D4.1 Effective Nuclear Charge

Periodic trends in atomic properties can be predicted by applying these ideas about electron-nucleus attraction and electron-electron repulsion:

• Electron-density distributions are in shells that increase in size as the principal quantum number, n, increases. Electrons in larger shells are, on average, farther from the nucleus and less strongly attracted.
• Electrons repel other electrons, raising Coulombic potential energy. This partly counteracts the attractive force between an electron and the nucleus. Electrons are said to screen or shield other electrons from nuclear charge.

The electron density of a core electron (an electron in an inner shell) is, on average, closer to the nucleus than the electron density of a valence electron. Thus, core electrons can significantly counteract the effect of nuclear attraction. Consider a lithium atom (Li, 1s²2s¹), which has three protons in the nucleus. Because the 2s orbital is larger than the 1s orbital, the 1s electron density is mostly located between the nucleus and the 2s electron density. (Move the slider in the middle of the figure below to see how much of the 1s electron density lies between the nucleus and 2s electron density.) Thus, the two 1s electrons repel the 2s electron away from the nucleus, counteracting part of the 3+ charge of the nucleus.

To account for such electron-electron repulsions, we use effective nuclear charge, Z_eff, the positive nuclear charge (given by the atomic number) reduced by the repulsion of a specific electron by all the other electrons. In the case of the Li 2s electron, quantum mechanics calculate that the repulsions from the two 1s electrons reduce the nuclear charge by 1.72; that is, Z_eff for the 2s electron is 3 − 1.72 = 1.28. If all the electron density of the 1s electrons were between the nucleus and the 2s electron, Z_eff would be reduced to 1.