RealQM vs StdQM and DFT for H2

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 $EKIN$. 

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 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.

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 $\Psi_1(x)$ and $\Psi_2(x)$. 

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 $E=-2.903$.  

Density Functional Theory DFT coincides with RealQM with one-electron wave functions with disjoint support directly expressing electron density satisfying Pauli Exclusion Principle. 

DFT as a reduced variant of StDQM with overlapping one-electron wave functions meets a difficulty transforming the electron-correlation into dependence on common density.   

Let us sum up:

• StdQM depends on 6 spatial dimensions and introduces new physics in the form of the exchange-interaction term $EC$. 
• RealQM depends on 3 spatial dimensions and does not require new physics beyond the kinetic energy $EKIN$ eliminating possibility of electron particle nature. 
• 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.

Does Quantum Mechanics Explain the Periodic Table?

A common view among modern physicists is that the Periodic Table is explained by the modern physics of quantum mechanics based on a linear multi-dimensional Schrödinger equation. 

The ground state of the one-electron distribution of a Hydrogen atom can be computed analytically to be given by a spherically symmetric wave function decaying with the distance $r$ from the kernel like $\exp (-r)$, which is referred as a $1S$ state, with energy $-\frac{1}{2}$ Hartree in full agreement with observation.  

The ground state of Helium as the next element in the table with two electrons, is presented to be $1S^2$ in the form of two electrons with the same $1S$ wave function sharing space around the kernel (with different so called spin to coexist). The ground state energy computes to $-2.75$ Hartree, while observed energy is $-2.903$ Hartree. The state $1S^2$ is thus not the ground state. 

When confronted with this fact, ChatGPT gives the following response, after consulting the literature:

• The $1S^2$ state is not fundamentally correct as the ground state of helium because it fails to account for essential electron correlation and yields an incorrect energy. While it can serve as a useful approximation and educational tool, the true ground state is a correlated quantum state that cannot be reduced to $1S^2$. Thus, any claim of its "conceptual correctness" must be qualified and limited to its role as a crude first-order approximation.
• The $1S^2$ concept can be misleading if presented as the full description of helium's ground state. While it serves as a useful approximation and pedagogical tool, it oversimplifies the true quantum nature of the system. To avoid misconceptions, it’s crucial to contextualize $1S^2$ as a starting point in understanding helium, not the definitive or complete description.

We understand that presenting $1S^2$ as the ground state of He is incorrect and so can be seriously misleading, yet it is supposed to be the quantum mechanical explanation of the second element in the periodic table! What then about the rest of the table based on excited states of Hydrogen? 

RealQM presents a different ground state for Helium consisting of two electrons occupying two separate half-spaces through the kernel, for which the energy agrees with observation. More generally, RealQM explains the periodic table as a packing problem of non-overlapping one-electron charge densities meeting at a Bernoulli free boundary, which appears as a better explanation of the period table than the accepted standard.  RealQM is based on a new Schrödinger equation in the form of classical continuum mechanics in 3 space dimensions.

Connecting to the previous post, the idea of a Helium ground state as $1S^2$ came out from the new quantum mechanics emerging 100 year ago as way to understand the atomic world, which today according to ChatGPT however appears to have missed the target. Instead of increasing understanding it fosters mis-understanding. A nice feature of ChatGPT compared to a living scientist heavily invested in a certain school of thought, is that hiding truth/information is not securely built in. 

More precisely: The ground state of Helium as $1S^2$ of the accepted view with incorrect energy, should be compared with the conceptually fundamentally different ground state of RealQM with correct energy. What is your choice? 

  

Computing Without Understanding in 2025

                              Understanding or non-understanding of chemical bonds?

Scientific Prospect for 2025.

My high school chemistry teacher Ido Leden (later prof at University of Lund) told us that chemical bonds are either covalent (sharing electrons), or ionic (non-sharing), or a mixture of both, which made some sense but lacked both physics and precise numbers. 

Today the numbers are more precise as the output of computational quantum chemistry, but understanding of the physics remains vague as shown in this post and this article and the many diverging aspects in books on quantum chemistry (DFT, QATIM, MO, VSEPR, Hartree-Fock...).

Recall the credo of the famous computational mathematician Richard Hamming:

• The purpose of computing is not numbers but understanding.       (1)

Compare with that of the famous quantum physicist Paul Dirac (or Richard Feynman):

• Shut up and calculate.                                                        (2)

Computational quantum chemistry is an off-spring of quantum physics following (2) with lots of numbers but little understanding and so far from (1).

But science is all about understanding, and so a modern chemist will have to make sense of quantum physics in one way or the other to be a scientist. This is also necessary to guide computations using the basic model of quantum physics in the form of the Schrödinger equation, because in basic form it is uncomputable due to its many spatial dimensions. 

A computational chemist thus has to reduce Schrödinger's equation to computational form, and that requires some form of understanding. This means that (2) leads nowhere: To calculate requires understanding. 

The prospects for advancement during 2025 do not seem to be good. The trend is that AI can take over, with the caveat that AI understanding is hidden to human understanding.  

But there is hope: Real Quantum Chemistry is based on a new form of the Schrödinger equation as RealQM, which

• is understandable as classical continuum physics
• is parameter free
• is directly computable without additional modeling
• agrees with observation. 

Maybe 2025 will be the year when chemists open their minds to RealQM, one may be enough to open a door.