Sunday, June 24, 2012

The trouble with using the aufbau to find electronic configurations

Eric Scerri
Department of Chemistry & Biochemistry
Los Angeles, CA 90095


The Periodic Table and the Aufbau
One of the biggest topics in the teaching and learning of chemistry is the use of the aufbau principle to predict the electronic configurations of atoms and to explain the periodic table of the elements.  This method has been taught to many generations of students and is a favorite among instructors and textbooks when it comes to setting questions.  In this blog I am going to attempt to blow the lid off the aufbau because it is deeply flawed, or at least the sloppy version of the aufbau.  The flaw is rather subtle and seems to have escaped the attention of nearly all chemistry and physics textbooks and the vast majority of chemistry professors that I have consulted on the subject.

The error comes from what may be an innocent attempt to simplify matters or maybe just an understandable slip as I will try to explain.  Whatever the cause there is no excuse for perpetuating this educational myth as I will try to explain.

So what’s the problem?
The aufbau method was originally proposed by the great Danish physicist Niels Bohr who was the first to bring quantum mechanics to the study of atomic structure and one of the first to give a fundamental explanation of the periodic table in terms of arrangements of electrons (electronic configurations).  Bohr proposed that we can think of the atoms of the periodic table as being progressively built up starting from the simplest atom of all, that of hydrogen which contains just one proton and one electron.  The other atoms differ from hydrogen by the addition of one proton and one electron.  Helium has two protons and two electrons, lithium has three of each, beryllium has four of each, all the way to uranium which at that time, (1913), was the heaviest known atom, weighing in at 92 protons and 92 electrons.  Neutron numbers vary and are quite irrelevant to this story incidentally.

The next ingredient is a knowledge of the atomic orbitals into which the electrons are progressively placed in an attempt to reproduce the natural sequence of electrons in atoms that occur in the real world.  Oddly enough these orbitals, at least in their simplest form, nowadays come from solving the Schrödinger equation for the hydrogen atom but let’s not get too sidetracked for the moment.

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 kinds of orbitals.  Here is the well-known scheme,

First shell contains                            1s orbital only
Second shell contains                       2s and 2p orbitals
Third shell contains                          3s, 3p and 3d orbitals
Fourth shell contains                        4s, 4p, 4d and 4f orbitals and so on.

The next part is that one needs to know how many of these orbitals occur in each shell.  The answer is provided by the simple formula 2(l+ 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 are potentially one s orbital, three p orbitals, five d orbitals, seven f orbitals and so on for each shell.

So far so good.  Now comes the magic ingredient which claims to predict the order of filling of these orbitals and here is where the fallacy lurks.  Rather than filling the shells around the nucleus in a simple sequential 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 orbitals filling.

And this is finally the point at which the aufbau diagram, which I am going to claim lies at the heart of the trouble, is trotted out. 

The order of filling is said to be obtained by starting at the top of the diagram and following the arrows pointing downwards and towards the left-hand margin of this diagram.  Following this procedure gives us the order of filling of orbitals with electrons according to this sequence,

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d …

This recipe when combined with a knowledge of how many electrons can be accommodated in each kind of orbital and the number of such available orbital 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.  Again I don’t want to get side-tracked and so will concentrate on one of the far more numerous regular configurations.

Some examples
To see how this simplified and ultimately flawed method works let me consider a few examples.  The atom of magnesium has a total of 12 electrons.  Using the method above this means that we obtain an electronic configuration of,

1s2, 2s2, 2p6, 3s2

in beautiful agreement with experiments which can examine the configuration directly through the spectra of atoms.  Let’s look at another example, an atom of calcium which has 20 electrons.  Following the well-known 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 very next atom, namely scandium with its 21 electrons.  According to the time honored aufbau method the configuration should be,

1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d1

and indeed it is.  But many books proceed to spoil the whole thing by claiming, not unreasonably perhaps, that the final electron to enter the atom of scandium is a 3d electron when in fact experiments point quite clearly to the fact that the 3d orbital is filled before the 4s orbital.  The correct version can be found in very few textbooks but seems to have been unwittingly forgotten or distorted in many cases by generations of instructors and textbook authors as I mentioned at the outset.  How can such an odd situation arise?

Why the mistake occurs
But how can such an apparently blatant mistake have occurred and taken such root in chemical education circles?  The answer is as interesting as it is subtle.  First of all there is the fact that the overall configuration is in fact correctly given by following the sloppy approach.  But if one asks questions about the order of filling the sloppy approach gives the wrong answer as I have been pointing out.  But even worse, it has led many teachers and textbooks to invent all kinds of contorted schemes in order to explain why even though the 4s orbital fills preferentially (as it does in the sloppy version) it is also the 4s electron that is preferentially ionized to form an ion of Sc+.  Since these contortions are pure inventions I will not waste the reader’s time by looking into them.  They are quite simply incorrect since as a matter of fact, the 4s orbital fills last and consequently, as simple logic dictates, is the first orbital to lose an electron on forming a positive ion. 

What’s the evidence? 
But how can I be so confident in claiming that the vast majority of chemistry teachers, professors and textbook authors have erred in presenting the sloppy version.  The answer is that one can just consider the experimental evidence on the ions of any particular transition metal atom such as scandium, 

Sc3+ (tri-positive ion)                        1s2, 2s2, 2p6, 3s2, 3p6, 3d0, 4s0
Sc2+ (di-positive ion)                         1s2, 2s2, 2p6, 3s2, 3p6, 3d1, 4s0
Sc1+ (mono-positive ion)                  1s2, 2s2, 2p6, 3s2, 3p6, 3d1, 4s1
Sc     (neutral atom)                            1s2, 2s2, 2p6, 3s2, 3p6, 3d1, 4s2

On moving from the Sc3+ ion to that of Sc2+ it is plain to see that the additional electron enters a 3d orbital and not a 4s orbital as the sloppy scheme 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 titatium, vanadium, chromium(with further complications), manganese and so on.

Psychological factors
I have been thinking about what psychological factors contribute to the retention of the sloppy aufbau.  As I already mentioned it does give the correct overall configuration for all but about 20 atoms that show anomalous configurations, such as chromium, copper, molybdenum and many others.

Another factor is that it gives chemistry professors the impression that they really can predict the way in which the atom is built-up starting from a bare nucleus to which electrons are successively added.  Presumably it also gives students the impression that they can make similar predictions and perhaps convinces them of the worthiness of the aufbau and scientific knowledge in general. 

The fact remains that it is not possible to predict the configuration in any of the transition metals, and indeed the lanthanides, or if it comes down to it even the p-block elements.  Let’s go back to scandium.  Contrary to the sloppy aufbau that is almost invariably taught, the 3d orbitals have a lower energy than 4s starting with this element.  If we were to try to predict the way that the electrons fill in scandium we might suppose that the final three electrons after the core argon configuration of

1s2, 2s2, 2p6, 3s2, 3p6

would all enter into some 3d orbitals to give,

1s2, 2s2, 2p6, 3s2, 3p6, 3d3

The observed configuration however is,

1s2, 2s2, 2p6, 3s2, 3p6, 3d1, 4s2

What’s really happening?
This amounts to saying that all three of the final electrons enter 3d but two of them are repelled into an energetically less favorable orbital, the 4s, because the overall result is more advantageous for the atom as a whole.  But this is not something that can be predicted.  Why is it 2 electrons, rather than one or even none?  In cases like chromium and copper just one electron is pushed into the 4s orbital.  In an analogous case from the second transition series, the palladium atom, the competition occurs between the 5s and 4d orbitals.  In this case none of the electrons are pushed up into the 5s orbital and the resulting configuration has an outer shell of [Kr]4d10

None of this can be predicted in simple terms from a rule of thumb and so it seems almost worth masking this fact by claiming that the overall configuration can be predicted, at least as far as the cases in which two electrons are pushed up into the relevant s orbital.  To those who like to present a rather triumphal image of science it is too much to admit that we cannot make these predictions.  The use of the sloppy aufbau seems to avoid this problem since it gives the correct overall configuration and hardly anybody smells a rat.

But why do electrons get pushed up into the relevant s orbital?
Finally, it is natural to now ask why it is that one or two electrons are usually pushed into a higher energy orbital, other than the answer I already gave which is to say that doing so produces a more stable atom overall.  The answer lies in the fact that 3d orbitals are more compact than 4s to consider the first transition, and as a result any electrons entering 3d orbitals will experience greater mutual repulsion. 

The slightly unsettling feature is that although the relevant s orbital can relieve such additional electron-electron repulsion different atoms do not always choose to make full use of this form of sheltering because the situation is more complicated than the way in which I have described it.  After all there is the fact that nuclear charge increases as we move through the atoms.  At the end of the day there is a complicated set of interactions between the electrons and the nucleus as well as between the electrons themselves.  This is what ultimately produces an electronic configuration and contrary to what some educators would wish for, there is no simple qualitative rule of thumb that can cope with this complicated situation.    

Bottom line
There is absolutely no reason for chemistry professors and textbook authors to continue to teach the sloppy version of the aufbau.  Not only does it give false predictions regarding the order of electron filling in atoms but it also causes authors and instructors to tell further educational lies.  They are forced to invent some elaborate explanations in order to undo the error in an attempt to explain why 4s is occupied preferentially (which it is not) but also preferentially ionized which it is.
The sloppy version also implies that the 4s orbital has a lower energy than 3d for all atoms which is not the case, or that the 5s orbital has a lower energy than 4d which is not the case for all atoms and so on.  Similar issues arise in the f-block elements.
It is high time that the teaching of aufbau and electronic configurations were carried out properly in order to reflect the truth of the matter rather than taking a short-cut and compounding it with a further imaginary story.

The following references are among the few that give the correct explanation;

S-G. Wang, W. H. E. Schwarz, Angew. Chem. Int. Ed. 2009, 48, 19, 3404–3415.

S. Glasstone, Textbook of Physical Chemistry, D. Van Nostrand, New York, 1946.

D.W. Oxtoby, H.P. Gillis, A. Campion, Principles of Modern Chemistry, Sixth Edition,
Thomson/Brooks Cole, 2007.

General Reference on the Periodic Table
Eric Scerri, A Very Short Introduction to the Periodic Table, Oxford University Press, 2011


  1. Thank you for this information. I teach on this topic and use the sloppy version (sorry to admit it). I find that with the amount of information necessary to cover in a semester I must make some generalizations. The information given here, while interesting to me, would confuse my students who are struggling with basic vocabulary and definitions. The sloppy aufbau at least gives them something to learn that is a rule of thumb. Perhaps the more detail could be learned in an upper level class.

  2. Eric,

    for me Aufbau is not about the order in which electrons enter the orbitals, but the order in which new electrons are introduced with respect to the atomic numbers. Calcium is characterized by 4s2 electron and Scandium is characterized by 3d1. The difference between Sc and all other elements lighter than Scandium is 3d1 electron, irregardless if 3d1 position is filled last or not.

    There are two popular ways to write electron configurations: 1) in order of ascending shell numbers and 2) in order of filling of the orbitals. Again, for Scandium the configuration can be written as 1s2 2s2 2p6 3s2 3p6 3d1 4s2, or as such: 1s2 2s2 2p6 3s2 3p6 4s2 3d1. I happen to agree with you that second method is flawed, or it is incorrectly named "in order of the orbital filling".

    The electron shells are real, while the order of orbital filling as presented by Aufbau is superficial. However, there is a merit behind n+l rule, it is the order of the introduction of novel electrons with regard to the atomic numbers. That is why ADOMAH periodic table, where vertical columns correspond to the electron shell numbers, is an advance if compared with Janets' LSPT as demonstrated at this web site: Use of ADOMAH allows writing of electron configurations both ways, while LSPT is stuck with "order of orbital filling" concept.

    Best regards,


  3. A couple typos: rater, electros.

    Correct answers should be reasonably short and fit in a reference section of a text. Or perhaps common shells could be in a table, and a hand held calculator program based on Quantum Mechanics could provide any outliers. QM can predict this sort of thing, right? As i understand it, you need QM to get enzyme chemistry right. It takes some fairly serious computing...

    I'm all for a mnemonic, but not for teaching a principal that's actually wrong. Standing waves describe electron behavior better than the orbits of planets. A lone planet around a star can have a stable orbit nearly anywhere. Not true for hydrogen and an electron. The orbit model was dropped in 1905. So, drop "orbits" and "orbitals" from the terminology. Just because we had to learn this nonsense, doesn't mean we need to punish generations to come forever. The cost calculation: finite cost now vs infinite cost in the future - which is less?

  4. Yea, I often run into these problems and simply thought there are exceptions to rules. Good to have it straightened out.

    I never really liked the Aufbau Principle anyway because it give students the impression that it is easy to transmutate (alchemy) simply by adding one more electron and increasing the atomic number by one. I like to point out that doing this makes a huge change in chemical properties.

    The other thing is that orbitals above H atom do not exist. Also you have written down the whole electron configs including the core electrons.

    I always write down the noble gas core, [He], [Ne] etc because those electrons are so tightly bound they do not engage in most chemical processes and all those tightly filled orbitals are much more distorted than the valence ones. Besides the noble gas cores are completely spherical.

  5. Here is the problem: when one writes electron configurations using noble gas core, as [Ar] or [Ne] for example, Aufbau Principle is presumed. Using above example, the configuration of Scandium then becomes [Ar] 4s2 3d1. Similarly, configuration for Germanium becomes [Ar] 4s2 3d10 4p2, unless it is written as [Ar] 3d10 4s2 4p2. Like it or not, Aufbau Principle is lurking there.

  6. Thanks Valery,

    I am not disputing that the sloppy aufbau gives the correct overall configuration. It's just that the sloppy version makes no sense when trying to explain ionization energies of transition metal atoms.

    If you like, the sloppy aufbau, whereby 4s fills first even for transition metals, gives the correct overall configuration for the wrong reason.

    eric scerri

  7. Hi Eric

    Interesting blog. I am not sure if this is a real problem. How many people actually use the sloppy aufbau principle? I always emphasise that the principle is always correct (Orbitals are filled in order of energy with the lowest energy orbitals being filled first) but that the sequence of atomic orbitals is not fixed and "may vary for some atoms because the energy of electrons in orbitals is affected by nuclear charge, the presence of other electrons in the same orbital or in the same sub-level, and the overall charge" (Constable and Housecroft). No need to get into details about multi electron wave functions, simply to indicate that the sequence is a guide but not always written in stone, with the classic exceptions being the 4s,3d levels. This is also discussed in mind-numbing detail in Gerloch and Constable



  8. Thanks Ed.

    Actually it is a problem. I am only aware of two US general chemistry textbooks that DO NOT use the sloppy version. But I will be glad to add your books to the list of "good guys" in any future publications on this. I look forward to seeing the detsils you mention in Gerloch and Constable.

  9. P.S. The 2 US textbooks that do it properly are Oxtoby as cited in my blog and also Munowitz, an excellent book published in 2000.

    1. I love Munowitz, a rare and very fine book that sadly was not widely adopted because it was "different." Few authors deal with the fundamental principles as well as Mike does here. I still use it in my classes for some topics; this being one of them.

  10. Hi Eric, good post. It leaves me very confused, though.

    Why do Ca and Sc+ have different electronic configurations?

    The only difference is the number of protons, right? Shouldn't the number of protons just stretch the orbitals, not reorder them? I have always thought that an atoms with X electrons would have identical configurations, regardless of their proton number (assuming that all electrons are bound).

    1. It would appear that this assumption is incorrect. The number of protons does clearly seem to affect the configuration. As you say Ca and Sc+ have different configurations. It might be better to doubt the assumption rather than the spectroscopic facts.


  11. HI again Ed,

    I have been looking at Housecraft and Constable and I find that you do actually state that 4s orbitals fill before 3d. THis first appears on p. 84, in a genera section on the aufbau in which you also refer to a more detailed treatment for transition metals in your chapter 16.

    In chapter 16, the claim that 4s fills before 3d is repeated again. (p. 822).

    I am working from an edition published in 1997.

    Would still be interested in your comments if any.

  12. Dear Eric

    I am teaching Inorganic Chemistry in Yogyakarta State University, Indonesia. Many chemistry textbooks (in Indonesian) for Senior High School and teachers as users misunderstood by following the (n+l) rule of aufbau as ordering energy of orbitals for all atoms. Consequently, the energy of (n-1)d orbital is always mistakenly understood to be higher than that of ns orbital, and therefore the electronic configurations of transition elements should be written as [Ar] 4s1-2 3d1-10 for the 1st serie. However, the electronic configuration can also be understood on the basis of the ordering number of shell (n), and thus it becomes [Ar] 3d1-10 4s1-2. In other words it is said that the electronic configuration can be written in two ways, one on the basis of the ordering energy of orbital following aufbau, and the other on the basis of the ordering quantum number of n. Of course, it is a big serious misconception, and unfortunately, no one does care of it, and the table of electronic configurations constructed merely on the basis of aufbau for all elements are found in many textbooks.
    While aufbau predict correctly the number of electrons in each orbital for almost all elements, the ordering energy of orbitals is only true for the first twenty elements, and beyond these do not obey aufbau. Thus I consider [Ar] 3d1-10 4s1-2 is the only correct electronic configuration based on the ordering energy, and consequently [Ar] 4s1-2 3d1-10 should be wrong configuration. Thus, (n+l) rule of aufbau does not reflect the true ordering energy of orbitals for all atoms.
    What I want to have confirmation from you is: how do you write down the electronic configurations of elements with atomic number greater than 20? Do yo consider correct for both configurations?

    Kristian H. Sugiyarto
    Dept. of Chemistry Education
    Faculty of Mathematics and Science
    Yogyakarta State University

  13. Dear Krystian,

    Thank you for your question. I agree 100% with what you have written. Given that there is clear experimental evidence that 3d orbitals fill before 4s for transition metal atoms it is more correct to write their configurations in such a way that 3d appears before 4s.
    But doing this renders the Madelung or n+l rule, or what I call the "sloppy aufbau" incorrect and books and teachers are very reluctant to abandon this nice little mnemonic. The mnemonic does give the overall correct configuration but not the order of filling and consequently predicts the incorrect order of ionization. So I agree with you we should insist on the correct listing in the way that a configuration is stated, even if the sloppy aufbau may still be used to arrive at the overall configuration. After all there could be a footnote to the use of the Madelung diagram to say "but reverse the order of the ns and (n-1)d orbitals when stating transition metal configurations. It would also need a further footnote for lanthanoids and actinoids of course. What do you think of this 'compromise' version?

  14. dear eric,
    it was an enlightening article for sure. im relieved to find that at last, someone did try to give an explanation for the highly dubious exceptions in the sloppy aufbau an A level student in India, and an nobody exactly seemed to know the proper explanation.

  15. Thanks for your comment Chirag. If you send me an E-mail I would be happy to send you some further articles on this topic.

    also see

    for other resources.

  16. Hi Eric,

    Thank you very much for posting this article it has helped clear up many issues that I had encountered in the past couple of months in many attempts to understand the aufbau principle. I'm currently in my first year of studies at university and this has probably been one of the biggest complications I've had to face thus far.

    The main issue had arisen when teaching the aufbau concept was reaching scandium in the periodic table. The concept seemed to work up to calcium but I was not convinced for the properties of ionisation energies and atomic radii for the transition elements.

    For the matter of atomic radii in period 4, potassium has the largest radius which then decreases when you move to calcium. The transition block is then is shown on a graph to contain elements with very similar atomic radii to each other. The way this was explained to me was that the 3d orbitals poorly penetrate the nucleus therefore the 4s provide good shielding of the 3d orbitals which then means the effective nuclear charge barely changes. This seems feasible at first but the concept of ionisation energy complicates this.

    The concept of ionisation energy does remove a 4s electron before the 3d electrons. Now this made absolutely no sense to me as the explanation of the trend in atomic radius for the transition block elements, that was given to me, suggests that the 3d orbitals are the outermost shells as the 4s electrons sheild the 3d. If this was indeed the case wouldn't this result in the 3d electrons being removed first in ionisation? And the atomic radius of scandium being larger than that of calcium?

    Filling the 3d shell before the 4s when arriving at scandium makes much more sense in the greater scheme of things. I am still quite confused about the whole concept and I feel fairly confident that I was right in suspecting something was wrong. Would it at all be possible to send some further information on the 3d and 4s orbitals?

    Thanks again.

  17. Hi Eric, I am so confused about which electrons shells are filled first and second in the d shell, for example for the element Titan, is the 3d orbital filled first or the 4s orbital?
    Can you please write after electrons order, the electron configuration for Vanadium and Chromium?

  18. In every transition metal starting with scandium and then titanium, the 3d orbital fills first followed by 4s. In the case of Cr, only one electron occupies the 4s orbital because this is a more stable arrangement than having two in 4s.

  19. Eric you said that your experimental evidence is the existence of the specific electron configurations. Are these being calculated or measured using spectra? If spectra, what complications arise from having such similar states for the electron. Do you see a lot of say 4s1 3d0 and 4s0 3d1 but more of one than the other?