Editing Ionization energy (section)

==Quantum-mechanical explanation==
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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>