Effective Nuclear Charge Calculator (Z-eff via Slater's Rules)
Select an element (Z = 1 to 118) and the target orbital to calculate the effective nuclear charge experienced by one of its electrons. The calculator applies Slater's rules, shows every shielding contribution by electron group, and walks through the full worked solution. Results update instantly as you change inputs.
What is effective nuclear charge?
An electron in a multi-electron atom does not feel the full attraction of all Z protons in the nucleus. Electrons closer to the nucleus partially block, or shield, the nuclear charge felt by outer electrons. Effective nuclear charge (Z-eff) is the net positive charge that a specific electron actually experiences after all of this shielding is taken into account. It is calculated as Z-eff = Z - sigma, where Z is the atomic number and sigma is the shielding constant summed over all other electrons. A valence electron in sodium (Z=11) has a Z-eff of only about 2.2, meaning most of the nuclear charge is cancelled by the ten inner electrons. Z-eff governs atomic radius, ionization energy, electron affinity, and electronegativity: a higher Z-eff pulls electrons closer, shrinks the atom, and raises ionization energy.
How Slater's rules work
Slater's rules, published by John C. Slater in 1930, give a practical recipe for estimating sigma without solving the Schrodinger equation. First, write the ground-state electron configuration and arrange orbitals into Slater groups: [1s][2s,2p][3s,3p][3d][4s,4p][4d][4f][5s,5p][5d][5f][6s,6p][6d][7s,7p]. Electrons in groups to the right of the target electron contribute nothing (their shielding of an inner electron is zero). For an s or p target electron, each electron in the same group contributes 0.35 (except within 1s, where it is 0.30), each electron in the n-1 shell contributes 0.85, and each electron in shells n-2 and below contributes 1.00. For a d or f target electron, electrons in the same group still contribute 0.35, but all electrons in any lower group contribute 1.00, reflecting the poor penetration of d and f orbitals to the nucleus. Sum all contributions to get sigma, then subtract from Z to get Z-eff.
Periodic trends explained by Z-eff
Atomic radius decreases across a period because each successive element adds one proton and one valence electron. The additional proton raises Z, but the same-shell electron adds only 0.35 to sigma, so Z-eff increases by about 0.65 per step and draws the valence shell inward. Down a group, principal quantum number n increases, which more than offsets the growing Z-eff and makes atoms larger. Ionization energy follows Z-eff closely: the greater the Z-eff, the more energy is needed to remove the valence electron. The irregular dip in ionization energy between nitrogen and oxygen is not predicted by simple Z-eff, but arises from electron-electron repulsion within the half-filled 2p subshell, illustrating that Slater's rules are a useful approximation rather than an exact model.
Limitations and alternatives
Slater's rules use fixed, empirical shielding constants and group electrons into broad shells, so they cannot reproduce the fine structure seen in high-resolution spectroscopy or the variation of shielding within a single subshell. The model treats all electrons at n-1 identically, regardless of whether they are s, p, or d. For more accurate values, the self-consistent field (SCF) approach of Clementi and Raimondi (1963) fitted shielding constants from Hartree-Fock calculations orbital by orbital, and produces Z-eff values that match measured ionization energies much more closely. Despite its simplicity, Slater's model is widely taught because it provides chemically useful insights with nothing more than an element's atomic number and a single table of coefficients.
Slater shielding constants at a glance
| Target orbital | Electrons in same Slater group | Electrons in shell n-1 | Electrons in shells n-2 and below |
|---|---|---|---|
| 1s | 0.30 | - | - |
| ns, np (n >= 2) | 0.35 | 0.85 | 1.00 |
| nd | 0.35 (same group) | 1.00 (all inner) | 1.00 (all inner) |
| nf | 0.35 (same group) | 1.00 (all inner) | 1.00 (all inner) |
Shielding factor contributed by each electron group toward the target electron, as defined by Slater's rules (1930).
Frequently asked questions
What is the effective nuclear charge formula?
The formula is Z-eff = Z - sigma, where Z is the atomic number (number of protons) and sigma is the shielding constant calculated by summing the contributions of all other electrons using Slater's rules. For example, for the 2p electron in oxygen (Z=8): sigma = 5 x 0.35 + 2 x 0.85 = 1.75 + 1.70 = 3.45, so Z-eff = 8 - 3.45 = 4.55.
Why is the shielding constant for the 1s orbital 0.30 and not 0.35?
The 1s orbital holds at most two electrons, and the two electrons repel each other significantly because they occupy the same compact orbital very close to the nucleus. Slater found empirically that the actual shielding each 1s electron provides to the other is slightly less than 0.35, and 0.30 better matches experimental data. For all other orbitals the same-group coefficient remains 0.35.
How does Z-eff change across period 2 (Li to Ne)?
Lithium (Z=3) has Z-eff 1.30 for its 2s electron; neon (Z=10) has Z-eff 5.85 for its 2p electrons. Each step across the period adds one proton (+1 to Z) and one same-shell electron (+0.35 to sigma), giving a net increase of 0.65 in Z-eff per element. This is why atomic radius decreases and ionization energy generally increases from left to right across a period.
Why do d and f electrons shield less effectively than s and p electrons?
Orbitals with higher angular momentum quantum number l have a node at the nucleus and their radial probability density is concentrated further from the nucleus. This poor penetration means d and f electrons are rarely found between the nucleus and the outer electrons, so they barely reduce the nuclear attraction that outer electrons feel. Slater's rules encode this by using a factor of 1.00 for all electrons inner to a d or f target, while also assigning the outermost s and p electrons (n = N-1) a factor of 0.85 rather than 1.00, because they still have some penetration.
What is the effective nuclear charge for sodium (Na)?
Sodium has Z = 11 and configuration 1s2 2s2 2p6 3s1. For the 3s valence electron: 8 electrons at n = 2 contribute 8 x 0.85 = 6.80, and 2 electrons at n = 1 contribute 2 x 1.00 = 2.00, giving sigma = 8.80 and Z-eff = 11 - 8.80 = 2.20. The single outermost electron therefore feels only about 2.2 out of 11 nuclear protons, which explains why sodium ionizes very easily.
How accurate are Slater's rules compared to real quantum calculations?
Slater's rules give Z-eff values that are accurate enough for qualitative comparisons and for explaining periodic trends. For quantitative predictions, SCF calculations by Clementi and Raimondi (1963) are much more accurate. For example, Slater gives Z-eff 3.45 for carbon 2s and 3.22 for carbon 2p, while Clementi-Raimondi give 5.67 and 3.14 respectively - the difference arises because Slater groups 2s and 2p together, but their actual penetration to the nucleus is quite different.
What is the difference between shielding and penetration?
Shielding (or screening) describes how inner electrons reduce the nuclear charge felt by an outer electron. Penetration describes how well an outer orbital manages to reach close to the nucleus despite the inner electrons in its path. An orbital with high penetration (like 2s versus 2p) spends more time close to the nucleus and therefore feels a higher effective nuclear charge, as well as shielding outer electrons more effectively. The two concepts are closely related: high-penetration orbitals are hard to shield.