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.

Uncertainty Principle vs Real Quantum Mechanics

Heisenberg’s Uncertainty Principle UP is a cornerstone of quantum mechanics stating that there is a limit to the precision both position and velocity of a particle like an electron, can be determined. The standard hand-waving argument is that precise measurement of position changes velocity and vice versa, and so precise measurement of both to arbitrary precision is impossible. This makes quantum mechanics fundamentally different from classical mechanics, where there is no such limit to measurement precision, in principle.

The other cornerstone is Schrödinger's wave function, which does not contain UP even if it is supposed to tell everything there is to say about the system. UP is thus an add-on to standard Quantum Mechanics stdQM based on Schrödinger’s equation, and then connected to measurement.

Is there any UP in RealQM? We recall that RealQM describes an atomic system as a collection of non-overlapping extended electronic charge densities which do not have particle character. For an extended body (on macro or microscale) there is no unique point position and velocity describing the state of the body and so there is a certain fuzziness or certain uncertainty depending on size if only one point, such as the center of gravity is to be used. RealQM thus, just like classical continuum mechanics for extended elastic bodies, comes with a form of UP depending on size, but this is perfectly normal and no mystery.

In RealQM there is no built-in limit to possible measure precision. The charge densities of an atom in ground state are stationary in space and shift around under radiation and so come with the usual fuzziness of extended bodies, which on microscale of course can be very real making it very difficult to point-wise determine the electron charge density in an atom, but there is no mystery like that of Heisenberg.

No Split between Classical and Quantum Mechanics

The scales of the World without split.

À standard answer to a question about the difference between classical and quantum mechanics is that the former concerns macroscopic physics and the latter microscopic physics, and of course size makes a huge difference, right? But more precisely, what difference does size make? 

We know that an ant faces different conditions/forces than an elephant, but the same basic laws of physic apply. We know that the Moon follows the same all-size law of gravitation as ants and elephants. 

A standard physicist would now expand on the macro-micro split by informing us that macroscopic physics is deterministic while microscopic physics is non-deterministic or probabilistic, and that is huge difference: Atoms play dice while the macroscopic world is predictable or deterministic. 

You could react on this message with skepticism from a wealth of experience that a complex macroscopic world is unpredictable or non-deterministic, while an atom as an entity of utter simplicity must be expected to be predictable deterministic. So if determinism is to be what makes macro different from micro, it would be micro/simple which is deterministic and macro/complex which is non-deterministic, in direct opposition to standard physics. 

Let us now seek the reason why standard physics insists that microscopic physics is non-deterministic while macroscopic physics as meanvalues of microscopic physics is deterministic, even if that contradicts sound logic. We shall see that it all depends on the probabilistic interpretation of the wave function of quantum mechanics by Max Born, which has captured modern physics. Born proclaimed in 1926 that the square of the wave function would express the probability of the occurrence of the configuration acting as argument for the wave function as a function over configuration space, as the new buzz word which gave him the Nobel Prize in 1954 with a near 30 year delay. Let us now seek the origin of the concept of a wave function over a configuration space measuring configuration probabilities as the hall mark of quantum mechanics.  

The motion in 3d space over time $t$ of a classical mechanical system of $N$ particles is described by Newton's Laws of Motion or alternatively in terms of Lagrangian mechanics based on the Principle of Least Action for a Lagrangian

• $L(x_1,x_2,...,x_N,\dot x_1,...\dot x_N)$       (1)

where $x_i (t)$ is the position and $\dot x_i (t)\frac{dxi}{dt}$ the velocity of particle $i=1,...,N$ as function of time $t$ as a description of the evolution of the state of the system over time,  taking the form  of Newton/Lagrange's Laws of Motion. Various coordinates for 3d space may be used, such as Euclidean, spherical, cylindrical et cet.

It is here common to refer the set of all possible states $(x_1,x_2,...,x_N,\dot x_1,...\dot x_N)$ as configuration space, which thus has $6N$ dimensions. It is important to understand that all particles exist in the same 3d space and that the $x_i(t)$ with $i=1,2,,,N$ simply describe the locations of all the particles at some time $t$. The setting is a common 3d space for all particles and the $6N$ dimensional configuration space is a formality. In short (1) makes perfect sense. Out of all possible states in configuration space a specific trajectory $X(t)=(x_1(t),...,x_N(t)$ with coordinates $x_i(t)$ is determined by Lagrange's equations as a function of time $t$ only. 

We now turn to quantum mechanics with its wave function for $N$ electrons of the form

• $\Psi (x,t)=\Psi (x_1,x_2,...,x_N,t)$                            (2)

where now the $x=(x_1, ...,x_N)$ are coordinates for $S=R^3\times R^3\times ...\times R^3$ as $N$ different independent copies of 3d space $R^3$ with coordinate $x_i$. Superficially (2) resembles (1) and we may even speak of a $S$ as a configuration space for (2) with coordinates $x=(x_1,x_2,...,x_N)$.

The wave function $\Psi (x,t)$ satisfies Schrödinger's equation $i\dot\Psi =H\Psi$ with $H$ the Hamiltonian differential operator acting on the wave function $\Psi$ depending on $(x,t)$-coordinates with $x=(x_1,...,x_N)$ $3N$-dimensional. 

The hope of Schrödinger was that the generality of Lagrangian mechanics in terms of number of particles (and choice of coordinates), would allow a direct generalisation of his stunningly successful equation for the Hydrogen atom to atoms/molecules with many electrons. What could go wrong if (2) was similar to (1) both with formally a high-dimensional configuration space?

Let us compare (1) and (2) or more specifically $X(t)$ as a classical trajectory satisfying Lagrange's equations as a function depending on $t$, and $\Psi (x,t)$ satisfying Schrödinger's equations as a function of $(x,t)$. Ok, we see a clear difference in the coordinate dependence: $X(t)$ depends on a time coordinate, while $\Psi (x,t)$ depends on a time coordinate and a multidimensional spatial coordinate $x=(x1,...,xN)$. 

That is a monumental difference. Classical mechanics describes actual trajectories $X(t)$ in $R^3$ selected by Lagrange's equations from a pool of all possible trajectories, which is never covered.

On the other hand, the wave function $\Psi (x,t)$ with its multidimensional space variable $x$ freely sweeping configuration space $S$ independently of $t$, thus describes possiblities and not actuality and there are immensely many more possibilities than actualities. The wave function $\Psi (x,t)$ is thus a monster supposed to describe the probability of all possible configurations, which is such a huge undertaking that it swamps every conceivable effort by its complexity.

To reduce complexity the variation of $\Psi (x,t)$ is (ad hoc) reduced to consist of linear combinations of products of single electronic wave functions $\psi_i(x_i,t)$ each one depending on a single spatial coordinate $x_i$, typically in the form of so called Slater determinants, but even that is like seeking to emptying an ocean with a spoon as an uncomputable problem. 

On the other hand, in RealQM a different Schrödinger equations is formulated for a wave function of the form 

• $\Psi (y,t) = \sum_i\psi_i(y,t)$          

as a sum of one-electron wave functions $\psi_i(y,t)$ all depending on the same common physical 3d coordinate $y$, which is computable in the same sense as a classical deterministic continuum mechanics solid/fluid problem. 

In RealQM the physics on macro and microscope is the same, which is to expect since a split cannot be made.  

The split macro-micro of standard quantum mechanics is an artefact of the introduction of a configuration space of all possible configurations which is way too big to handle.

Another strange aspect of the standard Schrödinger equations are linear thus allowing superposition of states as something unexpected. Newton/Lagrange's equations in general are non-linear and superposition a rare effect of linearity only in very special cases, like a small amplitude vibrating string. The Schrödinger equation of RealQM is non-linear, just like the majority of similar equations in solid/fluid mechanics. 

In short, there are many serious reasons to consider the standard multi-dimensional linear Schrödinger equation as a too easily formed mathematical construct without physics. As compensation $\Psi$ is invited to belong to a mathematical Hilbert space $\mathbb{H}$ of maximal prestige and so the world can very neatly be described by a $\Psi$ in a $\mathbb{H}$, but without physics. 

One can argue that classical physics concerns real physics using concrete Calculus, while standard quantum physics rather is is a form of formal mathematics without real physics. 

Quantum Mechanics: Probability or Not?

The main mystery of quantum mechanics in its proclaimed consensus version (whatever it is), is the interpretation of the wave function in terms real physics/ontology as the 

• Probability of finding electrons at some specific locations in an atom.      (*)

When asking a physicist (or chatGPT) about the meaning of "finding an electron at some specific location in an atom", you get the answer that such a thing is impossible. Locating an electron experimentally in an atom is impossible. An electron is both everywhere and nowhere at the same time. So finding an electron in an atom is impossible and so (*) has no physical meaning. 

We must then conclude that it cannot be possible to make experimental statistics out of impossible findings and so (*) cannot be experimentally verified. It is like being prevented from tossing a specific coin in an investigation of the statistical characteristics as possible non-symmetry and bias of that specific coin, and so have to be satisfied with a message that a theoretical coin certainly is perfectly symmetric without any bias, saying nothing specific about a specific coin. Not very illuminating as concerns a real coin or atom. In short: It is impossible to give (*) a physical/ontological meaning.  

The proclaimed probabilistic nature of the wave function for an atom is in sharp contrast to very precise measurements of atomic spectra spectrum without any sign of randomness. 

If then only deterministic qualities of an atom (like spectrum) are possible to observe, and very sharply so, why are physicists still insisting on (*) while admitting that experimental verification is impossible?

The answer is simple, yet hidden to the general audience: The reason is the multi-dimensional nature of the wave function $\Psi (x1,x2,...,xN)$ depending on $N$ 3d spatial coordinates $x1, x2,...,xN$ for an atom with $N$ electrons, thus altogether $3N$ spatial coordinates, and so e g 6 spatial dimensions for Helium with two electrons. It is like each electron is given a separate 3d space to live in, and no common 3d space for all electrons. Only for Hydrogen with 1 electron does the wave function have a real physical meaning in physical 3d space as an electronic charge distribution in a classical continuum mechanical sense without any stroke of randomness. 

Schrödinger kick-started quantum mechanics in 1925 by writing down the Schrödinger equation for the Hydrogen atom guided by some heavenly inspiration, but directly stumbled on the Helium atom with two electrons. A formal mathematical extension from one to many electrons was possible with a stroke of pen without thought by introducing a multi-dimensional wave function. But Schrödinger was not happy with the result, because it lacked physicality or "Anschaulichkeit".  

For $N>1$ the multidimensional wave function $\Psi (x1,x2,...,xN)$ lives outside real 3d space and then can only be interpreted as possibility rather actuality/reality. It was Max Born who invented this probabilistic interpretation, which so upset Schrödinger that he gave up quantum mechanics saying in 1926 to Bohr:  

• If all this damned quantum jumping were really here to stay, I should be sorry, I should be sorry I ever got involved with quantum theory. 

The proclaimed probabilistic nature of atomic physics is thus a consequence of a mathematical theoretical ad hoc assumption devoid of physics making generalisation to many electrons an easy/trivial catch, but then coming with many pseudo-problems without physics and answers filling physics books. It is like postulating that all celestial motion is circular or circle-upon-circle following Aristotle, because from mathematical point of view circles are perfect, which delayed scientific progress two millennia. What progress delay has been caused by (*) over the century it has served as road block? 

It is here that RealQM in the spirit of Schrödinger comes in with a different deterministic generalisation from Hydrogen to $N>1$ with direct extension to molecules and chemistry,  100 years later. Take a look. No statistics! Instead real physics!

The Role of Differentiation and Integration in Physics

This is a further reflection on the idea of Physics as Computation in the previous post with focus on the mystery of instant action at distance (there are many posts on this topic). 

Mathematical models of physics typically take the form of differential equations such as Poisson’s equation 

• $\rho=\Delta\Phi$        (1)
• $\Delta\Phi = \rho$      (2)

connecting gravitational/electric potential $\Phi (x,t)$, depending on a space coordinate $x$ and time coordinate $t$ coordinate, to mass/charge density $\rho (x,t)$, and $\Delta$ is the Laplacian differential operator involving second order differentiation.  

In a Hen-Egg setting $\Phi$ represents Hen and $\rho$ Egg, either as local differentiation/assignment $\rho =\Delta\Phi$ as Hen-laying-Egg,  or solution of $\Delta\Phi = \rho$ by global integration/summation as Egg-generating-Hen by instant action at distance. 

How to choose between (1) and (2)? Local differentiation or global integration/instant action at distance? 

If you are a (pure) mathematician, you would without hesitation say that there is a method for symbolic differentiation and so (1) is in a sense trivial. On the other hand there is no method for symbolic solution of (2), which is the non-trivial problem of the symbolic Calculus of Leibniz/Newton. 

Mathematicians know that if symbolic solution fails, because it has no method, it is always possible to resort to numerics as a form of trivial work-horse, which case-by-case can compute solutions by number crunching. So is mathematics split into symbolic/analytical mathematics and numerical mathematics (in descending prestige) with essentially different basic elements: symbols or numbers.

As an example, symbolic differentiation is trivial while numerical differentiation is a delicate subject because a derivative $\frac{dx}{dt}$ involves the quotient of small numbers requiring precision. In general differentiation is a delicate process because precise identification is needed. So what can be trivial in symbolic mathematics can be non-trivial in numerical mathematics. 

On the other hand, symbolic integration is non-trivial while numerical integration is trivial as it is just a form of summation. 

So the world of symbolic mathematician and numerical mathematics is very different, since what is trivial or non-trivial can be opposite. 

We now turn to real physics as something real existing in the real world (ontology). What is the relation of real physics to symbolic mathematics and to numerical mathematics? 

Since symbolic mathematics works with symbols rather than numbers it has a connection to epistemology. We now ask if numerical mathematics is closer to ontology/real physics and so if we can learn anything about real physics from numerical computation.

In particular, we seek the real physics of (1) vs (2) as the Hen-Egg question posed above. We recall that numerical solution of (2) is trivial as simply global summation, while (1) is non-trivial as delicate differentiation. 

If we believe that physics is non-trivial,  then (1) represents physics: Hen lays Egg as a delicate non-trivial local operation, but not asking for instant action at distance in a global solution process. 

On the other hand, for (2) to represent physics requires instant action at distance as instant global summation. 

We are thus led to the conclusion that (1) represents real physics as a local differentiation process. The apparent instant action at distance in (2) would then represent non-real fiction.

We thus find support of an idea that computation and real physics are closely connected, while the connection of symbolic mathematics to real physics can be difficult to assess.  

The gravitational potential generates mass by local differentiation. Mass does not (have to) generate gravitational potential by global instant action at distance. 

It seems to make sense to say that physics differentiates in the sense of evaluating force differences, while physics integrates by time stepping creating motion, which opens to physics without mysterious instant action at distance. In numerics differentiation (1) can be traded with integration (2) with fictitious instant action at distance.

Another aspect is that you can see mass but not gravitational potential itself only its effect, and you may be tempted to believe that what you can see is primary and what you cannot see is secondary. But that depends on your senses of perception and so may not tell the true story from an objective physical point of view. You see a person getting smaller receding from you, but you know it is an illusion.

PS It is also possible to give up the cause-effect aspect in the potential-mass connection and like Leibniz say that potential and mass are connected in Perfect Harmony or Best of Worlds, which has been ridiculed...maybe it is time for Leibniz to come back...in any case he laid the mathematical foundations to the digital world as a world combining ontology with epistemology... 

But of course it is possible to turn the argument around as follows: Consider Newton's 2nd Law

• $\frac{dv}{dt} = f(t)$     (3)

where $v(t)$ is velocity and $f(t)$ is force. The standard view is that the force $f(t)$ is given and causes the acceleration $\frac{dv}{dt} = f(t)$ as (2). But we may also view $v(t)$ to be given and $f(t)=\frac{dv}{dt}$ simply the force required as in (1). This describes a situation where the nature of the force is unclear, while velocity/motion is very real. This is the case with the Coriolis force and of course centrifugal force. Einstein tried to get rid of gravitational force (and aether) altogether, but did not succeed…

Numerical solution of (3) is done by time stepping $dv=f(t)*dt$ updating velocity with input from force, which has direct physical meaning as motion as summation, thus with numerics in tandem along with (2).   

 

Conclusion: We may say that numerics can connect to both (1) and (2), while the role of symbolic math in physics remains to be made precise with the physical meaning of the symbolic wave function $\Psi$ of quantum mechanics, as solution to Schrödinger’s equation, after 100 years of constant brooding, still being a complete mystery in its standard so called Copenhagen interpretation. On the other hand, RealQM offers a physical meaning in classical continuum mechanics terms with the kinetic energy of the electrons appearing as a form of elastic energy preventing the electron to fall into the kernel by an elastic force balancing kernel attraction, just like the centrifugal force of motion prevents a planet to fall into its Sun. Both forces appear as necessary conditions for maintenance of certain states (Lagrange multipliers) as virtual forces without concrete physical origin: Planets move the way they do because forces balance, and electrons assemble around the kernel because forces balance. Leibniz would agree, I am sure!

Recall that the wave function $\Psi (x)$ for the ground state of the Hydrogen atom minimises the total energy E as "kinetic" energy + potential energy: 

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

over all real-valued functions $\psi (x)$ with $\int\psi^2dx=1$, which can alternatively be interpreted as the state of a normalised elastic cloud subject to an elastic central force with the kinetic energy appearing as "elastic" energy.  The terminology "kinetic" energy is misleading (motivated by some deep symbolism) since no motion is involved, while "elastic" energy has a concrete physical meaning as a measure of elastic compression suggesting some form of electronic charge compression for the atom.   

Physics as Computation at John Chappell's Natural Philosophy

This is an intro to a live video talk I will give on Febr 3 2024 on John Chappell's channel Natural Philosophy: Where Critical Thinking Challenges Theory (directly connecting to the slogan of this blog). If you feel that this must be crackpot science, take a look at my arguments before deciding and remember that established physics can be crackpot science.

Digital computation, with AI (or even AGI) as latest achievement, is today reshaping human conditions and it is natural to ask if also the science of physics as the inner core of existence is transformed.

Classical physics is based on mathematical models in the form of differential equations expressing balance (of forces) in some system, such as Euler’s equations for fluid mechanics and Maxwell’s equations for electro-magnetics, while modern atomic physics is based on Schrödinger’s equation. 

The equations express system forces while solutions of the equations represent evolution in time of systems under given conditions. The task of determining solutions is thus central and here digital computation opens entirely new perspectives with computational complexity or computability as key element. 

Uncomputable systems keep their information hidden to inspection, with prime example Schrödinger’s equation which in its standard multidimensional form is beyond the capacity of any thinkable digital computer. On the other hand, computing solutions to Euler’s equations resolves the enigma of turbulence, as will be shown in the talk.

It is natural to view the evolution in time of a physical system as a form of analog finite precision computation as the action of forces takes the system over small time steps from one state to the next, which can be modeled by finite precision digital computation: 

• Physics as Analog Computation as Digital Computation.

The key elements of computability are (i) finite precision and (ii) stability/wellposedness as a measure of precision required to make computational model output reliable. Forward-in-time evolution then shows to be computable because it is stable, while backward in time evolution is uncomputable because it is unstable, which can be seen to be the essence of the 2nd Law. 

Physics as Computation offers solutions to open problems of (i) turbulence and (ii) atomic physics through new computable forms of Euler's and Schrödinger’s equations, which are the subjects of the talk: 

• Secret of Flight   (Real Euler: Computational Turbulent Incompressible Flow)
• Real Quantum Mechanics

Real here directly connects to computability. A real physical system computes its own evolution forward-in-time and so is analog computable and a mimicing digital computable model can be viewed to be a real model:

• Real models are digital computable because reality is analog computable. 

The standard multidimensional Schrödinger equation is an uncomputable model without real physical meaning (only statistical). RealQM is computable and has a real physical meaning as a collection of non-overlapping interacting charge densities.

Real Euler computes real turbulent flow, and RealQM computes real atoms/molecules, which opens entirely new perspectives on physics: Physics as Computation. 

Real Euler gives an explanation of the 2nd Law (Computational Thermodynamics) as forward-in-time computability and backward-in-time uncomputability. See the book The Clock and the Arrow for a general audience.

There is a connection to Wolfram’s Computational Foundations for the Second Law of Thermodynamics in the sense that computation is central, but the essence is different: For Wolfram it is computational irreducibility, while I favor finite precision+stability.  

Instant Action at Distance in Atom Physics/Quantum Mechanics

Instant action at distance is a fundamental element of both macro-scale gravitational mechanics and micro-scale quantum mechanics in the form of Newton’s Law of gravitation and Coulomb’s Law of electrostatics. 

The idea is that the presence of a mass/charge at one point in physical space without time delay generates a force at all other points decaying with the inverse square of distance, as the fundamental force of both classical and modern physics of Newton/Einstein and Heisenberg and Feynman as the golden boys of quantum mechanics, and of course Schrödinger.  

It also formed the foundation of the now forgotten, but once great, physicist Joseph Boscovich (1711-1787) as expressed in his monumental "A Theory of Natural Philosophy reduced to one unique Law of forces that exist in Nature" stating that the World is the result of instant action at distance of attractive and repulsive forces on both small and large scales. This a nothing but a Grand Unified Theory and what remains is to fill in details about the forces and in particular to explain how instant action at distance is realised, which has remained a fundamental mystery of physics. See the book Roger Boscovich-The Founder of Modern Science, by Stoiljkovic.

One way to summarise physics is to recall that both Newton's Law and Coulomb's Law take the form of Poissons’ equation: 

• $\Delta \phi (x) = \rho (x)$                                (1)

where $\Delta$ is the Laplacian acting in 3d space with coordinates $x$, $\phi (x)$ is  gravitational/electric potential and $\rho (x)$ is mass/charge density. This is a consequence of in the equation (1) viewing $\rho (x)$ as a locally given source generating the potential $\phi (x)$ globally as a solution to Poisson's equation which can be seen as a form of instant integration/summation process sending local source information instantly around globally as instant action at distance. Forces are generated as $\nabla\phi (x)$.

Boscovich's Theory that all force is instant action at distance contradicted the classical idea that forces are transmitted by contact, adding the explanation that there is always some little distance between different material bodies including atoms maintained by ever-present repellation thus reducing physics to one unique Law. See the book Roger Boscovich- The Founder of Modern Science by Stoiljkovich. 

It is natural to consider (1) as a limit of the following time dependent heat/wave equations:

• $\epsilon\dot\phi -\Delta \phi = -\rho$,     (2)
• $\ddot\phi -\Delta\phi = -\rho$,                  (3)

where the dot indicates differentiation with respect to time $t$, and $\epsilon >0$ is small constant formally reducing (2) and (3) to (1) when tending to zero. The expanded models require some form of heat conduction or wave propagation medium/ether giving physics to action at distance with finite speed. 

On the other hand (1) could be argued to not require any medium, since force transmission is replaced by instant action at distance, but then again without explanation. 

I have argued that that there is a way out of this dilemma by shifting the conception of the meaning of the equation (1) to a view with rather the potential $\phi (x)$ as primary source from which both force $\nabla\phi (x)$ and mass $\rho (x)=\Delta\phi (x) $ are generated through the local action of differentiation by the Laplacian differential operator. 

In this view potentials are primary from which everything (force/mass/charge) is generated by local differentiation. In particular it gives a new view on the quantum mechanics of an atom, where the primary concepts are the kernel and electron potentials, and the atom with kernel and electrons is generated by the Laplacian and then required to satisfy Schrödinger's equation. 

In physics it is natural to search for sources generating effects in a cause-effect setting, but the precise mechanism of generation may be difficult to pin down, e g exactly how differentiation generates mass from gravitational potential, or how instant action at distance comes about.

This connects to Leibniz' idea of a Pre-established Harmony beyond human inspection. The gravitational potential-mass harmony expressed by (1) may be of this kind. 

You find more under Labels.

  

 

Bond and Lattice Dissociation Energies of NaH by RealQM

Let us now check if RealQM readily computes the bond dissociation energy in gaseous phase of the molecule NaH as separation into Na and H atoms, and the lattice dissociation energy in solid phase as separation into Na+ and H- ions. 

In stdQM this is viewed to be very difficult, if possible at all, which is not strange because stdQM is uncomputable (as we know). Instead, an empirical Born-Haber cycle is used. 

RealQM on the spot produces the following predictions in close agreement with reference values:

Bond dissociation energy =  E(Na) + E(H) - E(NaH)  = 0.08 Hartree  (ref 0.0765) (code) 

Lattice dissociation energy = E(Νa+) + E(H-) - E(NaH) = 0.38 Hartree (ref 0.307) (code) 

where E(X) indicates ground state energy of X, and in the code examples the distance (2*D in code) between atoms/ions is varied to capture separation while the number of iterations is kept the same for fair comparison. 

We see that the energy of a covalent bond without full electron transfer is much smaller than the lattice energy with full electron transfer from Na to H, naturally a consequence of different spatial "filling of electrons" in the sense of RealQM.  

In stdQM the mystery is deeper since the spatial presence of electrons is mysterious.

Carbon and Graphene by RealQM

A carbon atom C has up to 4 valence electrons and forms a very large variety of compounds by connecting to 1 up to 4 other atoms as CO, CO2 and CH4... 

Graphene is a 2d hexagonal pattern of carbon atoms each atom connecting to 3 other atoms, thus involving 3 valence electrons.

We now let RealQM compute the energy of C in the following shell configurations:

• 2+4                     (2 electrons in 1st shell, 4 in 2nd shell)                                (code)
• 2+2+2                (2 electrons in 1st shell, 2 in 2nd shell and 2 in 3rd shell)    (code)
• 2+3+1                (2 electrons in 1st shell, 3 in 2nd shell and 1 in 3rd shell)    (code)

and get close to the a reference ground state value of -37.7 Hartree in all three cases. We also compute ionization energies in 2+2+1 (code) and 2+3 (code) configuration in agreement with a reference value of 0.4 Hartree.

Altogether, RealQM gives a picture of C with 3 possible ground states of about the same energy available to form molecules with 1-4 bonds.

The view of standard QM stdQM is that the ground state configuration of the C atom is $1s^22s^22p^2$ with 2 electrons in spherically symmetric orbits in 1st shell, 2 electrons in spherically symmetric orbits together with 2 electrons in p-states in 2nd shell, which suggests that 1-2 valence electrons are available, but the 3 in graphene.

To get 3 valence electrons in stdQM various hybrid sp-orbital states are introduced to give the following picture of the 2d hexagonal pattern of a graphene sheet with each carbon atom connected to 3 other atoms involving 3 sp2 valence electrons and the 4th electron hoovering above and under (or around) the sheet: 

We can connect the RealQM 2+3+1 configuration to this picture with 3 electrons centered in a plane and a 4th out-of-plane electron as 3d enclosure. 

RealQM approaches the electron distribution in atoms and molecules as a real physical packing problem in 3d. RealQM is ab initio. RealQM is computable at iPad power. 

StdQM uses a formal approach based on molecular orbits which have no real physical meaning. StdQM involves a set of ad hoc rules (Pauli, Hund's rules 1-3...) and is not ab initio. StdQM fills the books on Quantum Chemistry with a message that it works, even if stdQM itself is a mystery. StdQM is uncomputable on any thinkable digital computer. 

RealQM 2nd Row of Periodic Table: Na to Argon

We now proceed to the second row of the periodic table where to speed up computation the electrons in inner shells of an atom are homogenised to a common charge density and we keep individual electrons only in the outermost shell acting as the valence shell in formation of molecules by interaction with other atoms. 

RealQM gives the following ground state energies with display of electron distribution over shells from inner to outer:  

• Ne (2+8):                   -128      (-128.5) (code)
• Ne (2+4+4):               -128      (-128.5) (code)
• Na (2+4+4+1):         -161.8    (-162.4) (code)
• Mg (2+4+4+2):         -199.4   (-200.3)  (code)
• Al  (2+4+4+2+1):     -242.4   (-242.7) (code)
• Si  (2+8+4):               -290      (-290)   (code)
• P (2+8+4+1):            -342      (-342)  (code)
• S  (2+8+4+2):          -396      (-399)   (code)       
• Cl (2+8+4+3):         - 461      (-461.4) (code)
• Ar (2+8+8):             -528       (-529)   (code) 
• K (2+8+8+1):          -601      (-602)   (code)
• Xe (2+8+18+18+8) -7458    (-7438) (code)

We see good agreement between RealQM and reference values in parenthesis. 

We see that the number of valence electrons in the outermost shell ranges from 1 to 4  (except for the noble gases Xe), which is different from stdQM with 1 to 8 valence electrons according to the "octet rule". 

We see that the 2+4+4 configuration for Neon gives about the same energy as a 2+8 configuration  suggesting that 4+4 for inner shells is the same as 8.

We note that atoms with 1-2 valence electrons are metals/metalloids and those with 3 are non-metals. This gives a very simple non-standard classification. Metals with 1-2 valence electrons have small ionization energies and those with 3 large ionization energies. 

Molecules naturally form by combining metals with non-metals such as NaCl, which will be explored in an upcoming post. 

In the above computations we have kept a full 3d resolution of all shells, with spherical charge homogenisation of inner shells to speed up. A further speed up opening to large molecules will be made by resolving inner shells in spherical symmetry and only valence shells in full 3d.  

We note that the RealQM is very simple (3 lines essentially) and and as such is essentially ab initio.  We compare with stdQM computations using Hartree-Fock or Density Functional Theory which are very complicated and thus not ab initio. 

PS It is not yet clear exactly when to stop iterations and so the number of iterations can be used to arrive exactly at reference values, if desired. Further study of stop criterion is needed. 

Helium/Neon 1st Excited RealQM

We now let RealQM compute the first excited state of Helium with one electron moving from 1st shell to 2nd shell thus giving the excited state a 1+1 configuration. We get the following results: 

• Excited state with 1st el 1st shell, 2nd el 2nd shell = -2.16 (ref -2.145) (code)
• Reference with same code but only 1 el                  = -2.00  (ref -2.000) (code)
• Reference Helium ground state                                = -2.903
• Excitation energy                                                      = 0.758  Hartree    (20.61 eV)

We see good agreement. The difference between RealQM and stdQM is that electrons do not overlap in RealQM and meet with homogeneous Neumann condition (zero flux) and charge density continuity, while in stdQM in the 1s2s configuration the outer 2s electron overlaps with the inner 1s electron. 

Spin plays no role in RealQM, while in stdQM the two electrons are assumed to have different spin (whatever the physics of spin may be). 

For Neon RealQM gives about the same excitation energy as the difference between a 2+4+4 and 2+4+3+1 configuration with 1 electron in a new outer shell (code), in accordance with observation (0.8 Hartree).

Ionization: Helium to Neon…

Ionization energies (observed) for the 2nd row of the periodic table starting with Helium and ending with Neon follows the following pattern (energy in kJ/mol with 1 Hartree = 2625 kJ/mol):

RealQM gives the following energies (with shell configuration given and list values corresponding to the above graph):

• He 2             (code)  = -2.90       (list -2.903)
• He+ 1           (code)  = -2.00      (list -2.00)
• Li 2+1          (code)  = -7.47      (list -7.48)
• Li+ 2            (code)  = -7.17      (list -7.28)
• Be 2+2          (code) = -14.6       (list -14.5)
• Be+  2+1       (code) = -14.0       (list -14.2)
• B  2+2+1       (code) =  -24.4        (list -24.5)
• B+  2+2         (code) =  -24.2      (list -24.2)
• C 2+4            (code)  = -37.7      (list -37.7)
• C 2+2+2        (code)  = -37.8       (list -37.7)
• C+ 2+2+1     (code)  = -37.3       (list -37.3)
• O 2+4+2       (code)  = -74.9       (list -74.8)
• O+ 2+4+1     (code) = -74.4        (list -74.3)
• N 2+2+3       (code) = -54.4       (list -54.4)
• N+ 2+4         (code)  = -53.7      (list -53.9)
• F 2+4+3        (code) = -99.6       (list -99.4)
• F+ 2+4+2      (code) = -98.8       (list -98.7)
• Ne 2+4+4      (code) = -128.6     (list -128.5)
• Ne+ 2+4+3    (code) = -127.5     (list -127.7)
• Na 2+4+4+1  (code) = -162     (list -162)

Fairly good agreement on 50^3 mesh with list values in the graph capturing the big ionization energies for He and Ne and steady increase from Li with kernel charge/pull, modulo the kinks Be-B and N-O. In RealQM terms the decrease Be-B can be connected to the 1 electron valence of B compared to 2 for Be, and the 2 electron valence of O compared to the 3 electron valence of N. Continue on next row starting with Na on 100^3 mesh…

Ionization: Carbon1+ and Carbon2+

Let us now test RealQM on ionization where one or more electrons are ripped off an atom at some energy expense, starting with the example of Carbon with 6 electrons. 

We have seen that a shell configuration of 2+4 (2 electrons in 1st shell and 4 electrons in 2nd shell) gives a total energy of -37.7 Hartree in correspondence with list value (code). This configuration matches CH4 (code)

On the other hand, the configuration 2+2+2 has about the same energy (code), while the electrons in the 3rd shell are less tightly bound to the kernel than the ones in the 2nd shell of the 2+4 configuration, and so would require less energy to be ripped off. 

We thus let RealQM compute the energy of Carbon1+ with one electron removed in a 2+2+1 configuration to get a total energy of -37.3 Hartree in agreement with list value (code), with thus an ionization energy of 0.4 Hartree. 

We continue with Carbon2+ with configuration 2+2 and get -36.4 with ionization energy 0.9 again in agreement with list value (code).  

Note that to compute ionization energy by subtracting total energies of atom and ion, requires three correct decimal places and so is a bit delicate, because of the required mesh size cut-off of the singular kernel potential. 

We see that both 2+4 and 2+2+2 configurations for Carbon have about the same total energy and so may both be possible, while as concerns ionization the 2+2+2 configuration in agreement with list value seems to preferred, because the ionization to 2+3 requires much bigger energy (code).

Next objective is to test if RealQM can make sense of the following pattern for 1st ionisation energies:

   

Perspective on ElectroNegativity

Let us now give more perspective on the electronegativity explored by RealQM in the previous post as the decrease of energy achieved by hypothetically adding one electron to a given atom with kernel charge Z assuming the electron configuration of the next element in the periodic table with charge Z+1. 

For example, RealQM computes a decrease of about 4 Hartree when an electron is added to Fluorine with Z=9 with electron shell configuration 2+4+3 to obtain the configuration 2+4+4 of Neon with Z=10 as the ion F-. 

In a similar way we obtain energy decrease of 0.8 Hartree for Helium- (Z=2), 1.2 for Lithium- (Z=3), 1.7 for Beryllium- (Z=4) and 2.3 for Boron- (Z=5) increasing to 4 for F- (Z=9) as the maximal electronegativity for all elements. 

RealQM gives the very small value 0.06 for H- in opposition to an accepted value of 2.  

We next ask under what conditions the ion F- will be created from F by incorporation of one electron at an energy decrease of 4 Hartree? It directly connects to the nature of the bond of  molecule HF as ionic or covalent. In an ionic bond the F atom would fully capture the electron of H with a decrease of energy of more than 3 Hartree. This is very substantial and would correspond to a dissociation energy of HF of more than 3 Hartree which is 10 times bigger than that observed.

We have earlier seen that a HF with a covalent bond has a dissociation energy in accordance with observation.

We conclude that F- appears to be hypothetical and in particular does not combine with H+ to form HF by an ionic bond. In other words, it is not clear what role electronegativity has to play if bonds are rather covalent than ionic. Any idea? Recall that direct measurement is viewed to be impossible, which gives support to a suspicion that electronegativity is more fiction than reality.

PS The accepted electronegativity of H of 2 Hartree stands out as very singular/strange:

 

 

ElectroNegativity by RealQM

Electronegativity (or rather electron affinity, see this post) of an atom measures the decrease of total energy arising from adding an electron. Pauling suggested a scale to measure electronegativity addressing the following values to the elements in the periodic table:

We see in the 2nd row electronegativity increase from 1.0 for Lithium to 4.0 for Fluorine as the maximum over all elements. 

RealQM gives the following electron affinity values measured in Hartree:

• H-    0.04  (code)
• He-   0.8    (code)
• Li-    1.2    (code)
• Ber-  1.7    (code)
• B-     2.3    (code)
• F-      5.0    (code)

We see that the the 2nd row Pauling scale matches the RealQM values in Hartrees, which makes sense to Pauling's scale. 

We note that (i) Helium is missing in the Pauling scale, and (ii) the values for H- differ fundamentally.

The reason the Pauling scale does not take up He is probably the preconceived idea of standard quantum chemistry that He as a noble gas has no incentive at all to catch an electron. RealQM tells a different story, connecting to the previous post showing that He can form a He2 molecule. 

On the other hand, RealQM gives H a very small desire to catch an electron, thus supporting the common idea that H acts as an electron donor, in particular when forming the HF molecule by combining with F with maximal electronegativity in an ionic bond.  

The Pauling value of 2.0 for H- stands out as strange and in conflict with the idea of ionic bond in HF.  

H can form H2 molecule in a covalent bond even with small electronegativity, because no entire capture of an electron is needed, only sharing. RealQM captures the difference in capturing and sharing of electrons, which standard QM does not appear to do.