Effective Nuclear Charge Calculator — Slater's rules, term by term

The effective nuclear charge calculator above applies Slater's rules to work out how much of the nuclear charge an individual electron actually experiences once the other electrons in the atom have screened part of it. It builds the electron configuration for the element and charge you specify, sorts those electrons into Slater's groups, applies the screening coefficients group by group, and shows the resulting shielding constant broken down so you can see where each contribution came from.

Arb Digital publishes free calculators for the steps that are conceptually simple and fiddly in practice. Slater's rules are exactly that. The arithmetic is addition and one subtraction, but the grouping convention catches people out constantly, particularly the way 3d electrons form their own group and the way a d electron is screened differently from an s or p electron in the same shell.

What This Effective Nuclear Charge Calculator Does

It returns Z_eff, the effective nuclear charge, for whichever electron you nominate. The supporting figures are the shielding constant S that produced it, the total electron count of the species, Slater's effective principal quantum number n* for the shell in question, and an estimated orbital energy computed from the hydrogen-like expression using Z_eff and n*. The bar breakdown underneath shows how much each Slater group contributed to S, which is where the intuition actually lives.

Ions are handled by adding or removing electrons from the highest occupied group. That matters for questions like why the second ionisation energy of an element is so much larger than the first, or why a cation is smaller than its parent atom: removing an electron reduces shielding, so every remaining electron feels a stronger pull.

A boundary worth stating. The live electron configuration calculator gives the ground-state configuration, orbital diagram, valence count and block for an element, including the measured exceptions to aufbau filling. This page takes a configuration as a means to an end and never reports it as the product; what it produces is a screening calculation. If you want the configuration itself, that page is the right one and this one will not add anything.

How to Use It

1. Pick the element from the list, or type an atomic number directly if you prefer.
2. Set an ion charge if you are studying a cation or anion. Positive removes electrons, negative adds them.
3. Choose the group the electron sits in. The list rebuilds to show only the groups that are actually occupied for that species.
4. Read Z_eff from the hero, and the shielding constant that produced it from the grid.
5. Look at the bar breakdown to see which electrons did the screening. That is usually more instructive than the final number.

The Rules and How They Are Applied

Effective nuclear charge is defined simply as Z_eff = Z − S, where Z is the number of protons and S is the shielding constant. Slater's contribution was a set of empirical rules for estimating S. First the electrons are written in groups in this order: (1s) (2s,2p) (3s,3p) (3d) (4s,4p) (4d) (4f) (5s,5p) and so on, with s and p of the same shell sharing a group while d and f each get their own.

For an electron in an s or p group, electrons in groups further out contribute nothing, each other electron in the same group contributes 0.35 (or 0.30 if the group is 1s), each electron in the shell with principal quantum number one lower contributes 0.85, and each electron further in contributes 1.00. For an electron in a d or f group, other electrons in the same group still contribute 0.35, but every electron in any group to the left contributes a full 1.00, with no 0.85 tier at all.

Working the default example, potassium with Z = 19: the configuration gives 2 electrons in (1s), 8 in (2s,2p), 8 in (3s,3p) and 1 in (4s,4p). For that 4s electron there are no others in its own group, so 0. The n = 3 shell contributes 8 × 0.85 = 6.80. The ten electrons in n = 1 and n = 2 contribute 10 × 1.00 = 10.00. That gives S = 16.80 and Z_eff = 19 − 16.80 = 2.20. Compare that with the 3d electron of zinc, where the ten 3d electrons give 9 × 0.35 = 3.15, the eighteen electrons to the left give 18.00, the two 4s electrons give nothing, S = 21.15 and Z_eff = 8.85.

Why 4s Electrons Are So Poorly Held

The potassium result above is the whole story of the alkali metals in one number. Nineteen protons, and the outermost electron feels the pull of about 2.2 of them. That is why potassium ionises so easily, why it is so reactive, and why its atomic radius is enormous compared with the noble gas that precedes it. The screening is nearly complete because eighteen inner electrons sit between the nucleus and that lone outer electron.

Now walk across the fourth period. Calcium at Z = 20 has two 4s electrons, so the second one screens the first by only 0.35, and Z_eff for a 4s electron rises to 2.85. Every step to the right adds a proton but adds an electron that screens by only 0.35 in the same group, so Z_eff climbs by 0.65 per element. That steady climb is the reason atomic radius contracts across a period while the shell number stays the same, and it is the reason ionisation energy rises across a period.

Going down a group the opposite happens. The added shell puts the valence electron much further out and the new inner shell screens almost fully, so Z_eff barely changes while the distance grows. Radius increases, ionisation energy falls. Both of the classic periodic trends fall directly out of this arithmetic, which is why Slater's rules survive in teaching long after better methods existed.

What the Transition Metals Reveal

The transition series is where the grouping convention earns its keep. Across the first row, electrons are being added to 3d while the 4s electrons are the outermost. Those added 3d electrons screen a 4s electron by 0.85 each, nearly fully, so Z_eff at 4s rises only slowly across the series. That is why the transition metals resemble one another far more than the main-group elements of a period do, and why their atomic radii change so little from titanium to copper.

Meanwhile Z_eff for the 3d electrons themselves rises quickly, by 0.65 per element, because they screen each other by only 0.35. The 3d orbitals contract sharply across the row as a result. That contraction is behind the increasing difficulty of removing d electrons later in the series and behind the shift in preferred oxidation states.

The same effect at larger scale gives the lanthanide contraction. Fourteen 4f electrons screen poorly for anything outside them, so by the time you reach hafnium the accumulated increase in Z_eff has pulled the outer shells in enough that hafnium and zirconium have nearly identical radii despite a full extra shell. It is one of the most consequential facts in inorganic chemistry, and it is visible in this arithmetic.

Where Slater's Rules Break Down

These are fitted rules, published in 1930 and tuned to reproduce measured energies for the elements known well at the time. They were never a derivation from quantum mechanics, and they should not be expected to behave like one. Several specific limitations are worth knowing.

They do not distinguish s from p within a shell, so they cannot explain why 4s fills before 3d or why the 2s electron of an atom is more tightly bound than its 2p electron. They handle the heavier elements poorly, since the coefficients were fitted to light atoms and no relativistic effect is included, which becomes significant well before the sixth period. They give a single number per group with no angular dependence, so the shapes that drive real chemistry are absent. And the exceptions to aufbau filling, such as chromium and copper, are not represented in the standard configuration this page builds, so a Z_eff computed for those elements is based on the idealised filling rather than the measured ground state.

Refined screening constants published later fit spectroscopic data more closely, and modern computation replaces the whole approach. For measured ionisation energies and energy levels to check any estimate against, the NIST Atomic Spectra Database is the reference, and critically evaluated atomic property data is collected in the NIST periodic table of the elements.

Estimating an Orbital Energy From Z_eff

Slater paired the screening rules with a second idea: an effective principal quantum number n* that replaces n in the hydrogen-like energy expression. For n = 1, 2 and 3 it equals n, but for n = 4, 5 and 6 it takes the values 3.7, 4.0 and 4.2. The orbital energy is then approximately −13.6 × (Z_eff / n*)² electronvolts, and this page reports that figure in the grid.

Treat it as an order-of-magnitude guide rather than a prediction. For the potassium 4s electron the estimate is close to the measured first ionisation energy; for many other cases it is off by a substantial fraction. Its real value is comparative. If you want to know whether removing an electron from one species is easier or harder than from another, the two estimates side by side usually get the direction right even when neither magnitude is accurate. If you need photon energies for a spectroscopic comparison, the photon energy calculator converts wavelengths into electronvolts.

Common Mistakes to Avoid

• Putting 3d in the same group as 3s and 3p — Slater gives d and f their own groups, and this changes the answer substantially.
• Applying the 0.85 tier to a d electron — for d and f targets, every electron in a group to the left screens by a full 1.00.
• Counting the electron itself in its own group — only the other electrons in the group screen, so the count is one less.
• Using 0.35 inside the 1s group — the coefficient there is 0.30, a small special case that is easy to forget.
• Treating the result as a measurement — these are fitted empirical rules that reproduce trends, not spectroscopic values.

Related Free Tools From Arb Digital

For the ground-state configuration itself, including the aufbau exceptions, use the electron configuration calculator. The electronegativity calculator covers the other great periodic trend and the bond polarity that follows from it, the Coulomb's law calculator handles the electrostatic force this whole idea rests on, and the photon energy calculator connects energies to spectra. For mass rather than charge, see the average atomic mass calculator and the molar mass calculator. The full free online tools hub lists everything else.

Frequently Asked Questions

This calculator is provided for education and general reference. It applies a published empirical scheme to values you supply and does not replace spectroscopic data or computational chemistry results.