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...