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.