Electron configuration: the principles

Now that we're familiar with the orbitals and their locations within the atoms, we're ready to put electrons into the orbitals. There are three principles that we have to follow.

Aufbau principle

This principle says electrons would always fill up the orbitals with the lowest energy available, just like water flows to the lowest point on the ground.

But the problem is, which orbital has the lowest energy. The answer is that: in the same energy level, the energy of the orbitals in s sublevel is lower than that in p sublevel, which is lower than d sublevel, while f sublevel is the highest in energy. In short, we'll say the energy of orbitals is in the following sequence:

\[ s \lt p \lt d \lt f \]

However, there is one important exception, which says that empty 4s orbital has a lower energy than 3d orbitals.

Let's make it more clear with the help of an energy diagram.

In the diagram above, each horizontal line represents an orbital, while the vertical axis is the energy of the orbitals. Generally, orbitals in a higher energy level have higher energy than those in a lower energy level, while in the same energy level, they follow the sequence of \( s \lt p \lt d \lt f \), with the exception that \( 4s \lt 3d \).

How about those orbitals in the same sublevel? Well, they have the same energy, and we call them degenerated. Degenerated orbitals have the same energy. Thus you'll see orbitals in 2p, 3p, 3d, 4p, and 4d are at the same level, respectively (e.g. the green lines of 2p sublevel).

Therefore, when we fill up the orbitals with electrons by following the Aufbau principle, we would accomplish it in this sequence:

\[ 1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \]

Fret not. We'll show you a super simple way of memorizing it later on. So keep reading.

Pauli's exclusion principle

This is the second principle that we're going to follow while filling up electrons into the orbitals.

This principle says each orbital can accomodate a maximum of TWO electrons with opposite spins.

Let's explain what spin means. We'll show it in the illustration below.

In quantum mechanics, we said that electrons have both the characters of a wave and a particle. When deriving the wave functions, we mainly focused on the wave character of the electron. But we did forget, electrons behave as particles as well.

When a charged particle is rotating, just like an electron does (it spins), it generates a magnetic field.

If two electrons spin in the same direction, they would generate two identical magnetic fields, which results in significant repulsion between the two N poles and the two S poles. And don't forget these two electrons are already experiencing repulsive forces due to their negative charges. This is so unstable that they couldn't fit into the same orbital.

The only way to put two electrons into the same orbital is to have opposite spins which eliminates the repulsion between the magnetic poles and generates an attractive force instead.

Therefore, a single orbital could hold a maximum two electrons with the opposite spin.

The wave function explanation

This could be explained using the wave function as well. Remember that we mentioned there're four quantum numbers, but we only learned \( n \) for energy levels, \( l \) for sublevels, and \( m \) for orbitals.

The last quantum number, spin quantum number \( m_s \), which has a value of either \( -\frac{1}{2} \) or \( +\frac{1}{2} \), is used to describe the electrons rather than orbitals.

So the Pauli's exclusion principle could be illustrated using a more scientific way: there must NOT exist two electrons with all four quantum numbers identical in the same atom.

If two electrons are in the same orbital (\( n \), \( l \), and \( m \) are the same), they must have different \( m_s \) values, which indicates opposite spins of the electrons.

Hund's rule

The third rule that we have to obey when filling up electrons into the orbitals is the Hund's rule, which says electrons would fill up degenerated orbitals singly before they pair up.

So for example, there're 3 degenerated orbitals in 2p sublevel, how do you allocate 4 electrons into these orbitals?

Like:

\[ \displaylines{ \boxed{ \text{ } \upharpoonleft \downharpoonright \text{ } \mid \text{ } \upharpoonleft \downharpoonright \text{ } \mid \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } } \newline \text{2p} } \]

Or:

\[ \displaylines{ \boxed{ \text{ } \upharpoonleft \downharpoonright \text{ } \mid \text{ } \upharpoonleft \text{ } \mid \text{ } \upharpoonleft \text{ } } \newline \text{2p} } \]

The answer would be the second following Hund's rule.

It is always recommended that you fill up electrons in this way: put electrons into the boxes (orbitals) one by one from the left to the right, only one electron in each box, then with extra electrons, repeat from left to right again (see the numbers below).

\[ \boxed{ \text{ } {\scriptstyle \enclose{circle}{\kern .06em 1\kern .06em}} \upharpoonleft \text{ } {\scriptstyle \enclose{circle}{\kern .06em 4\kern .06em}} \downharpoonright \text{ } \mid \text{ } {\scriptstyle \enclose{circle}{\kern .06em 2\kern .06em}} \upharpoonleft \text{ } \mid \text{ } {\scriptstyle \enclose{circle}{\kern .06em 3\kern .06em}} \upharpoonleft \text{ } } \]

In the next post, we'll discuss in more details how to write electron configuration. We'll explain more on the box diagram shown above.

So, think about the question above, and let's move on.

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