Schrödinger's Equation in Physical Space as Physics of Chemistry

This connects to this recent post asking if chemistry is well explained by quantum mechanics.

The novelty of modern physics of the 20th century vs classical physics of the 19th century was a  mechanics for atoms and molecules in its canonical form Standard Quantum Mechanics StdQM based on Schrödinger's equation SE (1926), which was fundamentally different from classical continuum mechanics of macroscopic objects based on Newton's equations.  

Modern physicists are educated to claim that the physics of an atom (kernel + electrons) is described by StdQM and so will naturally argue that chemistry is simply physics of molecules and that so also chemistry can in principle be described by StdQM, as famously stated by the famous physicist Dirac in 1933. Modern chemists will say that chemical bonding as the essence of chemistry is not well described by StdQM and so chemists still have a role to play, then referring to Dirac's follow up that Schrödinger's equation is uncomputable and so in practice chemistry cannot be reduced to physics. 

Newton's equation's for a collection of macroscopic objects are computable since computational complexity grows linearly or quadratically with the number of objects. But the computational complexity of Schrödinger's equation grows exponentially with number of atoms/electrons making it uncomputable even for small molecules. The reason Schrödinger's equation is uncomputable is that it has new multi-dimensional form with a separate full 3d coordinate for each electron demanding computation in $3N$-dimensional space for an atom with $N$ electrons with exponential growth in $N$.

Over the 100 years since 1926, many attempts have been made to compress the multi-dimensionality of SE to computable form, most drastically in Density Functional Theory with a reduction to a single electronic charge density depending on a single physical 3d coordinate. I have come up with a less drastic reduction in the form of Real Quantum Mechanics RealQM based on non-overlapping electron densities. My hope is that RealQM can bring Dirac's idea of chemistry as physics into more practice. 

StdQM and RealQM both start from SE for the Hydrogen atom with one electron based on a Coulomb Hamiltonian of the form:

• $H=-\frac{1}{2}\Delta-\frac{1}{\vert x\vert}$     

where $\Delta$ is acting on an electronic wave function $\psi (x)$ depending on a 3d coordinate $x$. The corresponding total energy $E$ is given by 

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

as a sum of kinetic energy measured by $\vert\nabla\psi (x)\vert^2$ and potential energy measured by $\psi (x)^2$ weighted with the kernel potential $-\frac{1}{\vert x\vert}$ for a positive unit kernel/proton at the origin. We understand that the kinetic energy has nothing to do with motion in space but measures spatial variation of $\psi (x)$, which is a source of great confusion.

But StdQM and RealQM generalise to an atom with more than one electron in different ways. 

The Coulomb Hamiltonian $H_{std}$ for an atom with kernel of positive charge $Z$ at the origin of a 3d Euclidean coordinate system $R^3$ surrounded by $N=Z$ electrons, takes the form 

• $H_{std}= \sum_{i}(-\frac{1}{2}\Delta_i -\frac{Z}{\vert x_i\vert}) +\sum_{j<i}\frac{1}{\vert x_i-x_j\vert}$ for $i=1,2,...,N$,                                           

where each $x_i$ is a 3d coordinate for a copy of $R^3$ and $\Delta_i$ the Laplacian differential operator with respect to $x_i$. The Hamiltonian $H_{std}$ acts on a wave function $\psi (x_1,x_2,...x_N)$ depending on $3N$ spatial variables. Compared to classical mechanics in physical 3d space, this is a new strange construction with $N$ versions of $R^3$ so to speak stacked upon each other into a product space

$R^{3N}$ of $N$ versions of $R^3$ which are separated but yet share the same $R^3$ in the electronic repulsion potential $\frac{1}{\vert x_i-x_j\vert}$. The result is that $H_{std}$ has no interpretation in real physical space $R^3$, only a statistical invented by Born under protests from Schrödinger who never accepted $H_{std}$ as physics.

The Coulomb Hamiltonian of RealQM takes the same mathematical expression as that of StdQM above, but with a different meaning of the $x_i$ given as follows: Physical space $R^3$ is partitioned into non-overlapping domains $\Omega_i$ with $x_i$ being the coordinate $x$ in $R^3$ restricted to $\Omega_i$. The corresponding Coulomb Hamiltonian $H_{real}$ acts on a wave function $\psi (x)$ appearing as a sum of one-electron wave functions $\psi_i(x)$ with $x\in\Omega_i$ thus with non-overlapping supports, all depending on the same space coordinate $x$. 

It seems that such a Hamiltonian is not described in the literature, although it appears as the most natural generalisation of the Hamiltonian for the hydrogen atom, much more reasonable than that of StdQM. Maybe the reason is the Bernoulli free boundary condition of RealQM allowing electron densities to meet with continuity and zero normal derivative at boundaries between different domains $\Omega_i$, in forms of mathematical analysis of free boundary problems pursued from 1950s connecting to classical mechanics. 

Sum up:

• StdQM works with electronic wave functions with overlapping global supports (not natural).
• RealQM works with electronic wave functions with non-overlapping supports (natural).
• StdQM works with electrons without identity (not natural).
• RealQM works with electrons with identity by spatial occupancy (natural).
• StdQM is formulated in abstract multi-dimensional space (not natural).
• RealQM is formulated in real physical space (natural)
• StdQM is uncomputable (not reasonable).
• RealQM is computable (reasonable).
• StdQM cannot explain chemistry (not useful).
• RealQM has a potential to explain chemistry (possibly useful).

Will Trump Get Nobel Prize in both Peace and Economics?

In his election campaign Donald Trump promised the World to end US wars and bring never before seen economic prosperity to the working class, all in careful planning to get the Nobel Prize in both Peace and Economy, never seen before.

But after taking office Trump has reversed these policies into new US wars to boost war industry and the rich while delivering economic ruin to the working class. 

Trumps trade tariffs aim at bringing production of goods back to the US, for example jogging shoes to replace those now efficiently produced in Vietnam. 

This connects to the experience of my grandfather Edwin Johnson, who 18 years old emigrated to the US in 1888 with a three-year basic schooling, to start his working life making rubber boots in a factory in Millville Mass, up to 12 a day for 12 dollars. But Edwin had ambitions and evolved to become a respected engineer at General Electric inventing the famous "Johnson Breaker", to be recruited back to Sweden in 1908 by Sigfrid Edström the director of ASEA/ABB 1903-33, and then starting his own business in Gothenburg competing with ASEA. 

The rubber boot factory in Millville will now get a new start under the Trump tariffs as if that will bring prosperity to the US working class. Edwin would have said today that this is not a very good idea and that this would not be worth a Nobel Prize in Economics. Nor would starting new wars give a Peace Prize.

But wait: Obama got the Nobel Peace Prize and then started the US proxy war against Russia in Ukraine, and so the Norwegian Nobel Committee may well consider giving the Prize to Trump for finishing off the war by defeating Russia. And maybe the Swedish State Bank Nobel Prize in Economy will find Trump a worthy candidate for reinventing the so successful US trade tariff system 1861 - 1933 to give the West back its role as Leader of the World to a degree never seen before.

PS Later today Trump gave up his grand idea to change World economy by tariffs in a grand flash of intuition saving the World from economic catastrophe, which after all could be worthy of the Prize in Economy. Will a corresponding U-turn from war to peace appear in a flash of insight?

Schrödinger's View of Standard Quantum Mechanics 1952

Max Born was chief architect of what became the ruling form of the new physics emerging from Schrödinger's equation for the Hydrogen atom in 1926 as Standard Quantum Mechanics. Let me cite from the preface of the book Schrödinger's Philosophy of Quantum Mechanics by Bitbol: 

• Max Born was more closely involved than anyone else in the debate with Schrödinger. 
• In November 1952, he was due to hold a series of lectures at the university of London, and he expected Schrodinger to be one of the main participants in the public discussion. As it turned out, Schrodinger was unable to attend, due to ill health, but the elements of the controversy were recorded in two articles published in Born's edition of the Born-Einstein letters. 
• Let me first try to summarize Born's account of Schrödinger's position:

1. It is an essentially "conservative attitude towards quantum mechanics"; an attempt to recover the "classical physics of clearly comprehensible events",
2. It tends to dismiss the "statistical concept of quantum mechanics" and to reinstate determinism, in agreement with Einstein's views.
3. It leads one to the discarding of the very concept of a particle, to asserting that "there are no particles and there are no energy quanta".
4. Schrödinger considers that "particles are narrow wave packets",
5. Schrödinger insists that there is something behind the phenomena, the sense impressions, namely waves moving in a still scantily explored medium; he tends to forget the multi-dimensional character of the Psi-functions and to insist on waves in ordinary 3-dimensional space, which are supposed to rescue the "Anschaulichkeit" (picturability) of the theoretical description; 
6. He believes that his waves constitute the final deterministic solution.

This is essentially the picture I try to fill with RealQM. After recalling Schrödinger's position, Born went on to refute it once and for all, and Schrödinger passed away, but maybe to reappear at the upcoming centennial celebrations of his masterpiece from 1926. Schrödinger was awarded the Nobel Prize in physics  in 1933 for his wave equation, while Born had to wait to 1954 after his refutation of the same thing.

Compare with what chatGPT has to say.

Molecule Formation by RealQM

This post gives a summary with more details of recent posts exploring the potential of RealQM for simulation of molecule formation as a collection of atoms find a joint configuration with lower total energy than separate atoms. The main ingredients are: 

1.  Each atom is represented by an outer shell of negative valence electron charge density surrounding an inner shell of positive charge density representing the atom kernel shielded by remaining electrons. Charge densities do not overlap. 
2. Potentials appear as solutions to Poisson equations with charge density input. Each charge density interacts with the potentials from the other densities but not with its own.
3. The total energy $E$ is the sum of kinetic energy as measure of charge density spatial gradients and potential energy as measure of charge densities weighted by potentials.
4. Configuration of valence electrons for given inner shells/kernels are computed by gradient minimization of $E$ over valence charge densities meeting at a Bernoulli free boundary with continuity of charge density and zero normal derivative.
5. Inner shell configurations identified by kernel positions are for given valence charge densities updated from gradients of potentials acting on inner shell charges. Inner shell charges are kept constant over spheres of constant radius (for simplicity).  
6. The computer code consists of essentially three lines for update of (i) charge densities, (ii) potentials and (iii) free boundary followed by update of kernel positions based on gradients of potentials. 

We may compare with StdQM molecule dynamics, where for given kernel positions electron configurations are computed by DFT and then kernel positions are updated from computation of total energy gradients altogether using very complex computer codes.  

You can test RealQM for a 2d model with three atoms like Oxygen O forming an O3 ozon molecule by running this code with this output during iteration to minimum of total energy:

We see three atoms, each atom consisting of an inner shell with atom kernel shielded by electrons surrounded by an outer shell of valence electron charge density (red), which meet at a free boundary between the kernels. We see cross-cut through lower kernels of electron charge density (green), global potentials for lower inner shells (blue) with gradients (light-blue) indicating directions of update of kernel positions. We see kernels slowly moving in the directions given by light-blue gradients which tend to zero in approach to energy minimum.  

This is an exercise to show qualitative performance of RealQM for simulation of formation of a molecule froms atoms coming together in a process of energy minimisation. The computational complexity scales linearly with the total number of valence electrons, and so opens to simulations of formation of large molecules.

PS Here is description in mathematical notation;. The wave function $\Psi (x)$ with $x$ a 3d Euclidean space coordinate for the system/molecule takes the form of a sum of atom wave functions $\psi_i(x)$ and $\psi_{i+N}(x)$ for atom $i=1,...,N$,

• $\Psi (x) = \sum_{i=1}^{2N}\psi_i(x)$ 

where $\psi_1(x),...\psi_N(x)$ represent valence electrons of negative charge, $\psi_{N+1},...\psi_{2N}(x)$ represent inner shells of positive charge and all wave functions have disjoint support with total charge 1: 

• $\int\psi_i^2dx = 1$ for $i=1,...N$ and $\int\psi_i^2dx = -1$ for $i=N+1,...,2N$. 

The total energy $E(x)$ of the valence electrons is given as a sum of kinetic energy and potential energy: 

• $E(x)=\frac{1}{2}\sum_{i=1}^N\int\vert\nabla\psi_i(x)\vert^2dx+\sum_{i=1}^NP_i(x)\psi_i(x)^2dx$

where $P_i(x)$ for $i=1,..,N$ is the total potential acting on valence electron $i$ with contribution from other valence electrons and inner shells through solution to the Poisson problem 

• $-\Delta P_i =\frac{1}{2} \sum_{j=1,j\neq i}^N\psi_j^2 - \sum_{j=N+1}^{2N}\psi_j^2$, 

where the factor $\frac{1}{2}$ balances double count of electrons. 

The formation of the molecule is realised by minimisation of $E(x)$ over valence electrons and kernel positions with valence charge densities meeting at a Bernoulli free boundary. 

To guide motion of kernel positions we compute gradients of the potentials $P_i(x)$ for  $i=N+1,...2N$ acting on inner shells as solutions to  

• $-\Delta\Psi_i =\sum_{j=1}^N\psi_j^2 - \frac{1}{2}\sum_{j=N+1,j\neq i}^{2N}\psi_j^2$.

 

On Chemical Bonding

Let us cite from the Preface of The Chemical Bond Fundamental Aspects of Chemical Bonding (eds Frenking an Shaik) 2014) :

• One of the fundamental territories of chemistry is the chemical bond, the glue from which an entire chemical universe is constructed. 
• The bond is becoming again a central intellectual arena. This ‘‘return of the bond’’ has prompted the two editors to edit these two volumes on bonding, and it is only fitting that their publication date is close to the centenary of the Lewis seminal paper on electron-pair bonding.
• The first revolution is the ‘‘compositional revolution’’ that started with Lavoisier and culminated in the atomic hypothesis of Dalton.
• Starting in the late 1920s, the new quantum theory has entered chemistry and ushered the third revolution through the works of Heitler and London and subsequently by Pauling, Slater, Mulliken and Hund, Huckel, and so on. 
• Here, we have to digress for a moment and remark about the current tendency to classify chemistry as a service science and a sub-branch of physics.
• Covalent bonding involves a synergism between several interactions with quite different physical attributes and quantitative characteristics. 
• On the one hand, the lowering of the energy that establishes the bond is the result of a variational competition between the kinetic energy and potential energy. 
• On the other hand, there occurs an intricate interplay between various intra-atomic and interatomic interactions. 
• These basic agents have, moreover, to accommodate electron correlation. 
• It emerges that, in all cases, the driving force of covalent bond formation is the lowering of the kinetic energy gained by the delocalization of electronic waves over more than one atom. 
• This observation is only superficially discordant with the virial theorem which, as mentioned earlier, requires the molecule to have a higher total kinetic energy than the separated atoms. 
• The in-depth accounting of all interconnections between the various interactions shows that the information disclosed by the actual total kinetic and potential energies per se is insufficient for drawing any inferences regarding the origin of covalent bonding.

We learn from this exposition that the mystery of covalent chemical bonding has not been clarified by the new quantum theory and so awaits an explanation on physical grounds. This is were RealQM appears to open a window to new understanding. 

Molecule Formation by RealQM: Basic Test

To illustrate the potential of RealQM for simulation of the formation of molecules in covalent bonding, let as here consider a 2d model of an X3 molecule formed by three X atoms each represented by a negative electron valence charge outside an inner shell of positive charge with the following initial state:

We see the valence charge in red around an inner shell. We run this code (uniform 100 x 100 mesh) to follow the formation of the molecule as valence charges evolve to meet at a free boundary.

We see cross-cut through lower atoms in green of valence and inner shell charges, together with potentials acting on inner shells in blue. We decrease the inner shell charge from 2 to 1.8 to get using this code:

Here the average potential gradient acting on inner shell charge is displayed in light-blue showing a net attraction signifying negative valence charge accumulation between the atoms overpowering positive charge repulsion. Changing to 2.2 we get with this code:

We see potential gradients change sign with now positive charge repulsion overpowering negative charge attraction. We conclude that equilibrium is reached for C around 2. 

We understand that the secret of molecule formation is the accumulation of negative charge between the atoms without increase of kinetic energy because electron charges meet at a Bernoulli free boundary with non-zero density. This is the essential new physics brought by RealQM, which appears to unlock the secret of covalent bonding.

We here change the inner shell charge rather than the distance between atoms, for display simplicity.

The purpose of this exercise is to show the potential of RealQM for simulation of molecules with many electrons. The computational cost on a given mesh scales linearly with number of valence electrons involved thus with number of atoms.  

Compare with earlier 3d examples under tag Real Quantum Chemistry.vs 

Let us compare RealQM and StdQM vs the key ingredients of (1) kernel potential energy from accumulation of electron density between kernels (-), (2) electron kinetic energy (+) and (3) electron repulsion energy (+), where we indicate sign of contribution to total energy. Here RealQM has an advantage concerning (2) because of the Bernoulli free boundary and (3) because non-overlapping charge densities have smaller electron repulsion energy than overlapping densities. 

Evidence of (3) is given by StdQM energy for Helium of -2.75 with overlapping Hydrogen electron densities, while RealQM gives -2.87. 

We conclude that RealQM has a better chance to capture the secret covalent bonding than StdQM. 

Here is a first version of RealQM code for dynamic molecule formation with kernel geometry determined by electron potentials.

Here is a conversation with chatGPT showing that covalent bonding is not well understood even today 100 years after the advent of quantum mechanics. This is mind boggling…

PS Specifically, we compare

• $R1= \int_{R3}\int_{R3}\frac{\exp(-\vert x\vert)\exp(-\vert y\vert )}{\vert x -y\vert } dxdy$ 
• $R2= 4\int _{R31}\int_{R32}\frac{\exp(-\vert x\vert)\exp(-\vert y\vert )}{\vert x -y\vert }dxdy$ 

where $R3$ is all of 3d space, $R31$ is the half space $x_1<0$ and $R32$ is the half space $y_1>0$ with

$x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$, and find that $R2<R1$. Non-overlapping gives smaller electron repulsion energy than overlapping, as expected. 

Schrödinger vs Real Quantum Chemistry

Schrödinger was not happy with mainstream quantum mechanics, nor his cat.

The foundation of modern physics in the form of quantum mechanics is Schrödinger's equation formulated by Erwin Schrödinger in 1926 for the Hydrogen atom H with one electron as a partial differential equation of simple form in 3d Euclidean space with coordinate $x$ allowing analytical solution with eigenvalues in full agreement with the spectrum of H. The ground state electronic wave function as eigenfunction was given by the exponential function $\psi (x)=\frac{1}{\sqrt{\pi}}\exp(-\vert x\vert )$ decaying with distance $\vert x\vert$ from the H kernel at $x=0$ with $\psi (x)^2$ representing charge density with total charge $\int\psi(x)^2dx=1$.  

Schrödinger was very happy with his equation for the H atom since its eigenfunctions had a clear physical meaning as charge densities in a classical continuum mechanics sense. 

It remained to extend Schrödinger's equation to atoms with many electrons and this work was done in cooperation with other physicist including Born and Pauli, in the form of a purely formal trivial extension adding a new set of 3d spatial variables for each new electron into model in $3N$ dimensions for an atom with $N$ electrons to form Standard Quantum Mechanics StdQM. This development made Schrödinger very unhappy, since the eigenfunctions of the multi-dimensional model no longer had any real physical meaning, only a statistical meaning invented by Born. Schrödinger insisted that this could not be accepted but was overpowered and found no place in the further development until making a last fruitless effort in the 1950s to bring realism back to quantum mechanics. 

RealQM offers a new form of Schrödinger's equation as different generalisation of the one-electron model in the form of a system of non-overlapping charge densities with direct physical meaning. This model satisfies Schrödinger's demands of realism. 

Michel Bitbol gives in Schrödinger's Philosophy of Quantum Mechanics a detailed review of Schrödinger's successive positions from start to end, including the information that from 1928 Schrödinger resigned himself from teaching mainstream quantum mechanics. 

The lack of realism of StdQM is accepted by physicists, but poses problems for a chemist used to think of molecules as real physical structures in 3d kept together in chemical bonding. RealQM offers consolation for a chemist troubled to fit such a realist view with formalist StdQM.   

Bitbol writes:

• Pauli went as far as denouncing Schrodinger's "neurotic" regression and accusing him of entertaining the dream of a way back, back to the classical style of Newton-Maxwell, that is hopeless, off the way, bad taste and not even a lovely dream.
• Many physicists rejected Schrodinger's proposals, without even bothering go examine his arguments with any care. 
  

Can Chemistry be Reduced to Standard QM?

The mantra of modern physics is that Chemistry fundamentally is a branch of Quantum Physics based on Schrödinger's equation of Quantum Mechanics QM as Standard QM, which appears to be accepted in principle by a majority of chemists as reductionists, under the agreement that the reduction is yet to be realised, the big obstacle being that Schrödinger's equation is uncomputable.

There are some prominent chemists who have expressed clear anti-reductionist views pointing to inabilities of QM to describe fundamental aspects of chemistry:

• Hans Primas: emergence  
• Guy Woolley: molecular structure
• Norman Hunter: chemical bonds and reactivity 

and Roald Hoffman viewing chemistry as creative science beyond QM. It may be that in fact most chemists share such views and think of chemistry rather in terms of molecular structure in 3d with chemical bonds formed under chemical reactions, than as abstract StdQM.

RealQM is based on a new form of Schrödinger's equation, which has direct physical interpretation in terms of non-overlapping charge densities, closely connects to chemists conception of molecules as formed by atomic kernels surrounded by electrons carrying individuality by occupying different regions of space. RealQM thus opens to a form of Real Quantum Chemistry explored under this tag, where there is no gap between theory and chemistry reality since the Schrödinger equation of RealQM is readily computable. 

• Quantum chemistry is but a narrow subfield of theoretical chemistry and numerical quantum chemistry is nothing but a powerful tool. (Hans Primas in Chemistry, Quantum Mechanics and Reductionism).
• All that has happened (in the 20th century) is that the 19th century rupture between chemistry and physics has been patched over in framework of quantum chemistry. (Guy Woolley in Foundations of Molecular Quantum Electrodynamics) 

Secret of Covalent Chemical Bonding by RealQM

RealQM offers an easily understandable explanation on clear physical grounds of covalent chemical bonding, which has remained a mystery within Standard text book QM (see PS below).

To exhibit the physical mechanism forming a covalent bond, let us here as an exercise consider a 2d model of a X2 molecule formed by two X atoms, each with a +1 kernel surrounded by -1 electron charge density. We can think of X as 2d model of an H atom. The model is described in this post and is realised in this code. The real case in 3d is considered in this post.  The essential physics is Coulomb interaction between charges of same/different sign. The question is how a Coulomb chemical bond can be formed between two neutral atoms? 

We start running the code with the atoms separated (D=50) to get this output:

We see two essentially non-interacting 2d charge densities (red) together with mid line crosscuts of charge density (red), kernel potentials (blue) and electron potentials (light-blue, different sign) with a total energy of -0.9068. We see that the electron potentials (light-blue) are roughly half of the kernel potentials (blue), reflecting two-way interaction between electrons and one-way interaction between kernel and electron. The 2d energy here of -0.9068 is not the 3d energy -1.

We next run the code with smaller kernel distance (D=20) to get a smaller total energy  -1.05443:

We see that the left and right electron wave functions/charge densities now interact and meet at line orthogonal to the line between the kernels with continuity (and small/zero derivative). We understand that the decrease of total energy has two sources:

• Decrease of kinetic energy from the fact the electron charges meet with non-zero value at the free boundary, which does not require decay to zero with kinetic energy cost.
• Decrease of potential energy from the shaded region between the left kernel potential (blue) and left electron potential (light-blue) in interaction with the right electron charge density (red) and vice versa.

We decrease the distance further (D=15) to get essentially the same total energy as minimum, as the kernel repulsion takes over, running this code:

Altogether, RealQM offers an explanation on clear physical grounds in the form of Coulomb potentials of covalent chemical bonding. The essence is the Bernoullli free boundary between electron charges allowing concentration of electron charge between kernels without cost of kinetic energy. The secret is the decrease of potential energy from the above shaded area combined with large charge density at the free boundary. We understand that the bond is formed from a subtle interplay between electron charge distributions of substantial width with kernels of much smaller width. This is a new revelation by RealQM. Text book explanations struggle with “charge concentration”combined with “charge decentralisation” which appears contradictory.

PS chatGPT admits that there is no explanation in real physical terms of covalent bond.

 

Can an Unphysical Theory Explain Real Physics?

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

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

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

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

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

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

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

Discussion with Eric Scerri on the Periodic Table

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

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

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

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

RealQM Molecule Model 2D

To illustrate the basic feature of the new Schrödinger equation of Real Quantum Mechanics RealQM, let us  reduce from 3d to 2d, to get the following formulation for a molecule with $N$ electrons (like graphene):

Find the wave function $\Psi (x)$ with $x$ a 2d Euclidean coordinate, of the form 

• $\Psi (x) = \Psi_1(x) + \Psi_2 (x) +    + \Psi_N(x)$

where the $\Psi_n(x)$ for $n=1,2,...,N$ are one-electron wave functions with non-overlapping supports meeting a Bernoulli free boundary $\Gamma$, which minimizes the total energy 

• $E(\Psi ) = \frac{1}{2}\sum_{n=1}^N\int \vert\nabla\Psi_n\vert^2dx-\int P(x)\Psi^2dx$  (kinetic + potential energy)

with $P(x)$ a potential with contribution from electron and kernel charges, under the side condition 

• $\int\Psi_n^2 dx = 1$ for $n=1,2,...,N,$

and the Bernoulli free boundary condition:

• $\Psi (x)$ is continuous and 
• the normal derivative of $\Psi_n(x)$ vanishes on $\Gamma$ for $n=1,2,...,N.$  

This minimisation problem is solved with a gradient method realised as an explicit update consisting of three lines for iterative update of (i) wave function, (ii) level set function for $\Gamma$ and (iii) potential $P(x)$ realised in this code.

We consider a molecule consisting of two H atoms in green and one Beryllium atom with valence shell consisting of two electrons as two "half-shells" in red and blue. We start the iteration with the electrons concentrated into disks without overlap:

 

We find the following configuration after energy minimisation using this code to see the formation of a free boundary between electron wave functions:

We see that electron wave functions meet at a free boundary with continuity and zero normal derivative displayed in a horizontal cross-cut in yellow and vertical in magenta. We find a total energy of -12.34.

We compare with the atoms well separated with a total energy of -5.803 after energy minimisation.  

We understand that the reason the energy is lower for electrons in Bernoulli free boundary contact is that the charge density is concentrated between the kernels thus decreasing potential energy without increase of kinetic energy measured by $\nabla\Psi_n(x)$, since RealQM wave functions are not required to tend to zero like the global wave functions of stdQM.

Does Quantum Mechanics Explain Chemical Bonding?

This is a follow up of the recent post posing the question if quantum mechanics explains the Periodic Table connecting to the work by Eric Scerri as a world authority on the chemistry of the Periodic Table. In this article from 2023 A commentary on Robin Hendry’s views on molecular structure, emergence and chemical bonding Scerri starts out:

• There is a long-standing problem in the philosophy of chemistry arising from the fact that molecular structure cannot be strictly derived from quantum mechanics. 
• Two or more compounds which share a molecular formula, but which differ with respect to their structures, have identical Hamiltonian operators within the quantum mechanical formalism. 
• As a consequence, the properties of all such isomers yield precisely the same calculated quantities such as their energies, dipole moments etc. 
• The only means through which the difference between the isomers can be recovered is to build their structures into the quantum mechanical calculations, something that is carried out by the application of the Born-Oppenheimer approximation. 
•  I agree that chemists view bonding in a more realistic fashion and may consider bonds to be in some senses real, while physicists may consider bonding in more abstract energetic terms. 

We learn from Scerri that quantum mechanics does not distinguish between isomers like C2H5OH (ethanol)  CH3OCH3 (dimethyl ether) with vastly different chemistry from different bonding and so must be complemented by some form of expertise, which differs between chemists who seek explanation in some real terms and physicists satisfied with something purely abstract.

RealQM is a new form of quantum mechanics in terms of classical continuum mechanics as a system of non-overlapping charge densities with direct physical meaning in 3d space, which meets a chemist's demand of a reality. In RealQM the formation/bonding of a molecule can be simulated which can make a distinction between isomers without need of expertise. The window opened by RealQM is explored in posts on Real Quantum Chemistry. 

Non-Overlapping Wave Functions/Charge Densities

This is a follow up of the recent post on The Secret of Covalent Bonding with further computations comparing the non-overlapping wave functions/charge densities of RealQM meeting with (i) continuity and zero derivative, with a hypothetical case of instead (ii) zero density. 

To pin-point the essential aspect, we consider the following 1d model problem of an atom with N electrons: 

Find the function $\Psi (x)$ on the interval $[0,1]$ which minimises the energy $E=E_k + E_p$ with

• $E_p=\frac{1}{2}\int_0^1D\psi^2dx$             (kinetic energy)
• $E_k =-\int_0^1P(x)\psi (x)$           (potential energy)

over wave functions $\psi (x)$ with $D\psi =\frac{d\psi}{dx}$ the derivative, with total charge 

• $\int_0^1\psi^2dx = N$,

where $-P(x)$ is a given potential.

The Ansatz of RealQM is to seek $\Psi (x)$ on the form 

•  $\Psi (x) = \Psi_1(x) + \Psi_2(x) + .... + \Psi_N(x)$,

where the $\Psi_i(x)$ are one-electron wave functions with disjoint supports which meet on a Bernoulli free boundary with continuity and zero derivative. Running this code in a case with $N=4$ and $P(x)$ the potential from two kernels, we get the following result for case (i): 

We see in red 4 non-overlapping wave functions meeting at a free boundary marked by green with continuity and zero derivative. The total energy is -247 with kinetic energy 30. 

We run the same code but require wave-functions to vanish at meeting points and get for case (ii):

We see total energy larger total energy -231 with much larger kinetic energy 308. 

We understand that requiring wave functions to vanish as in (ii) when meeting, gives much larger kinetic energy than asking only for continuity and zero derivate as in (i), and that the net effect is larger total energy.

The purpose of this exercise is to exhibit the possibility opened by RealQM of electron concentration between kernels decreasing potential energy without balancing increase of kinetic energy, which appears as the holy grail of covalent bonding. 

In StdQM wave functions have global support, which may be overlapping, and so like the functions in (ii) above have to pay a price of added kinetic energy from charge concentration which is less favourable for binding than RealQM.

The finding of this post agrees with the real case considered in previous posts.

 

Does Quantum Mechanics Explain the Periodic Table?

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

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

Here are the key points of Scerri's position:

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

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

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

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

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

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

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

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

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

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

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

Stupid Swedish Foreign Policy

All political parties in Sweden from left to right have come to a historic consensus on foreign policy:

1. Trump is bad even if what he now does is to continue the politics of Biden: Genocide in Gaza, proxy war against Russia in Ukraine and bombing of the Houtis. 
2. Putin is bad and the only way to stop him from invading Sweden is to participate in the US/NATO proxy war against Russia in Ukraine with maximal military support.  

1. means that Trump will not come to help Sweden in case of Russian retaliation because of 2. Worst possible combination.

This is leading into catastrophe for the Swedish people, but opposition to this stupid policy is suppressed by very effective propaganda.

A better alternative is to come to a peaceful agreement with Putin by stoping sending Swedish weapons to Ukraine. This would open to rightful criticism of Trump without ruining Swedish society.

Form Classical to Modern to Post-Modern Physics

Classical physics in the form Newtonian mechanics emerged during the scientific revolution of the 18th and 19th centuries from an Enlightenment of rational logic/mathematics combined with observation of reality in a fundamental shift away from religion scholastics. The basic idea was rational mechanics as physics, which was (more or less) understandable and not only magical.  

In the late 19th century classical physics incorporated electro-magnetics made understandable through Maxwell's equations.  

But the modern physics of relativity theory and quantum mechanics emerging in the beginning of the 20th century signified a return to magical thinking. 

All the great physicists of the 20th century Einstein, Bohr, Schrödinger, Dirac, Feynman, Gell Mann, Weinberg admitted that quantum mechanics cannot be understood, while suggesting that this only adds to its beauty as the prime achievement of human intellect. 

To teachers of quantum mechanics this created a problem: How to teach a subject that is not understood by even the sharpest minds? The only way out for the average teacher was to pretend to understand and refer to the admissions of the top physicists of non understanding, as only a sort of teasing: Of course quantum mechanics is well understood and so can be presented in text books to students expected to come to understanding by diligent study. 

The computer is often presented as an off-spring of quantum mechanics thus showing its power, but the first realisations of the idea of a computer was made in rational mechanical terms as the Analytical Engine by Babbage with all its wheels and gears. A realisation in terms of quantum mechanics would be a quantum computer but the question is if there will ever be such a thing. 

Is it possible to imaging a form of post-modern physics with a return to the rationality of classical physics. My answer in all modesty is RealQM.  

Ab Initio Chemistry: StdQM vs RealQM

Discovering the World from a desk.

The ideal mathematical model of some physics is according to Einstein a model without parameters. Newton's Law of Gravitation is the prime example within classical physics/mechanics taking the from

• $\Delta\phi (x)  = \rho (x)$

where $\phi (x)$ is gravitational potential, $\rho$ mass density and $\Delta$ the Laplacian differential operator with respect to a Euclidean coordinate $x$. 

The prime example within modern physics/quantum mechanics is Schrödinger's equation for the hydrogen atom taking the form of an eigenvalue problem with eigenvalue $E$ 

• $-\frac{1}{2}\Delta\psi (x) -\frac{\psi (x)}{\vert x\vert} = E\psi (x)$  

where $\psi (x)$ is a wave-function and $-\frac{1}{\vert x\vert}$ is a kernel potential. No parameter appears if units are properly chosen.

The amazing property of a parameter-free model is to give information about physics without other physics input than a mathematical model without parameters. This allows the theoretical physicist at his desk to make predictions about the world without any input from the world. Such a prediction is said to be ab initio. Amazing! A priori knowledge in the sense of Kant.

In general, however, mathematical models of some physics contain parameters which have to be known from experiment or other models, in order to get anything out of the model. The Standard Model of particle physics contains 19 parameters which cannot be normalised by choice of units. It is very difficult to determine these parameters by experiment. 

Let us now compare StdQM with RealQM as concerns ab initio simulation of chemical bonding. 

For StdQM we turn to chatGPT and get this response stating that this is impossible because the Schrödinger equation of StdQM, which is parameter free, is uncomputable and thus has to be modified or prepared to deliver any result and the preparation builds on massive input of know-how and parameters/experiments.

In contrast the new Schrödinger equation of RealQM is parameter-free and readily computable and thus delivers ab initio simulation capability of e g chemical bonding.  See posts on Real Quantum Chemistry with links to everything about RealQM.

PS Here is what Eric Scerri says about ab initio StdQM in Selected Papers on the Periodic Table:

• Whereas most chemists and educators seem to believe that all is well, I think that there is some benefit in pursuing the question of how much is strictly explained from the theory. 
• It is indeed something of a miracle that quantum mechanics explains the periodic table to the extent that it does at present. 
• But we should not let this fact seduce us into believing that it is a completely ab initio explanation. We have not yet arrived at the super - ab initio phase of quantum chemistry and nor are we even close. 
• If anything, the compromises that have been struck with the acceptance of parametrization as well as the mixing of wavefunction and DFT approaches begin to question the earlier promise of ab initio quantum chemistry.

The Secret of Covalent Chemical Bonding

Recent posts have exhibited the fact that the physics of chemical bonding still is debated as a fundamentally unresolved problem. Consensus appears to be that covalent bonding results from some form of "sharing of electrons", which decreases kernel potential energy by electron charge concentration between kernels without full compensation of increase of kinetic energy.

It is further agreed that this picture can be given support by quantum mechanics with the caveat that full solution of Schrödinger's equation for systems with several electrons is impossible. The idea is that somehow "delocalisation" of electrons over an entire molecule as a purely quantum mechanical effect, will make electron charge concentration possible without full compensation of increase of kinetic energy. But the quantitative details appear evasive.

RealQM offers a different account of the physics of covalent bonding which we here illustrate in a generic 1d molecule with two atomic kernels and two electrons. The crucial feature of RealQM is decomposition of the total electron wave function U(x) = U1(x) + U2(x) into one-electron wave functions with non-overlapping supports  meeting at a free boundary X with continuity and zero (normal) derivative. (Bernoulli condition). Running this code we get this result with kernel potential in blue and electron wave functions in red and green: 

We see a concentration of electron charge densities between the kernels meeting with non-zero joint value and zero (normal) derivates at the free boundary X. The total energy is -65.258 with kinetic energy 26.771.

We compare with a model with overlapping electron densities as in StdQM using this code:

We find a higher total energy -63.144 with substantially higher kinetic energy 35.111. 

We understand that the reason the non-overlapping wave functions above has smaller kinetic energy than the overlapping below, is that the non-overlapping meet at a free boundary with non-zero value and zero derivative allowing dU1/dx*dU1/dx and dU2/dx*dU2/dx to be smaller,  since overlapping wave functions have to meet zero end point values. The overlap region of StdQM is thus in RealQM replaced by a free boundary effectively decreasing the kinetic energy while keeping the potential energy and thus decreasing the total energy into bonding. 

Conclusion: 

RealQM offers an explanation in clear physical terms of covalent chemical bonding emerging from concentration of charge density between kernels without compensating increase of kinetic energy thus with lower total energy than separated atoms. The above experience in the basic model problem is reflected in the full molecule models reported in these posts. If this analysis is correct, it asks for major revision of the standard physics of covalent bonding.

Why Philosophy of Chemistry?

The recent post on the talk The Nature of Chemical Bonding by Eric Scerri, connects to an emerging field of Philosophy of Chemistry promoted by him. Asking chatGPT about the role of philoposphy in science we get the message that 

• Philosophy of Quantum Mechanics is a very active field since 100 years. 
• Philosophy of Newtonian Mechanics has played out its role since "science is settled".
• Philosophy of Chemistry is a new field formed during the last 20 years. 

We are led to an idea that if a philosophy of a certain scientific discipline is currently an active topic, like philosophy of quantum mechanics and chemistry, it expresses that "science is not yet settled".  It signals that there are foundational questions which cannot be resolved within the discipline itself using the tools of the discipline, and a resort to some form of metaphysics or philosophy is brought in to help.  

Beginning in the later 19th century and culminating in the 1930s, a search for the foundations of mathematics engaged many famous mathematicians like Cantor, Russell, Hilbert and Brouwer without ever coming to any agreement and the interest then slowly faded away when computer science gave its resolution, leaving pure mathematicians to explore new ideas without worrying about the true nature of in particular the infinite set of real numbers and the set of all square integrable functions of real numbers. Giving them names was decided to be enough. Today philosophy of mathematics is largely viewed as "settled science".

2nd Trump Talk to Putin

Who has the upper hand?

On March 18 Trump called Putin for the 2nd time since he won the November election on a promise to end the US/NATO proxy war against Russia in Ukraine in 24 hours. See also previous post.

To get a into a position of strength before the 2nd talk, T resumed military support to the proxy war, after a short stop to show good will to peace. In the talk P repeated his condition for progress towards peace as a return to a halt of military support from the US. 

P agreed to a proposal from T to have a 30-day cease-fire limited to energy infrastructure, and directly ordered Russian implementation, as a sign of good will. T did not say he was willing to stop military support, which could have been a sign of good will.

T suggested that the war possibly could be settled in ice-hockey matches between US and Russia, which was met positively by P.  But T did not say he was willing to stop military support, which would  have been welcomed by P as a sign of good will. 

T prides himself to be a man of quick action, but the talks with P follow a very slow schedule. It is still unclear if T really wants  peace or just seeks a way to prolong the war indefinitely giving in to neo-cons.

Swedish Defense Minister Pål Jonsson declares that Sweden will continue to send weapons to Ukraine independent of any peace talks between US and Russia, and that peace will only create even bigger problems for Sweden and Europe. The stocks of Saab sky-rocket. Anything which is good for Russia like peace, is bad for Sweden. Russia won over Sweden in Ice-hockey WC in 1967, 1969, 1970, 1973, 1981 and 1986.

Former Swedish Prime Minister Magdalena Andersson says that it absurd that T and P talk peace over the head of Sweden/Ukraine.

Sweden stands strong in its war against Russia. Finland lost to the Russian Empire in 1809 must be brought back under the Swedish Kingdom.

The Still Unknown Nature of Chemical Bonding

Eric Scerri named the 2nd most influential chemist for the decade 2010-2020 gave a plenary talk at 26th Conference of the ISPhil of Chemistry in 2022 on The Nature of Chemical Bonding starting out presenting two views according to Robin Hendry:

• Structural as actual "bonds".
• Energetic as "bonding". 

Scerri continues recalling Coulson's words (1955):

• A bond does not exist: no-one has ever seen it, no-one ever can. It is a figment of our own imagination.

Scerri adds a statement by his colleague Seifert from an email conversation:

• Neither the structural nor the energetic provide "suitable characterization of chemical bonding". 

This is not a good start in 2022 in view of Gilbert Lewis clarification in 1916: 

• In the mind of the organic chemist the chemical bond is no mere abstraction; it is a definite physical reality. a something that binds atom to atom.  

Scerri then proceeds to scrutinise of how chemical bonding is described in text books of chemistry concluding that there is no satisfactory explanation of in particular the covalent bond somehow emerging from "sharing of electrons" as the dominating theme (together with the ionic bond as simply Coulomb attraction between ions). 

Scerri struggles to make sense of the textbook explanations following the very natural idea that somehow a bond can form if electrons can find a configuration close to kernels with decreasing potential energy which is not balanced by increase of kinetic energy supposedly emerging from two conflicting actions: 

• concentration of electron charge distribution decreasing potential energy
• delocalisation of electrons charge distribution avoiding increase of kinetic energy.   

But Scerri stops there: The Nature of Chemical Bonding is still a mystery, since it is the title of hus talk in 2022. But it must have a nature in physical terms, it cannot just be imagination. The World consists of molecules as atoms held together by chemical bonds.

It seems to me that Scerri is saying that only modest progress has been made over more than 100 years, despite the giant progress of modern physics in the form of quantum mechanics supposedly describing all of the physics of atoms and molecules including chemical bonding. How can this be? How is it possible that the nature of chemical bonding is still basically a mystery?

Here RealQM comes in offering a physical explanation how electrons can concentrate between atomic kernels to form a "glue" from decrease of potential energy without increase of kinetic energy. The secret is revealed in this plot as print-out of this code from a previous post from a collection of posts on H2:

We see two H kernels in black and a cross-cut along a line through the kernels of two electron wave functions with non-overlapping supports meeting at a plane orthogonal to the line midway between the kernels (as a free boundary) in yellow with plots of characteristic functions of the supports in black.

Computations are 3d on a fixed mesh with 1 million mesh points with mesh sise 0.1 in atomic units. The code is fully explicit consisting of 3 lines for update pf wave functions, potentials and free boundary. The computed energy  (-1.146) is in fair agreement with observation (-1.17) with further improvement under additional iterations. 

The essence is that in RealQM electron wave functions can meet with non-zero joint values at a free boundary which makes concentration of electron charge between the kernels possible without increase of kinetic energy. 

RealQM offers a new theoretical explanation of chemical bonding which agrees with previous intuition without theory.  Here are posts on Real Quantum Chemistry.

Geometry of the H2O Molecule by RealQM

This is a follow up of the previous post with now special attention to the non-symmetric geometry of the  H2O (or HOH) molecule with the two H atoms on one side the O atom, thus forming a dipole with positive charge on the side of the H atoms and negative on the other. 

An O atom has 8 electrons with 2 electrons in a first shell and the remaining 6 with the following thinkable electron distributions in 2nd + 3rd shell:

1.  6
2.  4 + 2
3.  3 + 3
4.  2 + 4   

We compute using RealQM in spherical symmetry using this code to find that 3 matches the observed ground state energy. We conclude that the O atom has an electron shell distribution as 2+3+3 and understand that 3 electrons in the outer shell invites to non-symmetry. 

We now turn to RealQM in 3d with 3 electrons in a third shell out of which 2 bind to the 2 electrons of the H atoms on one side leaving 1 electron free on the other side. We compute dissociation energy using this code with output: 

We see the 2 valence electrons of O (red and yellow) bonding with the 2 electrons of the H atoms (green) leaving 1 O electron on the other side (blue). We see that a somewhat bigger angle than 90 degrees of the HOH molecule is compatible with the 3 O electrons (red, yellow and blue), in qualitative agreement with observed 104.5 degrees.  

We see that RealQM offers an explanation of the non-symmetric geometry of the HOH molecule in direct physical terms of electron charge distribution.  This is different from that offered by StdQM in terms of 2 2 valence electrons and 2 lone-pair electrons in a shell with 6 electrons without direct physics. 

Towards Understanding of Chemical Bonding

RealQM based on a new form of Schrödinger's equation in terms of non-overlapping electron charge densities meeting at a Bernoulli free boundary, gives a new view on the physics of formation of molecules by groups of atoms coming together into states of lower energy than when apart by establishing chemical bonds.  

StdQM describes chemical bonds as either covalent by "sharing of electrons" or ionic with "transfer of electrons" creating Coulomb bonds between charged ions, but in general as some sort of mix.  

RealQM gives a different description as basically covalent where the "sharing of electrons" is realised through the Bernoulli free boundary, where electrons meet with continuity of charge density with corresponding wave functions having vanishing normal derivatives. "Sharing" is thus replaced by "meeting". This allows electrons to meet between atom kernels and so decrease potential energy, without increase of kinetic energy because the "meeting" does not require vanishing charge density only normal derivative of wave function. 

The basic case is the formation of a H2 molecule from two H atoms considered e g in this post .

Let us now consider molecules of the form HXH with X an atom with two valence electrons in two forms as the 2nd column of the periodic table: Be, Mg, Ca, Sr, Ba,Ra, and 5th column: O, S, Se, Te, Po, Lv. Here the 2nd column is linear molecule, while the 5th including HOH is a bent molecule. 

RealQM offers a reduced model of X in the form of two valence electrons outside an inner shell of certain radius R representing the net effect of the other electrons + kernel as a +2 kernel. Running this code we get for a molecule bent to 90 degrees we get for R=0.5 a dissociation energy of 0.3178 Hartree (compared with 0.354 for HOH) with this output (code 3d on regular mesh with mesh size 0.1 atomic unit thus about 20 mesh points between kernels):

For a linear molecule like HBeH, we get with this code 

We see in these pictures as sections of 3d output the following:

• X atom electrons in red connecting to two H atoms in green, in bent and linear configurations.
• H atom meeting X in a narrow overlap region from level set representation of free boundary.
• Line section through kernels show wave function in yellow with in particular continuity of non-zero charge density/wave function at the free boundary as expression of the binding effect of the meeting between kernels. 
• We see small dissociation energy for large radius R of inner shell. 

More precisely we may compare to the following list values of dissociation energies:

• HOH   0.35 Hartree
• HSH    0.247
• HSeH  0.221
• HTeH  0.193
• HPoH  0.176
• HBeH  0.17
• HMgH 0.10
• HCaH  0.079
• HSrH   0.070
• HBaH  0.064
• HRaH  0.059

to see that R = 0.5 corresponds to HOH and R = 1 to HMgH. 

It remains to explain why HOH is bent but HBeH not. We expect this to depend on the different electron configurations of O and Be, with two electrons in the outer shell for Be, but possibly up to 6 for O. Will return with details. This connects to the dipole quality of HOH with the outer part of O negative and those of H positive. 

Bottom line so far:

• In RealQM electrons meet at a free boundary between kernels which decreases potential energy without increase of kinetic energy, as something easy to understand.  
• In StdQM electrons are "shared" which is a complex subject. 

We may view electrons as rivals competing to come close to kernels while repelling each other. An idea of meeting of rivals as a form of coexistence keeping integrity, may seem to be more natural than an idea of "sharing" between rivals with loss of integrity.  

Why Trump Does Not Dare to Speak to Putin

This is a follow up of the post from Nov 28 2024 asking When Will Trump Call Putin?  

Since then Trump has had one telephone conversation with Putin (on Febr 12 2025). 

On March 12 Trump proposed to Russia an unconditional 30-day cease-fire of the war in Ukraine. Putin was given the ball.

On March 13 Putin responded that a cease-fire would be fine, but not unconditional allowing US/NATO to rearm for continued war after 30 days. Trump was given the ball.

On March 14 Trump responded that he hopes Putin will show good will. The ball went over the fence.

Trump does not lift the telephone to speak directly to Putin. Trump sends his buddy real estate mogul Witkoff to Moscow to do the talking. Witkoff left Moscow after a Thursday late night meeting with Putin. 

P nows the facts, T does not and so does not dare to talk directly to P again. The time for mutual congratulation to excellency of political leadership is over. Now it is time to talk, and T has nothing to say.

On March 15 T send P a message asking for mercy for the Ukrainian troops including mercenaries from the West encircled in Kursk. What dos that say about the weight of T vs P?