# Effective nuclear charge

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Effective Nuclear Charge Diagram

The effective nuclear charge (often symbolized as $Z_{\mbox{eff}}$ or Z*) is the net positive charge experienced by an electron in a multi-electron atom. The term "effective" is used because the shielding effect of negatively charged electrons prevents higher orbital electrons from experiencing the full nuclear charge by the repelling effect of inner-layer electrons. The effective nuclear charge experienced by the outer shell electron is also called the core charge. It is possible to determine the strength of the nuclear charge by looking at the oxidation number of the atom.

## Calculating the effective nuclear charge

In an atom with one electron, that electron experiences the full charge of the positive nucleus. In this case, the effective nuclear charge can be calculated from Coulomb's law.

However, in an atom with many electrons the outer electrons are simultaneously attracted to the positive nucleus and repelled by the negatively charged electrons. The effective nuclear charge on such an electron is given by the following equation:

$Z_{eff} = Z - S$

where

Z is the number of protons in the nucleus (atomic number), and
S is the average number of electrons between the nucleus and the electron in question (the number of nonvalence electrons).

S can be found by the systematic application of various rule sets, the simplest of which is known as "Slater's rules" (named after John C. Slater). Douglas Hartree defined the effective Z of a Hartree–Fock orbital to be:

$Z_{eff} = \frac{_H}{_Z}$

where

<r>H is the mean radius of the orbital for hydrogen, and
<r>Z is the mean radius of the orbital for an electron configuration with nuclear charge Z.

## Example

Consider a sodium cation, a fluorine anion, and a neutral neon atom. Each has 10 electrons, and the number of nonvalence electrons is 2 (10 total electrons - 8 valence) but the effective nuclear charge varies because each has a different atomic number:

$Z_{eff}(F^-) = 9 - 2 = 7+$
$Z_{eff}(Ne) = 10 - 2 = 8+$
$Z_{eff}(Na^+) = 11 - 2 = 9+$

So, the sodium cation has the largest effective nuclear charge, and thus the smallest atomic radius.

## Values

Updated values of screening constants were provided by Clementi et al.[1][2]

 2p 33.039 34.030 35.003 35.993 36.982 37.972 38.941 39.951 40.940 41.930 42.919 43.909 44.898 45.885 46.873 47.86 48.847 49.835 3s 21.843 22.664 23.552 24.362 25.172 25.982 26.792 27.601 28.439 29.221 30.031 30.841 31.631 32.42 33.209 33.998 34.787 35.576 3p 21.303 22.168 23.093 23.846 24.616 25.474 26.384 27.221 28.154 29.020 29.809 30.692 31.521 32.353 33.184 34.009 34.841 35.668 4s 12.388 13.444 14.264 14.902 15.283 16.096 17.198 17.656 18.582 18.986 19.865 20.869 21.761 22.658 23.544 24.408 25.297 26.173 3d 21.679 22.726 25.397 25.567 26.247 27.228 28.353 29.359 30.405 31.451 32.540 33.607 34.678 35.742 36.8 37.839 38.901 39.947 4p 10.881 11.932 12.746 13.460 14.084 14.977 15.811 16.435 17.140 17.723 18.562 19.411 20.369 21.265 22.181 23.122 24.03 24.957 5s 4.985 6.071 6.256 6.446 5.921 6.106 7.227 6.485 6.640 (empty) 6.756 8.192 9.512 10.629 11.617 12.538 13.404 14.218 4d 15.958 13.072 11.238 11.392 12.882 12.813 13.442 13.618 14.763 15.877 16.942 17.97 18.974 19.96 20.934 21.893 5p 8.47 9.102 9.995 10.809 11.612 12.425