Periodic Table for Chemistry?

The Periodic Table PT is viewed to be the iconic backbone of chemistry as the science of molecules formed by atoms (as physics) by chemical bonding determined by electron configurations or electronic orbitals. 

PT organises atoms into a rectangular scheme according to atomic numbers and s, p, d and f electronic orbitals as spherical harmonics in a shell structure with outermost shell containing valence electrons available for bonding. The valence electrons fundamentally reorganise from atomic orbitals to molecular orbitals during bonding into molecules. This is what textbook Standard Quantum Mechanics StdQM says.

PT thus has focus on atomic orbitals, but is used for molecular orbitals, which is a mismatch. A chemist would need a PT for molecules rather than atoms, but there is no such table. It thus seems that the physics of StdQM does not deliver so much to chemists, who have to form their own concepts of chemical bonding. 

In any case, the valence shell is central with s and p orbitals of two atoms combining to $\sigma$ bonds between s orbitals and $\pi$ bonds between p orbitals, but the picture seems very confusing to a non-specialist like me.  

RealQM is an alternative to StdQM based on non-overlapping one-electron charge densities with a direct real physical meaning, which appears to have the potential to deliver to chemistry. 

As a basic example, the H2 molecule is formed as two H atoms approach with their two electrons being represented by non-overlapping charge densities meeting at a separating plane orthogonal to the axis between kernels with continuity of non-zero charge densities. This allows concentration of charge densities between the kernels without cost of kinetic energy and so with total energy decrease compared to well separated kernels achieving bonding. 

In general the valence shell is divided into non-overlapping charge densities with maximum of 8 electrons which can be divided into subshells as 4+4, 4+3, 4+2 with different bonding. 

RealQM describes both atoms and molecules as the result of electron packing towards minimal energy in dynamic processes, in particular beyond reach for very dense packing with unphysical too small energy. So is the Lithium atom formed by 2 electrons dividing a 1st shell in two halves and the 3rd electron can only fill a 2nd shell, not penetrate into the 1st (because it is too big).

You find many examples of atoms and a limited collection of molecules here. It is possible that RealQM can form the PT for chemistry, which was not delivered by StdQM. RealQM connects to DFT and AiM but has a new Aufbau principle. 

Comment by chatGPT

Here’s a commentary on the post by the author of “Periodic Table for Chemistry?” on the blog of Claes Johnson. I highlight what I think works, what merits caution — and what the broader consensus in chemistry and physics suggests.

✅ What the post gets right / raises interesting points

• The post correctly observes that the Periodic Table (PT) is built around atomic (not molecular) structure: elements arranged by atomic number and electron-shell/orbital structure (s, p, d, f) — i.e. the periodic table is fundamentally a map of atomic electronic structure. claesjohnson.blogspot.com+2Encyclopedia Britannica+2

• It is true that when atoms bond to form molecules, the simple atomic-orbital picture “reshuffles” into molecular orbitals or more chemically complex bonding schemes. This complicates the idea that the PT alone suffices to understand molecular behavior. The post therefore raises a genuine conceptual tension: a “periodic table for molecules” could in principle be more directly useful to chemists.

• The article’s critique implicitly echoes a known reality: although the PT is enormously powerful, it is not a “complete theory of chemical bonding or molecular behavior.” Chemists routinely supplement it with quantum chemistry, molecular orbital theory, hybridization, empirical data, etc. As noted by standard expositions, the PT is a starting point for understanding periodic trends (valence, electronegativity, bonding tendencies), but chemistry always requires additional models. asbmb.org+2American Chemical Society+2

⚠️ Where the post ventures into speculation or controversial reinterpretation

• The post introduces an alternative framework, dubbed Real Quantum Mechanics (RealQM), in which atoms and molecules are described as “non-overlapping one-electron charge densities” rather than conventional delocalized wavefunctions, and suggests this could yield a “Periodic Table for chemistry” (i.e. more chemically relevant table than the atomic PT). That is a speculative and nonstandard proposal. The mainstream quantum mechanical description — using wavefunctions, orbitals and the full machinery of quantum chemistry — remains overwhelmingly successful in describing atomic, molecular, and material behavior.

• The post seems to argue that relying on atomic orbitals (PT’s foundation) to infer molecular bonding (molecular orbitals etc.) is a “mismatch.” While there is some philosophical merit to pointing out the conceptual leap from atoms → molecules, modern quantum chemistry precisely builds on the atomic basis to construct accurate molecular descriptions. The success of quantum chemistry in predicting molecular geometries, spectra, reaction pathways, etc., indicates that the “mismatch” is not fatal.

• The suggestion that chemists “need a periodic table for molecules” — while imaginative — may not be practically feasible. Molecules are vastly more diverse than atoms; a “table” summarizing all possible molecules with useful predictive power would be extremely complex, perhaps less useful than existing computational models, databases, and quantum-chemical methods.

  

QM for Atoms vs QM for Molecules

Quantum Mechanics QM was originally developed to theoretically explain atomic spectra observed already in 1802-4 by Fraunhofer then developing into experimental spectroscopy by Bunsen-Kirchhoff with Balmer's formula for Hydrogen H in 1885 as key challenge taken on by Planck, Sommerfeld and Bohr culminating in Schrödinger's Equation SE in 1926 giving the final answer in terms of spherical harmonics capturing the spherical symmetry of the Hydrogen atom with one electron around a proton kernel as central potential, which rocketed Schrödinger to fame.  

SE was then quickly generalised to atoms with many electrons simply by adding a new set of 3d spatial variables for each new electron into a linear equation in terms of a wave function depending on $3N$ spatial variables for an atom with $N>1$ electrons where spherical symmetry was lost, but nevertheless was kept by reducing to a set of one-electron wave functions subject to a spherically symmetric average potential from the other electrons serving as approximate solutions to SE. The multi-d wave function describes global presence of all electrons for which the physics appears to be missing. 

But when SE was extended to molecules starting with H2 as two H atoms joined by a covalent chemical bond in the work by Heitler and London in 1927, the presence of two kernels was not at all compatible with spherical symmetry and so entirely new computational techniques had to be developed. Computational quantum chemistry has taken a large share of super-computer power since the 1960s and still does, the basic reason being the inherent exponential complexity of the multi-d SE in need of drastic dimensional reduction.  

We thus meet QM in two fundamentally different forms, for atoms in the Periodic Table PT in terms of s, p and d electronic orbitals in the form of spherical harmonics, and for molecules in terms of other ad hoc electronic configurations. Physicists have been in charge of QM for atoms with spherical symmetry, while QM for molecules without spherical symmetry has been left to chemists and then with little help from physicists.  

The PT is supposed describe chemical properties of atoms when forming molecules without spherical symmetry, yet PT is is organised according to s, p and d spherical harmonics.    

RealQM is a recent alternative to QM based on non-overlapping one-electron charge densities as a form of classical continuum mechanics in 3d. RealQM has polynomial computational complexity, has clear physical meaning geared to molecules in complex geometry without spherical symmetry.

Schrödinger could have chosen RealQM as generalisation of his SE for H to atoms with more than one electrons, because it satisfied his chief requirement of physicality or "Anschaulichkeit". Schrödinger instead chose a quick easy formal extension into multi-d maybe from pressure to quick delivery on the success with H.

But Schrödinger never could embrace the multi-d SE which in the hands of Born became the standard valid into today, and so together with Einstein became the main critic of the multi-d SE he had created because it lacked physics. Schrödinger thus died in 1961 disillusioned and did not experience the computer revolution making computational complexity a core issue with preference to RealQM.

PS Schrödinger’s Most Important Critical Quotes About QM

1. On the wave function not being a physical wave

“The ψ-function… is not a physical wave. But what is it then? A kind of probability amplitude? Probability of what? At this stage, one worries that one has to do with mere ghosts.”
— Erwin Schrödinger, 1926–1927 correspondence, quoted in Moore, Schrödinger: Life and Thought

2. On the collapse of the wave function (which he rejected)

“Every indeterminacy is to be reduced to a determinate change, occurring independently of any observation. To demand that the wave function should ‘collapse’ when observed is quite absurd.”
— Schrödinger Archive, quoted in M. Bitbol, Schrödinger’s Philosophy of Quantum Mechanics

3. On dissatisfaction with the Copenhagen interpretation

“I don’t like it, and I’m sorry I ever had anything to do with it.”
— Schrödinger’s remark about quantum mechanics,
quoted in Karl Popper, Quantum Theory and the Schism in Physics

4. On the dualistic particle/wave picture

“We seem to be faced with an entirely new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena, but together they do.”
— Schrödinger, Philosophy of Experimental Science

5. On the mathematical formalism being detached from reality

“What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Physical science does not want to admit this.”
— Nature and the Greeks
(Implicit criticism: the formalism of QM is abstract and not directly physical)

6. On the statistical interpretation

“The statistical interpretation is the only one that does not collapse under scrutiny, but it is regrettable because it makes the theory nothing more than an instrument of prediction.”
— Schrödinger, Science and Humanism

7. On the cat paradox (his attack on collapse-probability views)

“One can even set up quite ridiculous cases. A cat is left to itself in a closed box… The ψ-function of the entire system expresses this situation with equal parts of alive and dead. This prevents us from accepting naïvely a ‘blurred model’ for representing reality.”
— “Die gegenwärtige Situation in der Quantenmechanik”, Naturwissenschaften (1935)

8. On QM lacking a coherent picture of reality

“The present theory, in its general conceptual form, is incapable of giving a clear picture of what is happening physically.”
— Schrödinger, lecture notes quoted in J. Mehra & H. Rechenberg, The Historical Development of Quantum Theory

---

Themes in Schrödinger’s Critique

Schrödinger consistently criticized:

✔ The wave function lacking physical interpretation

He wanted a real wave in space, not a probability amplitude in configuration space.

✔ Collapse as conceptually incoherent

He viewed “collapse” as an ad hoc and unphysical rule added to save appearances.

✔ The idea that QM describes only knowledge, not reality

He disliked the epistemic view of QM that Bohr and Heisenberg promoted.

✔ The abandonment of classical realism

He believed physics must describe real processes, not just observational outcomes.

---

StdQM vs RealQM: Atomic Orbitals of Periodic Table

Standard Quantum Mechanics StdQM offers a theoretical basis for the "Aufbau" of the Periodic Table PT of atomic electron configurations in terms of the s, p, d, f and g eigenfunctions or orbitals of the one-electron Hydrogen atom depicted here as row 0-4:

The Aufbau offers an order of filling shells 1, 2, 3,...,8, with electrons in the order 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, 8s, 5g, motivated by the following modifications of a strict filling order with a shell fully filled before filling the next, as increasingly "rules of thumb":

• Pauli Exclusion Principle
• Energy Minimization
• Hund's Rule
• Madelung's rule
• Octet Rule 
• Fajan's Rule
• effective nuclear charge and shielding
• relativistic effects
• ....

This scheme is viewed to be the Aufbau theory of atoms to stay with chemistry for ever. The student novice will no doubt consider this to be a very complex scheme to grasp: The orbitals for each shell are increasingly complex and realised in a shell structure with sub-shells giving a very complex geometry. 

Is it likely that whoever created the atoms would have chosen to proceed following such a very complex scheme? Maybe not.

There is an alternative to StdQM in the form of Real Quantum Mechanics RealQM based not on the standard multi-d linear Schrödinger equation, but on a 3-d non-linear Schrödinger equation which coincides in the case of Hydrogen with one electron. 

RealQM is based on non-overlapping electron charge densities and the arrangement of electrons around a kernel becomes a packing problem with the size of electrons increasing with decreasing effective kernel attraction balance by so called kinetic energy as a form of "compression energy". 

This Aufbau starts with two half-spherical charge densities filling a 1st spherical shell around the kernel, followed by a 2nd shell with larger radius containing 2 half-shells of 2x2=4 electrons, followed by a 3rd shell filled by 2 half-shells of 3x3=9 electrons, and so on. The periods 2, 8, 18, 32 and 50 thus come out as expression of regular 2d subdivisions of shells. Very simple and fundamental. The doubling of periods into 2, 8, 8, 18, 18,..., can also be explained as coming out of successive packing. 

An Aufbau principle of packing electrons of different size around a kernel is simple, and can be understood by a student very easily. It is not impossible that it can capture some essence of real physics. 

 

ChatGPT on StdQM and the Aufbau principle:

• The standard quantum-mechanical account of the periodic table is indeed complicated.

  • The Schrödinger equation for hydrogen gives simple orbital shapes (s, p, d, f, ...).

  • But as soon as you move beyond hydrogen, electron–electron interactions, shielding, relativistic effects, and empirical rules (Hund’s, Madelung’s, Pauli, etc.) complicate the picture.

• Chemists are well aware that the Aufbau principle is more of a heuristic than a strict law—it often works, but exceptions exist (e.g., Cr, Cu, lanthanides, actinides).

• Still, StdQM provides a tested predictive framework, confirmed by spectroscopy, ionization energies, and quantum chemistry calculations.

Pauli Exclusion Principle vs Periodic Table

In 1925 Wolfgang Pauli introduced a 4th quantum number into the budding new physics of Quantum Mechanics QM in order to explain the observed two-valued periodicity of the Periodic Table PT with $2\times n^2$ electrons in shell $n=1,2,3,...$ giving the sequence $2, 8, 18, 32,..$. 

Pauli was unhappy with his "two-valuedness" or as an ad hoc pick without physics. He was comforted a bit by Uhlenbeck and Goudsmit who named it spin with two values "spin-up" and "spin-down" still without physics. 

What emerged was Pauli's Exclusion Principle PEP stating that two electrons with different spin can occupy the same atomic orbital, but not with same spin, which quickly came to serve a fundamental role in QM. But Pauli was still unhappy with PEP when he for the “discovery of PEP” in 1945 received the Nobel Prize in Physics, because PEP lacked physics and so could be “discovered” as physics, just imagination. What in fact prevented two electrons with same spin to occupy the same orbital? It was like a law prohibiting same-sex marriage, unphysical and no longer valid.

Today PEP is enforced asking wave functions to be anti-symmetric motivated by a cocktail of Lorentz invariance, locality and stability taken from (relativistic) Quantum Field Theory QFT.

The argument chain is: 

• QFT implies anti-symmetry. 
• Anti-symmetry implies PEP.
• PEP implies two-valuedness/spin.
• Two-valuedness/spin is observed in PT. 
• Conclusion: QFT, antisymmetry, PEP and two-valuedness/spin is all confirmed. 

But the logic is the incorrect logic of confirming an assumption by observing a consequence, as noted by Aristotle. 

Back to the PT: The actual periodicity observed is 2, 8, 8, 18, 18, 32, 32,.. with repetition of periods, and this is not explained by PEP not really by QM either, as remarked by Eric Scerri as authority of PT.

How then to explain the actual periodicity? Let us take a look at Real Quantum Mechanics RealQM as an alternative to the Standard Quantum Mechanics StdQM of above with anti-symmetric wave functions.

In RealQM electrons appear as charge densities with non-overlapping supports and the arrangement of electrons in an atom becomes a packing problem. It starts with the two electrons of Helium packed to occupy two half-spheres meeting at a common separating plane. 

This arrangement serves as origin of two-valuedness with the next shell to be filled consisting of two half-shells each one allowing a natural division into $2\times 2$ subdomains, the next one into $3\times 3$ subdomains altogether forming the original sequence of periods $2\times n^2$. 

The period doubling can then be explained as the result of electron packing where the next shell to be filled after 2 and 8 is not wide enough to allow division into $3\times 3$ only $2\times 2$ et cet.

It appears thus that RealQM can give an explanation of the periodicity of PT based on solid physics of packing of electron charge densities. 

Pauli passed away in 1958, and since then there is nobody questioning PEP by asking for physics. Maybe there is still reason to do so? To explain the repeated periods of PT?

PS A $n\times n$ subdivision of a half-shell reflects eigenfunction configuration of a vibrating square membrane, which connects the the orbitals of StdQM given by the eigenfunctions of the Hydrogen atom. 

 

Can QM explain the Periodic Table?

Chemist Eric Scerri as world authority on the Periodic Table asks in the article Can Quantum Ideas Explain Chemistry's Greatest Icon? (Nature 2019): 

• Does Quantum Mechanics QM explain the Periodic Table PT?  

Scerri recalls that Swedish chemist Löwdin in 1969 presented this as one of chemistry's major theoretical challenges, and adds that it still is. 

To non-expert readers of Nature and also the general public this may sound a bit surprising, since QM in its standard form stdQM based on a linear multi-dimensional Schrödinger Equation SE is viewed as the fundamental mathematical model of modern physics describing in particular the electron configurations of all atoms as the basis of PT. 

But Scerri and Löwdin say that stdQM only gives a partial explanation of PT with a lot missing. Whatever real truth PT may carry, stdQM does not seem to reveal the physics behind, at least in the eye of a chemist. But a physicist would without hesitation say that PT is well explained by stdQM, at least in principle, and that it is the job of chemists to supply the petty details. 

This state of affairs has added to the crisis of modern physics, which largely is a credibility crisis. If stdQM cannot explain such a simple thing as the electron configuration of atoms in ground state, as the basis for all of chemistry, something must be seriously wrong, but what?

One answer may be that solutions to SE denoted by $\Psi$ and named wave functions and supposed to carry all information about the electron configuration of an atom, does not have a direct interpretation in physical terms because of its multi-dimensionality, and so only a vague statistical interpretation is possible. Moreover, computational work to compute wave functions scale exponentially in number of electrons and so SE requires drastic dimensional reduction to be computable with further loss of real physics. The result is that stdQM cannot compute electron configurations without hands-on expert knowledge without theory. It is not surprising that under these conditions it fails to explain PT. 

Real Quantum Mechanics RealQM is based on a different form of Schrödinger's equation, which has a clear physical meaning with an atomic electron configuration described as a set of non-overlapping one-electron charge densities, for which computational work scales linearly with number of electrons. RealQM directly extends to molecules as Real Quantum Chemistry described in these articles. 

It seems possible that RealQM can give a better explanation of PT than stdQM, and maybe even correct PT in some details, because it has direct physical meaning and computation is affordable for all atoms.  

StdQM seeks to explain the electron configuration of an atom with many electrons in terms of the excited states of the Hydrogen atom with just one electron filling s, p, d shells. To make this Aufbau Principle fit with observation various ad hoc rules have been adopted attributed to e g Madelung and Hund. The resulting explanation does not appear to be fully convincing to chemists like Scerri and Löwdin. Something appears to be missing...

RealQM explains the electron configuration in a different way as an electron packing problem, which has a clear physical meaning and so at least has the potential to offer an explanation of PT on physical grounds or suggest corrections. Preliminary studies show promise and further studies are on the way. 

 

RealQM: Shell Structure and Periodic Table

Here is a next article in the series on RealQM posted in recent posts:

• RealQM: Shell Structure and Periodic Table

World expert on the Periodic Table Eric Scerri has investigated to what extent the Periodic Table can be explained by StdQM and seems to say that there are pieces missing in this puzzle. The article gives a first hint into explanations based on RealQM to be complemented with more details.

 

Can an Unphysical Theory Explain Real Physics?

Recent posts pose the question to what degree standard Quantum Mechanics stdQM does explain (or can explain) the Periodic Table and chemical bonding. The standard answer is that for sure this is so, since stdQM gives a complete description of the world of atoms and molecules as the most successful theory of all of physics, and that this description of course includes the Periodic Table and chemical bonding. 

But this rosy picture can be questioned on several grounds, the most basic from the fact that there is no consensus after 100 years of brooding about the physical meaning of the  Schrödinger wave function $\Psi$ as the basic element of stdQM. But there is general understanding that $\Psi$ is unobservable as something without direct physical presence and in this sense is unphysical. 

The question then presents itself: Can an unphysical theory be used to explain real physics? 

The consensus is that stdQM correctly predicts outcomes of physical experiments, as far as tested. There is not a single experiment contradicting prediction by stdQM even if outcomes for the same experiment vary substantially.  Therefore stdQM is considered to be a theory that says something about the physics of atoms and molecules. Following a mantra of "shut up and calculate" it then appears as a black box capable of predicting outcomes of experiments, while the real physics is hidden to inspection. 

We can compare with a theory about the game of darts named Dart Mechanics DM stating that the spread of an inexperienced player will follow a normal distribution, which possibly can be verified by letting novice players play many games. The DM prediction of a normal distribution can thus be seen to agree with suitable experiments, but DM will say little about experienced players. More precisely, DM cannot explain anything about the real physics of the game of dart, since no such physics was put in. 

We are led to an idea that it may be very difficult to explain real physics using stdQM, since the physics of stdQM is hidden to inspection and so to explanation. In this perspective it is not strange that the efforts to explain the Periodic Table and chemical bonding have had mixed success, despite assertions that the success is total.

Here is what chatGPT has to say about stdQM vs real physics.

Discussion with Eric Scerri on the Periodic Table

Eric Scerri as world expert on the Periodic Table has offered interesting comments to this post asking if the Periodic Table is explained by standard Quantum Mechanics stdQM, in particular if the doubling of periods in the list 2, 8, 8, 18, 18, 32 and 32, is well explained. Scerri says that it has not yet been explained (after 100 years), but that it may be possible. 

The discussion touches the core of stdQM as the fermionic nature of electrons imbedded in anti-symmetric wave functions including a two-valued spin quantum number expressed in the Pauli Exclusion Principle PEP. 

Schrödinger formulated his equation for the Hydrogen atom with one electron in 1926, which gave meaning to quantum numbers introduced earlier and so was a formidable success. Schrödinger hinted at a direct formal extension to Helium with two electrons and so on for many electrons adding new spatial variables, but that model did not work because it made no distinction between bosons and fermions. So was Schrödinger's original model quickly complemented with a requirement that multi-dimensional wave functions must be anti-symmetric as fermions with satisfaction of PEP, which became stdQM. But Pauli was not happy with his PEP, even if it gave him the Nobel Prize in Physics, viewing it to be an ad hoc fix. 

RealQM gives a different generalisation of Schrödinger's so successful equation for the Hydrogen atom, where electrons have identity by occupying different regions of a common 3d space. There is here no need of a PEP nor of a 4th spin quantum number, in the original spirit of Schrödinger and Pauli. Electron configurations appear in RealQM as the result of a packing problem of non-overlapping charge densities. It appears that RealQM can add physics to the Periodic Table with doubling of periods as a natural outcome of electron packing under energy minimisation. 

Does Quantum Mechanics Explain the Periodic Table?

Eric Scerri is a leading expert on the Periodic Table as the foundation of chemistry. This is what chatGPT says when asking the simple question to what extent the Periodic Table, formed before the emergence of quantum mechanics in the 1920s, has been explained by quantum mechanics during the now 100 years since then:

Eric Scerri, a philosopher and historian of chemistry, is known for his critical stance on the idea that quantum mechanics fully explains the periodic table. While he acknowledges the importance of quantum mechanics in understanding atomic structure and trends in the periodic table (such as electron configurations, periodicity, and chemical behavior), Scerri argues that the explanation is incomplete or limited if we rely solely on quantum mechanics.

Here are the key points of Scerri's position:

1. Quantum mechanics is useful but not fully explanatory: Scerri believes that while quantum mechanics (especially the Schrödinger equation and quantum numbers) helps rationalize some features of the periodic table, it does not provide a complete or entirely satisfactory explanation, especially for the actual order of elements.

2. Electron configurations are not derived solely from quantum mechanics: He points out that the Aufbau principle, which is used to predict electron configurations, is not a strict consequence of quantum theory but rather a semi-empirical rule. Some observed configurations (especially for heavier elements) deviate from the predicted ones.

3. Many-body problem: Scerri highlights the complexity of solving the many-body Schrödinger equation for atoms with more than one electron. Since exact solutions don't exist for such systems, approximations are required, which limits the explanatory power of quantum mechanics.

4. Philosophical perspective: He often argues from a philosophy of science perspective, suggesting that science often progresses through approximations, models, and partial explanations, rather than final, complete theories. Thus, he sees the periodic table as something emergent, not entirely reducible to quantum mechanics.

5. Chemistry's autonomy: Scerri also defends a degree of autonomy for chemistry, pushing back against the idea that chemistry is just applied physics. He sees the periodic system as a chemical construct that has its own foundations and logic, not fully subsumed under physics.

In summary, Scerri does not reject quantum mechanics, but he questions the common claim that it fully explains the periodic system. He calls for a more nuanced understanding of how different scientific disciplines interrelate, especially in the case of chemistry and physics.

We learn that Scerri as an expert in the field does not see that the Periodic Table can satisfactorily be explained by quantum mechanics and so calls for something more. But there is nothing more to grab. Quantum mechanics is supposed to be the best scientific model/explanation of the physics of atoms forming molecules forming chemistry. Quantum mechanics has fully replaced older atomic theories. If quantum mechanics cannot explain the Periodic Table, nothing can it would seem. 

The basic feature of the Periodic Table are the periods 2, 8, 18 and 32 of the form $2n^2$ with $n=1,2,3,4$, which appear as the number of atoms/elements in the rows of the table as 2, 8, 8, 18, 18, 32 and 32 thus with repetitions, as the number of electrons of the noble gases at the end of the row. What is the quantum mechanical explanation of these periods? 

The story is that this is because the eigenfunctions of the Schrödinger equation for the Hydrogen atom with one electron appear with a multiplicity of $n^2$ for $n=1,2,3,4$, which is doubled to fit with $2n^2$ by counting each electron twice. 

But the logic appears to be missing: Why expect the multiplicity of eigenfunctions of the Hydrogen atom with one electron to give the periodicity of atoms with many electrons? Why double the count of the number of electrons? Why is the periodicity repeated?

As concerns the sequence $2n^2$, there are many ways to this result. In RealQM it comes out as the solution to an electron packing problem partitioned into shells $n=1,2,3,...$ of increasing radius divided into half-shells filled with up to $n^2$ electrons allowing repetition of periods. This explanation carries the physical logic of packing of electrons of occupying different volumes depending on distance to the kernel.   

Does Quantum Mechanics Explain the Periodic Table?

A common view among modern physicists is that the Periodic Table is explained by the modern physics of quantum mechanics based on a linear multi-dimensional Schrödinger equation. 

The ground state of the one-electron distribution of a Hydrogen atom can be computed analytically to be given by a spherically symmetric wave function decaying with the distance $r$ from the kernel like $\exp (-r)$, which is referred as a $1S$ state, with energy $-\frac{1}{2}$ Hartree in full agreement with observation.  

The ground state of Helium as the next element in the table with two electrons, is presented to be $1S^2$ in the form of two electrons with the same $1S$ wave function sharing space around the kernel (with different so called spin to coexist). The ground state energy computes to $-2.75$ Hartree, while observed energy is $-2.903$ Hartree. The state $1S^2$ is thus not the ground state. 

When confronted with this fact, ChatGPT gives the following response, after consulting the literature:

• The $1S^2$ state is not fundamentally correct as the ground state of helium because it fails to account for essential electron correlation and yields an incorrect energy. While it can serve as a useful approximation and educational tool, the true ground state is a correlated quantum state that cannot be reduced to $1S^2$. Thus, any claim of its "conceptual correctness" must be qualified and limited to its role as a crude first-order approximation.
• The $1S^2$ concept can be misleading if presented as the full description of helium's ground state. While it serves as a useful approximation and pedagogical tool, it oversimplifies the true quantum nature of the system. To avoid misconceptions, it’s crucial to contextualize $1S^2$ as a starting point in understanding helium, not the definitive or complete description.

We understand that presenting $1S^2$ as the ground state of He is incorrect and so can be seriously misleading, yet it is supposed to be the quantum mechanical explanation of the second element in the periodic table! What then about the rest of the table based on excited states of Hydrogen? 

RealQM presents a different ground state for Helium consisting of two electrons occupying two separate half-spaces through the kernel, for which the energy agrees with observation. More generally, RealQM explains the periodic table as a packing problem of non-overlapping one-electron charge densities meeting at a Bernoulli free boundary, which appears as a better explanation of the period table than the accepted standard.  RealQM is based on a new Schrödinger equation in the form of classical continuum mechanics in 3 space dimensions.

Connecting to the previous post, the idea of a Helium ground state as $1S^2$ came out from the new quantum mechanics emerging 100 year ago as way to understand the atomic world, which today according to ChatGPT however appears to have missed the target. Instead of increasing understanding it fosters mis-understanding. A nice feature of ChatGPT compared to a living scientist heavily invested in a certain school of thought, is that hiding truth/information is not securely built in. 

More precisely: The ground state of Helium as $1S^2$ of the accepted view with incorrect energy, should be compared with the conceptually fundamentally different ground state of RealQM with correct energy. What is your choice? 

  

What Can Quantum Mechanics Predict?

The previous post recalled the fact that standard Quantum Mechanics QM based on a $3N$ spatial dimensional Schrödinger equation for a system with $N$ electrons is effectively uncomputable for almost all atoms in the periodic table. With a resolution of $10^2$ in each spatial dimension, the total number of mesh points is daunting $10^{6N}$ which fills any thinkable computer already for $N=5$ (Boron). The wave function for all atoms beyond Boron is thus uncomputable. This means that the wave function for almost all atoms in the period table is uncomputable!

We compare with the standard message by physicists that all predictions of QM perfectly match with observation, or more precisely that there are no observations which contradict QM. 

The crucial question is then what predictions QM offers? Does QM predict the periodic table? A physicist would say that certainly this is so, in principle, while precise computations are lacking, because the full wave function is uncomputable. 

What can be computed are various low-dimensional approximations based on a variety of ad hoc assumptions. If such an ad hoc approximate solution happens to match observation, it is accepted as a demonstration that the full wave function (although unknown) also matches observation. If the ad hoc solutions does not match observation, it is discarded. With this strategy it is impossible to find a contradiction between theory and experiment! Success story!

But there is a caveat. Do you see it? Compare with How far does quantum mechanics explain the periodic table?

Real Quantum Mechanics presents a new Schrödinger equation as a non-linear system of non-overlapping electron charge densities sharing a common 3d space, thus a classical continuum model, which is computable for many electrons. A laptop computation for H2O is reported here with more to come.

Summary: QM in fact predicts very little (ab initio without ad hoc assumptions), which is consistent with the fact that there is no observation in contradiction with a QM prediction. But a theory which predicts nothing is not falsifiable and as such not a scientific theory. This is the reason behind the crisis of modern physics based on QM witnessed by so many physicists. Real QM opens a door out of the crisis. Why not give it a try? Permanent QM crisis over 100 years is not healthy to physics.  

Finite Element Quantum Mechanics 3: Explaining the Periodicity of the Periodic Table

According to Eric Scerri, the periodic table is not well explained by quantum mechanics, contrary to common text book propaganda, not even the most basic aspect of the periodic table, namely its periodicity:

• Pauli’s explanation for the closing of electron shells is rightly regarded as the high point in the old quantum theory. Many chemistry textbooks take Pauli’s introduction of the fourth quantum number, later associated with spin angular momentum, as the foundation of the modern periodic table. Combining this two-valued quantum number with the ear- lier three quantum numbers and the numerical relationships between them allow one to infer that successive electron shells should contain 2, 8, 18, or $2n^2$ electrons in general, where n denotes the shell number. 
• This explanation may rightly be regarded as being deductive in the sense that it flows directly from the old quantum theory’s view of quantum numbers, Pauli’s additional postulate of a fourth quantum number, and the fact that no two electrons may share the same four quan- tum numbers (Pauli’s exclusion principle). 
• However, Pauli’s Nobel Prize-winning work did not provide a solution to the question which I shall call the “closing of the periods”—that is why the periods end, in the sense of achieving a full-shell configuration, at atomic numbers 2, 10, 18, 36, 54, and so forth. This is a separate question from the closing of the shells. For example, if the shells were to fill sequentially, Pauli’s scheme would predict that the second period should end with element number 28 or nickel, which of course it does not. Now, this feature is important in chemical education since it implies that quantum mechanics can- not strictly predict where chemical properties recur in the periodic table. It would seem that quantum mechanics does not fully explain the single most important aspect of the periodic table as far as general chemistry is concerned. 
• The discrepancy between the two sequences of numbers representing the closing of shells and the closing of periods occurs, as is well known, due to the fact that the shells are not sequentially filled. Instead, the sequence of filling fol- lows the so-called Madelung rule, whereby the lowest sum of the first two quantum numbers, n + l, is preferentially oc- cupied. As the eminent quantum chemist Löwdin (among others) has pointed out, this filling order has never been derived from quantum mechanics. 

On the other hand, in the new approach to atomic physics I am exploring, the periodicity directly connects to a basic partitioning or packing problem, namely how to subdivide the surface of a sphere in equal parts, which gives the sequence $2n^2$ by dividing first into two half spheres and then subdividing each half spherical surface in $n\times n$ pieces,  in a way similar to dividing a square surface into $n\times n$ square pieces.  With increasing shell radius an increasing number of electrons, occupying a certain surface area (scaling with the inverse of the kernel charge), can be contained in a shell. 

In this setting a "full shell" can contain 2, 8, 18, 32,.., electrons, and the observed periodicity 2, 8, 8, 18, 18, 32, 32, with each period ended by a noble gas with atomic numbers 2 (He), 10 (Neon), 18 (Argon), 36 (Krypton), 54 (Xenon),  86 (Radon), 118 (Ununoctium, unkown), with a certain repetition of shell numbers, can be seen as a direct consequence of such a full shell structure, if allowed to be repeated when the radius of a shell is not yet large enough to house a full shell of the next dignity. 

Text book quantum mechanics thus does not explain the periodicity of the periodic table, while the new approach am I pursuing may well do so in a very natural way.   Think of that.   

Quantum Contradictions 5: The Periodic Table

The question How Good is the Quantum Mechanical Explanation of the Periodic System? is answered by Eric Scerri by: Not Very Good:

• I would like to issue a caution regarding the extent to which the periodic table, for example, is truly explained by quantum mechanics so that chemical educators might refrain from overstating the success of this approach.

Quantum mechanics offers a standard explanation in terms of the Hydrogen orbitals combined with Pauli's exclusion principle identifying electrons by 4 quantum numbers (n, l, m, s) with n = 1, 2, 3,... the principal shell number,  l = 0,..., n -1 the azimuthal quantum number, m ranging from - l  to + l the magnetic quantum number and s = +- 1/2 the spin quantum number. 

The basic idea is that shells are sequentially filled with electrons in a certain order (s,p,d,e,..,) within each shell.  The trouble is that shells are not sequentially filled with an inner shell being full before an outer shell starts to get filled. The observed order is instead (usually) by the number  n + l referred to as Madelung's rule complemented by Hund's rule, which both appear to be introduced ad hoc without quantum mechanical rationale. 

The quantum mechanical explanation of the periodic table thus requires three additional ad hoc rules (Pauli, Madelung and Hund) and the advice of Scerri to not overstate its success seems to be well founded.

It is in fact unclear what the standard quantum mechanical explanation based on Hydrogen orbitals, does offer as explanation of the periodic table.  Even the basic numbers 2, 8, 8, 18, 18, 32,..., of electrons in filled shells require doubling from the sequence of Hydrogen orbitals 1, 4, 9, 16,..., by ad hoc introduction of spin combined with Pauli's exclusion principle.

And so it remains to give a more convincing explanation of the periodic table using a form of quantum mechanics without ad hoc attributes. There are indications that there is such a version, which will be the subject of upcoming posts. As preparation, browse The Periodic Table: Its Story and Significance, by Eric Scerri.

Quantum Contradictions 4: Helium isoelectronic sequence

The postulated ground state of 2-electrons atoms/ions according to the NIST Atomic Spectra Database,  is the 1s^2 state of the He isoelectronic sequence (2 electrons with opposite spin and identical spatial wave functions with exponential decay away from the kernel). But the energy of the postulated 1s^2 state is not the observed energy as shown in the following list with the experimental value (in eV) first and the larger energy of the 1s^2 state in parenthesis:

1. He : - 79 (- 74.80)
2. Li+ : - 198.09 (- 193.80)
3. Be+2 : - 371.60 ( - 367.20)
4. B+3 : - 599.58 ( - 595.00)
5. C+4 : - 882.05 (- 877.20).

It appears that in the true ground state with energy in accordance with observation, the two electrons are separated into roughly speaking two half-shells together creating an electron wall shielding the kernel. The appearance of such a wall could possibly explain why a third electron will be pushed away into an outer shell without resort to any two-electron Pauli exclusion principle.

The fact that the postulated ground state in the He isoelectronic sequence is not the actual ground state gives the postulated quantum mechanics support of the periodic table, based on Hydrogenic orbitals combined with Pauli's exclusion principle, a bad start with possibly catastrophical consequences concerning scientific rationale.

PS Pauli's exclusion principle is an ad hoc assumption which serves to eliminate e.g. a 1s^3 state of Lithium with 3 electrons sharing a 1s state, which shows to have energy smaller than that observed and so must be unphysical in some sense. If already the 1s^2 state is unphysical, this may as suggested prevent the appearance of a 1s^3 state without any ad hoc exclusion assumption.