Effective nuclear charge refers to the charge felt by the outermost (valence) electrons of a multi-electron atom after taking into account the number of shielding electrons that surround the nucleus. The formula for calculating the effective nuclear charge for a single electron is "Zeff = Z - S", where Zeff is the effective nuclear charge, Z is the number of protons in the nucleus, and S is the average amount of electron density between the nucleus and the electron for which you are solving.
As an example, you can use this formula to find the effective nuclear charge for an electron in lithium, specifically the "2s" electron.
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The calculation for effective nuclear charge is Zeff = Z - S. Zeff is the effective charge, Z is the atomic number, and S is the charge value from Slater's Rules.
Determine the value of Z. Z is the number of protons in the nucleus of the atom, which determines the nucleus's positive charge. The number of protons in the nucleus of an atom is also known as the atomic number, which can be found on the periodic table of elements.
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In the example, the value of Z for lithium is 3.
Find the value of S by using Slater's Rules, which provide numerical values for the effective nuclear charge concept. This can be accomplished by writing out the electron configuration of the element in the following order and groupings: (1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d), (4f), (5s, 5p), (5d), (5f), etc. The numbers in this configuration correspond to the shell level of the electrons in the atom (how far away the electrons are from the nucleus) and the letters correspond to the given shape of an electron's orbit. In simplified terms, "s" is a spherical orbital shape, "p" resembles a figure 8 with two lobes, "d" resembles a figure 8 with a doughnut around the center, and "f" resembles two figure 8s that bisect each other.
In the example, lithium has three electrons and the electron configuration looks like this: (1s)2, (2s)1, meaning there are two electrons on the first shell level, both with spherical orbital shapes, and one electron (the focus of this example) on the second shell level, also with a spherical shape.
Assign a value to the electrons according to their shell level and orbital shape. Electrons in an "s" or "p" orbit in the same shell as the electron for which you're solving contribute 0.35, electrons in an "s" or "p" orbital in the shell one energy level lower contribute 0.85, and electrons in an "s" or "p" orbital in shells two energy levels and lower contribute 1. Electrons in a "d" or "f" orbital in the same shell as the electron for which you're calculating contribute 0.35, and electrons in an "d" or "f" orbital in all lower energy levels contribute 1. Electrons in shells higher than the electron for which you're solving do not contribute to shielding.
In the example, there are two electrons in the shell that is one energy level lower than the shell of the electron for which you're solving, and they both have "s" orbitals. According to Slater's Rules, these two electrons each contribute 0.85. Do not include the value for the electron for which you are solving.
Calculate the value of S by adding together the numbers you assigned to each electron using Slater's Rules.
For our example, S equals .85 + .85, or 1.7 (the sum of the values of the two electrons we're counting)
Subtract S from Z to find the effective nuclear charge, Zeff.
In the example using a lithium atom, Z equals 3 (the atomic number of lithium) and S equals 1.7. By changing the variables in the formula to the correct values for the example, it becomes Zeff = 3 - 1.7. The value of Zeff (and thus the effective nuclear charge of the 2s electron in a lithium atom) is 1.3.