Effective Nuclear Charge Calculator

Effective nuclear charge (Zeff) is the net positive charge experienced by a particular electron in a multi-electron atom, after accounting for the shielding (screening) effect of the other electrons. It is calculated as Zeff = Z - S, where Z is the actual atomic number and S is the shielding constant estimated from Slater's rules.

Slater's rules, developed by physicist John C. Slater in 1930, are a set of empirical rules for estimating the shielding constant S experienced by an electron in a multi-electron atom. Electrons are grouped into shielding groups (1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)... and each group of electrons contributes a different amount of shielding depending on its position relative to the electron being examined.

Electrons occupying the same subshell group as the electron of interest are, on average, at a similar distance from the nucleus, so they only partially block the nuclear charge from reaching each other. Slater's rules assign 0.35 per same-group electron (0.30 specifically for the 1s group, since there is no lower shell to compare against).

Electrons in the shell immediately below the target electron (n-1) spend most of their time closer to the nucleus and are quite effective, but not perfectly effective, at blocking nuclear charge from the target electron, since there is still some orbital overlap. Slater's rules assign 0.85 per electron in this n-1 group.

Electrons in shells two or more levels below the target electron (n-2 and lower) are, on average, so much closer to the nucleus that they effectively block the full charge of one proton each from reaching the target electron, so Slater's rules assign them a full shielding factor of 1.00 per electron.

For a target electron in an nd or nf subshell, Slater's rules treat every electron in any lower (more inner) group, including the s and p electrons of the same principal shell, as contributing a full 1.00 shielding factor, while electrons in outer groups contribute nothing. This reflects that d and f orbitals penetrate less toward the nucleus than s and p orbitals of the same shell.

Moving left to right across a period, each new electron is added to the same outer shell (the same Slater shielding group), where it only shields the other valence electrons by 0.35 each, while the nuclear charge Z increases by a full unit each time. The net effect is that Zeff for the valence electron increases fairly steadily across the period, which is why atoms get smaller left to right.

Moving down a group, each new period adds a complete new shell of electrons, and the underlying core electrons shield the new valence electron almost as effectively (0.85 or 1.00 per electron) as the nuclear charge increases, so Zeff for the outermost electron changes only modestly down a group. This is why the atomic radius increases fairly smoothly down a group instead of shrinking.

No. Slater's rules are a simplified empirical approximation, not an exact quantum mechanical calculation. They give reasonably accurate Zeff estimates for teaching periodic trends, but more precise effective nuclear charge values come from self-consistent field (Hartree-Fock or later) calculations, which account for orbital shapes and penetration more rigorously.

Nuclear charge (Z) is simply the number of protons in the nucleus, the same for every electron in the atom. Effective nuclear charge (Zeff) is smaller than Z and is specific to each individual electron, since it accounts for how much the other electrons in the atom shield that particular electron from the full nuclear charge.

A higher effective nuclear charge on the valence electron means that electron is held more tightly, requiring more energy to remove, which raises ionization energy. This is why ionization energy generally increases across a period (as Zeff rises) and decreases down a group (as the valence electron moves further from the nucleus despite roughly constant Zeff).