New Year Rubber Model of Real Quantum Mechanics

A popular model of a planetary system is formed by a small ball (Earth) moving on a horizontal elastic rubber sheet suspending a heavy ball (Sun) at the origin $x=0$:

The slope of the rubber sheet gives the force on the little ball balancing centripetal acceleration to make it circle the big ball. The heavy ball suspended by the elastic rubber sheet is a model of gravitational force created by a massive object arising from minimisation of a functional of the form 

• $G(\psi ) =\frac{1}{2}\int\vert\nabla\psi (x)\vert^2 dx - \psi (0)$          (1)

over functions $\psi (x)$ vanishing for $x$ far from the origin. The first term of $G(\psi )$ represents elastic energy of compression/expansion $\nabla\psi (x)$ and the -1 factor of $\psi (0)$ is the heavy ball load potential. 

A similar model can be used to describe a hydrogen atom with a proton kernel surrounded by an electronic charge density $\psi (x)$ as minimisation of 

• $E(\psi ) =\frac{1}{2}\int\vert\nabla\psi (x)\vert^2 dx -\int\frac{\vert\psi (x)\vert^2}{\vert x\vert }dx$     (2)

over functions $\psi (x)$ with $\int\vert\psi (x)\vert^2dx=1$. The first term of $E(\psi )$ is usually named "kinetic energy", which lacks physics since no motion in time is involved. In view of (1) it can better be seen as a form of electronic compression/expansion energy again measured by $\nabla\psi (x)$. The second term involves the kernel potential $\frac{1}{\vert x\vert }$. 

We thus see a close similarity of (1) and (2) with both Solar system and Hydrogen atom captured by the same elastic rubber sheet model. The classical Bohr model of an atom as a small planetary system may thus in fact be closer to reality than a probabilistic modern quantum model (with the electron charge density corresponding to the elastic rubber and the proton the central heavy ball). 

In any case that is the basic idea of RealQM as a hope for the New Year out of the 100 year mystery of stdQM…

New Years Gift: You can play with this interactive code to explore how the energy of two hydrogen atoms depends on distance to find the molecule H2 as the configuration with minimum energy. 

   

From Possibility to Actuality in the Quantum World 2024?

As the New Year is approaching bringing possibilites into actualities it may be worth while to contemplate the difference between modern physics of indeterminism and classical physics of determinism.  

After 100 years of brooding there is still no consensus about the physical meaning of the multi-dimensional Schrödinger equation in multi-dimensional configuration space as the basic mathematical model of Quantum Mechanics as the incarnation of modern physics. It is not likely that this will change in the future.

The big trouble is the multi-dimensional configuration space, of dimension $3N$ for a system with $N$ electrons, as representation of possibilities or probabilities, while physics in 3d space is all about actuality. 

This is expressed in the so called multi-verse interpretation of QM, where all possibilities are viewed as actualities, but this is convincing to only a few. Even more troublesome, the multi-d Schrödinger equation is uncomputable for $N>4$ as noted by Nobel Laureate Walter Kohn, who developed Density Functional Theory DFT as a reduced computable model.  

The break-through of QM in the 1920s came from a deterministic prediction of the spectrum of the hydrogen atom with $N=1$, thus in 3d space in stunning agreement with observation. The door was then opened to $N>1$ by a direct purely formal mathematical generalisation borrowed from linear algebra in any number of dimensions, however at the loss of physicality in a step from actuality to possibility. 

And there we are 100 years later with an unfathomable space of possibilities hiding actuality in the form of standard QM without physicality. But is it not true that stdQM can accurately predict the spectrum of any atom by clever approximate solution of the multi-d Schrödinger equation such as DFT as an actuality leaving out all other possibilities as being of little interest? Yes maybe so, but everything depends on a clever reduction of the multi-d Schrödinger equation, which can always be done so as to match observation, and so pull out actuality from possibility/probability. 

But, what is then the role of the multi-d Schrödinger equation if it is uncomputable and so anyway must be reduced to computable form? Why not instead seek to give some reduced model a physical meaning leaving the multi-d model to endless speculations by philosophers of quantum mechanics in the spirit of  medieval scholastics?

Recall that the by stdQM postulated erratic stochastic probabilistic unpredictable behavior of quantum particles like electrons around an atom being nowhere and everywhere at the same time, is nothing which can be verified by experiments, since tracking of individual particles by postulate is impossible. It is thus not possible to verify the validity of the multi-d model experimentally, and so it cannot be elevated from pure speculation based on mathematical convenience. You can thus choose to believe in this model, or not , without any notable practical consequence, just as the angels on knives edge by the scholastics.

This leads to RealQM, as an alternative to stdQM, which has a direct physical interpretation in terms of non-overlapping electron charge densities, and which is  readily computable and so shows very good agreement with observation. Maybe 2024 will open to a breakthrough of RealQM, as a possibility becoming reality?

PS1 After a lengthy conversation with ChatGPT, it is agreed that atomic spectra in experiments appear fully deterministic and so cannot be used as verification of the basic postulate of stdQM of probabilistic origin of atom physics. ChatGPT as a language model follows logic admitting that there is a difference between deterministic and probabilistic and that atomic spectra are deterministic and not probabilistic. This logic would be hard for a real modern physicist to accept following the cryptic illlogic of Bohr viewing contradictions simply "complementary" in a "duality" without contradiction: 

• A great truth is a truth whose opposite is also a great truth.
• Contraria sunt complementa. Opposites are complementary.
• We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough.

What a sad story.When logic breaks down, reason breaks down and so rational science, and what remains is fiction science, which is not even science fiction (which like rational science is based on reason). 

PS2 About stdQM:

• Quantum mechanics is weird (Weinstein, Rovelli, Hossenfelder 2022)
• Quantum mechanics isn't as weird you think; it is weirder (Scientific American 2023)

Note that the Scientific American article first gives 3 examples where stdQM after all is not weird at all, and then concludes with something claimed to be really really weird, in order to maintain the official image that stdQM is so weird that more public funding must be given to physicists to clear up the mess. A clever tactic which has worked fine for 100 years.

 

Why Is the H2O Molecule Bent?

Standard Quantum Mechanics stdQM presents the following geometry of the water molecule H2O:

We see covalent (2-electron) bonds between O and H leaving O with two lone electron pairs (depicted by dots) on top of O in the picture. The resulting accumulation of negative charge on one side of O is then claimed to cause the bending by repelling the H atoms towards the other side.

Is this a convincing argument? Not really, since the accumulation of the two lone pairs on one side instead of a symmetrical on opposite sides with a linear unbent configuration, is not explained. 

In any case, if a bent configuration is preferred before a linear, it must be because it has lower energy and this is also what RealQM shows (compare code linear and code bent), with a difference of about 0.1 Hartree. Also compare code with free boundary update.

We are then led to a different explanation of the lower energy for the bent configuration understanding that it brings the H atoms closer into a covalent bond without changing the bonds to O. Recall that the energy of H2 is 0.17 Hartree lower than that of two separate H atoms.

In RealQM the configuration of O is 2+4+2 with two valence electrons interacting with the two H atoms. 

In stdQM the configuration of O is 2+6 with thus six valence electrons of which only two bond with H, and the remaining four form the two lone pairs (with their mysterious bending effect).

PS Here is what ChatGPT has to say based on reading all texts of stdQM:

• The bent or angular shape of the water (H2O) molecule is a result of the arrangement of its atoms and the distribution of electron pairs around the central oxygen atom. Water consists of one oxygen atom and two hydrogen atoms. Oxygen has six valence electrons, and in the case of water, two of these electrons form covalent bonds with the hydrogen atoms, leaving four electrons as two lone pairs on the oxygen atom. 
• According to the Valence Shell Electron Pair Repulsion (VSEPR) theory, electron pairs, whether bonding or non-bonding (lone pairs), repel each other. In water, the repulsion between the lone pairs of electrons is greater than that between the lone pairs and the bonding pairs. As a result of this increased repulsion, the two lone pairs of electrons position themselves closer together, pushing the bonding hydrogen atoms closer to each other. 
• This repulsion causes a distortion in the molecule’s shape, resulting in a bent or angular geometry where the H-O-H bond angle is approximately 104.5 degrees. Therefore, the bent shape of the water molecule arises from the repulsion between the lone pairs of electrons, which affects the arrangement of the hydrogen atoms, leading to a non-linear molecular structure.

We understand that this is not very convincing, which indicates the state of the art.   

RealQM for H2, H3 and H3+

We now let RealQM compute the ground state energy E of the following molecules formed by the hydrogen atom H (of energy -0.5):

• H2      E = -1.17  (stable)        (code with interactive version) 
• H3      E = -1.67 (unstable)     (code)
• H3+    E = -1.16  (stable)        (code)

which agree with reference values. 

Here the cation H3+ is of particular interest as one of the most abundant molecules in the Universe. Also compare with the previous post on HeH+ believed to be the first molecule to form after Big Bang.

RealQM can capture the formation of H3+ as the two electrons of H2 are attracted by a nearby proton (top in image) into the following 2-electron distribution (red/magenta) over a triangle of protons:

Adding an electron to the H3+ cation formally gives an H3 molecule with E = -1.67, which can dissociate to H2 and H, and so is unstable. 

On the other hand, both H2 and H3+ are stable since their energies are substantially smaller than the energy -1 of two separate H atoms. We see that the energies for H2 and H3+ are almost the same, while the electron configurations are very different. 

We see that RealQM can capture the dynamic physical process of redistribution of electrons from changing kernel configuration since electrons do not overlap and keep identitity.

From He Atom and Proton to He+H Cation as First Molecule

Following up the previous post we now let RealQM simulate the formation of the He+H molecule believed to be the first chemical compound created after the Big Bang (article in Nature). 

The general idea is that Big Bang created the Helium atom as a +2 proton kernel surrounded by two electrons as well as free protons, and that the compound He+H subsequently was formed when one of the two electrons of He was somehow transferred to a proton to form an H atom.

We now let RealQM model this process in 3d (run code yourself also finer resolution), and thus start with a Helium atom of ground state energy E =-2.903 Hartree combined with a proton within the range of its electrons, as depicted in this mid section plot (the colors represent the supports of the electrons):

We see here the characteristic feature of RealQM for Helium with the two electrons occupying two separate non-overlapping half lobes meeting at a plane together with the +2 kernel in the middle, and we also see a proton inserted on the left. The energy of the system is the ground state energy E = -2.903 of He. We watch the dynamics when the left electron is attracted to the proton and the right electron gradually takes over the Helium kernel with energy increasing to E -2.546:

and discover the following final state with energy E = -2.490:

We understand that -2.500 is the energy of separate He+ (-2.000) and H atom (-0.500) and so the final state of He+H can easily dissociate into separate He+ and H, thus altogether forming H from He. 

We also understand that the formation of He+H from He and proton requires input of energy (from -2.90 to -2.50), which we (as maybe expected) find to be equal to the repulsion energy of 0.40 between the kernels (see updated code), that is the energy required to bring a proton close to the He atom (somehow supplied by Big Bang).

RealQM thus let us follow the (protonation) process when He delivers one of its electrons to an inserted proton to create the molecule He+H which can separate into He+ and H. We understand that the initial He atom with two electrons separated in space is instrumental in the subsequent passage of the left electron to surround the proton preparing formation of an H atom, while leaving the Helium kernel to the right electron.  

In stdQM electrons overlap and separation as above seems more farfetched, right?

Recall the shooting off an electron from He requires 0.903 Hartree, which is a lot, while the above process with instead a proton capturing an electron involves only 0.403 Hartree from -2.903 to -2.500.

Also recall the stdQM gives a completely different picture with the energy of He+H ranging from -2.93 to -2.97 indicting that He+H forms from He and proton under release of energy. We thus have a clear case to compare RealQM with stdQM. What is your verdict?

PS The experimental discovery of HeH+ in 2019:

RealQM Confirms Detection of HeH+ as First Compound after Big Bang

The Guardian 2019 sent the message that the Helium Hybrid cation HeH+ long thought to be the first stable chemical compound to have appeared after the big bang, has now for the first time been detected in space:

• The positively charged molecule HeH+ known as helium hydride is believed to have played a starring role in the early universe, forming when a helium atom shared its electrons with a hydrogen nucleus, or proton (thus more precisely forming He+H).
• Not only is it thought to be the first molecular bond, and first chemical compound, to have appeared as the universe cooled after the big bang, but it also opened up the path to the formation of molecules of hydrogen. 
• The lack of evidence of the very existence of helium hydride in the local universe has called into question our understanding of the chemistry in the early universe. The detection reported here resolves such doubts.

The idea is that He+H forms when one of the two electrons of Helium is passed over to a proton thus forming an H atom in a compound with He+ as the cation He+H, which when combining with an electron added to He+ can form the compound HeH. 

So everything started when He combined with a proton to form He+H and then combine with an electron into HeH, which can be dissociated into He and H thus "opening a path to form molecules of hydrogen".   

From the point of view of standard quantum mechanics/chemistry stdQM this appears contradictory, since He as a noble gas is not supposed to form compounds with anything, and this is also the message sent by Computational Chemistry Comparison and Benchmark DataBase where atomisation energies for HeH and HeH+ are lacking as if these compounds do not exist! 

So stdQM says that the molecules HeH+ and HeH do not exist, which is in contradiction with their basic role after Big Bang now supported by observation.  

RealQM gives the following positive atomisation energy E

• HeH+     E = 0.065 Hartree     (code) 
• HeH       E = 0.153 Hartree     (code)    (this will be revised in later post)

in conformity to the detected existence of these compounds, the first to appear after Big Bang!

Does this give evidence that RealQM has something to offer beyond stdQM?

In the next post we will follow the dynamic dramatic event when one of the two electrons of He  is passed over to a proton to form an H atom. Spectacular!

Towards Big Molecules with RealQM

Here is an application of the reduced models of the previous post to linear 3-atom molecules with 1 or 4 valence electrons showing good agreement to experimental data:

• H2O (or HOH)     (code)  (O 2 valence)
• CH2 (or HCH)     (code)   (C 2 valence)
• CO2 (or OCO)     (code)
• BeH2 (or HBeH)  (code)  (Be 2 valence)
• LiOH                    (code)  (alt code)
• NaOH                   (code)  (alt code)
• N2                         (code)  (N 3 valence)
• NH3                      (code)  
• CO                        (code) (alt code) (C 4 valence)
• H2CO                   (code)  (C 4 valence)
• OH                        (code)  (O 2 valence)

It thus seems possible to represent an atom in RealQM in a reduced model defined by an effective radius R and number of valence electrons by fitting to experimental atomisation energy data, and then build molecule models from such reduced atomic models as in the above examples. 

The computational complexity for a system with $N$ kernels will then scale with at most $N^2$ like a particle system in classical mechanics with all particles connected (or even $N$ with only local connections). RealQM modelling of big molecules thus seems to be possible with a window to ab initio protein folding.  

  

It may be that the number of valence electrons effectively is at most 4, and not up to 8 (as in the octet rule of stdQM) with O having 6,  and so that the above examples in fact covers a wide range of molecules.   

Also recall Real Atom Simulator allowing you to explore atoms in spherical symmetry. 

 

Atomisation Energy for X2 Molecules Valence Bond

As an important step in reducing computational cost in ab initio molecular dynamics presented in the previous post, we let RealQM compute atomisation energies E of X2 valence bond molecules with the X atom represented by 1 or 2 valence electrons surrounding a +1 or +2 kernel of certain radius R for a range of X and R and compare with experimental observations:

For +2 we get (code):

• R = 0     E = 0.38   compare with postulated E<0 for He2 and discussion below
• R = 0.5  E = 0.29   compare with 0.23 for C2
• R = 1     E= 0.19    compare with 0.18 for O2
• R=1.2    E= 0.14
• R=1.5    E=0.029  compare with 0.004 for Be2

We see increasing E with decreasing R as the full X kernel charge increases. Comparing with observations we can fit a proper R to X to be used in valence bond computations of molecules including X. 

We use these values to compute atomisation energies for 

• HeH     (code)      
• BeH     (code)
• CH       (code)
• OH       (code)

with fair agreement with observation. Recall that HeH does not form from He+H since He+H dissociates into He+ and H. An upcoming post will investigate if HeH can form directly from He and H. So even if HeH has lower energy than He and H separate, it does not guarantee that it will form, since a physical path to lower energy is required.  

 

For +1 we get (code):

• R = 0      E = 0.17    compare with 0.17 for H2
• R = 0.5   E = 0.09    compare with 0.11 for B2
• R = 1      E = 0.05   compare with 0.04 for Li2
• R = 1.5   E = 0.002  compare with 0.003 for Na2

We see the same pattern of increasing E with decreasing R with H2 to be compared to Na2. For 3 valence electrons represented by N see next post.

What stands out is the atomisation energy of He2 to be compared with that of H2 both with R=0 and maximal E. H2 with E=0.17 is the accepted observed standard value, while it is commonly claimed that He2 cannot form because E<0. 

RealQM gives E=0.38 for He2 indicating that He2 can form (maybe under pressure). 

So we have here a neat case to test RealQM: Does He form a He2 molecule, or not? 

Molecular Dynamics with RealQM

Molecular dynamics describes the internal motion of a single molecule (or set of molecules) as a collection of atomic kernels surrounded by electrons determined by Newtonian mechanics from a potential $V(R)$ depending on the geometric configuration of the kernels represented by $R$, from which kernel forces are determined as the gradient $\nabla_RV(R)$ with respect to $R$.

The potential $V(R)$ for a specific configuration $R$ is determined as the corresponding quantum mechanical electronic ground state energy $E(R)$, assuming that electrons quickly adjust to a new configuration, so that kernels move on a slower time scale than electrons.  

In particular, a stationary ground state of a molecule is determined as a configuration $R$ with minimal $E(R)$ or $\nabla_RV(R)=0$, the search of which only requires at path of $R$ over configurations.

In stdQM the cost of computing the potential $V(R)$ for many configurations is prohibitive, because already the cost for a single configuration scales with $100^{3N}$ where $N$ is the number of electrons, thus beyond any thinkable computer for $N>10$. Ab initio computation of $V(R)$ is thus unthinkable in stdQM and instead various reduced models have been tried such as Carr-Parrinello.  

Here RealQM appears to open entirely new possibilities because the cost of ab initio computation of  $V(R)$ for a single configuration instead scales with $N\times 100^3$, allowing computation of $V(R)$ over a wide range of $R$ with readily available computer power, and so directly $\nabla_RV(R)$ as difference quotient. 

As an example, which you can test yourself running this code and changing the parameter D, is the hydrogen molecule H2 as 2 +1 kernels each surrounded by 1 electron, which computes the following potential $V(R)$ depending on the distance $R$ between the kernels (in atomic units):  

• $V(1.0) = -1.040$
• $V(1.2) = -1.158$
• $V(1.4) = -1.170$
• $V(1.6) = -1.170$
• $V(1.8) = -1.157$
• $V(2)  = -1.145$
• $V(2.2) = -1.127$
• $V(3) = -1.106$
• $V(4) = -1.102$
• $V(5) = -1.013$

We see a minimum of $-1.170$ for $R=1.4-6$ in agreement with observations. Each 

computation is 3d and takes seconds on an iPad and so RealQM delivers the full potential function $V(R)$ for H2 in a minute. Similarly the potential function for other molecules covered in previous posts can be computed. 

It may be that RealQM can open a new window to ab initio molecular dynamics, simply because RealQM is computable while stdQM is not.

RealQM for Molecules: Valence Electron Reduction

RealQM for molecules can be reduced to detailed 3d model only for the valence electrons by modelling each atom minus its outer valence electron as spherical charge density of a certain radius R + kernel interacting with the valence electrons. 

For example, Sodium Na with 2+8 electrons in two inner shells and 1 valence electron in an outer shell, can be reduced to a net +1 spherical charge density surrounded by a -1 valence electron, thus a reduction from 11 electrons to 1 electron. 

Or Oxygen O can be reduced to a +2 spherical charge density surrounded by 2 valence electrons. An O2 molecule can then be modeled with 2+2 interacting valence electrons.  

What distinguishes an atom is then the radius of the inner spherical electron charge density. 

Dissociation energy (or atomisation energy) of a molecule XY composed of an X and a Y atom can be computed by varying the distance between X and Y from large with the total energy the sum of the energies of X and Y as atoms, to a minimum total energy as the energy of the molecule XY, and the difference between these energies being the dissociation energy. Dissociation energies typically from 0.1 to 0.3 Hartree.

We get the following dissociation energies for XX=X2 molecules with 2 or 1 outer valence electrons with R=0 and typical value R>0: 

• He2 +2kernel  R = 0:           0.1 Hartree    (ref 0.1)      (code)
• O2   +2kernel  R = 1:           0.19 Hartree (ref 0.19)     (code)  
• H2   +1kernel  R = 0:           0.17 Hartree (ref 0.17)     (code)
• Li2  +1kernel  R = 1:           0.05  Hartree (ref 0.04)     (code)
• Na2 +1kernel  R = 1.5:        0.03  Hartree (ref 0.03)     (code) 

In general good agreement, which indicates that indeed full 3d modelling required only for the valence electrons with inner electrons homogenised to spherical charge density + kernel.    

Electron Shielding: RealQM vs stdQM

A neutral atom with a kernel of positive charge $Z$ surrounded by $Z$ electrons in some (shell) configuration can attract an outside electron thus forming a negatively charged ion at the release of energy, referred to as (negative) electron affinity. 

For example the electron affinity of Lithium (Z=3) with 2 inner-shell electrons and 1 outer-shell electron has an observed electron affinity of - 0.028 Hartree and Fluorine (Z=9) -0.125 Hartree. 

Ok, so if a Lithium atom can attract a negative electron under release of energy, the kernel must exercise some attraction outside the formally neutral atom, which can be thought of as an effective charge $Z_{eff}$ resulting from incomplete shielding of the kernel by the surrounding electrons. This is referred to as electro-negativity as a qualitative property on a certain empirical scale, see post on electro-negativity.

Because of the physical shell structure of RealQM with non-overlapping electrons, RealQM directly informs that the shielding effect of $N$ inner-shell electrons on outer-shell electrons is $0.5\times N$, so that $Z_{eff} = Z-0.5\times N$ and in particular outside the atom $Z_{eff}=0.5\times Z$, which conforms with about 3 times larger affinity for Fluorine compared to Lithium. The precise shielding effect is directly computable by RealQM. 

StdQM does not deliver any theoretical prediction with the excuse that such a thing must be a very very complicated problem.  

What stdQM offers is Slater's empirical rule to calculate the shielding effect of inner electrons on outer electrons in an atom, which e g says that the shielding effect of the 2 inner electrons of Lithium on the outer electron comes with an empirical factor $0.85$ so that $Z_{eff} =3-2\times 0.85 = 1.30$, while for Fluorine Slater gives $Z_{eff}=4.55$. But Slater says nothing about the shielding effect outside the atom.

On the other hand, RealQM says $Z_{eff} = 3-2\times 0.5=2$ for the outer electron of Lithium and $9-0.5*8=5$ for Fluorine, that is a bit less effective shielding than Slater's rule and so more outside attraction available for electron attraction.  

The reason RealQM gives a direct answer to the shielding effect is its physical shell subdivision without the electron overlapping confusing the picture for stdQM. 

For Helium (Z=2) RealQM gives $Z_{eff}=2-0.5\times 2=1$ and so the anion He- can form even if He is a so called noble gas (with an energy release of $0.8$ Hartree according to code), in agreement with observation. Even He2- with additional energy release of $0.5$ Hartree (code) appears to be possible, but realisation may require very special conditions. 

In stdQM a distinction is being made between electron affinity and electronegativity, with electron affinity the energy release when an single atom adds an electron, and electronegativity rather capacity to share electron in a covalent bond. It is not clear that this distinction is essential and does not appear to be so within RealQM.