What Is the Shape of an Atom?

Standard Quantum Mechanics StdQM presents a picture of an atom as being built from electron configurations/orbitals represented by the eigenfunctions for the Hydrogen atom H as the excited states of its one and only electron according to the following partial list:

There is no real rationale for this choice of representation for the electron configuration of an atom with many electrons, other than the completeness of the set of Hydrogen eigenfunctions like any other set. 

The idea is thus to build many-electron configurations as combinations of the above list. Is it a good idea? Does it appear to be complicated? Is it likely that real physics is built this way from some storage of Hydrogen eigenfunctions? It has the same ad hoc character as the Linne's Sexual System classifying  plants into groups based on the arrangement of stamens and pistils, today completed in the Standard Model of particle physics.

We see in the above collection a presence of a shell structure which connects to a different representation of a many-electron configuration as organized in an expanding sequence of shells with at most $2*n^2$ electrons in shell $n=1,2,,$. Here is this shell structure for Iron complemented by electron orbitals below

RealQM is based on a new form of Schrödinger equation for a collection of one-electron wave functions with non-overlapping supports meeting at a Bernoulli free boundary, which for an atom has a shell structure carrying the supports and where the actual electron distribution comes out as a packing problem under energy minimisation. You can inspect the representation of a Gold atom in this code by pressing start.   

The basic shape of an atom in RealQM in spherical symmetry thus consists of a sequence of non-overlapping spherical shells with a certain number of equally distributed electrons in each shell, like a spherical onion built from spherical shells:

In this model there is no need of complicated Hydrogen electron orbitals from the above lists. Using Ockham's razor we can thus leave these out and make life much simpler as concerns mathematical modeling of atoms. This model meets the requirement of Anschaulichkeit demanded by Schrödinger, but not met by StdQM.

For molecules, RealQM offers a richer set of shapes formed by shell interaction of outermost shells of several atoms. See label RealQM Quantum Chemistry.

Swedish Stupidity

Sad to say the following video all too well describes the Stupidity of Swedish Foreign Policy of today:

In March 7 2024 Sweden gave up its 200 years of neutrality by joining NATO in its proxy war against Russia to expand NATO into Ukraine. This happened without debate over night and since then the Swedish people has been led by massive propaganda to prepare for war with opposition to war efficiently suppressed. 

Bonhoeffer says that stupidity has no moral, which in Sweden today is expressed in the mantra that the death of 1 million Ukrainian soldiers is justified because it has saved Sweden from Russian invasion.  

Bonhoeffer’s Argument on Stupidity

1. Stupidity is more dangerous than evil – While evil can be confronted and exposed, stupidity is much harder to counter because a stupid person is often unaware of their own ignorance.
2. Stupidity is not an intellectual defect but a moral one – Bonhoeffer did not see stupidity as a lack of intelligence but rather as a failure to think critically or ethically. It arises when people allow themselves to be manipulated.
3. People become stupid in groups – He observed that individuals often lose their ability to think independently when they conform to mass movements, propaganda, or authoritarian regimes. Stupidity thrives in environments where people surrender their autonomy to power.
4. Stupid people are resistant to facts – Reasoning with a stupid person is almost impossible because they refuse to listen to reason. Instead, they cling to slogans and ideologies fed to them by those in power.
5. The only way to counter stupidity is through inner liberation – Bonhoeffer believed that true wisdom comes from a deep ethical and spiritual awakening, not just education. A society can only free itself from stupidity when people take responsibility for their thinking.

Weak Reactivity of Gold Explained by RealQM

This is a follow up on the two previous posts letting RealQM explain the weak reactivity of Gold Au as compared to the strong reactivity of Caesium Cs both in 6th row of the periodic table having one valence electron. The same argument applies to Silver Ag vs Rubidium Rb (5th row) and Copper Cu vs Potassium K (4th row). 

Recall the following electron configurations (number of electrons in expanding sequence of shells)

• Au: 2+8+18+32+18+1    79 electrons in  6 shells
• Cs: 2+8+18+18+8+1      55  electrons in 6 shells   
• Ag: 2+8+18+18+1          47  electrons in 5 shells
• Rb: 2+8+18+8+1            37   electrons in 5 shells
• Cu: 2+8+18+1                 29   electrons in 4 shells
• K: 2+8+8+1                     19   electrons in 4 shells

We observe that the electron sequences for Au, Ag and Cu end with 18+1, while Cs, Rb and K end with 8+1. 

We compute using RealQM in a spherical symmetric form to get the following radius R in atomic units of the outermost shell containing the valence electron 

• Au: R = 2.94  (run code by pressing start)
• Cs: R = 3.36   (code)
• Ag: R = 2.45  (code)
• Rb: R= 2.85    (code)           

We observe that the radius for Au is somewhat smaller than that of Cs although Au contains 22 more electrons than Ce. The same pattern for Ag vs Rb and Cu vs K. 

In the previous post we gave explanations using RealQM along the following lines of thought:

1. Weaker reactivity by more tightly bound valence electron for 18+1 than 8+1.
2. Weaker reactivity down the column of Cu, Ag and Au because binding by electron sharing is increasingly counteracted.    

Here 1. expresses common insight, while 2. is new insight brought by RealQM, which can be illustrated as follows:

Small radius of inner shells (red): Binding (green)

  

Large radius of inner shells (red): Small binding (green).

Recall that StandardQM offers an explanation based on an idea that a Gold atom has inner electrons reaching half the speed of light with increased mass binding the valence electron. It is a mind-boggling explanation. More precisely, the value of relativity theory is that it can explain any observation outside mainstream science as a relativistic effect.  That the relativistic effect is very strong for Au(79) but very weak for Cs (55) is hard to accept, since they can differ by at most a factor $(55/79)^2\approx 0.5$. 

Here is an illuminating chat with GPT. In particular the seemingly contradictory fact that Francium Fr (87) has stronger relativistic effect and stronger reactivity is discussed.

Here are the basic (3) lines of the RealQM code: update of electrons, potential and free boundary:

//Update of shell-electron wave functions

for (var q=1;q<Q+1;q++){

  for (var i=M[q-1]+1;i<M[q]-1;i++){

  u[i]=u[i]+0.5*dt*(u[i+1]-2*u[i]+u[i-1])/pow(h,2)+0.5*dt*(u[i+1]-u[i-1])/(h*i*h) + dt*K[i]*u[i] -       dt*2*P[i]*u[i];

 if (q<Q}

  u[M[q]]=u[M[q]+1];

  u[M[q]-1]=u[M[q]-2];

}

}

//Normalisation of shell-electron charge

  normu[q]=0;

  for (var i=M[q-1];i<M[q];i++){ 

  normu[q]= normu[q] + pow(u[i]*(i*h),2)*h;

  }

  for (var i=M[q-1];i<M[q];i++){ 

  u[i]=sqrt(E[q])*u[i]/sqrt(normu[q]);

}

//Update of shell-electron potentials

for (var q=0;q<Q+1;q++){

  for (var i=M[q]+1;i<M[q+1]+1;i++){

  P[i]=0;

  for (var j=1;j<N+1;j++){

  P[i]=P[i]+0.5*pow(u[j],2)*pow(j*h,2)*h*min(1/(i*h),1/(j*h));

  if (j>M[q] && j<M[q+1]){

  P[i]=P[i]-0.25*pow(u[j],2)*pow(j*h,2)*h*min(1/(i*h),1/(j*h));

  }

  }

  }

}

//Update of free boundary

for (var q=1;q<Q;q++){

  if (u[M[q]]>u[M[q]-1]+diff){  

  M[q]=M[q]-2;

  }

  if (u[M[q]]<u[M[q]-1]-diff){

  M[q]=M[q]+1;

  }

}

 

RealQM: Ionization Energies for Atoms with 1 Valence Electron

We will now let RealQM compute the 1st ionization energy E in for atoms with 1 valence electron and compare with the following data from NIST Atomic Data base including electron configuration with number of electrons in a sequence of shells with increasing radii: 

• H        1                                          E = 0.5000 Hartree
• Li       2+1                                      E =  0.198
• Na      2+8+1                                  E = 0.188
• K       2+8+8+1                              E = 0.159 
• Cu     2+8+18+1                            E = 0.213
• Rb     2+8+18+8+1                        E = 0.154
• Ag     2+8+18+18+1                      E = 0.205
• Ce     2+8+18+18+8+1                  E = 0.143
• Au     2+8+18+32+18+1                E = 0.339
• Fr      2+8+18+32+18+8+1            E = 0.145 (est)

We see two remarkable features of E: 

1. Steady decrease H, Li, Na, K, Rb, Ce, Fr all with shell sequence ending with 1 or 8+1.
2. Higher energy for Cu, Ag and Au all ending with 18+1.

We now compare with RealQM recalling that electrons have non-overlapping supports meeting at a free boundary and so form a sequence of distinct spherical shells with a certain number of electrons in each shell with the electron wave functions meeting at a free boundary between shells with continuity and zero normal derivative. 

We can thus model an atom with one valence electron as a +1 kernel surrounded by one electron wave function with support outside some distance R to the kernel, with thus the void with distance smaller than R representing the net of the ion without the valence electron. We compute the following energies for different R in atomic units using this code (with print out for R = 2 below):

• R = 0            E = 0.500
• R = 0.4         E = 0.426
• R = 0.6         E = 0.368
• R = 0.8         E = 0.320
• R = 1.0         E = 0.280
• R = 1.2         E = 0.247
• R = 1.4         E = 0.219
• R = 1.6         E = 0.196
• R = 1.8         E = 0.177
• R = 2.0         E = 0.161
• R = 2.2         E = 0.149
• R = 2.4         E = 0.141

We see a steady decrease of E with increasing R in the code, which we can match to the above NIST data as follows:

• Li     R = 1.6
• Na    R = 1.8
• K      R = 2.0
• Rb    R = 2.2
• Ce    R = 2.4

This is to be expected since R increases with number of shells as long as the electron configuration ends with 8+1. 

We now turn to Cu, Ag and Au having larger E which matches to smaller R. The radius of Cu thus comes out as smaller than that of K, that of Ag smaller than Rb, and that of Au smaller than Ce. We can see this as an effect of the 18+1 ending of these atoms with electrons densely packed into an 18 shell.

Altogether RealQM matches with observation as concerns 1st ionization of atoms with one valence electron. The relative success can be attributed to the fact that in RealQM electrons only have support in one shell, to be compared with Standard Quantum Mechanics where electrons have presence in all shells.  

R=2.0 Electron in red/yellow around void in white.

I will complement with RealQM for full shell configurations giving information on relevant R for each atom.

Test of RealQM vs StdQM: Reactivity of Gold

The electronic configuration for Gold Au is supposed to be 2 + 8 + 18 +  32 +  18 +  1 with a single electron in shell 6. Gold is very weakly reactive despite its single valence electron.

We compare with Sodium Na with configuration 2 + 8 + 1 with a single valence electron in shell 3. Sodium is highly reactive. Same for Lithium Li 2 + 1 and Potassium Ka 2 + 8 + 8 + 1.

What is the difference between Au and Li, Na and Ka, all with one single valence electron?

Standard Quantum Mechanics StdQM says that the weak reactivity of Gold is a relativistic effect coming from the 2 electrons in shell 1 moving at speed more the half the speed of light increasing the mass of the inner shell causing contraction of shell 6. This is a mind boggling explanation. To buy it requires a vivid fantasy. Physicists/chemists/scientists generally seek to come up with an explanation of some observed phenomenon rather than admitting that they have no clue, unless the phenomenon is of such grand scale that human understanding is beyond reach, like Big Bang. The weak reactivity of Gold has small scale and should be possible to understand. 

Let us compare the answer offered by RealQM. We then model a Gold atom Au as a +1 kernel surrounded by an electron which occupies a region around the kernel with distance bigger than R to the kernel representing all shells but the last. We seek to determine if two Au atoms can combine to form a molecule Au2 as a test of reactivity of Au. 

We run RealQM varying R and the distance between the two kernels using this code and get the following results for the energy E1 in Hartree (as minimal energy) at small distance D1 in atomic units and E2 as energy at large distance D2:

• R = 0     E1 =-1.17  at D1 = 1.4       E2 = -1     at  D2 = 4
• R = 0.5  E1 = -0.87  at D1 = 2         E2 = -0.85 at D2 = 4
• R = 1     E1 = -0.51  at D1 =2          E2 = - 0.63 at D2 =4

The case R = 0 corresponds to H2 with energy release 0.17 under formation of molecule.

The case R = 1 corresponds to Au2 which needs input of 0.12 to form.

The case R = 0.5 is intermediate with very small energy release 0.02.

The radius R increases with number of shells along with weakening of reactivity.   

We see that RealQM can explain the weak reactivity of Gold as an effect of increasing radius as an effect of increasing number of shells. To understand the essence of the argument, take a look at this output of RealQM for R = 1 at D1 = 2.1 showing the regions representing all the shells except the last, preventing accumulation of electron charge density between the kernels an so preventing chemical binding:

It is a simple picture, easy to understand and it has nothing to do with electrons inside atoms at relativistic speeds. The essence is that accumulation of charge density between the kernels (red) with binding effect is reduced with increasing R (white). 

PS RealQM in the above setting computes the 1st ionization energy of Gold to 0.33 Hartree in accordance with observation.

Sweden Continues War Against Russia

The Swedish Prime Minister Ulf Christerson is together with European leaders in Kiev to give support to continued war against Russia on Ukrainian land expressed in the following joint statement Febr 24 together with the Presidents of Finland, Latvia, Lithuania and Ukraine and the Prime Ministers of Denmark, Estonia, Iceland, Norway:

• We unequivocally condemn Russia's ongoing illegal, unprovoked and unjustified full-scale military invasion of Ukraine. 
• Our main priority is to strengthen Ukraine. We will further bolster our support. 
• Europe needs to demonstrate its leadership .... to achieve peace through strength.
• We pledge to provide more support to Ukraine, including air defence and ammunition. 
• We will continue to put maximum pressure on Russia, including through sanctions and measures against the shadow fleet to constrain Russia. 
• We underline that just and lasting peace will not be possible without accountability for Russia’s international crimes.
• Together, we also continue to support Ukraine on its irreversible path to full Euro-Atlantic integration, including NATO membership.

This is nothing but a declaration of war against Russia by Sweden &Co. Peace is War. War is Peace.

US will seek an agreement with Russia and leave the fighting to Sweden &Co. Sweden has less than 10.000 soldiers to do the dirty work, including 600 young men and women sent to Latvia for 6 month service.  

Why is the Swedish Prime Minister leading the Swedish people and society into full destruction?  

Why does not the Swedish People stand up and say No to this destruction? 

Here is perspective from the side of Russia which must be part of a serious discussion seeking peaceful cooperation and not war.  The Swedish Prime Minister is not aware of such a perspective by listening only to Swedish Big Media.

ChatGPT on Covalent Chemical Bond

Here is a transcript of a conversation with chatGPT 4.0 on the physics of covalent bond as the main form of chemical bond supposedly arising from "sharing of electrons" between atoms in a molecule as a main theme of recent posts. I am informed that 

• Our current understanding of chemical bonding is one of the great successes of quantum mechanics.
• At a conceptual level, we now know that chemical bonds arise from the interplay between electron kinetic energy and the Coulombic attractions between electrons and nuclei.
• In summary, while the core physics of chemical bonding is well established and successfully applied in many areas, key challenges remain.
• There is, in fact, a broadly accepted quantum mechanical view of chemical bonding. The core idea is that bonds—especially covalent bonds—arise from the sharing of electrons through the constructive interference of atomic orbitals, which results in an increased electron density between nuclei and a lowering of the system’s energy.
• So while the overall quantum mechanical picture of bonding is well established and widely accepted, the “extra” details—how exactly to describe, partition, and interpret a bond in a way that aligns perfectly with our classical intuition—remain active areas of research. 
• The broadly accepted view is that chemical bonds form when electrons are not confined to individual atoms but instead occupy molecular orbitals that extend over two or more nuclei. This “sharing” of electrons—lowers the total energy of the system and creates stability.
• ...the detailed picture of exactly how this sharing occurs is described in multiple, often complementary, ways. 
• So, while everyone agrees on the overall principle that bonds form due to electron sharing, the “fine print”—the precise partitioning, the weighting of different interactions, and the best way to capture these details in a simple picture—remains subject to ongoing investigation and debate. 
• The full quantum mechanical description of chemical bonds is governed by the many‐electron Schrödinger equation—a set of precise, well‐defined mathematical rules. In principle, this description is unambiguous and exact. However, because the resulting many-electron wavefunction is extremely complex and high-dimensional, it isn’t something we can “inspect” or visualize directly in simple, chemical terms.
• In summary, there is broad agreement on the precise underlying quantum mechanics, but because the full description is not directly accessible to our intuition, different models are used to interpret it, leading to differences in the way we talk about chemical bonds.

What I learn from this conversation with chatGPT, the only available partner since real chemists stay silent, is that there is a precise quantum mechanical description of covalent bond, but unfortunately this description is behind a pay wall for human intelligence. Is this a true description of state of the art of the  physics of molecules? What if it is true?

Key Test: Why Does not Helium He form He2 Molecule?

The crown jewel of modern physics is Schrödinger's equation as a mathematical model describing the physics of atoms and molecules in a new form of mechanics named quantum mechanics fundamentally different from classical mechanics of macroscopic objects, which we refer to as Standard Quantum Mechanics StdQM. 

Schrödinger's equation was formulated in 1926 coming with the expectation that it would also cover chemistry as the physics of atoms and molecules, in particular the physics of chemical bonding when atoms combine to form molecules under release of energy on a path towards atomic energy minimization.  

But concrete results did not come easy because the Schrödinger equations showed very difficult/impossible to solve because of its many dimensions, and so could not really deliver new understanding of how chemical bonds are formed and so chemists continued to rely on heuristic models such as octet rule, Lewis Structure, Valence Shell Electron Pair Repulsion VSEPR, molecular orbital hybridization, and electronegativity. 

It appears that today the physics of chemical bonding, covalent or ionic or a mixture, is still not uncovered in the form of solutions of Schrödinger's equation, still impossible to solve without drastic simplification coming with unknown errors.

RealQM is a new alternative to StdQM as a form of classical continuum mechanics in 3 space dimensions, which is computable for atoms/molecules with many electrons. 

Let us now see what StdQM and RealQM have to say about the physics of a covalent bond between two Helium He atoms each with two electrons around a kernel of charge +2, formed by interaction between the electrons into a He2 molecule.  

A covalent bond will form if the total energy of He2 at some kernel distance of about 1 Å or 2 atomic units with substantial spatial presence of electrons from both atoms, is less than that of two separate He atoms.

StdQM tells that a covalent bond will not form, because it is prevented by strong repulsion between the electrons of one atom and those of the other, with reference to the Pauli Exclusion Principle. 

RealQM tells that there is no reason for a covalent bond to form since the energy of two He atoms stays essentially the same under approach until kernel repulsion makes it increase. 

To make a verdict between StdQM and RealQM, let us recall the following observations of release of energy in Hartree when elements in the 2nd row of the periodic table form diatomic molecules:

• Li2    E = 0.04
• Be2   E = 0.004
• B2     E = 0.061
• C2     E = 0.23
• N2     E = 0.360
• O2     E = 0.19
• F2      E = 0.059

We see a decrease towards zero as we move left in the table (up in the list) from maximal energy 0.360 for N2 to very small for Be2 and Li2, which extrapolates to 0 for He2. This fits with RealQM showing no change of energy of He2 under atomic approach as outlined in the two previous posts. Extrapolation down the list suggests 0 for Neon2. 

But it contradicts StdQM claiming strong electronic repulsion asking for strong input of energy.  

In short: RealQM gives no net electronic repulsion for He2, while StdQM gives strong electronic repulsion (of size 0.37-0.44 Hartree according to chatGPT web search).

Direct observation of the energy of He2 under approach is said to be difficult because of the assumed strong repulsion, and so is not available. But if there is no repulsion, then it should be possible to measure that as a null result.

We find a situation where observation gives indirect support to RealQM as no repulsion, rather than StdQM as strong repulsion. 

A direct experiment could give direct support to no repulsion, but no such experiment has been performed with the motivation that it would be impossible because of strong repulsion.  The experiment that could refute StdQM is thus deemed to be impossible, and so has not even been attempted. Or so it seems...

 

Can we ask experimenters to measure the electronic repulsion between He atoms? It does not appear to be difficult... 

Uncovering Secrets of Chemical Bonds

In the previous post we used RealQM to uncover the formation of a H2 molecule as two Hydrogen H atoms at a kernel distance of 1.4 atomic units au find a electron configuration of minimal energy 0.17 Hartree below that of two separate atoms, thus forming a chemical bond with dissociation energy 0.17 Hartree.

RealQM also uncovered why two Helium He atoms do not form a chemical bond to a He2 molecule, that is why they find no kernel distance with lower energy than separate atoms. 

RealQM thus offers an explanation why H2 forms but not He2 and so reveals some of the secret of the chemical bond, which appears to be hidden in Standard Quantum Mechanics StdQM. 

Recall that the essence of the RealQM explanation of chemical bond is the presence of electron density accumulation between the kernels with two electrons meeting with non-zero density thus allowing decrease of potential energy forming a bond without increase of kinetic energy. 

In this post we let RealQM compute the dissociation energy E in Hartree of an X2 molecule with X an atom with 2 valence electrons outside a sphere around the kernel of varying radius R in au: 

• R = 0         E = 0
• R = 0.5      E = 0.09
• R = 1         E = 0.33
• R = 1.2      E = 0.28
• R = 1.5      E = 0.17 

which we can compare with the following experimental data for 2nd row of the periodic table

• Li2            E = 0.04
• Be2           E = 0.004
• C2             E = 0.23
• O2             E = 0.19

and for the 3rd row

• Na2           E = 0.027 
• Mg2          E = 0.00011
• Si2            E = 0.073
• S2             E = 0.11   

We see a qualitative match between RealQM and observation with dissociation energy starting at 0 for He slowly increasing to a maximum for C2 (R=1) and then decaying to O2 (R=1.2), with the same pattern in the 3rd row.

Let us now summarise RealQM for X2 molecules with 2 valence electrons recalling the previous post and starting by inspecting the following output (3d ab initio computation on uniform grid of mesh size 0.1 au with code link above):

We see two atomic kernels in black each surrounded by a 2-electron cloud/wave function in red (2d plot on plane trough the kermels), each 2-electron wave function  consisting of 2 half-spherical shell 1-electron wave functions meeting at a atomic free boundary slightly shifted outward from each kernel. We also see the inner half-spherical electron shell of each 2-electron meeting at a free boundary midway between the kernels. The yellow curve is a cross-cut of electron wave function with the dips showing the free boundaries. The radius of the inner shell system of X is 1 with 2 valence electrons outside which forms a chemical bond with dissociation energy 0.33 Hartree as shown above.

After proper inspection of the image, we can now understand the physics of chemical bonding of atoms with 2 valence electrons, as follows:

• Electron accumulation between the kernels add to binding.
• Electron accumulation between the kernels push the interatomic free boundary outwards to reach continuity with outer half-shell wave function, which subtracts from binding.
• There is a net binding effect which decreases proceding to the left in each row of the periodic table towards 0 for He and Neon. 
• In short: The presence of inner shells allow outer 2 valence shells to form a bond. 
• He does not have any inner shell and so does not bond to He2. 

The binding of atoms with 1 valence electron follows the physics of the H atom. We have thus reached some understanding of chemical binding of diatomic molecules with 1 or 2 valence electrons. 

It remains to consider the case with 3 valence electrons (Nitrogen N as 2+2+3) and 4 (Carbon C as 2+4), which can be expected to be similar to the case of 2 just considered. Together it will form a new theory of chemical bonding based on a new atom model in the form of RealQM. 

PS Concerning the state of the art of understanding of chemical bonding, let me quote from the recent article The Quantum Origin of the Periodic Table - The Role of Electrons Dynamics and Energy Degeneracy in Atomic Reactivity and Chemical Bonding by prof Sture Nordholm University of Gothenburg:

• It would be reasonable to also expect quantum chemistry to produce better understanding and more refined physical models of bonding but, unfortunately, this development has lagged behind high expectations. The access to rigorous and accurate quantum chemical methods has certainly helped understand but not greatly revise and improve, the simple models above, nor resolved the historical debate over the best physical interpretation of the computational results.

In short, chemical bonding does not appear to be well understood by expert chemists of today.

Why H Forms Molecule H2, while He Does not form He2

Why does the Hydrogen atom H form a H2 molecule, while the Helium atom He does not form a He2 molecule?

Standard Quantum Mechanics StdQM offers a vague explanation of physics:

• The two electrons of H2 combine to form a bond.
• Out of the four electrons of He2, two electrons combine to form a bond and the other two electrons form an anti-bond, altogether a no-bond. 

StdQM computation (without explanation of physics) agrees with observation:  

• H2 exists as a molecule with minimal energy -1.17 Hartree at kernel distance 1.4 atomic units au, compared to the energy -1 of separated H atoms. The dissociation energy of H2 is thus 0.17 Hartree.
• He2 does not form a molecule since the energy of He2 for kernel distances smaller than 5 atomic units is not lower than that of separated He atoms.  

StdQM computations for He2 is viewed to be very difficult because of the 12 spatial dimensions involved for 4 electrons and so results are very scarce if any.  

Let us now show that RealQM as a new alternative to StdQM delivers (i) results in agreement with StdQM computation and (ii) explanation of physics more convincing than StdQM. 

RealQM computations with this code on a uniform 3d grid with mesh size 0.1 au gives the following energies E depending on kernel distance D normalised to zero for maximal distance D=4 au:

•  D = 4       E = 0.00
•  D = 3.2    E = -0.04
•  D = 2.4    E = 0.003
•  D = 1.6    E = 0.20

We see essentially no energy drop from 4 to 1.6 au giving the message that two He atoms do not form a chemical bond into He2. The total energy of two He atoms is -5.806 and so the drop of 0.04 for D=3.2 represents a relative drop less than 0.01 on a 0.1 au grid in accordance with second order convergence,

RealQM offers the following explanation of the physics of bond for H2 and no-bond for He2:

The secret of the bond for H2 is hidden in the electron configuration of RealQM as consisting of non-overlapping electron densities. As two H atoms approach the two electron densities meet at a free boundary which is a midway normal plane to the line between the kernels with equal non-zero density together with vanishing normal derivative on both sides of the plane. This means that electron density can accumulate between the kernels thus decreasing kernel potential energy by profiting from both kernels, without increase of kinetic energy, thus forming a bond. This explanation is different from that given by StdQM based on bonding orbitals, and is based on the new electron configuration of RealQM. 

You can follow the physics by running this code. Very educating.

With the explanation for H2 in mind we now turn to He and meet the question why apparently the same explanation does not work for He. We recall that the RealQM electron configuration of a He atom consists of two half-spherical distributions meeting at a plane through the kernel as free boundary. Let us now imagine two He atoms approaching with the free boundary planes normal to the line between the kernels, which thus two half-spherical electron distributions meeting at a free boundary between the kernels and the other two outside. Again there will be an accumulation of charge density between the kernels, which  will now create a jump in electron density between inner and our electrons, which will push the inter-atom free boundary outwards and so act to decrease the inter-kernel electron accumulation and disable bonding. Computation shows that this effect is real. 

In short: The new atom/molecule model of RealQM offers an explanation based on physics of both the bond of H2 and the no-bond of He2.

RealQM for He2 at kernel distance 2.4 au with energy 0.003 Hartree above two He atoms indicating no-bond. Notice shift of free boundary beyond kernel to reduce accumulation of charge between kernels allowing yellow curves to take on same value at inter-atom free boundary. 

   

Sweden Has Lost Again

Swedish Foreign Minister Maria Stenergard repeated Sweden's steadfast support to Ukraine/NATO in its war with Russia made clear in the previous post, by today expressing:

• Sweden stands strong.
• It is important to secure ourselves.
• This is a genuine Swedish interest.
• We take on our responsibility.
• We invest in our military defence. 
• We are now NATO members and we feel more safe than ever before.
• A prime interest of Sweden is to strengthen our support to Ukraine in order the get an agreement which prevents any further Russian aggression.   

Swedish Prime Minister Ulf Christerson adds that Sweden may send troops to Ukraine.

Both ministers have missed that Ukraine/NATO has lost the war and that talks between Russia and USA are now under way towards a peace deal where the winner sets the conditions. 

This means that the massive Swedish military support to Ukraine has been meaningless and has only contributed to destruction. A harsh reckoning by the people of those responsible for this disaster, now will start. 

Swedish Prime Minister Febr 19:

• More weapons to Ukraine.
• I do not think Russia wants any peace negotiations.
• Sweden is in a dangerous situation (since we are at war with Russia).

Sweden vs Russia New Match

In the Foreign Policy Debate today in the Swedish Parliament Swedish Foreign Minister Maria Stenergard opened by forcefully declaring:

• Sweden is in the midst of long term confrontation with Russia.
• Support to Ukraine is the Government's top foreign policy priority. 
• Our task is inescapable. 
• Sweden never stands alone.
• We will constrain Russias ability to do us harm, particularly through our support to Ukraine.
• We are guided by our belief in a free and strong Ukraine.
• Sweden has increased its support to Ukraine each year of the war, totalling 70 billion SEK.
• Russia's goal is to impose a sphere of influence with series of vassals and satellites.
• The war is determined on the battle field.
• For Sweden support to Ukraine is a moral obligation and an indispensable investment in our own security.
• Swedish land forces are now part of Nato's forward forces in Latvia.
• It is up to Ukraine if and when negations will start.
• Our unequivocal support....to make Ukraine's position stronger.
• Pressure on Russia's war economy must increase.
• Our military support to Ukraine must be strengthened... now its 18th military support package the largest to date.
• The only sustainable peace is one that Ukraine achieves through strength.
• Negotiations from a position of weakness would only further Russian aggression.
• Sweden will continue military support to Ukraine as long as it takes.

So Sweden is again at war again with Russia to be added to this long list of wars from 1164 to 1809 when Sweden lost Finland in the Finnish War between the Kingdom of Sweden and Russian Empire.  

But this is forgotten by the Swedish Government, which instead lives in a fantasy of repeating the glorious victory in the battle at Narva in 1700 when a Swedish army of 10.000 soldiers lead by warrior King Karl XII overpowered a Russian army of 40.000. And Sweden won Hockey WM against Russia in 1957 and 1987.

Short story: The Swedish Government is not well informed, because of lack of Intelligence?

Computational Chemistry vs Solid Mechanics

Computational Chemistry CC may take up to 50 % of supercomputer resources, while Computational Solid Mechanics CSM may take less than 10%. 

The world of molecules built from atoms and electrons can be seen as a microscopic analog of a macroscopic world built from beams of steel such as a bridge,  and so we may expect to find CC as a microscopic analog of CSM, at least in a perspective of classical continuum mechanics.  

But this is not what the above numbers show: CC requires vastly more computational work than CSM. Why is that?

Consider a macroscopic object like a bridge composed of $N$ elements, which could be the finite elements in a discretisation of the object in CSM based on Navier's equations of elasticity. The computational work can scale with $N$, that is linearly in $N$ since each element interacts with just a few neighbouring elements. Finite element codes with $N=10^6$ can be run on a lap top.

The situation in CC is vastly different. This is because CC is based on Schrödinger's equation as the basic model of Standard Quantum Mechanics StdQM, which for a molecule with $N$ electrons involves $3N$ spatial dimensions, a full 3d space for each electron. This means that the computational work increases exponentially with $N$ which makes even $N=10$ beyond the power of any thinkable computer, and so only simplified versions of Schrödinger's equation are used in practice. Full solutions named wave functions then appear as pieces of conversation, which have no precise quantitative form. 

In Density Functional Theory DFT as the current work horse of CC, Schrödinger's equation is draconically reduced into a 3d equation in a single electrons density. The CSM analog would be to compress a complex bridge construction into one simple beam, where all element individuality is erased. 

Electron individuality is thus destroyed in DFT, asking for some form of recovery as electron exchange-correlation, which has shown to be difficult to realise.

RealQM is new methodology starting from a new form of Schrödinger's equation i terms of a collection of a non-overlapping one-electron densities keeping individuality of electrons by spatial occupation. RealQM can be seen as a refined form of DFT with one-electron densities maintaining individuality. 

RealQM in principle scales linearly with $N$ just like CSM. Below you can compare a bridge and a molecule and ask yourself why the molecule in CC with StdQM requires vastly more computational work than the bridge in CSM, and so get motivated to take a look at RealQM for which the work is comparable.

Molecule.

Bridge.

First Molecule HeH+ by RealQM and DFT

Crosscut 3d showing one electron (red) moved from He++ kernel to H+ kernel and remaining electron (yellow) around He++ kernel. Note free boundary developed between electrons starting from initial vertical cut through He++ kernel. Run code below to follow dynamics.

This is a follow up on previous post on the first molecule formed after a Big Bang when one of the two electrons of a Helium atom He joins with an approaching proton H+ to form a helium hydrid ion molecule HeH+ (or rather He+H) built by a cation He+ and a Hydrogen atom H. The energy count in Hartree is as follows:

• Energy of He atom = -2.903
• Energy of He+ and H separated = -2.000 - 0.500 = -2.500
• Energy E of HeH+ molecule = -2.592 
• Dissociation Energy of HeH+ into He+ and H = 0.092  observed
• Energy for formation FE of HeH+ from He and H+ = 0.311 

Let us compare RealQM and DFT as concerns prediction of the observed FE = 0.311. Notice the difference between He plus H+ and He+ plus H. Check by noticing that -2.903 = - 2.500 - 0.311 - 0.092. Notice that the bulk of FE is supplied by exterior forcing to make H+ approach the He++ kernel. 

RealQM code gives FE = 0.301 from E = -2.602. You can follow the transfer of one electron from He to H by running the code starting from two electron half-lobes around the He kernel with supports displayed on red and yellow. You can test a different location of H+ vs He electron split by running this code. In both cases see how one electron dynamically shifts from He to H+ forming a molecule of He+ and H starting from He and H+. 

DFT gives according to chatGPT:

DFT Functional	Predicted Dissociation Energy (Hartree)	Error Trend
LDA (Local Density Approximation)	~ -0.35	Overbinds HeH⁺ (too stable)
GGA (PBE, BLYP)	~ -0.33 to -0.34	Still overestimates bond strength
Hybrid (B3LYP, PBE0)	~ -0.30 to -0.32	Closest to exact (-0.311)
High-Accuracy (CCSD(T), FCI)	-0.311	Exact value

• LDA and GGA functionals overestimate binding, leading to a more negative dissociation energy (~ -0.34 to -0.35 Hartree).
• Hybrid functionals (B3LYP, PBE0) improve accuracy, but they still may predict a slightly too strong bond.
• Post-HF methods (CCSD(T), FCI) match experimental values (-0.311 Hartree).

We see that the precision with standard DFT is not better than RealQM, rather the opposite. It is not clear that DFT can model the dynamics of the shift of one electron from He to H+.

Notice that we are here dealing with the simplest possible problem in quantum mechanics, a molecule with only two electrons as the first molecule formed in the early universe, with H2 coming only later after dissociation of HeH+ into He+ and H (and then formation of H2 under release of energy). Would you expect that DFT after 50 years of massive investment would deliver a very convincing result? Did we get that?

Electron Affinity RealQM vs DFT

Chart of electron affinity from 0 to 0.133 Hartree with grey zero affinity.

Electron affinity is a measure of the drop in total energy in energy when a neutral atom A captures an  electron under release of energy forming a negatively charges anion A-  named negative affinity.

An atom with zero affinity has no tendency to capture another electron.  

We consider two basic cases one with zero and one with negative affinity:

• Helium with 2 electrons filling the 1st shell as a noble gas with zero electron affinity.
• Fluorine with negative electron affinity by filling the 2nd shell from 7 to 8 electrons.  

Observed negative electron affinities range from 0.1- 0.3 Hartree with 0.12 for Fluorine. The total energy of Fluorine is -99.7 Hartree, and so to recover a change of 0.1 Hartree in computation requires a precision of 4 correct decimals.  

Here you can run RealQM as essentially a 3-line parameter-free code in 3d with only input the kernel charge giving the following total energies in Hartree:

• Helium    -2.903
• Helium-   -2.906
• Fluorine   -99.4
• Fluorine-  -99.8

We see that RealQM recovers zero affinity for Helium and the trend of negative affinity for Fluorine, if not the exact value with the present resolution of a $50^3$ grid. 

Density Functional Theory DFT as a very complex code, typically gives positive affinity for Helium, and can give values in the range of 0.12 for Fluorine under suitable adjustments of the code. 

PS This is what chatGPT has to say about the role of DFT in years to come:

• DFT will continue to be the dominant method for simulating chemical systems in the foreseeable future. Despite its limitations, it offers the best trade-off between accuracy, computational cost, and scalability. However, machine learning (ML) is emerging as a potential competitor—and in some cases, it might even surpass DFT.

The question is if RealQM can take over this role as a new form of DFT with a collection of one-electron densities instead of just one common density. The investment in DFT has been massive over a period of 50 years, while  RealQM is a spin-off of computational mechanics realised with little manpower. It is thus of interest to compare RealQM and DFT on basic tasks.

Self-Interaction in DFT vs RealQM

A main difficulty of Density Functional Theory DFT as working with a single electron density $\rho (x)$ depending on a 3d spatial coordinate $x$ representing all electrons, is that electron self-interaction is present and has to be eliminated. 

Without correction DFT gives a much too small effective net electric potential outside a neutral atom as the net potential from kernel and electrons, for which the true net potential is $-\frac{1}{r}$ with $r$ the distance to the kernel for any atom, the same for all atoms as that of the Hydrogen atom. This is the case without van der Waal dipole effects.    

Real Quantum Mechanics RealQM is a new alternative to StandardQM StdQM with DFT, works with a collection of non-overlapping one-electron charge densities. 

In RealQM  there is no self-interaction since electron Coulomb potentials contribute to the total energy only for pairs of distinct electrons. 

RealQM thus gives the correct effective potential $-\frac{1}{r}$ simply because for an atom with kernel charge $Z$ and $Z$ electrons, each electron interacts with $Z-1$ other electrons with net 1 as the charge in the effective potential $-\frac{1}{r}$ of the Hydrogen atom.

In StdQM the effective potential of $-\frac{1}{r}$ is viewed to be the result of incomplete shielding of the kernel by the surrounding electrons always leaving a net potential of $-\frac{1}{r}$ even if the total net charge is $0=Z-Z$. But why the shielding effect is precisely $Z-1$ is not so obvious with the typical overlapping electron orbitals used in Hartree-Fock and DFT based on Hartree-Fock. 

StdQM/DFT: 

• works with globally overlapping electron densities without boundaries,
• has to struggle to remove effects of non-physical electron self-interaction.

RealQM: 

• works with nonoverlapping electron densities meeting at a free boundary,
• has no electron self-interaction. 

A simple test of consistency of any atom model is to check if the net potential outside the atom is $-\frac{1}{r}$. RealQM directly passes this test, while basic DFT has to be modified to pass the test by eliminating non-physical effects of electron self interaction.

 

Standard Quantum Mechanics as Classification without Physics vs RealQM

This is a continuation of the previous post but can be enjoyed independently.

Science and religion both arise from human minds seeking to find mental interpretations of the Word as the creation of superhuman mind. Quantum Mechanics QM as the mechanics of atoms and molecules came out from a perceived shortcoming of classical Newtonian physics in the late 19th century, so immensely successful in describing the macroscopic world, to capture the microscopic world of atoms and molecules. 

The German physicist Erwin Schrödinger in 1926 took the first step out of a deadlock into the modern physics of QM by formulating a mathematical model of the Hydrogen atom with one electron surrounding a proton kernel in the form of an eigenvalue problem for a partial differential equation for a negative charge density subject to Coulomb attraction from a positive kernel. 

The success was complete since the  eigenvalues precisely agreed with the observed spectrum of excited states of the Hydrogen atom with corresponding eigenfunctions as wave functions representing vibrational spatial modes of the electron taking the following form:

to be compared with those of e g a circular membrane    

The complete success of the Schrödinger equation capturing the spectrum of the Hydrogen atom with one electron demanded generalisation to atoms with more than one electron, which was accomplished  by formally adding a new set of 3d coordinates for each electron into a linear Schrödinger equation (S) in $3N$ spatial dimensions for an atom/molecule with $N>1$ electrons like a Gold atom with 79 electrons in $237$ spatial dimensions, as a partial differential equation of a completely new form with solutions named wave functions for which the physics had to be invented as probabilities of electron configurations. 

(S) has come to serve as the foundation of the modern physics in the form of Standard QM StdQM filling text books of modern physics. 

The first task for (S) was to explain the "Aufbau" of the  Periodic Table in terms of wave functions and then the following strange idea came up: 

• Describe ground state electron configurations of atoms/molecules with $N>1$ as linear combinations of excited states of the one electron of the Hydrogen atom.                                  

To see the strangeness compare with an idea of

• describing a complex system as a copy of a simple component building the system
• preformatism of the 17th century as a tiny human ("homunculus") inside the egg of a female. 

We understand that this is illogical: A complex system cannot be copy of its simple parts. Nevertheless it serves as a fundamental principle of StdQM: Excited states of a Hydrogen atom describe all atoms/molecules. It defies logic but is state of the art, still 100 years after conception. It does not help to recall that the eigenfunctions of the Hydrogen atom form a complete orthonormal system and so can describe anything. 

The fact that StdQM comes out from a formal extension from one to many electrons, means that StdQM appears as an (ad hoc) classification system rather than as model of real physics. This is evident by looking at the rich classification imposed by StdQM culminating in the Standard Model:

• symmetric and antisymmetric wave functions as bosons and fermions
• electrons with spin-up and spin-down
• paired and unpaired electrons 
• Hund's rule, Madelung rule, octet rule
• s, p, d, f states in atoms 
• quarks, leptons, gluons, weak force, strong force...

It is natural to compare with the plant classification system of Linné (18th century) based on the number of stamens and pistils in flowers, which is artificial and not reflective of natural relationships, but anyway still is used.  

RealQM offers an alternative to StdQM as a atom/molecule model based on Coulomb interaction between atomic kernels and non-overlapping electron densities as a natural system without need of elaborate classification. 

We can put StdQM ve RealQM into a broader perspective of idealism and realism with StdQM imposing atomic form and RealQM uncovering natural atomic form. RealQM gives a different rationale of the Periodic Table as an electron density packing problem. 

If you force a metal through a square die in an extrusion process, it will come out with quadratic cross section. If you force Hydrogen eigenfunction form upon general atomic states, everything will look Hydrogenic but you will violate natural physics.  

Triumph of Mathematics: Newton and Schrödinger

In the previous post we discussed the danger of being carried away by mathematical beauty into the fantasy land of modern physics in the form of Standard Quantum Mechanics, Standard Model and String Theory. Compare with Sabine Hossenfelder's Lost in Math and Max Tegmark's Mathematical Universe.

Let us recall the greatest successes of mathematical thinking about physics: 

• Newton's Theory of Gravitation                                    (N)
• Schrödinger's equation for the Hydrogen Atom.          (S)
• Maxwell's equations for electromagnetics.                   (M)

Newton's Theory of Gravitation is mathematics because it is captured in the following equations

•  $\Delta\phi =\rho$                                                         (N1)
• $F=-\nabla\phi$                                                         (N2)

where $\rho$ is mass density, $\phi$ is gravitational potential and $F$  is gravitational force all depending on a Euclidean space coordinate $x$ with corresponding Laplacian $\Delta$ and gradient $\nabla$ differential operator. 

These equations can be derived mathematically from a principle of conservation of energy and force. A point mass at $x=0$ then comes with a gravitational potential $-\frac{1}{\vert x\vert}$ and gravitational force scaling with $\frac{1}{\vert x\vert^2}$ as Newton's power two law. 

Newton's Model (N1) + (N2) capturing all of celestial mechanics through the differential operators $\Delta$ and $\nabla$, is the most formidable success of mathematical thinking all times. There was simply no  alternative for a celestial Creator, which allowed humans to get insight into the creation process by pure mathematical thinking. Amazing, right?  

Schrödinger's equation formulated in 1926 describes the charge density $\psi$ of a Hydrogen atom as the minimiser of the total energy $E_{tot}=E_{kin}+E_{pot}$ with 

• $E_{kin}=\frac{1}{2}\int\vert\nabla\psi\vert^2dx$
• $E_{pot}= -\int\frac{\psi^2}{\vert x\vert}dx$ 

under the side condition 

• $\int\psi^2dx =1$, 

as the solution of the eigenvalue problem 

• $-\frac{1}{2}\Delta\psi +V\psi = E\psi$                                  (S)

where $V(x)=-\frac{1}{\vert x\vert}$ is the electric Coulomb potential of the proton kernel, and $E$ is an eigenvalue. The connection between Coulomb potential and charge density is the same as between gravitational potential and mass density.

This model is not derived from basic principles like Newton's model, but is itself a first principle.

The success is that its eigenvalues match the observed spectrum of the Hydrogen atom to high precision, with a smallest eigenvalue $-\frac{1}{2}$ as the minimum of  $E_{tot}$ attained by a charge distribution finding the optimal combination of 

• being compressed around the kernel making $E_{pot}$ small
• paying a compression cost of $E_{kin}$

and so taking the simple form 

• $\psi (x) = \exp (-\vert x\vert )$

as a density decaying exponentially with the distance the kernel. 

Schrödinger's model of the ground state of a Hydrogen atom with a proton kernel surrounded by an electron charge density, can be compared with Newton's model of a Sun surrounded by an orbiting planet finding a balance of kinetic energy of motion and gravitational potential energy. 

The "compression energy" $E_{kin}$ of the electron then corresponds to the kinetic energy of the planet scaling with $\frac{1}{2}p^2$ with $p$ momentum . This gives a formal connection between $\nabla$ and $p$ as motivation of the name kinetic energy given to $E_{kin}$. 

The electron of a Hydrogen atom cannot be a point particle orbiting the kernel like a planet orbiting a sun, because a moving electron would radiate energy and so collapse into the kernel. The electron thus must have extension is space and it makes sense to associate $\nabla\psi$ with some form of "compression" or "strain" as in an elastic body. 

It is thus possible to argue that (S) is a most natural model of a Hydrogen atom as a kernel surrounded by an electron charge density finding an optimal solution to small potential energy at a compression cost. One can argue that the Creator had no choice when complementing celestial mechanics with Hydrogen atoms. 

Sum up so far: The macroscopic world of mechanics captured by (N) and the microscopic world of the Hydrogen atom captured by (S) as the result of pure mathematical thinking without need of observational input (Kant a priori), is a formidable success. Add to that (M).

But the success is not complete as concerns Schrödinger's equation since only the Hydrogen atom is covered. What then about a Schrödinger equation for atoms or molecules with many electrons? 

This was the question confronting Schrödinger and the world of modern physics in 1926 and the route that was taken has come to form modern physics all the way into our time. The idea was again to rely on pure mathematical thinking and so generalise (S) formally from one to many electrons simply by adding a new spatial coordinate for each new electron to arrive at a Schrödinger equation for an atom/molecule with $N$ electrons taking form in $3N$ spatial dimensions. No physics was involved in the generalisation, only mathematical formality. The mathematician von Neumann took control with his Mathematical Foundations of Quantum Mechanics 1932 leading physics into a strange universe of wave functions evolving in Hilbert spaces according to von Neumann Postulates (without physics), and physicists had no choice but to follow. 

The resulting Schrödinger equation was a multi-dimensional monster which did not describe any physics and in addition showed to be uncomputable except for very small $N$. The equation was easy to write down on a piece of paper just increasing the number of spatial dimensions, but real physics in three space dimension was present only in the case $N=1$. To save the situation the model was given a statistical interpretation by Max Born as all possibilities instead of particular realities as Standard Quantum Mechanics StdQM. Schrödinger never accepted StdQM.

RealQM offers a different generalisation in the original spirit of Schrödinger in the form of non-overlapping electron charge densities interacting by Coulomb potentials each coming with compression cost in total energy minimum of ground state. RealQM is a deterministic model of physics in three space dimensions and as such readily computable. 

Final sum up: (N) and (S) represent astounding complete successes of pure mathematical thinking, while StdQM appears as a monumental failure of too much belief in power of mathematical formality. RealQM gives hope to continued success of more modest mathematical thinking. The idea of a mathematical universe is great but should be combined with modesty according to the Greeks.

Paul Dirac claimed that very advanced mathematics was required to design the universeas, maybe as an excuse that his equation for the electron was very complicated beyond human understanding leading to Dirac's famous "Shut up and calculate". 

I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.  (von Neumann)