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. 

 

RealQM Configuration of Neon as 2+4+4

A basic pillar of moden chemistry (supposedly with some support from standard Quantum Mechanics) is the octet rule, which says in particular that the electrons of the Neon Ne atom are arranged in two spherical shells with an inner shell around the +10 charge kernel formed by 2 electrons and an outer (valence) shell with 8 electrons, with the message that the valence shell is "full" and so Neon does not want to interact with other atoms through a covalent or ionic bond.

On the other hand, Flourine F with +9 charge kernel with 7 electrons in the outer valence shell is very keen to attach another electron to "complete the octet" and so readily forms eg an HF molecule with an H atom in a covalent/ionic bond. 

RealQM gives a different picture with Ne in 2+4+4 configuration and F as 2+4+3 with a third shell harboring 4 or 3 electrons acting as the valence shell. This means that that F combines with H to form a HF molecule, while Ne like He does not. You can interact with simulations in p5js-code here:

• Fluorine
• Neon
• HF    

For Neon in a 2+8 configuration RealQM gives a much too small energy compared to observation, which thus appears to not be attained since it requires somehow squeezing 8 electrons into one shell.

The reason H can bond to F in a 2+4+3 configuration, rather than in 2+7, appears to be that the size of an H atom better fits with the size of electrons in a valence shell of 3 electrons instead of 7.

The from observations estimated radius of an F atom is about 1 atomic unit au (50 pm). The thickness of atomic shells scales with 2/Z with Z the reduced charge reaching a shell, with thus 2/10 + 2/8 + 2/4 about 1 au for a 3-shell 2+4+4 configuration, while a 2-shell 2+8 configuration would give a too small radius compared to observation.

What can the difference be between C as 2+4, which bonds with H and O in particular, and Ne as 2+4+4, which does not want to bond with anything, when they have the same 4 electron valence  shell? It is thinkable that this is a geometric effect with the C valence shell being more compatible with those of H and O, which may be uncovered by further explorations with RealQM including the validity of the octet rule.