury (element)|mercury]]): All of these elements have extremely high ionization energies compared to the elements preceding them in their respective groups. The essence of this is due to the lanthanide contraction's influence on post lanthanides, in addition to the relativistic stabilization of the 6s orbital.
* Group 13:
** Gallium's IE is higher than aluminum's. This is once again due to d-orbitals, in addition to scandide contraction, providing weak shielding, and hence the effective nuclear charges are augmented.
** Thallium's IE, due to poor shielding of 4f electrons<ref name="Lang & Smith 2003">{{cite journal |last1=Lang |first1=Peter F. |last2=Smith |first2=Barry C. |title=Ionization Energies of Atoms and Atomic Ions |journal=Journal of Chemical Education |date=August 2003 |volume=80 |issue=8 |pages=938 |doi=10.1021/ed080p938 |bibcode=2003JChEd..80..938L }}</ref> in addition to lanthanide contraction, causes its IE to be increased in contrast to its precursor [[indium]].
* Group 14: [[Lead]]'s unusually high ionization energy ({{nuclide|Pb|&nbsp;}}: 7.42 eV) is, akin to that of group 13's thallium, a result of the full 5d and 4f subshells. The lanthanide contraction and the inefficient screening of the nucleus by the 4f electrons results in slightly ''higher'' ionization energy for lead than for [[tin]] ({{nuclide|Sn|&nbsp;}}: 7.34 eV).<ref>{{Cite web|date=2015-12-02|title=The Group 14 elements|url=https://www.webelements.com/nexus/the-group-14-elements/|access-date=2020-09-13|website=Chemistry Nexus|language=en-US}}</ref><ref name="Lang & Smith 2003"/>

== Bohr model for hydrogen atom ==
The ionization energy of the hydrogen atom ({{tmath|1= Z = 1 }}) can be evaluated in the [[Bohr model]],<ref name="bohr1">{{cite journal |last1=Bohr |first1=N. |title=I. On the constitution of atoms and molecules |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=July 1913 |volume=26 |issue=151 |pages=1–25 |doi=10.1080/14786441308634955 |url=https://zenodo.org/record/2493915 }}</ref> which predicts that the atomic energy level <math>n</math> has energy
: <math> E = - \frac{1}{n^2} \frac{Z^2e^2}{2a_0} = - \frac{Z^2 R_\text{H}}{n^2} = - \frac{Z^2 \times \mathrm{13.6\ eV}}{n^2}</math>

{{tmath| R_\text{H} }} is the [[Rydberg constant]] for the hydrogen atom. For hydrogen in the ground state <math>Z=1</math> and <math>n=1</math> so that the energy of the atom before ionization is simply {{tmath|1= E = \mathrm{-13.6\ eV} }}.

After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is {{tmath|1= I = E(\mathrm{H}^+) - E(\mathrm{H}) = \mathrm{+13.6\ eV} }}. This agrees with the experimental value for the hydrogen atom.

==Quantum-mechanical explanation==
{{Expand section
 | 1           = more calculation formulas for ionization energies
 | section     = 4
 | small       = yes
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According to the more complete theory of [[quantum mechanics]], the location of an electron is best described as a probability distribution within an [[electron cloud]], i.e. [[atomic orbital]].<ref>{{cite web|url=https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_CLUE_(Cooper_and_Klymkowsky)/2%3A_Electrons_and_Orbitals/2.6%3A_Orbitals%2C_Electron_Clouds%2C_Probabilities%2C_and_Energies |title=Orbitals, Electron Clouds, Probabilities, and Energies|date=May 23, 2019 |website=chem.libretexts.org|publisher=UC Davis ChemWiki |access-date=November 2, 2020}}</ref><ref>{{cite web|url=https://www.khanacademy.org/science/physics/quantum-physics/quantum-numbers-and-orbitals/a/the-quantum-mechanical-model-of-the-atom|title= Quantum numbers and orbitals- The quantum mechanical model of the atom|website=Khan Academy|access-date=November 2, 2020}}</ref> The energy can be calculated by integrating over this cloud. The cloud's underlying mathematical representation is the [[wavefunction]], which is built from [[Slater determinant]]s consisting of molecular spin orbitals.{{sfn|Levine|1991|p=315|ps=: "In the Hartree-Fock approximation, the wave function of an atom (or molecule) is a Slater determinant or a linear combination of a few Slater determinants"}} These are related by [[Pauli's exclusion principle]] to the antisymmetrized products of the atomic or [[molecular orbital]]s.

There are two main ways in which ionization energy is calculated. In general, the computation for the ''N''th ionization energy requires calculating the energies of <math>Z-N+1</math> and <math>Z-N</math> electron systems. Calculating these energies exactly is not possible except for the simplest systems (i.e. hydrogen and [[Hydrogen-like atom|hydrogen-like]] elements), primarily because of difficulties in integrating the [[electron correlation]] terms.{{sfn|Levine|1991|pp=290–291}} Therefore, approximation methods are routinely employed, with different methods varying in complexity (computational time) and accuracy compared to empirical data.  This has become a well-studied problem and is routinely done in [[computational chemistry]]. The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by [[Koopmans' theorem]], which involves the highest occupied molecular orbital or "[[HOMO and LUMO|HOMO]]" and the lowest unoccupied molecular orbital or "[[HOMO and LUMO|LUMO]]", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected.{{sfn|Levine|1991|p=475}} This means that the ionization energy is equal to the negative of HOMO energy, which in a formal equation can be written as:<ref>{{cite web |url=https://www.shodor.org/chemviz/ionization/students/background.html |title=Background Reading for Ionization Energy |date=2000 |website=shodor.org |publisher=The Shodor Education Foundation, Inc. |access-date=November 15, 2020 |quote=...&nbsp;The second method is called Koopman's Theory. This method involves the HOMO.}}</ref>

: <math>I_i=-E_i</math>

==Molecules: vertical and adiabatic ionization energy==
[[File:Franck-Condon-diagram.png|280px|thumb|'''Figure 1.''' Franck–Condon principle energy diagram. For ionization of a diatomic molecule, the only nuclear coordinate is the bond length. The lower curve is the [[Potential energy surface|potential energy curve]] of the neutral molecule, and the upper curve is for the positive ion with a longer bond length. The blue arrow is vertical ionization, here from the ground state of the molecule to the v=2 level of the ion.]]

Ionization of molecules often leads to changes in [[molecular geometry]], and two types of (first) ionization energy are defined – ''adiabatic'' and ''vertical''.<ref>{{cite web|url=http://cccbdb.nist.gov/adiabatic.asp|title=The difference between a vertical ionization energy and adiabatic ionization energy |work = Computational Chemistry Comparison and Benchmark Database | publisher= [[National Institute of Standards and Technology]]}}</ref>

===Adiabatic ionization energy===
The [[adiabatic theorem|adiabatic]] ionization energy of a molecule is the ''minimum'' amount of energy required to remove an electron from a neutral molecule, i.e. the difference between the energy of the [[molecular vibration|vibrational]] [[ground state]] of the neutral species (v" = 0 level) and that of the positive ion (v' = 0). The specific equilibrium geometry of each species does not affect this value.

===Vertical ionization energy===
Due to the possible changes in molecular geometry that may result from ionization, additional transitions may exist between the vibrational ground state of the neutral species and vibrational [[excited state]]s of the positive ion. In other words, ionization is accompanied by [[vibrational spectroscopy|vibrational excitation]]. The intensity of such transitions is explained by the [[Franck–Condon principle]], which predicts that the most probable and intense transition corresponds to the vibrationally excited state of the positive ion that has the same geometry as the neutral molecule. This transition is referred to as the "vertical" ionization energy since it is represented by a completely vertical line on a potential energy diagram (see Figure).

For a diatomic molecule, the geometry is defined by the length of a single [[bond length|bond]]. The removal of an electron from a bonding [[molecular orbital]] weakens the bond and increases the bond length. In Figure 1, the lower [[Potential energy surface|potential energy curve]] is for the neutral molecule and the upper surface is for the positive ion. Both curves plot the potential energy as a function of bond length. The horizontal lines correspond to [[Molecular vibration|vibrational levels]] with their associated [[Quantum harmonic oscillator|vibrational wave functions]]. Since the ion has a weaker bond, it will have a longer bond length. This effect is represented by shifting the minimum of the potential energy curve to the right of the neutral species. The adiabatic ionization is the diagonal transition to the vibrational ground state of the ion. Vertical ionization may involve vibrational excitation of the ionic state and therefore requires greater energy.

In many circumstances, the adiabatic ionization energy is often a more interesting physical quantity since it describes the difference in energy between the two potential energy surfaces. However, due to experimental limitations, the adiabatic ionization energy is often difficult to determine, whereas the vertical detachment energy is easily identifiable and measurable.

==Analogs of ionization energy to other systems==
While the term ionization energy is largely used only for gas-phase atomic, cationic, or molecular species, there are a number of analogous quantities that consider the amount of energy required to remove an electron from other physical systems.

===Electron binding energy===
[[File:Electron binding energy vs Z.jpg|thumb|500px|Binding energies of specific atomic orbitals as a function of the atomic number. Because of the increasing number of protons, electrons occupying the same orbital are more tightly bound in heavier elements.]]

Electron [[binding energy]] is a generic term for the minimum energy needed to remove an electron from a particular electron shell for an atom or ion, due to these negatively charged electrons being held in place by the electrostatic pull of the positively charged nucleus.<ref>{{cite web |url=https://radiopaedia.org/articles/electron-binding-energy#:~:text=The%20electron%20binding%20energy%20is,1.6%20x%2010-19%20J. |title=Electron binding energy |last1=Murphy |first1=Andrew |last2=Wong |first2=Monica |date=2019 |website=radiopaedia.org |publisher=Radiopaedia |access-date=December 7, 2020 |quote=The electron binding energy is the minimum energy that is required to remove an electron from an atom}}</ref> For example, the electron binding energy for removing a 3p<sub>3/2</sub> electron from the chloride ion is the minimum amount of energy required to remove an electron from the chlorine atom when it has a charge of −1. In this particular example, the electron binding energy has the same magnitude as the [[electron affinity]] for the neutral chlorine atom. In another example, the electron binding energy refers to the minimum amount of energy required to remove an electron from the dicarboxylate dianion <sup>−</sup>O<sub>2</sub>C(CH<sub>2</sub>)<sub>8</sub>CO{{su|b=2|p=−}}.

The graph to the right shows the binding energy for electrons in different shells in neutral atoms. The ionization energy is the lowest binding energy for a particular atom (although these are not all shown in the graph).

===Solid surfaces: work function===
[[Work function]] is the minimum amount of energy required to remove an electron from a solid surface, where the work function {{math|''W''}} for a given surface is defined by the difference<ref>{{cite book |last1=Kittel |first1=Charles |last2=McEuen |first2=Paul |chapter=Free Electron Fermi Gas |pages=133–162 |chapter-url={{GBurl|nNpVEAAAQBAJ|p=133}} |title=[[Introduction to Solid State Physics]] |date=2018 |publisher=John Wiley & Sons |isbn=978-1-119-45416-8 }}</ref>
:<math>W = -e\phi - E_{\rm F}, </math>
where {{math|−''e''}} is the charge of an [[electron]], {{math|''ϕ''}} is the [[electrostatic potential]] in the vacuum nearby the surface, and {{math|''E''<sub>F</sub>}} is the [[Fermi level]] ([[electrochemical potential]] of electrons) inside the material.

==Note==
{{Notelist}}

==See also==
* [[Rydberg equation]], a calculation that could determine the ionization energies of [[hydrogen]] and [[Hydrogen-like atom|hydrogen-like]] elements. This is further elaborated through this [https://socratic.org/questions/how-would-you-determine-the-ionization-energy-of-a-hydrogen-atom-in-kj-mol-if-th-1 site.]
* [[Electron affinity]], a closely related concept describing the energy released by ''adding'' an electron to a neutral atom or molecule.
* [[Lattice energy]], a measure of the energy released when [[ions]] are combined to make a compound.
* [[Electronegativity]] is a number that shares some similarities with ionization energy.
* [[Koopmans' theorem]], regarding the predicted ionization energies in [[Hartree–Fock]] theory.
* [[Ditungsten tetra(hpp)]] has the lowest recorded ionization energy for a stable [[chemical compound]].
* [[Bond-dissociation energy]], the measure of the strength of a chemical bond calculated through cleaving by homolysis giving two radical fragments A and B and subsequent evaluation of the enthalpy change
* [[Bond energy]], the average measure of a chemical bond's strength, calculated through the amount of heat needed to break all of the chemical bonds into individual atoms.

==References==
{{Reflist}}

==Sources==
* {{cite book |last1=Levine |first1=Ira N. |title=Quantum Chemistry |date=1991 |publisher=Prentice Hall |isbn=978-0-205-12770-2 }}

{{Authority control}}

{{DEFAULTSORT:Ionization Energy}}
[[Category:Ions]]
[[Category:Molecular physics]]
[[Category:Atomic physics]]
[[Category:Chemical properties]]
[[Category:Quantum chemistry]]
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