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The farther an electron moves away from the nucleus of an atom, the weaker their attraction is: discover why with our effective nuclear charge calculator.

Here you will learn what the effective nuclear charge is and how to calculate it using Slater's rules. You will see some examples and get a quick review of the quantum theory behind atoms — and finally, you will learn how to use our Slater's rules calculator. Ready?

A quick review of the nuclear structure

We need to take a quick look at the nuclear structure to understand what electron shielding is and how to calculate the effective nuclear charge.

Here are the basics of the atomic orbital model first! An atom is composed by:

A positively charged nucleus made of protons and neutrons. The number of protons defines the atomic number, uniquely identifying a chemical element.
A negatively charged electronic cloud. Each electron can be found in a set of defined regions of space around the nucleus called orbitals.

🙋 Quantum physicist here: orbitals are solutions to the Schrodinger equation, which describes the position of an electron in space and time — but remember, electrons are neither particles nor waves! Wavefunctions are complex quantities (in mathematical language) and bear no physical meaning: we need to take their squared modulus that, according to the rules of quantum mechanics, is proportional to the probability of finding an electron in a given set of coordinates.

An orbital is described by a set of discrete integer numbers called the quantum numbers. That's why we speak of quantized — "quanta" is a Latin word for "discrete quantity". Let's discover them:

The principal quantum number, $n$ , which gives an indication on the distance of the electron from the nucleus. The smaller the number, the closer the electron. The value for $n$ can be any integer, positive value:
$n = 0,1,2,3,\ldots$ .
The azimuthal quantum number, $l$ , which describes the shape of the region where it is possible to find the electron. Its values are related to the value of $n$ , being the integer numbers from $0$ to $n-1$ :
$l=0,1,\ldots,n-1$ .
The magnetic quantum number, $m$ , which is associated with the orientation of the orbitals in space. It varies according to the value of $l$ :
$m= -l,\ -l+1,\ \ldots,\ l-1,\ l$

Electrons with equal $n$ and $l$ but different values of $m$ have identical energy: we call the respective orbitals degenerate.

Each orbital is defined by a unique set of quantum numbers and can host at most two electrons, one for each value of spin — another quantum property that can assume one of two values, namely spin up or spin down.

We need to take a closer look at the various orbitals to understand how to calculate the effective nuclear charge. Let's proceed in order with the quantum numbers, starting with the electronic shell closest to the nucleus.

For $n=1$ , the other quantum numbers are $l=0$ and $m=0$ . There is a single orbital, called $1s$ , with a spherical shape.
For $n=2$ there are two possible values for $l$ : $l=0$ and $l=1$ .
- For $l=0$ , $m=0$ too, and we get another spherical orbital, $2s$ .
- For $l=1$ there are three possible values of $m$ : $m=-1,0,+1$ . We call these orbitals $2p$ , and to distinguish the three orientations we add a coordinate to them: $2p_x$ , $2p_y$ , and $2p_z$ . Instead of spherical, they have a dumbbell shape.
For $n=3$ we have the same orbitals we've just met, with the addition of the ones associated to $l=2$ , for which $m$ can assume five different values: $m=-2,-1,0,1,2$ . These are the $3d$ orbitals, and they assume various shapes. Our favorite of these shapes is a donut with a pear on each side.
For $n=4$ we add another set of orbitals, the $4f$ . They are described by the set of quantum numbers $n=4$ , $l=3$ , and $m=-3,-2,-1,0,1,2,3$ — for a total of seven orbitals. Their shapes are even more complex than the one of the $d$ orbitals.

Going up with the value of $n$ , we would meet even more complex orbitals, like $g$ , but they don't appear in the elements we know at this time: talking of them would be meaningless!

Shape of orbitals — The shape of some of the atomic orbitals. In the first line, the $1s$ orbital and its spherical shape. Notice also the small size compared to the others: it is in the vicinity of the nucleus. In the second row, there are the $2p$ orbitals, with their three orientations. Finally, in the bottom row, you can see the five shapes of the $3d$ orbitals.

The electron configuration

It is possible to identify each element using its electron configuration. This is a way to specify the occupation of the orbitals, progressively filling the periodic table.

Writing the electron configuration for an element is relatively simple — it gets challenging only for heavier elements that start to misbehave. Let's take a look at the configuration for hydrogen:

\text{H}=1s^1

Hydrogen above has a single electron in the first shell. On the other hand, helium has a full first shell with two electrons. Its configuration is:

\text{He}=1 s^{2}

It goes on like this, following the progression of the orbitals dictated by the following scheme:

Order of occupation in orbitas — Visual representation of the occupation order of orbitals.

Follow the blue arrow from top to bottom, writing down the orbitals in the orders they are crossed. Remember the number of electrons hosted in each shell:

$s\rightarrow 2$
$p\rightarrow 6$
$d \rightarrow 10$
$f \rightarrow 14$

Let's try with a heavier element, tellurium. Its atomic number is 52, which is also the number of electrons we need to fit in the configuration.

\begin{gather*} \footnotesize 1s^2,\ 2s^2,\ 2p^6,\ 3s^2,\ 3p^6,\ 4s^2, \\ \footnotesize 3d^{10},\ 4p^6,\ 5s^2,\ 4d^10,\ 5p^4 \end{gather*}

After you filled the electron configuration following the graphic rule, you can rewrite it in the most intuitive order:

\begin{gather*}\footnotesize 1s^2,\ 2s^2,\ 2p^6,\ 3s^2,\ 3p^6,\ 3d^{10},\\ \footnotesize 4s^2,\ 4p^6,\ 4d^{10},\ 5s^2,\ 5p^4 \end{gather*}

That's how they are commonly found in textbooks, online, and also on our effective charge calculator.

🔎 You can see that the electron configuration of heavy elements gets a little cumbersome. Chemists found a way to make it easier to write and remember! You can use as reference the electron configuration of noble gases and start writing the configuration from the last one in the periodic table, indicating it with the symbol of the element in square brackets. The electron configuration of the Tellurium would be
$\text{Te}=\left[\text{Kr}\right],\ 4d^{10},\ 5s^2,\ 5p^4$ . Easier, isn't it?

What is electron shielding?

Electrons feel the attraction of the nucleus since they have opposite charges. However, only a single electron would experience the attractive force in its entirety. For every added electron sharing the same orbital or occupying lower energy orbitals, the negative charge of those particles adds a repulsive component, which contributes to the shielding of the nucleus' electrostatic interaction.

🔎 The methods explained here are approximations that don't take into account the position of the electrons and other factors. However, they fit the observed data.

What is the effective nuclear charge?

We need to understand first what is the nuclear charge. It is, straightforwardly, the charge of the nucleus in units of elementary charge, the charge of electrons and protons (with opposite sign). The nuclear charge $Z$ thus coincides with the atomic number. Without electrons, that would be the potential energy centered in the nucleus.

When we consider the repulsive interaction of other electrons, however, we see that the farther we get from the nucleus, the lower the charge felt by an electron. We need to talk of effective nuclear charge. We denote it by $Z_\text{eff}$ .

For the first electron around the nucleus, the effective nuclear charge equals the nuclear charge: $Z_\text{eff} = Z$ .

The value of $Z_\text{eff}$ then decreases approaching $1$ for an infinite distance from the nucleus. This is the value of the potential energy experienced by the last electron added to the shell.

The effective nuclear charge has some distinctive trends across the periodic table. It increases following the groups from left to right, decreasing descending into the periods. The ratio ${Z_\text{eff}}/{Z}$ is smaller for the elements in the first group, decreasing for heavier elements (where the larger amount of electrons has a more substantial shielding effect).

How to calculate the effective nuclear charge: What are Slater's rules?

Now that you know what the effective nuclear charge is, it's time to learn how to calculate it. To do this, allow us to introduce Slater's rules. The concept at the core is that to calculate the effective nuclear charge we need to compute the overall contribution of the shielding electrons.

Slater's rules need the complete electron configuration of an element to be applied. We then choose an electron belonging to a specific orbital. At this point, we have a few different paths that tell us the shielding contribution of each electron in the configuration.

The electrons in orbitals to the right of the chosen one give a zero contribution to the shielding.
If you chose an electron from a $p$ or an $s$ orbital, with principal quantum number $n=N$ , then:
- Electrons from orbitals with the same principal quantum number have a shielding factor of $0.35$ apart from the electrons in $1s$ , which shield $0.30$ .
- Electrons from orbitals with $n=N-1$ shield $0.85$ .
- Electrons coming from orbitals with $n=N-2$ or less shield $1.00$ as they are close to the nucleus.
If you chose an electron from an orbital with $l$ associated to $d$ or $f$ , and again $n=N$ , then:
- Electrons from orbitals with $n=N$ , and equal or bigger $l$ (following $s<p<d<f$ ) shield $0.35$ .
- Electrons from orbitals with $n+N$ but smaller $l$ than the one of the chosen electron shield $1.00$ .
- All other electrons from orbitals with $n<N$ shield $1.00$ .

💡 When choosing the electron, it's not essential to specify which position in the orbital we are considering: the shielding effect is not affected by degeneracy.

How do we calculate the effective nuclear charge, then? Each shielding factor is multiplied by the number of electrons in the related orbitals, remembering to subtract one when it comes to the orbital to which the chosen electron belongs. The resulting contributions are summed.

The resulting shielding is called $\sigma$ and is used to calculate $Z_\text{eff}$ through:

Z_\text{eff}=Z-\sigma

🙋 There is an exception! Hydrogen, having a single electron, has effective nuclear charge equal to the nuclear charge — that poor electron can't shield itself!

Example of how to calculate the effective nuclear charge

Let's choose an element and an orbital. Selenium and $3p$ , you say? That's what we were thinking too!

This is the electron configuration of selenium, with the chosen orbital highlighted:

\footnotesize 1s^{2}, 2s^{2}, 2p^{6}, 3s^{2}, \textbf{3p}^\textbf{6}, 3d^{10}, 4s^{2}, 4p^{4}

Ignore the orbitals to the right!

\footnotesize 1s^2, 2s^2, 2p^6, 3s^2, \textbf{3p}^{\textbf{6}}, \sout{3d^{10}},\sout{4s^2}, \sout{4p^4}

We chose a $p$ electron, and it's time to apply the appropriate Slater's rules. There are $7$ other electrons in the same group, with $n=3$ ; each contributes with $0.35$ . In the orbitals with $n=2$ , $2s$ , and $2p$ , there are $8$ electrons, which contribute individually with $0.85$ . Lastly, with $n=3-2=1$ , we have the two electrons in $1s$ , which contribute with a full $1.00$ each.

Let's sum up the various contributions then:

\begin{align*} \small \sigma & =7\cdot 0.35+8\cdot 0.85+2\cdot 1.00 \\ & =11.25 \end{align*}

Selenium has a nuclear charge of $Z=34$ . The effective nuclear charge is obtained by subtracting from $Z$ the value of shielding $\sigma$ :

\small Z_\text{eff} = 34-11.25=23.75

How to use our effective nuclear charge calculator?

First thing, choose an element from the list. They are in order of increasing atomic number. Its electron configuration will appear just below. Choose the desired electron from the configuration and input the appropriate quantum numbers.

✅ You can't get things wrong with this calculator. If the chosen principal quantum number or the azimuthal quantum number is not available in the electron configuration, we will stop you until you insert a correct set!

At the bottom of the calculator, you will find the values of the total shielding and of the effective nuclear charge $Z_\text{eff}$ .

A final review!

This is all we needed to say about this topic. Now you should know how to calculate $Z_\text{eff}$ without any difficulty, but here is a quick refresher:

Choose an electron from the electron configuration.
Apply Slater's rules to calculate the total shielding.
Remember to ignore electrons from higher orbitals.
To calculate the effective nuclear charge, subtract the shielding from the nuclear charge.

That's it! If you want to learn more about atomic properties, check out our electronegativity calculator, the radioactive decay calculator, or visit the chemistry page at omnicalculator.com!

FAQ

What are Slater's rules?

Slater's rules are a set of rules used in physical chemistry to calculate the effective nuclear charge experienced by an electron around a nucleus. The rules assign a specific value of shielding to each electron according to its orbital.

Check out the Slater's rule calculator on omnicalculator.com to discover more about it!

How do I calculate the effective nuclear charge?

To calculate the effective nuclear charge:

First, compute the overall shielding effect of the electrons orbiting the nucleus.
Subtract this value from the nuclear charge (equal to the number of protons of the element).
Remember that the value of the effective nuclear charge depends on the orbital you are using in the calculations, since outer electrons don't contribute to the shielding!

What is the trend of the effective nuclear charge?

The effective nuclear charge experienced by the last electron in the negatively charged shell decreases as the period (the rows of the periodic table) increases. This is because a complete shell has been added below it (and according to Slater's rules, the contribution to the shielding is higher), increasing the distance from the nucleus. The effective charge increases within the same period because the contribution to the shielding for orbitals with equal n is less significant.

What is the effective nuclear charge for neon?

Considering the last electron in the electron cloud of neon, in the orbital 2p, we have seven electrons contributing 0.35 to the shielding. We then add the contribution of the two electrons in the orbital 1s with factor 0.85.

The total shielding is 7×0.35 + 2×0.85 = 4.15, and the effective nuclear charge is 10 − 4.15 = 5.85.

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Here are the basics of the atomic orbital model first! An atom is composed by:

A positively charged nucleus made of protons and neutrons. The number of protons defines the atomic number, uniquely identifying a chemical element.
A negatively charged electronic cloud. Each electron can be found in a set of defined regions of space around the nucleus called orbitals.

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