Aufbau Principle

Aufbau comes from the German word "Aufbauen" which means "to build". In essence when writing electron configurations we are building up electron orbitals as we proceed from atom to atom. As we write the electron configuration for an atom, we will fill the orbitals in order of increasing atomic number. The Aufbau principle originates from the Pauli’s exclusion principle which says that no two fermions (e.g., electrons) in an atom can have the same set of quantum numbers, hence they have to "pile up" or "build up" into higher energy levels. How the electrons build up is a topic of electron configurations.

The aufbau method was initially proposed by the Danish physicist Niels Bohr, who was the first person to use quantum mechanics to study atomic structure. He was also one of the first to fundamentally explain the periodic table in terms of arrangements of electrons (electronic configurations). Bohr proposed that the atoms of the periodic table can be thought of as being progressively built up one electron at a time: starting from the simplest atom of all, hydrogen with just one electron, moving onto helium with two electrons, lithium with three, all the way to uranium – which at that time (1913) was the heaviest known atom – with 92 electrons.

The next ingredient is a knowledge of the atomic orbitals into which the electrons are progressively placed. These orbitals, at least in their simplest form, nowadays come from solving the Schrödinger equation for the hydrogen atom.

The orbitals

The different atomic orbitals come in various kinds that are distinguished by labels such as s, p, d and f. Each shell of electrons can be broken down into various orbitals and as we move away from the nucleus each shell contains a progressively larger number of types of orbital: the first shell only contains a 1s orbital, the second shell 2s and 2p orbitals, the third shell 3s, 3p and 3d orbitals, the fourth shell 4s, 4p, 4d and 4f orbitals and so on.

Next, we need to know how many of these orbitals occur in each shell. The answer is provided by the formula 2l + 1, where l takes different values depending on whether we are speaking of s, p, d or f orbitals. For s orbitals l = 0, for p orbitals l = 1, for d orbitals l = 2 and so on. As a result there is potentially one s orbital, three p orbitals, five d orbitals, seven f orbitals and so on for each shell.

The flaw

Electrons are arranged in various atomic orbitals in the increasing order of their energy

So far, so good. Now comes the magic ingredient in the sloppy version of this principle that claims to predict the order in which these orbitals fill (and here is where the fallacy lurks): rather than filling the shells around the nucleus in a simple sequence, where each shell must fill completely before moving onto the next shell, we are told that the correct procedure is more complicated. But we are also reassured that there is a nice simple pattern that governs the order of shell and consequently of orbital filling. This is demonstrated using the aufbau diagram, which lies at the heart of the trouble.

The order of filling is said to be obtained by starting at the top of the diagram and following the arrows. This process gives the order of filling of orbitals with electrons according to this sequence:

1s > 2s > 2p > 3s > 3p > 4s > 3d > 4p > 5s > 4d

and so on.

This diagram, when combined with a knowledge of how many electrons can be accommodated in each kind of orbital and the number of such available orbitals in each shell, is now supposed to give us a prediction of the complete electronic configuration of all but about 20 atoms in which further irregularities occur, such as the cases of chromium and copper. But let’s not get sidetracked by these anomalies and instead concentrate on a far deeper problem with this approach.

Some examples

Let’s consider a few examples. The atom of magnesium has a total of 12 electrons. Using the aufbau diagram we obtain an electronic configuration of 1s2, 2s2, 2p6, 3s2 in beautiful agreement with experiments that can examine the configuration directly by looking at the spectra of atoms. Another example is calcium, which has 20 electrons. This method gives a configuration of 1s2, 2s2, 2p6, 3s2, 3p6, 4s2 and once again there is perfect agreement with experiments on the spectrum of calcium atoms.

But now let’s see what happens for the next atom, scandium, with its 21 electrons. According to the aufbau diagram the configuration should be 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d1 and indeed it is. But conventional wisdom claims that the final electron to enter the atom of scandium is a 3d electron, when experiments indicate that the 3d orbital is filled before the 4s orbital.

Why the mistake occurs

But how can this apparently blatant mistake have occurred and taken root in chemical education circles? The answer lies with the fact that the aufbau diagram gives the overall configuration correctly in all but about 20 cases (see Anomalous electronic configurations). It is only when one questions the order of filling that this approach gives the wrong answer.

Unfortunately, sticking to this way of teaching electronic configurations has led many teachers and textbooks to invent all kinds of contorted schemes to explain why even though the 4s orbital fills preferentially (as it does if you follow the aufbau diagram) it is also the 4s electron that is preferentially ionised to form an ion of Sc+. These explanations are all incorrect, since the 4s orbital actually fills last and consequently it is perfectly natural that it should be the first orbital to lose an electron on forming a positive ion.

Anomalous electronic configurations

Grouped by periods and shown in correct order of orbital filling

Chromium	[Ar] 3d⁵ 4s¹
Copper	[Ar] 3d¹⁰ 4s¹

Niobium	[Kr] 4d⁴ 5s¹
Molybdenum	[Kr] 4d⁵ 5s¹
Ruthenium	[Kr] 4d⁷ 5s¹
Rhodium	[Kr] 4d⁸ 5s¹
Palladium	[Kr] 4d¹⁰ 5s⁰
Silver	[Kr] 4d¹⁰ 5s¹

Lanthanum	[Xe] 5d¹ 6s²
Cerium	[Xe] 4f¹ 5d¹ 6s²
Gadolinium	[Xe] 4f⁷ 5d¹ 6s²
Platinum	[Xe] 4f¹⁴5d⁹ 6s¹
Gold	[Xe] 4f¹⁴5d¹⁰ 6s¹

Actinium	[Rn] 6d¹ 7s²
Thorium	[Rn] 6d² 7s²
Protactinium	[Rn] 5f² 6d¹ 7s²
Uranium	[Rn] 5f³ 6d¹ 7s²
Neptunium	[Rn] 5f⁴ 6d¹ 7s²
Curium	[Rn] 5f⁷ 6d¹ 7s²

Examining the evidence

One source of proof that the sloppy version of the aufbau principle is wrong comes from examining the experimental spectral evidence from the ions of transition metal atoms.2 Using scandium as an example:

    Sc3+ has the configuration 1s2, 2s2, 2p6, 3s2, 3p6, 3d0, 4s0
    Sc2+ is 1s2, 2s2, 2p6, 3s2, 3p6, 3d1, 4s0
    Sc1+ is 1s2, 2s2, 2p6, 3s2, 3p6, 3d1, 4s1
    Sc is 1s2, 2s2, 2p6, 3s2, 3p6, 3d1, 4s2

On moving from the Sc3+ ion to that of Sc2+ it is clear that the additional electron enters a 3d orbital and not a 4s orbital as the aufbau diagram dictates. Similarly on moving from this ion to the Sc1+ ion the additional electron enters a 4s orbital as it does in finally arriving at neutral scandium atom or Sc. Similar patterns and sequences are observed for the subsequent atoms in the periodic table including titanium, vanadium, chromium (with further complications as mentioned before), manganese and so on. Not only is it not possible to predict the configuration in any of the transition metals, but the aufbau diagram also falls down for the lanthanides, and even the p-block elements.

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