Chemistry can be particularly complicated, so any tool to make it easier for those new to it is welcome.

One of the most popular ways to learn about Madelung’s rule and the electron configuration of atoms is the Moeller diagram, a graphical mnemonic rule that makes it very easy to see which orbitals electrons are in.

Then **we will find out what the Moeller diagram consists of**, how it relates to Madelung’s rule, how it is applied through a few solved examples, and which chemical elements do not obey this strategy.

Table of Contents

## What is the Moeller diagram?

Moeller’s diagram, also known as the rain method or diagonal rule, is **a graphic and mnemonic method for learning Madelung’s rule, a technique for knowing and writing the electronic configuration of chemical elements**.

This diagram is characterized by the plotting of diagonals through the columns of orbitals, from top to bottom from right to left. Through the Moeller diagram, an order is defined for the filling of the orbitals, which will be defined by three quantum numbers: n and ml.

The Moeller diagram works by looking at the following:

Each column corresponds to a different orbital through which the electrons of an atom circulate, subatomic particles that have a negative charge. **The orbitals in question are: s, p, diff, each with a specific space to house electrons and, therefore, different energy levels.**.

If we draw the diagonals or arrows in the above direction, we have that the first orbital is 1s. The second arrow begins with the 2s orbital. The third arrow crosses 2p and 3s. The fourth diagonal is 3p and 4s. The fifth diagonal is 3d, 4p and 5s, and so on. The Moeller diagram is an introductory technique for those beginning to study the electronic configurations of the elements in the periodic table in chemistry.

## Madelung’s rule

From the Moeller diagram **is the graphic representation of Madelung’s rule** (also known as Klechkovsky’s rule in some countries), first we need to know what it is. According to this rule, the filling of the orbitals of an atom must obey the following two rules:

### Madelung’s first rule

**Orbitals with the lowest values of n + l are filled first, n being the principal quantum number, and the orbital angular momentum**.

For example, the 3d orbital corresponds to an = 3 yl = 2. Therefore, n + l = 3 + 2 = 5. On the other hand, the 4s orbital corresponds to an = 4 yl = 0, and therefore is n + l = 4 + 0 = 4. From this it is established that the electrons first fill the 4s orbital before the 3d, because 4s = 4 while 3d = 5.

### Madelung’s second rule

**If two orbitals have the same value of n + l, the electrons will first occupy the one with the lower value of n + l**.

For example, the 3d orbital has a value of n + l = 5, the same as that of the 4p orbital (4 + 1 = 5) but because the 3d orbital has the lowest value for n, it will be filled first than the 4p orbital.

From all these observations and rules, the following order can be achieved in the filling of atomic orbitals: 1s 2s 2p 3s 3p 4s 3d 4p. Although this order is fixed, memorizing it by heart is complicated, which is why there is the Moeller diagram which graphically represents its order.

## Steps to follow when using the Moeller diagram

As we saw in the previous section, Madelung’s rule uses the formula n + l to establish which orbitals are filled before and from there to determine what is the electronic configuration of a given element. However, the Moeller diagram already represents it graphically and simply, so you just need to follow the columns of the same diagram and draw diagonals to find out in what order the orbitals of each element are filled.

**To find out the electronic configuration of an atom and in which orbitals the electrons are located, you must first know its atomic number Z**. The number Z corresponds to the number of electrons in an atom, as long as this atom is neutral, or what is the same, which is not an ion, neither positive (cation) nor negative (anion).

So, knowing Z for a neutral atom, we already know how many electrons a neutral atom usually has from that element. With that in mind, we’ll start drawing the diagonals in the Moeller diagram. **We must keep in mind that each type of orbital has a different ability to receive electrons.**, which are:

- s = 2 electrons
- p = 6 electrons
- d = 10 electrons
- f = 14 electrons

It stops at the orbital where the last electron given by Z was occupied.

## Examples of the Moeller diagram

To better understand how the Moeller diagram works, we will see below some practical examples of establishing the electronic configuration of different elements.

### Beryllium

To establish the electronic configuration of a neutral beryllium (Be) atom, it must first be looked for in the periodic table, **an alkaline earth located in the second column and the second row of the table**. Its atomic number is 4, so Z = 4 and it also has 4 electrons.

With all of this in mind, we are going to use the Moeller diagram to see how the 4 electrons of this element are located. We start by making diagonals in the direction mentioned above, from top to bottom and from right to left.

When we fill the orbitals, **it is recommended to put the quantity of electrons found in each of them by exposing**. Since 1s is the first orbital and it occupies two electrons, we will write it:

Since we still have free electrons, we continue to fill the orbitals. **The next one is the 2s orbital and, as for 1, occupies 2 electrons**, so 2s2. As we already have all the electrons well located in the orbitals of the neutral atom of Be, we can say that the electronic configuration of this element is:

We make sure we have done it correctly by adding the exponents: 2 + 2 = 4

### Phosphorus

**The element phosphorus (P) is a non-metal that is found in the third row and column 16 of the periodic table.**, with Z = 15, it therefore has 15 electrons in total which must occupy the orbitals.

Having seen the previous example, we can move forward a bit and locate 4 of its electrons in the same orbitals that beryllium has for its 4 electrons, failing to locate 9 more electrons.

After the 2s orbital, the next diagonal enters the 2p orbital and ends at the 3s orbital. The 2p orbital can occupy 6 electrons, and in the case of 3s only 2. We would therefore have:

At the moment we have 12 well located electrons, but we still need 3 more. **We make another diagonal and this time we enter the 3p orbital according to the Moeller diagram, an orbital that has room for 6 electrons**, but since we only have 3 electrons left, this orbital will not be fully occupied, putting an exponent of 3. So, to finish with phosphorus, its electron configuration is as follows:

We make sure we have done it correctly by adding the exponents: 2 + 2 + 6 + 2 + 3 = 15

### Zirconium

**The element zirconium (Zr) is a transition metal located in column 4 and row 5 and has a Z = 40**. By shortening the path by taking advantage of the previous example, we can locate the first 18 electrons.

After the 3p orbital, the next ones to fill us guided by the Moeller diagram are the 4s, 3d, 4p and 5s orbitals, with capacities of 2, 10, 6 and 2 electrons respectively.

When you complete the first nine orbitals in the diagram, a total of 20 electrons are added, **leaving the remaining 2 electrons which lodge in the next orbital, the 4d**. Thus, the electronic configuration of the neutral zirconium element is:

We make sure have done it by adding the exponents: 2 + 2 + 6 + 2 + 6 + 2 + 10 + 6 + 2 + 2 = 40

### Oxygen

**Here we see a slightly more complicated example which is oxygen (O)**. This gas, located in column 16 and row 2 of the periodic table, is a nonmetal and has the atomic number 8.

So far, seeing the other examples, one might think that its Z = 8, but it is not so simple since this gas is of a special nature, being almost always in the form of an ion with a charge of -2.

This means that although an oxygen neutral atom has 8 electrons as indicated by its atomic number, the truth is that in nature it has more, if it is 10 (8 electrons + 2 electrons or, if it is preferred, – 8 electric charges -2).

So in this case, **the amount of electrons we need to place in the orbitals is not 8 but 10 electrons**, as if we were locating the electrons of the chemical element neon which has Z = 10.

Understood this, we only have to do the same thing we did in the previous cases only considering that we are working with an ion (anion):

We make sure we have done it correctly by adding the exponents: 2 + 2 + 6 = 10

### Soccer

**Something similar to oxygen happens to calcium (Ca), but in this case we are talking about a cation, that is, a positively charged ion.**.

This element is found in column 2, row 4 of the periodic table with the atomic number of 20, but in nature it usually occurs as an ion with a positive charge +2, which means that its charge electron is 18 (- 20 + 2 = 18; 20 electrons – 2 electrons = 18 electrons).

We make sure we have done it correctly by adding the exponents: 2 + 2 + 6 + 2 + 6 = 18

## Exceptions to the Moeller diagram and the Madelung rule

If the Moeller diagram is very useful for understanding Madelung’s rule and knowing where the electrons of different chemical elements are located, the truth is that it is not infallible. There are some substances whose composition does not obey what we have explained.

**Their electronic configurations differ experimentally from those predicted by Madelung’s rule for quantum reasons.**. Among these elements which do not respect the rules we have: chromium (Cr, Z = 24), copper (Cu, Z = 29), silver (Ag, Z = 47), rhodium (Rh, Z = 45), cerium (Ce, Z = 58), niobium (Nb; Z = 41), among others.

Exceptions are very common when filling orbitals and f. For example, in the case of chromium, which should have a valence configuration ending in 4s ^ 2 3d ^ 4 according to Moeller’s diagram and Madelung’s rule, it actually has 4s ^ 1 3d ^ 5. Another example strange is that of money, which instead of having the last 5s ^ 2 4d ^ 9 a 5s ^ 1 4d ^ 10.

Bibliographical references

- Mannequins. (2018). How to represent electrons in an energy level diagram. dummies.com. Retrieved from: https://www.dummies.com/education/science/chemistry/how-to-represent-electrons-in-an-energy-level-diagram/
- Gabriel Bolivar. (October 27, 2018). Moeller diagram: what it consists of and exercises solved. Lifeder. Retrieved from https://www.lifeder.com/diagrama-de-moeller/.
- Professor Susi [Susi profe] (January 6, 2021). ELECTRONIC CONFIGURATION Moeller Diagram and Diagonal Rule. [Video] Youtube. Retrieved from: https://www.youtube.com/watch?v=Gone3phF_48
- Ostrovski, VN (2005). “On the recent discussion of the quantum justification for the periodic table of the elements.” Fundamentals of chemistry. 7 (3): 235-39. doi: 10.1007 / s10698-005-2141-y
- Scerri, ER (2017). “On the rule of Madelung.” Inference. 1 (3).