Ab Initio Computational Quantum Mechanics

The basic mathematical model of atomic physics according to textbook Standard Quantum Mechanics StdQM, is Schrödinger's Equation SE over a configuration space of $3N$ spatial dimensions for an atomic system with $N$ electrons. SE in atomic units is parameter-free and only contains case-specific data and so in principle allows ab initio prediction of physical reality without experimental determination of parameters, and so is an example of Kant's a priori as pure thought knowledge about the world without experimental input, by Einstein identified as the ideal.

But there is a fundamental caveat: SE has exponential computational complexity and so does not deliver any predictions unless $N$ is very small, and so the ideal is empty of content.

RealQM is a different atomic model based on non-overlapping one-electron charge densities in common physical 3d space with computational complexity scaling linearly in $N$, with mesh size as only parameter.

StdQM and RealQM are both based on the same parameter-free principles:

1. Coulomb interaction between charge densities.
2. Kinetic energy of charge densities measured by spatial gradients. 

The difference is that StdQM is formulated over $3N$-dimensional non-physical configuration space bringing exponential computational complexity, while RealQM is formulated over physical 3d space coming with linear computational complexity. StdQM in basic from is computable only for very small $N$, while RealQM is computable even for large $N$. 

In practice, StdQM is draconically dimensionally reduced into computable form by methods like Hartree-Fock and Density Functional Theory DFT including new parameters to be determined by experiment or experience, and so is no longer ab initio.

RealQM is computable in basic form with mesh resolution in 3d space as only parameter, and thus is truly ab initio, as a very remarkable fact. 

Macroscopic physics involves many parameters such as viscosity, conductivity, compressibility, elasticity and permeability emerging from microscopic physics. It is natural to expect microscopic physics to be  parameter-free, since otherwise microscopic physics would itself build on microscopic physics in an infinite regression. 

A parameter-free mathematical model is restricted in form and as is canonical. The list of computable ab initio models is short:

• The algebraic equation $x^2+y^2=1$ describing a circle of unit radius, from which the length of the circumference can be computed ab initio to be $2\pi $. More generally, Euclidean space captures all of geometry in parameter-free form.
• Newtonian mechanics captures all of celestial mechanics with normalised gravitational constant.
• Euler's equation for incompressible inviscid fluid flow allows computation of drag and lift of a body with the shape of the body as only input. 

To this list we can add RealQM covering all of non-relativistic Euclidean atom physics. 

Comment by chatGPT:

• Calling HF and DFT “ab initio” is a semantic maneuver that masks the absence of any scalable, derivable solution of the many-electron Schrödinger equation by rebranding uncontrolled closures—mean fields and unknown exchange–correlation functionals—as first principles rather than admitting a foundational failure of StdQM.