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| {{Distinguish|Slater–Condon rules}}
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| In [[quantum chemistry]], '''Slater's rules''' provide numerical values for the [[effective nuclear charge]] concept. In a many-electron atom, each electron is said to experience less than the actual [[atomic nucleus|nuclear charge]] owing to [[Shielding effect|shielding]] or [[Electric field screening|screening]] by the other electrons. For each electron in an atom, Slater's rules provide a value for the screening constant, denoted by ''s'', ''S'', or ''σ'', which relates the effective and actual nuclear charges as
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| :<math>Z_{\mathrm{eff}}= Z - s.\,</math>
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| The rules were devised [[semi-empirical]]ly by [[John C. Slater]] and published in 1930.<ref name="slater57">{{cite journal|last=Slater|first=J. C.|year=1930|title=Atomic Shielding Constants|url=http://astrophysics.fic.uni.lodz.pl/100yrs/pdf/04/008.pdf|journal=Phys. Rev.|volume=36|issue=1|pages=57–64|doi=10.1103/PhysRev.36.57|bibcode=1930PhRv...36...57S}}</ref>
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| Revised values of screening constants based on computations of atomic structure by the [[Hartree-Fock method]] were obtained by [[Enrico Clementi]] ''et al'' in the 1960s.<ref>{{cite journal|last=Clementi|first=E.|coauthors=Raimondi, D. L.|title=Atomic Screening Constants from SCF Functions|journal=J. Chem. Phys|year=1963|volume=38|issue=11|pages=2686–2689|doi=10.1063/1.1733573|bibcode=1963JChPh..38.2686C}}</ref><ref name="clem67">{{cite journal|last=Clementi|first=E.|coauthors=Raimondi, D. L.; Reinhardt, W. P.|title=Atomic Screening Constants from SCF Functions. II. Atoms with 37 to 86 Electrons|journal=Journal of Chemical Physics|year=1967|volume=47|pages=1300–1307|doi=10.1063/1.1712084|issue=4|bibcode=1967JChPh..47.1300C}}</ref>
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| ==Rules==
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| Firstly,<ref name="slater57"/><ref>{{cite book|last=Miessler|first=Gary L.|coauthors=Tarr, Donald A.|title=Inorganic Chemistry|publisher=Prentice Hall|year=2003|pages=38|isbn=978-0-13-035471-6}}</ref> the electrons are arranged into a sequence of groups in order of increasing [[principal quantum number]] n, and for equal n in order of increasing [[azimuthal quantum number]] l, except that s- and p- orbitals are kept together.
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| :[1s] [2s,2p] [3s,3p] [3d] [4s,4p] [4d] [4f] [5s, 5p] [5d] etc.
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| Each group is given a different shielding constant which depends upon the number and types of electrons in those groups preceding it.
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| The shielding constant for each group is formed as the ''sum'' of the following contributions:
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| #An amount of 0.35 from each ''other'' electron within the ''same'' group except for the [1s] group, where the other electron contributes only 0.30.
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| #If the group is of the [s p] type, an amount of 0.85 from each electron with principal quantum number (n) one less and an amount of 1.00 for each electron with an even smaller principal quantum number
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| #If the group is of the [d] or [f], type, an amount of 1.00 for each electron inside it. This includes i) electrons with a smaller principal quantum number and ii) electrons with an equal principal quantum number and a smaller [[azimuthal quantum number]] (''l'')
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| In tabular form, the rules are summarized as:
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| {| class="wikitable"
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| |-
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| ! Group
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| ! Other electrons in the same group
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| ! Electrons in group(s) with [[principal quantum number]] n and [[azimuthal quantum number]] < ''l''
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| ! Electrons in group(s) with [[principal quantum number]] n-1
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| ! Electrons in all group(s) with [[principal quantum number]] < n-1
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| |-
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| | [1s]
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| | 0.30
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| | N/A
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| | N/A
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| | N/A
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| |-
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| | [''n''s,''n''p]
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| | 0.35
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| | N/A
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| | 0.85
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| | 1
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| |-
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| | [''n''d] or [''n''f]
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| | 0.35
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| | 1
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| | 1
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| | 1
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| |-
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| |}
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| ==Example==
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| An example provided in Slater's original paper is for the [[iron]] atom which has nuclear charge 26 and electronic configuration 1s<sup>2</sub>2s<sup>2</sub>2p<sup>6</sub>3s<sup>2</sub>3p<sup>6</sub>3d<sup>6</sub>4s<sup>2</sub>. The screening constant, and subsequently the shielded (or effective) nuclear charge for each electron is deduced as:<ref name="slater57"/>
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| :<math>
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| \begin{matrix}
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| 4s &: 0.35 \times 1& + &0.85 \times 14 &+& 1.00 \times 10 &=& 22.25 &\Rightarrow& Z_{\mathrm{eff}}(4s)=3.75\\
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| 3d &: 0.35 \times 5& & &+& 1.00 \times 18 &=& 19.75 &\Rightarrow& Z_{\mathrm{eff}}(3d)=6.25\\
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| 3s,3p &: 0.35 \times 7& + &0.85 \times 8 &+& 1.00 \times 2 &=& 11.25 &\Rightarrow& Z_{\mathrm{eff}}(3s,3p)=14.75\\
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| 2s,2p &: 0.35 \times 7& + &0.85 \times 2 & & &=& 4.15 &\Rightarrow& Z_{\mathrm{eff}}(2s,2p)=21.85\\
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| 1s &: 0.30 \times 1& & & & &=& 0.30 &\Rightarrow& Z_{\mathrm{eff}}(1s)=25.7
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| \end{matrix}
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| </math>
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| Note that the effective nuclear charge is calculated by subtracting the screening constant from the corresponding atomic number.
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| ==Motivation==
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| The rules were developed by John C. Slater in an attempt to construct simple analytic expressions for the [[atomic orbital]] of any electron in an atom. Specifically, for each electron in an atom, Slater wished to determine shielding constants (''s'') and "effective" quantum numbers (''n''*) such that
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| :<math>\psi_{n^{*}s}(r) = r^{n^{*}-1}\exp\left(-\frac{(Z-s)r}{n^{*}}\right)</math>
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| provides a reasonable approximation to a single-electron wave function. Slater defined ''n''* by the rule that for n = 1, 2, 3, 4, 5, 6 respectively; ''n''* = 1, 2, 3, 3.7, 4.0 and 4.2. This was an arbitrary adjustment to fit calculated atomic energies to experimental data.
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| Such a form was inspired by the known wave function spectrum of [[hydrogen-like atom]]s which have the radial component
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| :<math>R_{nl}(r) = r^{l}f_{nl}(r)\exp\left(-\frac{Zr}{n}\right),</math> | |
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| where ''n'' is the (true) [[principal quantum number]], ''l'' the [[azimuthal quantum number]], and ''f''<sub>''nl''</sub>(''r'') is an oscillatory polynomial with ''n'' - ''l'' - 1 nodes.<ref>{{cite book|last=Robinett|first=Richard W.|title=Quantum Mechanics Classical Results, Modern Systems, and Visualized Examples|publisher=Oxford University Press|location=New York|year=2006|pages=503|isbn=978-0-13-120198-9}}</ref> Slater argued on the basis of previous calculations by [[Clarence Zener]]<ref>{{cite journal|last=Zener|first=Clarence|year=1930|title=Analytic Atomic Wave Functions|journal=Phys. Rev.
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| |volume=36|issue=1|pages=51–56|doi=10.1103/PhysRev.36.51|bibcode = 1930PhRv...36...51Z }}</ref> that the presence of radial nodes was not required to obtain a reasonable approximation. He also noted that in the asymptotic limit (far away from the nucleus), his approximate form coincides with the exact hydrogen-like wave function in the presence of a nuclear charge of ''Z''-''s'' and in the state with a principal quantum number n equal to his effective quantum number ''n''*.
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| Slater then argued, again based on the work of Zener, that the total energy of a ''N''-electron atom with a wavefunction constructed from orbitals of his form should be well approximated as
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| :<math>E = -\sum_{i=1}^{N}\left(\frac{Z-s_{i}}{n^{*}_{i}}\right)^{2}.</math>
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| Using this expression for the total energy of an atom (or ion) as a function of the shielding constants and effective quantum numbers, Slater was able to compose rules such that spectral energies calculated agree reasonably well with experimental values for a wide range of atoms. Using the values in the iron example above, the total energy of a neutral iron atom using this method is -2497.2 [[Rydberg constant|Ry]], while the energy of an iron cation lacking a single 1s electron is -1964.6 Ry. The difference, 532.6 Ry, can be compared to the experimental (circa 1930) [[K-edge|K absorption limit]] of 524.0 Ry.<ref name="slater57"/>
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| ==References==
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| {{reflist}}
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| [[Category:Atomic physics]]
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| [[Category:Chemical bonding]]
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| [[Category:Quantum chemistry]]
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Friends contact her Claude Gulledge. Bookkeeping is what she does. To perform croquet is some thing that I've done for years. Delaware is our birth location.
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