LOG#114. Bohr’s legacy (II).

Dedicated to Niels Bohr

and his atomic model


2nd part: Electron shells,

Quantum Mechanics

and The Periodic Table


Niels Bohr (1923) was the first to propose that the periodicity in the properties of the chemical elements might be explained by the electronic structure of the atom. In fact, his early proposals were based on his own “toy-model” (Bohr atom) for the hydrogen atom in which the electron shells were orbits at a fixed distance from the nucleus. Bohr’s original configurations would seem strange to a present-day chemist: the sulfur atom was given a shell structure of  (2,4,4,6)  instead of 1s^22s^22p^63s^23p^4, the right structure being (2,8,6).

The following year, E.C.Stoner incorporated the Sommerfeld’s corrections to the electron configuration rules, and thus, incorporating the third quantum number into the description of electron shells, and this correctly predicted the shell structure of sulfur to be the now celebrated sulfur shell structure (2,8,6). However neither Bohr’s system nor Stoner’s could correctly describe the changes in atomic spectra in a magnetic field (known as the Zeeman effect). We had to wait to the complete Quantum Mechanics formalist to arise in order to give a description of this atomic phenomenon an many others (like the Stark’s effect, spectrum split due to an electric field).

Bohr was well aware of all this stuff. Indeed, he had written to his friend Wolfgang Pauli   to ask for his help in saving quantum theory (the system now known as “old quantum theory”). Pauli realized that the Zeeman effect could be due only to the outermost electrons of the atom, and was able to reproduce Stoner’s shell structure, but with the correct structure of subshells, by his inclusion of a fourth quantum number and his famous exclusion principle (for fermions like the electrons theirselves) around 1925. He said:

It should be forbidden for more than one electron with the same value of the main quantum number n to have the same value for the other three quantum numbers k [l], j [ml] and m [ms].

The next step was the Schrödinger equation. Firstly published by E. Schrödinger in 1926, it gave three of the four quantum numbers as a direct consequence of its solution for the hydrogen atom: his solution yields the (quantum mechanical) atomic orbitals which are shown today in textbooks of chemistry (and above). The careful study of atomic spectra allowed the electron configurations of atoms to be determined experimentally, and led to an empirical rule (known as Madelung’s rule (1936) for the order in which atomic orbitals are filled with electrons. The Madelung’s law is generally written as a formal sketch (picture):


Shells and subshells versus orbitals

In the picture of the atom given by Quantum Mechanics, the notion of trajectory looses its meaning. The description of electrons in atoms are given by “orbitals”. Instead of orbits, orbitals arise as the zones where the probability of finding an electron is “maximum”. The classical world seems to vanish into the quantum realm. However, the electron configuration was first conceived of under the Bohr model of the (hydrogen) atom, and it is still common to speak of shells and subshells (imagine an onion!!!)  despite the advances in understanding of the quantum-mechanical nature of electrons (both, wave and particles, due to the de Broglie hypothesis). Any particle (e.g. an electron) does have wave and particle features. The de Broglie hypothesis says that to any particle with linear momentum p=mv corresponds a wave length (or de Broglie wavelength) given by


Remark: this formula can be easily generalized to the relativistic domain by a simple shift from the classical momentum to the relativistic momentum P=m\gamma v, so

    \[\lambda =\dfrac{h\sqrt{1-\beta^2}}{mv}\]


    \[ \beta=v/c\]

An electron shell is the set of energetic allowed states that electrons may occupy which share the same principal quantum number   n (the number before the letter in the orbital label), and which gives the energy of the shell (or the orbital in the language of QM). An atom’s nth electron shell can accommodate 2n^2 electrons, e.g. the first shell can accommodate 2 electrons, the second shell 8 electrons, and the third shell 18 electrons, the fourth 32, the fifth 50, the sixth 72, the seventh 92, the eighth 128, the ninth 162, the tenth 200, the eleventh 242, the twelfth 288 and so on. This sequence of “atomic numbers” is well known

    \[ (2,8,32,50,72,92,128,162,200,242,288,...)\]

In fact, I have to be more precise with the term “magic number”. Magic number (atomic or even nuclear physics), in the shell models of both atomic and nuclear structure, IS any of a series of numbers that connote stable structure.

The magic numbers for atoms are 2,10,18, 36, 54, and 86, 118, 168, 218, 290, 362,… They correspond to the total number of electrons in filled electron shells (having ns^2np^6 as electron configuration ). Electrons within a shell have very similar energies and are at similar distances from the nucleus, i.e., inert gases!

The factor of two above arises because the allowed states are doubled due to the electron spin —each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin +1/2 (usually noted by an up-arrow) and one with a spin −1/2 (with a down-arrow).

An atomic subshell is the set of states defined by a common secondary quantum number, also called azimutahl quantum number, ℓ, within a shell. The values ℓ = 0, 1, 2, 3 correspond to the spectroscopic values s, p, d, and f , respectively. The maximum number of electrons which can be placed in a subshell is given by 2(2ℓ + 1). This gives two electrons in an s subshell, six electrons in a p subshell, ten electrons in a d subshell and fourteen electrons in an f subshell. Therefore, subshells “close” after the addition of 2,8,10,18, 36,50,72,… electrons. That is, atomic shells close after we reach ns^2np^6, with n>1, i.e., shells close after reaching the inert gas electron configuration.

The numbers of electrons that can occupy each shell and each subshell arise from the equations of quantum mechanics,in particular the Pauli exclusion principle: no two electrons in the same atom can have the same values of the four quantum numbers stated above. The energy associated to an electron is that of its orbital. The energy of any electron configuration is often approximated as the sum of the energy of each electron, neglecting the electron-electron interactions. The configuration that corresponds to the lowest electronic energy is called the ground (a.k.a. fundamental) state.

Aufbau principle and Madelung rule

The Aufbau principle (from the German word Aufbau, “building up, construction”) was an important part of Bohr’s original concept of electron configuration. It may be stated as:

a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy orbitals are filled before electrons are placed in higher-energy orbitals.
The approximate order of filling of atomic orbitals, following the sketch given above arrows from 1s to 7p. After 7p the order includes orbitals outside the range of the diagram, starting with 8s.

The principle works very well (for the ground states of the atoms) for the first 18 elements, then decreasingly well for the following 100 elements. The modern form of the Aufbau principle describes an order of orbital energies given by Madelung’s rule (also referred as the Klechkowski’s rule). This rule was first stated by Charles Janet in 1929, rediscovered by E. Madelung in 1936, and later given a theoretical justification by V.M.Klechkowski. In modern words, it states that:

A) Orbitals are filled in the order of increasing n+l.

B) Where two orbitals have the same value of n+l, they are filled in order of increasing n.

This gives the following order for filling the orbitals:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, (8s, 5g, 6f, 7d, 8p, and 9s)

In this list the orbitals in parentheses are not occupied in the ground state of the heaviest atom now known (circa 2019), the ununoctiom (Uuo) or oganesson (E118), an atom with Z=118 protons in its nucleus and thus, 118 electrons in its ground state.

The Aufbau principle can be applied, in a modified form, to the protons and neutrons in the atomic nucleus, as in the atomic shell model. The nuclear shell model predicts the magic numbers at Z,N=2, 8, 20, 28, 50, 82, 126 (and Z,N=184 and 258 for spherical symmetry, but it does not seem to be the case for “deformed” nuclei at high values of Z and N).

Shortcomings of the Aufbau principle

The Aufbau principle rests on a fundamental postulate that the order of orbital energies is fixed, both for a given element and between different elements; neither of these is true (although they are approximately true enough for the principle to be useful). It considers atomic orbitals as “boxes” of fixed energy into which can be placed two electrons and no more. However the energy of an electron “in” an atomic orbital depends on the energies of all the other electrons of the atom (or ion, or molecule, etc.). There are no “one-electron solutions” for systems of more than one electron, only a set of many-electron solutions which cannot be calculated exactly. The fact that the Aufbau principle is based on an approximation can be seen from the fact that there is an almost-fixed filling order at all, that, within a given shell, the s-orbital is always filled before the p-orbitals. In a hydrogenic (hydrogen-like) atoms , which only has one electron, the s-orbital and the p-orbitals of the same shell have exactly the same energy, to a very good approximation in the absence of external electromagnetic fields. (However, in a real hydrogen atom, the energy levels are slightly split by the magnetic field of the nucleus, and by the quantum electrodynamic effects like the Lamb shift).

Exceptions to Madelung’s rule

There are several more exceptions to Madelung’s rule among the heavier elements, and it is more and more difficult to resort to simple explanations such as the stability of half-filled subshells. It is possible to predict most of the exceptions by Hartree–Fock calculations, which are an approximate method for taking account of the effect of the other electrons on orbital energies. For the heavier elements, it is also necessary to take account of the effects of Special Relativity on the energies of the atomic orbitals, as the inner-shell electrons are moving at speeds approaching the speed of light . In general, these relativistic effects tend to decrease the energy of the s-orbitals in relation to the other atomic orbitals. The electron-shell configuration of elements beyond rutherfodium (Z=104) has not yet been empirically verified, but they are expected to follow Madelung’s rule without exceptions until the element Ubn (Unbinillium, Z=120). Beyond that number, there is no accepted viewpoint (see below my discussion of Pykko’s model for the extended periodic table).


from the Greeks to Mendeleiev and Seaborg

Atoms and their existence from Greeks to Mendeleiev have suffered historical evolution. In this section, I am going to give you a visual tour from the “ancient elements” until their current classifications via Periodic Tables (Mendeleiev’s being the first one!).

Some early elements and periodic tables:







Just for fun, Feng Shui elements are…


And you can also find today apps/games with elements as “key” pieces…Gamelogy! LOL…


Turning back to Chemistry…Or Alchemy (Modern Chemistry is an evolution from Alchemy in which we take the scientific method seriously, don’t forget it!)



After the chemical revolution in the 18th and 19th century, we also have these pictures (note the evolution of the chemical elements, their geometry and classification):









PT8MeyerPeriodicTablePT10oldelementNotations PT11mendeleievsConjectureInGerman















Some interesting pictures about “new tables” and geometries of some periodic tables and their “make-up” process:
















The periodic table of Astrobiology is also a cool table:


Extended periodic tables

and the island of stability

Seaborg conjectured that the 8th period elements were an interesting “laboratory” to test quantum mechanical and physical principles from relativity and quantum physics. He claimed that there could be possible that around some (high) values of Z, N (122, 126 in Z, and about 184 in N), some superheavy elements could be stable enough to be produced. This topic is yet controversial by the same reasons I mentioned in the previous post: finite size of the nucleus, relativistic effects make the nuclei to be deformed, and likely, some novel effects related to nonpertubative issues (like pair creation in strong fields, as Greiner et al. have remarked) should be taken into account. Anyway, the existence of the so-called island of stability is a hot topic in both theoretical chemistry and experimental chemistry (at the level of the synthesis of superheavy elements). It is also relevant for (quantum and relativistic) physics. However, we will have to wait to be able to find those elements in laboratories or even in the outer space!

Some extended periodic tables were proposed by theoretical chemists like Seaborg and many others:






Pykko’s model and beyond

The finnish chemist Pekka Pykko has produced a beautiful modern extended periodic table from his numerical calculations. He has discovered that the Madelung’s law is modified and then, the likely correct superheavy element included Periodic Table should be something like this (with Z less or equal than 172):



You can visit P. Pykko homepage’s here http://www.chem.helsinki.fi/~pyykko/I urge to do it. He has really cool materials! The abstract of his periodic table paper deserves to be inserted here:


and some of his interesting results from it are the modified electron configurations with respect to the normal Madelung’s rule (as I remarked above):




Indeed, Pykko is able to calculate some “simple” and “stable” molecules made of superheavy elements!


It is interesting to compare Pykko’s table with other extended periodic tables out there, like this one:


His extended periodic paper can be dowloaded here


and you can also watch a periodic table video by the most famous chemist in youtube talking about it here


We have already seen about the feynmanium in the last paper, but what is its electron configuration? It is not clear since we have up most theoretical predictions since NO atoms from E137 have been produced yet. Thus, Feynmanium’s electron configuration is assumed to be \left[Ms\right] 5g^{17}8s^2, but due to smearing of the orbitals due to the small separation between the orbitals, the electron configuration is believed to be \left[Ms\right] 5g^{11}6f^{3}7d^18s^28p^2. The hyperphysics web page also discusses this problem. It says:

“(…)Dirac showed that there are no stable electron orbits for more than 137 electrons, therefore the last chemical element on the periodic table will be untriseptium (137Uts) also known informally as feynmanium _{137}Fy. It’s full electron configuration would be something like …

1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^{10} 5p^6 6s^2 4f^{14} 5d^{10}6p^6 7s^2 5f^{14}6d^{10} 7p^6 8s^2 5g^{17}

or is it …

1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^{10} 5p^6 6s^2 4f^{14} 5d^{10}6p^6 7s^2 5f^{14}6d^{10} 7p^6 8s^1 5g^{18}

What is the right electron configuration? Without a synthesized element, we do not know…

Even more, you can have fun with this page and references therein http://planetstar.wikia.com/wiki/Feynmanium

There, you can even find that there are proposals for almost every superheavy element (SHE) name! Let me remark that today, circa 2019, 10th July, we have named every chemical element till Z=118. Feynmanium, neutronium, and any other superheavy element name are not oficial. The IUPAC recommends to use a systematic name until the discoverers have proposed the name and it is “oficially” accepted. Thus, feynmanium should be called untriseptium until we can produce it!

More Periodic Table limits? What about a 0th element with Z=0? Sometimes it is called “neutronium” or “neutrium”. More details here


Of course it is an speculative idea or concept. Indeed, in japanese culture, the void is the 5th element! It is closer to the picture we get from particle physics today in which “elementary particles” are excitations from some vacuum for certain (spinorial, scalar, tensor,…) field. We could see the “voidium” (no, it is no the dalekenium! LOL) as the fundamental “element” for particle physics. And yet, we have that only a 5% of the known Universe are “radiation” and “known elements”. What a shock!




Just for fun, again, the anime Saint Seiya Omega uses 7 fundamental “elements” (yes, I am a geek, I recognize it!)


The Seaborg’s original proposal was something like the next table:


And you see, it is quite a different from the astrological first elements from myths and superstitions:




And finally, let me show you the presently known elementary particles again, the smallest “elements” from which matter is believed to made of (till now, of course):


Remark: Chemistry is about atoms. High Energy Physics is about elementary particles.

Final questions:

1st. What is your favorite (theoretical or known to exist) chemical element?

2nd. What is your favorite elementary particle (theoretical or known to exist in the Standard Model)?

May The Chemical Elements and the Elementary Particles be with YOU!

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LOG#114. Bohr’s legacy (II). — 4 Comments

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  2. My favourite element is Flerovium. I choose Ununquadium as a nick, because it was mainly unknown, a strange name… After I found out the thrilling story of the magic numbers and the stability island, so I am proud about the serendipity? of chosing 114 element 😉
    Maybe could be of your interest this video conference: Bohr no fue el primero http://www.eitb.tv/eu/bideoa/naukas–quantum-donostia/2708215333001/2728798194001/capitulo-13/. Also your article remind me the controversy with Madelung, maybe you can contribute in the comments here: http://ehfdquimica.com/2014/01/20/aufbauprinzip-o-el-leitmotiv-para-no-reducir-la-quimica-a-fisica-actualmente/.
    Thank you for sharing this nice article! I love the models of the nerd periodic tables, I want some to decorate my room! 😀

  3. Pingback: ¿Tiene límite la tabla periódica de los elementos? | Ciencia | La Ciencia de la Mula Francis

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