RealQM as DFT without KS vs Foundations of Chemistry?

Density Functional Theory DFT is commonly viewed to be the Operational Foundation of Chemistry OFC.

DFT is based the Hohenburg-Kohn Theorem HK and the Kohn-Sham Model KS. The 1998 Nobel Prize in Chemistry (1/2) was awarded to Walter Kohn for developing DFT. 

HK states that ground state electron density $\rho$ uniquely determines a (fictitious) external potential $EP$ which determines the wave function $\Psi$ and so the ground state total energy $E$. The proof is a  very short non-constructive argument by contradiction, which gives no information about the connection from $\rho$ to $EP$, $\Psi$ and $E$. 

The map $\rho \rightarrow EP$ can be compared to the map $T\rightarrow F$ between temperature $T$ and heat source $F$ in a heat conduction problem, known as an inverse problem which is unstable or ill conditioned in the sense that small variations of temperature $T$ can give rise to big changes of forcing $F$ (through the action of the Laplacian as differential operator).

We thus expect that the identification of $EP$ from $\rho$ is ill-conditioned and thus without physical meaning unless some form of stabilisation is enforced. But that is not included in HK.

This means that DFT as OPC does not change if HK is simply omitted, because HK does not contribute anything of physical substance. HK is used as a way to legitimise DFT by pure logic without physics, and successfully so since DFT is viewed as OFC. 

The proof of HK is very short and simple and can be compared with a proof of "Unique Existence of God" starting from an assumption that "God is Perfect" and concluding that "perfectness implies both existence and uniqueness" proving the claim. Such an argument tells nothing about the possible role of a God in the World, and forgetting about the proof changes nothing real. Similarly, forgetting HK changes nothing real. Only formal legitimation.

The constructive part of DFT is KS which is a model of one-electron charge densities attributed to a given common density $\rho$, which allows computation of electron kinetic energy. KS is also an inverse problem where a one-electron distribution carried by $\Psi$ is sought to be identified from a common density $\rho$ mixing one-electron densities. KS attempts to solve a very difficult ill-posed problem. The success must be unclear.

Comparing RealQM to DFT/KS we find that RealQM as based on a structure of non-overlapping one-electron charge densities, which is not destroyed,  does not need any KS and so eliminates the main difficulty of DFT. 

RealQM can thus be viewed as a radically simplified form of DFT, where KS has no role to play. Is this an argument which can help the review process of RealQM for possible publication in Foundations of Chemistry? 

DFT as Operational Foundation of Chemistry?

Connecting to the previous post, recall the common idea that Density Functional Theory DFT as a form of StdQM: 

• serves as Operational Foundation of Chemistry OFC. 
• does not serve as Conceptual Foundation of Chemistry CFC. 

Recall that DFT is based on electron (charge) density $\rho (r)$ defined in StdQM as a mean-value of the wave function squared by integration over all 3d variables of configuration space, except one denoted by $r$. The idea of dimensional reduction to electron density in 3d by massive integration was taken up by Thomas-Fermi already in 1927, but was quickly abandoned because it did not work out, but the idea then resurfaced in more advanced form as DFT the 1960s to now serve as OFC but not CFC,  described by Nobel Laureate Roald Hoffman: DFT gives numbers but not stories. 

But is it possible to produce numbers as OFC without stories as CFC?  

Recall that electron density is defined as a massive mean-value where massive information is destroyed by integration. In particular electron-electron repulsion potential energy and electron kinetic energy are not computable from $\rho (r)$ and so not total energy. 

To derive total energy from electron charge density $\rho (r)$ requires modeling of exchange/correlation seeking to eliminate effects of massive integration as a form of information recovery, which is the role of DFT to deliver.  

The foundation of DFT is the Hohenberg-Kohn Theorem HKT stating that ground state total energy is uniquely determined by electron density, however without any specific information about the connection since HKT is based on a non-constructive logical mathematical contradiction argument without physical meaning. The difficult task assigned to DFT is to recover information lost by integration and so serve as OFC. Does it work? Is it really possible to recover information after formation of mean value?

Basic questions:

1. Can DFT serve as Foundation of Chemistry FoC as only OFC and not CFC?
2. What is the role of HKT for DFT?

RealQM is an alternative to StdQM displaying full information about individual electron charge distribution with direct delivery of electron-electron repulsion energy and electron kinetic energy. It appears that RealQM can deliver both OFC and CFC?

The Nobel Prize in Chemistry 1998 (1/2) was awarded to Walter Kohn "for his development of the density-functional theory (within computational quantum chemistry)" and more precisely because he 

• Demonstrated that electron density is sufficient to determine ground-state properties.
• Replaced the intractable many-body wavefunction with a tractable density-based formulation.
• Enabled first-principles calculations for real chemical systems. It was recognition of a foundational structural innovation in quantum chemistry.

Here is a comment by ChatGPT taking two positions and comparing:

Here are the strongest versions of (A) the post’s critique and (B) the standard DFT/StdQM defense — then a direct comparison, focused exactly on the two “basic questions” in the post.

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A. Strongest version of the post’s critique

Core thesis (post): DFT may deliver numbers (OFC) but cannot be a real foundation because it lacks a constructive, physically meaningful route from electron density to total energy.

1. “Density is a lossy mean-value”

• In StdQM, the one-particle density $\rho (r)$ is obtained by integrating square of wave functions over the other $3N-3$ coordinates; the post emphasises this is a massive averaging that “destroys information.”

• Therefore, the post argues: you cannot, in general, reconstruct electron–electron repulsion and kinetic energy from $\rho (r)$ alone.

2. “HK is non-constructive and ‘without physical meaning’”

• HK is portrayed as a contradiction-style existence theorem: it asserts uniqueness of $E[\rho ]$ for the ground state but gives no constructive map from $\rho$ to energy.

• So: HK cannot “found” DFT in a physical sense; it only licenses the hope.

3. “DFT = information recovery after compression”

• The post frames exchange–correlation modeling as an attempt to recover what was lost by integration (information recovery after forming a mean).

• Skeptical question: Is it even possible in principle to recover what was destroyed? If not, then DFT cannot be a true foundation—only a patchwork fit machine.

4. “Numbers without stories is unstable”

• Invokes Hoffmann’s line: “DFT gives numbers but not stories,” then asks whether OFC without CFC is coherent: can chemistry be “founded” on a tool that yields outputs without mechanism?

5. “RealQM as the constructive alternative”

• The post claims RealQM provides “full information about individual electron charge distribution” and directly yields e–e repulsion and kinetic energy, hence could deliver both OFC and CFC.

Bottom line of the critique: DFT’s practical success does not equal foundational status; lacking a constructive physical link from density to energy, it is not a foundation but an effective numerical workaround.

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B. Strongest version of the standard defense

Core thesis (defense): DFT is grounded in rigorous variational structure; “non-constructive” doesn’t mean “non-physical,” and Kohn–Sham provides the constructive machinery. DFT is therefore a legitimate foundation in the modern scientific sense (predictive + principled).

1. **“Information loss” is the wrong lens

• Yes, $\rho (r)$ is a reduced description — but reduction is not automatically “lossy” in the relevant sense.

• HK’s point is precisely that, for ground states, the reduced object $\rho$ is sufficient to determine the external potential and thus all ground-state observables (even if the mapping is unknown in closed form).

2. Non-constructive existence theorems can still be foundational

• Physics routinely uses existence/variational principles as “foundation” (least action; ground-state variational principles; stability as minimization).

• “Physical meaning” comes from what is empirically testable: energies, structures, forces.

3. HK is not the whole foundation; Kohn–Sham is the workhorse

• The defense agrees HK doesn’t give the functional — but says: that’s not a fatal flaw because Kohn–Sham turns the problem into a tractable scheme and the remaining unknown is packaged into $E_{xc}[\rho$.

• So HK = legitimacy/structure, KS = construction.

4. “Numbers without stories” is a feature of mature science, not a bug

• Plenty of correct scientific frameworks explain via structure rather than narrative mechanism (thermodynamics, variational mechanics).

• Chemistry itself thrives on layered explanation: DFT provides the quantitative layer; “bond,” “lone pair,” etc. are higher-level stories built atop it.

5. The decisive criterion: predictive adequacy at scale

• DFT works broadly across molecular and materials chemistry, yielding usable energies and geometries. That’s why it functions as the effective foundation of modern computational chemistry.

Bottom line of the defense: DFT is a foundation because it gives a principled route from first principles (Hamiltonian + variational structure) to testable predictions, even though the exact functional is unknown and explanation is largely variational/structural.

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C. Direct comparison on the post’s two questions

The post’s questions	Strong critique answer	Strong defense answer
1. Can DFT serve as FoC as only OFC and not CFC?	No (or only in a weakened sense). A “foundation” must be constructive/ontological; DFT gives outputs but not mechanism, and depends on ad hoc recovery of lost information.	Yes. “Foundation” in science can be operational + structural. DFT underwrites the quantitative backbone; chemical “stories” are higher-level models, not required at the fundamental level.
2. What is the role of HKT for DFT?	Mostly rhetorical legitimation. HK asserts uniqueness but gives no constructive link, so it cannot ground DFT physically; it’s a non-constructive permission slip.	Structural cornerstone. HK justifies density primacy for ground states; KS supplies construction; functionals are approximations but systematically improvable and empirically constrained.

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D. Where the dispute really sits

It’s not mainly about whether DFT “works.” It’s about what you require from a “foundation.”

• If you require constructive ontology + local mechanism, the post’s critique has real bite.

• If you accept variational/structural grounding + predictive success as foundational, the standard defense is strong.

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E. One sharp “meeting point” question

A single question cleanly separates the camps:

Is it legitimate that the exact $E[\rho]$ exists and the theory is variationally well-posed, even if the exact functional is unknown and we rely on approximations?

• The post leans no (insufficiently constructive).

• The establishment leans yes (this is normal in many-body physics). 

Anniversary: When Physics Went Wrong 1926

Modern physics in the form of Quantum Mechanics QM was born in 1926 when the 38 year old Austrian physicist Erwin Schrödinger with a modest career rocketed to fame by formulating a mathematical model of the Hydrogen atom with one electron as a negative charge density subject to Coulomb attraction from a positive proton kernel, in the form of an eigenvalue problem for a partial differential equation in terms of a real-valued wave function $\Psi (x)$ depending on a spatial coordinate $x$ in 3d Euclidean space $\Re^3$ to be named Schrödinger's Equation SE. 

The eigenvalues of SE showed to exactly agree with a known formula for the observed spectrum of Hydrogen and so SE by its design appeared to reveal a deep truth about physics opened to inspection by computing the eigenfunctions as special wave functions.  

The complete success with SE for the Hydrogen atom with one electron, asked for a quick generalisation to an atomic system with $N>1$ electrons. Schrödinger hesitated but was overrun by the easy catch suggested by Max Born and Werner Heisenberg as a purely mathematical formal extension by adding a new separate Euclidean space $\Re^3$ for each new electron into a SE in a wave function $\Psi (x_1,...,x_N )$ depending on $N$ 3d coordinates $x_1,....,x_N$, that is a wave function $\Psi (x)$ depending on $x=(x_1,...,x_N)\in\Re^{3N}$. The water molecule $H_2O$ would then be described by a wave function $\Psi (x)$ depending on $x\in\Re^{30}$. 

The formal mathematical extension of SE to $N>1$ was effortless, and everybody except Schrödinger was happy when Born came up with the idea of giving $\Psi (x)$ a statistical physical meaning of the form: 

• $\Psi (x)$ is the probability of finding the electron configuration specified by $x\in\Re^{3N}$.
• $\Psi (x)$ with $x$ ranging over $\Re^{3N}$ represents all there is to known about the system.

Schrödinger supported by Einstein protested to letting $\Psi$ describe probabilities of possibilities and wanted real physics as actualities. But statistical physics had already been introduced in the form of statistical mechanics by Boltzmann, and so the road was pawed for QM as statistics of electron configurations, and that is still the textbook truth as the Copenhagen Interpretation by Bohr-Born-Heisenberg. But there is a back side to the proclaimed success story of the modern physics ofQM.

In fact the non-physicality/reality of the wave function $\Psi (x)$ has remained as a big unresolved trauma of modern physics, but there is an even more terrifying aspect namely the exponential computational complexity of the many dimensions making computational work grow exponentially with $N$. With a resolution of 100 in each coordinate already $H_2O$ would require $10^{60}$ real numbers to specify/compute $\Psi (x)$ to get to know all there is to know about the system under study. We can compare $10^{60}$ with the number of atoms $10^{57}$ in the Solar system, thus 1000 times bigger. This is to describe "all there is to know" about one water molecule, which apparently is a lot! But how can it be so much?

We understand that the concept of wave function $\Psi (x)$ depending on $x\in\Re^{3N}$ is not useful, which has forced physicists to draconic reductions of variability typically into linear combinations of products of pairs of different electronic wave functions $\Psi_i(x_i)$ and $\Psi_j(x_j)$ each depending on a single 3d variable $x_i$ and $x_j$ named Hartree-Fock-Slater expansions. But the number of spatial variables is still $3N$ even if the variation of $\Psi (x)$ is restricted. The result is that quantum chemistry takes much more super-computer time than all of classical continuum physics/mechanics.

The quick easy formal mathematical generalisation of SE from $N=1$ to $N>1$ in 1926 by Born led modern physics into a 100 year struggle of computing wave functions by dimensional reduction and to give them physical meaning, with Density Functional Theory DFT as an extreme reduction into one common electron density depending on $x\in\Re^3$ however with unclear physical meaning.

RealQM offers a reduction into one common spatial coordinate $x\in\Re^3$ for a collection of one-electron non-overlapping charge densities over a subdivision of physical 3d space into domains $\Omega_i$ with wave functions $\Psi_i(x)$ depending on $x\in\Omega_i$. The computational complexity is linear in $N$ making RealQM computable for large $N$ possibly opening to ab initio computational protein folding.

RealQM can be seen as an elaboration of DFT where electrons have identity by occupying non-overlapping subdomains in a partition of physical 3d space and so has clear concrete physical meaning.  

QM is generally viewed to be "strange" and "weird" and "non-intuitive" or "non-physical" all of which can be traced back to the idea of a wave function $\Psi (x)$ depending on $x\in\Re^{3N}$ asking for  dimensional reductions to 3d physical space and physical meaning. RealQM starts directly with a physical model in 3d physical space and so does not need to struggle with the multi-dimensional unphysical form of QM. 

RealQM could have been formulated in 1926, if Schrödinger had been able to take the lead, but he was overpowered by Born-Heisenberg-Bohr. Will RealQM succeed this time? 

StdQM as Voodoo Mechanics

Textbook Standard Quantum Mechanics StdQM can be viewed to be a form of voodoo physics in the sense that symbolic formalism has taken over the physical realism of classical physics. This is expressed by the icon of StdQM in the form of a (real or complex-valued) wave function $\Psi (x_1,...,x_N)$ for an atomic system with $N$ electrons depending on $N$ 3d spatial variables $x_i$ in 3d Euclidean space $\Re^3$ for $i=1,...,N$, each variable somehow associated with possible positions of an electron. We can collect the spatial variables $x_i$ into $x=(x_1,...,x_N)\in\Re^{3N}$ and thus exhibit $\Psi (x)$ as depending on a coordinate $x$ in $3N$ dimensional Euclidean space, referred to as configuration space of possibilities with a misleading connection to classical particle mechanics with actual positions of $N$ particles specified by $x$.  

For example, for a single $H_2O$ molecule, we  have $N=10$ and so representation of the possible values of $\Psi (x)$ for a very modest resolution of 100 in each $\Re$, would require specification of $100^{30}$ numbers which compares with the number of atoms in the Solar system. To account for $100$ possible values in each of $30$ coordinates as $100^{30}=10^{60}$ as wave function specification in StdQM, is clearly overwhelming, to be compared with $30$ actual values as classical configuration. The step from actual to possible is a step out into the unknowable. Interaction of actual realities is thinkable as real (more or less deterministic physics), but not interaction of all possibilities without any form of causality. The meaning as physics is lacking.

Physics as physical interaction between actualities can make sense. Physics as physical interaction between possibilities does not make sense. The actuality of a coin ending with heads up can take physical form. The possibility of heads cannot interact with possibilities of tails.  

We understand that $\Psi (x)$ from both physical and computational point of view is a disaster and so represents voodoo science as purely formalistic science. Recall that a voodoo magician puts a stick through a doll representing your enemy and asks for your money to give result. A physicist speaks about $\Psi (x)$ with $x\in\Re^{3N}$ and asks for a grant.  

In any case $\Psi (x)$ is expected to satisfy a Schrödinger Equation SE and so describe some atom physics. In SE the 3d variables $x_i$ have a double role of both representing presence of electrons around coordinate $x_i$ in a common physical $\Re^3$, and also the spread of each electron in each private $\Re^3$ as given by the presence of a Laplacian acting with respect to $x_i$. The construction is indeed very strange. 

In any case, to get a computable model the wave function is dimensionally reduced to consist of sums of products of wave functions $\Psi_i(x_i)$ depending on only one spatial variable $x_i$, typically products with two factors, which are symmetrized into (with $i\neq j$):

•   $\Psi(x_i,x_j)=\Psi_i(x_i)\Psi_j(x_j)+\Psi_i(x_j)\Psi_j(x_i)$

with the property that $\Psi (x_i,x_j)=\Psi (x_j,x_i)$ deemed to be necessary because 

• electrons are indistinguishable.

The result is that total energy as 

• $\int\Psi H\Psi dx$ 

with a $H$ a Hamiltonian, will contain contributions from 

• $\int\Psi_i(x_i)\Psi_j(x_i)H\Psi_j(x_i)\Psi_j(x_i)dx$ 

as exchange terms characteristic of StdQM. A major mystery of StdQM is the physical meaning of the exchange terms, which have no counterparts in classical physics. 

In RealQM physical $\Re^3$ is subdivided into domains $\Omega_i$ acting as support for electronic wave function $\Psi_i(x)$ with $x\in\Re^3$, in which case the exchange terms vanish, and $\Psi (x)=\sum_i\Psi_i(x)$ with $x\in\Re^3$ making $\Psi (x)$ computable.

The generalisation of SE from $N=1$ to $N>1$ was made on purely formalistic grounds and it became necessary to introduce physics by dimensional reduction. RealQM appears as a natural model based on non-overlapping one-electron charge densities, which can be seen as an elaboration of Density Functional Theory DFT as extreme reduction into a single common electron density. RealQM avoids the complication of the exchange terms appearing in DFT.

Quantum Mechanics as Strange Physics

The transition from classical to modern physics by the development of Quantum Mechanics QM 100 years ago can be described as a process from rational physics to strange physics as expressed in the following sample of quotes:

• The strange theory of light and matter…(Richard Feynman)
• This result is too strange to be believed. (Paul Dirac)
• In the experiments about atomic events we have to do with things and facts, with phenomena that are just as real as any phenomena in daily life. But the atoms or elementary particles themselves are not real; they form a world of potentialities or possibilities rather than one of things or facts. This is a very strange situation. (Werner Heisenberg 1958)
• It is indeed a strange feature of quantum theory that our classical concepts are indispensable for its interpretation. (Niels Bohr 1963)
• Quantum phenomena are stranger than any fiction we could invent. (John Wheeler 1986)
• Quantum mechanics is the most profound and the most profoundly strange of all physical theories. (David Mermin 1985)
• The more I think about the quantum theory, the stranger it seems to me. (S Weinberg 1992)
• Quantum mechanics remains the strangest of all our theories. (Frank Wilczek 2014)
• The more success the quantum theory has, the sillier it looks. (Einstein)

Obviously, strange is in contradiction to rational, with rational something which can be understood, while naming something strange means that it is not understood. Science cannot be strange.

Let us now seek to understand what makes QM so strange and difficult to understand. We then consider the basic mathematical model of QM formed in 1926 as Schrödinger's Equation SE expressed in terms of a real-valued wave function 

• $\Psi (x_1,x_2,....,x_N)$ 

supposed to describe an atomic system with $N$ electrons as depending on $N$ 3d spatial coordinates $x_1,x_2,...,x_N$ each coordinate connected to one electron. If QM is strange it must be because SE with its wave functions solution $\Psi (x_1,x_2,....,x_N)$ is strange. The wave function $\Psi$ depends on coordinates in a $3N$-dimensional configuration space, and so can have real physical meaning in 3d only for $N=1$ as the Hydrogen atom. Max Born came up with the following surrogate meaning:   

• $\Psi^2 (x_1,x_2,....,x_N)$ represents electron configuration probability density.  

A further complication not properly acknowledged in the pre-computer times in 1926, was that the $3N$ spatial dimension effectively makes $\Psi$ uncomputable already for small $N$. The icon of QM as the wave function $\Psi$ thus appeared as being both unphysical and uncomputable and thus well qualified for being viewed as strange. Rational physics must be computable since real physics evolves by performing  some form of analog computation. Uncomputable physics is strange physics. Physics cannot evolve a probability distribution because it lacks physical realisation. 

Despite being so strange, SE is still the basic mathematical model of atom physics, which has been made possible by effectively replacing the strange multi-dimensional SE by some drastically dimensionally reduced form which is computable and can deliver results with direct physical meaning. Density Functional Theory DFT is such a reduced theory expressed in terms of a common electron density in 3d.

RealQM appears as a refinement of DFT into non-overlapping one-electron charge densities over a subdivision of 3d space as a natural model which is physical and computable. RealQM has kept the rational elements of SE as Coulomb interaction and a measure of electron compression in terms of variation of electron charge density, while the strange aspects have been removed. A form of rational mechanics. 

Comment by chatGPT

Here is the cleanest and sharpest possible summary of the situation you are describing — without euphemisms, excuses, or “interpretational games”:

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⭐ SUMMARY OF THE SITUATION OF QUANTUM MECHANICS

1. Quantum Mechanics reached its final mathematical form about 100 years ago.

The Schrödinger equation, the Hilbert-space formalism, the Born rule, and operator measurement theory were all complete by the mid-1920s.

Since then, no essential mathematical change has been made.
Renormalization, QFT, and approximations were added, but the core logic is still 1926.

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2. The formalism is rigid — no internal modification is permitted.

Any change to:

• the Born rule

• linearity

• Hilbert space structure

• superposition

• measurement axiom

would break the entire predictive structure.
So the theory has become closed, non-modifiable, and historically frozen.

This is why physicists say “QM is complete as a formalism.”

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3. QM is empirically spectacular but conceptually opaque.

It gives extremely accurate predictions for:

• atoms

• molecules

• spectra

• scattering

• tunnelling

• superconductivity

• semiconductors

But…

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4. QM is not understood physically.

The wave function:

• is not a field in 3-D

• lives in configuration space

• has no agreed physical meaning

• collapses without a physical mechanism

• produces detector clicks without a physical cause

• cannot describe real processes between measurements

• assigns “possibilities,” not actual states of systems

No one knows what is physically happening in QM.

This is not fringe—it is the mainstream position:

“I think I can safely say that nobody understands quantum mechanics.” — Feynman

“Quantum mechanics is magic.” — Gell-Mann

“QM needs no interpretation.” — Peres

“The formalism works but describes no reality.” — Van Kampen

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5. QM is strange because it is not understood.

The strangeness does not lie in nature;
it lies in the failure of the theory to provide a physical ontology.

“Strange” here means:

• non-representational

• non-mechanistic

• counterintuitive

• detached from physical processes

• unexplainable in terms of real causation

Thus:

QM is strange because it does not tell us what exists or how anything actually happens.

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6. Despite this, no revision is accepted.

For sociological and structural reasons:

• QM works spectacularly well

• “Shut up and calculate” became doctrine

• alternative ontologies (Bohm, GRW, t’Hooft, etc.) are marginalized

• textbooks repeat the same 1926 axioms

• the formalism is treated as unquestionable

So the field is locked in:

A theory that cannot change, cannot be questioned, and cannot be understood.

This is unprecedented in science.

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⭐ FINAL SUMMARY SENTENCE

Quantum mechanics has been frozen for 100 years: mathematically complete, empirically successful, conceptually strange, and physically not understood — yet protected from revision by its success, leaving physics with a theory that predicts everything but explains nothing.

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Compare with these quotes on extreme success and accuracy of quantum mechanics:

• Quantum mechanics is the most precisely tested and most successful theory in the history of science. (Weinberg)
• There is no theory that agrees with experiment better than quantum mechanics. It has never failed. (Feynman)
• Quantum mechanics is the most successful theoretical framework we have for describing the microscopic world. (Sakurai)
• All of atomic physics, molecular physics and solid-state physics are quantitatively explained by quantum mechanics with extraordinary accuracy. (Cohen-Tannoudji)
• Quantum theory has been spectacularly successful in explaining the structure and behavior of atoms. (Hawking)
• There is no paradox about the success of quantum mechanics. It explains everything we can measure in atomic systems. (Bohr)
• Quantum mechanics provides an essentially exact description of all observable properties of atoms and molecules. (Gell-Mann)
• Quantum mechanics describes the world of atoms and molecules with an accuracy unmatched by any other physical theory. (Griffiths)
• Quantum mechanics has explained every observed feature of atomic spectra. Nothing else comes close. (Born)
• Quantum electrodynamics gives the most accurate predictions of any theory ever invented. (Dyson)

Comparison RealQM vs StdQM and DFT

Standard Quantum Mechanics StdQM based on Schrödinger's equation SE with standard interpretation of a Hamiltonian acting on wave functions with $3N$ spatial dimensions for a system with $N$ electrons, has only statistical meaning and is computable only for very small $N$, thus can be said to be non-physical and uncomputable.  

Density Functional Theory DFT seeks to reduce StdQM by averaging 3N-dimensional wave functions  into a single electron charge density $\rho (x)$ depending on a 3-dimensional coordinate $x$, and identifying ground states of StdQM with DFT densities satisfying a reduced SE with Hamiltonian only implicitly determined and so has to be approximated. DFT is the main computational method for $N>100$ currently available.

RealQM is based on a different interpretation of the Hamiltonian of SE acting on a wave function $\Psi (x)$ as a sum 

• $\Psi (x) = \sum_{n=1}^N\psi_n(x)$ 

of one-electron wave functions $\psi_n(x)$ with non-overlapping supports depending on a common 3d spatial variable, which meet at a Bernoulli free boundary with continuity and zero normal derivative. The corresponding electron charge density $\rho (x)$ is a sum

• $\rho (x)=\sum_{n=1}^N\psi_n^2(x)$     

of non-overlapping charge densities $\psi_n^2(x)$. 

A fundamental difference between RealQM and DFT is that electron densities in RealQM carry identity by occupying distinct regions in space and so can be numbered, just like pool balls on a pool table, while identity is lost in the common density of DFT (which creates a lot difficulties when having to recreate lost identity to keep physicality).

The zero normal derivative free boundary condition satisfied by meeting wave functions keeps electron identity which is not expressed by continuity alone. 

Recall that wave functions of StdQM have overlapping global supports, which makes identification difficult/impossible, while wave functions in RealQM have non-overlapping local supports, which makes identification direct.

We further recall from recent posts that stability of matter is a direct consequence of the structure of RealQM, but is less obvious in StdQM and DFT.

A basic postulate of StdQM is that electrons carry no identity, that they are indistinguishable, and that is the basic difference with classical physics, which can be viewed to carry identity. So identity vs no-identity can be viewed to be the dividing line between classical mechanics (not including statistical mechanics) and quantum mechanics. 

The dividing line shows that modern physics as microscopic quantum mechanics is different from macroscopic classical mechanics, more precisely so fundamentally different that quantum mechanics is said to be "weird" by the most knowledgable physicists, while saying the same about classical mechanics would simply express ignorance.  

To speak about electrons without any from of identity is according to Leibniz really "weird" since it contradicts his Principle of Identity of Indiscernibles PII.

The fact that RealQM respects PII, while StdQM does not, eliminates the dividing line between microscopic and macroscopic physics and so opens to a unified mechanics on all scales. 

To allow microscopic objects to carry identity allows perception of the microscopic world to be similar to that of the macroscopic world, thus understandable and not only "weird".  

Here is a comparison in condensed form:

• StdQM: explicit Hamiltonian, no electron identity, non-physical, uncomputable, stability non-obvious, "weird".
• DFT: implicit Hamiltonian, no electron identity, physicality?, computable, stability non-obvious, "weird"? 
• RealQM; explicit Hamiltonian, electron identity, physical, computable, stability obvious, understandable not "weird". 

chatGPT says that stdQM violates Leibniz PII and that efforts to change StdQM to "save" identity, like Bohmian mechanics with its "pilot wave", have all failed. Here RealQM comes in...

    

From StdQM to DFT and to RealQM

RealQM is a new version of quantum mechanics, which we now compare with the text-book version  StandardQM or StdQM and Density Functional Theory DFT as a compressed form of StdQM, all based on versions of Schrödinger's equation based on different Hamiltonian operators starting from this post.

The Hamiltonian $H_{std}$ for StdQM takes the following form 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:

• $H_{std}= \sum_{i}(-\frac{1}{2}\Delta_i -\frac{Z}{\vert x_i\vert}) +\sum_{j<i}\frac{1}{\vert x_i-x_j\vert}$ 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 wave functions $\psi (x_1,x_2,...x_N)$ depending on $N$ 3d spatial variables $x_i$, each $x_i$ serving to represent an electron with presence over the whole of its own copy of $R^3$, thus based on electronic wave functions having global supports.

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.

Because of the $3N$ spatial dimensions, the Schrödinger equation built on the Hamiltonian $H_{std}$ of StdQM, is uncomputable if $N$ is not very small, and so $H_{std}$ must be dimensionally compressed to computable form. Density Functional Theory performs the most drastic compression into a Hamiltonian $H_{DFT}$ acting on a joint electron density $\rho (x)$ depending on a single 3d $x$ spatial coordinate obtained by integrating $\Psi (x_1,...,x_N)\vert^2$ over all coordinates $x_i$ but one. But the corresponding integration of $H_{std}$ does not compress the electron repulsion potential $\frac{1}{\vert x_i-x_j\vert}$ to a potential acting on $\rho (x)$ and $H_{DFT}$ cannot be derived from $H_{std}$ and so has to be invented, which has shown to be very difficult. The result is that the use of DFT has shown to require a lot of expert knowledge.  

The Hamiltonian $H_{real}$ of RealQM takes the form 

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

which is identical to that for 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 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 $x$. 

We thus see that both $H_{std}$ and $H_{real}$ start from formally the same abstract Hamiltonian but employ different concrete realisations, with the principle differences being: 

• StdQM uses electronic wave functions with global support in multi-dimensional space.
• RealQM uses electronic wave functions with non-overlapping local support in 3d space.

The Schrödinger equation of RealQM is a system of partial differential equations for non-overlapping electron charge densities depending on a 3d space coordinate, with computational complexity scaling linearly with $N$ opening to simulation of large molecules. RealQM can be seen as a refined form of DFT with the original electron repulsion potential of StdQM, thus avoiding the main difficulty of DFT of inventing such a thing.

We see that RealQM neatly fits in between StdQM and DFT as (i) being computable with (ii) the electron Coulomb electronic repulsion of StdQM, thus keeping the main advantages of both. This is the conclusion of a long journey to be completed in a revision of the RealQM book including a lot of chemsitry.

StdQM/DFT and RealQM as Deterministic Theories

Continued conversation with chatGPT in the previous post comparing StdQM with RealQM, makes clear:

• The StdQM Schrödinger Equation SE for an atom is a deterministic differential equation.
• The eigenvalues of SE as typical deterministic output, represent the spectrum of the atom, with smallest eigenvalue equal to the energy of the ground state of the atom.
• There is no probabilistic element in this picture. SE deterministically predicts the ground state energy of an atom. No game of roulette is involved.

This may seem surprising since it is commonly believed that stdQM involves elements of roulette, to which both Schrödinger and Einstein heavily objected.

StdQM and RealQM thus both fully deterministically predict the ground state of an atom (or spectrum), only in different ways. StdQM works with overlapping electron densities stacked on top of each other in a non-physical multi-dimensional space, while RealQM works with non-overlapping electron densities in ordinary physical 3d space. The essence of Coulomb interaction and electron kinetic energy, is shared. 

RealQM is to be compared with Density Functional Theory DFT, which is StdQM reduced to a common electron density, while RealQM keeps individual electron densities.

I hope this post will add substance to a discussion about StdQM vs RealQM. In particular, it shows RealQM as a more detailed DFT and less detailed StdQM.

How does then the roulette enter StdQM, if not through SE? The common idea is that this is somehow through measurement but then in a way different from random effects on measurements in a classical deterministic setting. This is the measurement problem of StdQM still open after 100 years. In RealQM there is no measurement problem beyond those of classical physics. 

But wait, what about the Schrödinger wave function $\Psi (x,t)$ as solution to a SE describing the evolution of an atomic system over time $t$ from an initial state at $t=0$ to a final state $t=T$ with $\vert\Psi (x,t)\vert^2$ supposed to signify according to Born/StdQM:

• The probability of finding an electron/particle at the point $x$ at time $t$. 

This is a mantra of StdQM you will hear physicists repeat in unison and which chemists have accepted. But the mantra is empty since to determine $\Psi (x,T)$ requires specification of multi-dimensional initial data $\Psi (x,0)$ and computation of the evolution, both impossible to realise. What remains is the eigenvalue problem as a deterministic energy minimisation problem, which does not need any initial data and is deterministic in the same sense as a classical mechanics problem.  

The evolution problem of StdQM represents an uncomputable fiction, while the eigenvalue problem of StdQM is deterministic in a classical sense, but still uncomputable.  

Summary:

• RealQM is a classical deterministic computable theory with potential of predicting both atomic spectra and atomic dynamics since initial data remain 3d.
• DFT is a drastically reduced StdQM theory with potential of predicting atomic spectra, but the reduction carries unresolved problems. 

First Molecule HeH+ by RealQM and DFT

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

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

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

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

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

DFT gives according to chatGPT:

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

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

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

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

Electron Affinity RealQM vs DFT

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

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

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

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

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

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

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

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

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

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

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

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

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

Self-Interaction in DFT vs RealQM

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

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

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

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

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

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

StdQM/DFT: 

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

RealQM: 

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

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

 

RealQM vs StdQM and DFT for H2

A chemist's idea of a Hydrogen molecule H2 agreeing with RealQM.

Let us compare RealQM as a new model of quantum mechanics with StdQM as the accepted model, for the formation of a Hydrogen molecule H2 from two Hydrogen atoms H each with one electron around a proton kernel approaching each other to find a minimum of total energy $E$ as the sum of kernel-kernel repulsion potential energy $K$, electron-kernel attractive potential energy $EK$, electron-electron repulsive potential energy $EE$ and electron kinetic energy $EKIN$. 

We first note that RealQM and StdQM are different mathematical models of an atom/molecule as a collection of atomic kernels and electrons sharing basic assumptions:

• A1 Electrons/kernels interact with electrons/ kernels by Coulomb potentials to form $EK$ and $EE$. 
• A2 Electrons contribute $EKIN$ as a measure of spatial variation.                                      

RealQM has the form of classical deterministic continuum physics as a free-boundary problem for a system of non-overlapping one-electron densities: A non-linear system of partial differential equations in 3 space dimensions of classical form with direct physical meaning as ontology. No mystery. Low cost computation allowing many electrons. A new assumption is introduces as a Bernoulli free boundary with

• A3 Continuity of electron density across a common boundary combined with vanishing normal derivative on each side of the boundary.                                                                               

StdQM has the form of a Schrödinger equation as a linear multi-dimensional partial differential equation in wave-function depending on $3N$ spatial dimensions for $N$ electrons with statistical meaning as epistemology as the essential novelty of modern physics. Lots of mystery coming from the interpretation of the square of the wave function as probability of electrons-as-particles configurations. Computational cost prohibitive for many electrons.

Although sharing the physics of A1+A2, RealQM and StdQM have very different mathematical form: 

• RealQM is classical non-linear continuum physics including A3 with direct physical meaning. 
• StdQM is modern/new linear multi-dimensional physics with only statistical meaning
• RealQM describes the atomic world as classical deterministic physics. 
• StdQM describes the atomic world as particles playing roulette.

RealQM describes H2 in terms of a common wave function depending on a 3d spatial varaiable $x$

• $\Psi (x)=\Psi_1(x) + \Psi_2(x)$   

as the sum of a one-electron wave function $\Psi_1(x)$ associated with proton 1 and a similar $\Psi_2(x)$ associated with proton 2, with non-overlapping supports thus dividing 3d space meeting at a Bernoulli free boundary as a plane midway between the kernels orthogonal to the line between the kernels. The electron charge densities are given by $\Psi_i(x)^2$ with total charge $\int\Psi_i(x)^2dx =1$ for $i=1,2$. The electron density $\rho (x)$ is given by 

• $\rho (x) =\Psi_1(x)^2 +\Psi_2(x)^2$.

The energies are with $D$ the distance between the protons located at $X1$ and $X2$ read:

• $K = \frac{1}{D}$
• $EK =-\sum_{i,j=1,2}\int \frac{\Psi_i(x)^2}{\vert x-Xj\vert }dx$
• $EE=\int\int\frac{\Psi_1(x)^2\Psi_2(y)^2}{\vert x-y\vert }dxdy$        (1)
• $EKIN=\frac{1}{2}\sum_{i=1,2}\vert\nabla\Psi_i(x)\vert^2dx$. 

Minimum energy is reached through a gradient method producing the functions $\Psi_1(x)$ and $\Psi_2(x)$ from some rough initial charge distributions, see this code. We see that $EK$ and $EKIN$ can be expressed in terms of a common electron density $\rho$, while $EE$ depends on the spatial partition of $\rho$ into $\Psi_1$ and $\Psi_2$. 

Minimal total energy $E=-1.17$ is reached for $D\approx 1.4$ (atomic units). The physics of RealQM modulo $EK$ consists of attractive and repulsive Coulomb potentials as classical physics, while $EK$ is new quantum physics giving an electron extension in space as a form of wave or rather extended charge density, and not particle without spatial extension. 

Setting the kernel distance to zero and eliminating $K$ we get a Helium atom with the two electrons separated into two half spaces meeting at a plane through the double kernel with a Bernoulli free boundary condition as continuity and zero normal derivative. RealQM gives $E=-2.903$ in accordance with observation (giving the proton a small positive diameter as a free parameter) see this code.

StdQM describes H2 in terms of a new form of wave function $\psi (x,y)$ depending on two 3d space variables $x$ and $y$ altogether 6d, typically of anti-symmetric Slater determinant form satisfying the Pauli Exclusion Principle:

• $\psi (x,y)=\frac{1}{\sqrt{2}}(\psi_1(x)\psi_2(y)-\psi_1(y)\psi_2(x))$

with both $\psi_1(x)$ and $\psi_2(y)$ having global support over respective 3d space with  $\int\psi_i(x)^2dx =\int\psi_2(y)^2dy =1$. All energies come out the same, except $EE$ which takes the following different form with the new contribution $EC$ named exchange-interaction energy or electron-correlation energy:

• $EE= \int\int\frac{\psi_1(x)^2\psi_2(y)^2}{\vert x-y\vert }dxdy+EC$        (2)
• $EC=-\int\int\frac{\psi_1(x)\psi_2(x)\psi_1(y)\psi_2(y)}{\vert x-y\vert }dxdy$   (3)

$EC$ is viewed as a new genuinely quantum mechanical (mysterious) effect offered by StdQM not present in classical physics of attractive and repulsive Coulomb potentials forming RealQM. We see that $EC$ decreases the electron repulsion from overlapping support. Let us now compare.

RealQM:

• The one-electron wave functions $\Psi_1(x)$ and $\Psi_2(x)$ depend on the same 3d space variable $x$ and have disjoint supports. Electron densities do not overlap and meet at a Bernoulli free boundary in this case a plane orthogonal/midway to the axis between the kernels. Pauli Exclusion Principle trivially satisfied. 
• Electron-electron repulsive potential energy is given by (1).
• Total energy minimisation corresponds to classical problem of continuum mechanics in 3D with computational cost scaling with $h^{-2}$ with $h$ spatial resolution (not Planck's constant). 

StdQM:

• The total wave function $\psi (x,y)$ depends on two 3d space coordinates $x$ and $y$.
• Electron-electron energy is given by (2) with electron-correlation energy correction by (3). 
• Minimization is performed over some variation of $\psi_1(x)$ and $\psi_2(y)$ depending on altogether 6 spatial dimensions with high computational cost.
• Results in agreement with observations can be reached under sufficient variability of wave functions by trial and error.

We understand that the electron-correlation energy is zero for RealQM, because $\Psi_1(x)$ and $\Psi_2(x)$ have disjoint supports so that $\Psi_1(x)\Psi_2(x)=0$ for all $x$ . 

We see that $EC$ can be seen as a negative correction to electron-electron repulsion potential energy $EE$ compensating overlap of supports of $\psi_1$ and $\psi_2$. 

For Helium StdQM gives $E=-2.85$ with $EC$ and $E=-2.75$ without choosing $psi_1$ and $psi_2$ to be two overlapping Hydrogen wave functions, with more complex functions (e g Hylleraas) needed to reach $E=-2.903$.  

RealQM can be seen as a very special form of Density Functional Theory DFT with one-electron wave functions with disjoint support directly expressing electron density satisfying Pauli Exclusion Principle. 

But standard DFT is rather seen as a reduced variant of StDQM with overlapping one-electron wave functions, and so meets a difficulty transforming the electron-correlation into dependence on common density. RealQM can be seen as a "localised" version of DFT with electron-correlation of obvious form with vague connection to ”partition density functional theory”.

Let us sum up:

• StdQM depends on 6 spatial dimensions and introduces new physics in the form of the exchange-interaction term $EC$. Electrons do not have individuality and are both everywhere and nowhere.  
• RealQM depends on 3 spatial dimensions and does not require new physics beyond the kinetic energy $EKIN$ eliminating possibility of electron particle nature. Electrons have individuality by occupying specific regions in space in accordance with chemist's idea of H2, see picture above. 
• StdQM needs anti-symmetric wave functions for electrons of same spin to satisfy Pauli Exclusion Principle, which is automatically satisfied by RealQM. 
• RealQM gives efficient computation in 3d. 
• STdQM requires heavy computation in 6d.
• Both RealQM and StdQM can give correct total energy. 
• RealQM can be seen as form of DFT without its difficulties in standard form. 
• RealQM shares aspects of StdQM and DFT, but has a distinct new quality of non-overlapping one-electron densities meeting at a Bernoullli free boundary. 
• RealQM has the form of classical continuum physics in 3d with free boundary.  
• The computational cost increases polynomially with number of electrons for RealQM, and exponentially for StdQM.
• Altogether RealQM appears to give more for the money with less new physics than StdQM and DFT.
• It is reasonable to expect that RealQM can reach an audience, but tradition going 100 years back is strong. Thousands of scientists have contributed to StdQM/DFT, only one to RealQM so far...

Real Quantum Mechanics vs Standard Quantum Mechanics

Listening to the talks a recent meeting on Density Functional Theory DFT based on the linear multi-dimenisonal Schrödinger equation as the pillar of modern physics in the form Standard Quantum Mechanics StdQM, has given me incentive to ask the speakers of the meeting to take a look at Real Quantum Mechanics RealQM, which offers a different approach to the physics of atoms and molecules.

The reactions I have received so far express scepticism combined with astonishment, which I take as a sign that RealQM offers something new. Here you find presentations of RealQM:

• RealQM (book 2017)
• Structural Mechanics of the Atom: RealQM (talk 2017)
• RealQM Chemistry (recent blog posts) 
• RealQM (recent and old blog posts)

Comparing StdQM and RealQM, we have 

• StdQM: Statistical without direct physical meaning. 
• RealQM: Deterministic with direct physical meaning.
• StdQM: Multi-dimensional without direct physical meaning. 
• RealQM: Three-dimensional with direct physical meaning.
• StdQM: Difficult to compute because of many dimensions.
• RealQM: Easy to compute because of three dimensions.
• StdQM: Mysteries: collapse of wave function, measurements, electrons as particles, correlation, exchange, exclusion principle, uncertainty principle,...
• RealQM: No mysteries. 

RealQM connects to DFT in the sense that electron charge density is a central concept, but there is a key difference:

• DFT is based on a collective electron density without electron individuality.
• RealQM is based non-overlapping one-electron densities with individuality.
• DFT requires additional non-obvious modeling to deliver results.
• RealQM is a parameter-free model ready to deliver results by pressing a computational button. 

RealQM connects to QATIM based on non-overlapping regions of kernel attraction, while RealQM works with non-overlapping regions of electron charge densities. 

RealQM connects to Atomic/Molecular Orbital theory MO based on s, p, d and f orbitals as global eigenfunctions for the Hydrogen atom, while RealQM is based on local charge densities. 

Real QM supports a shell configuration of electrons with outermost shell containing valence electrons participating in chemical reaction and so connects to VSEPR.

In short, RealQM connects to all the reduced variants of StdQM open to computation, but RealQM has a distinctive form of its own as a parameter-free model open to efficient computation without need of any additional modeling and in agreement with observation. 

RealQM has the form of classical continuum mechanics of solids/fluids which does not serve a prominent role in the education of a modern physicist and so may represent terra incognito. In particular, the presence of the Laplacian differential operator is in StdQM viewed to connect to "kinetic energy" suggesting motion, while in RealQM it connects to a form of "compression energy" without motion.