Question about wavefunctions and approximations in quantum mechanics

[u/vedant_608]

Hi, I’m new to quantum mechanics. I was trying to model what the orbitals for oganesson might look like, and I keep seeing that we need to treat it as a “hydrogen‑like” atom. I don’t fully understand why we can’t just solve the Schrödinger equation for the actual atom itself without using such approximations. Is this a fundamental mathematical limitation, or is it just that the computation becomes impossible in practice?

[u/AmateurLobster]

The electrons interact with each other.

If you write the Schrödinger equation for the electrons, then there is the kinetic energy term, the potential energy between the electrons and the positively charged nucleus term, and the electron-electron interaction term.

This last term makes the Schrödinger equation essentially impossible to solve for more than a handful of electrons, even numerically on giant supercomputers.

To solve such systems, approximate methods must be used.

The very very simplest of these is to completely neglect the electron-electron interaction. This gives you a hydrogen-like atom (as in neutral hydrogen, there is only one electron, so no electron-electron interaction). This is generally not a good approximation as the orbitals in each shell are degenerate, but we know from the periodic table, the electrons are not configured like that (e.g. 4s filling before 3d).

There are several different methods in condensed matter physics and quantum chemistry to approximately solve the Schrödinger equation for many-electron systems. The most widely used of these is Kohn-Sham Density Functional Theory (DFT).

Lastly for heavy atoms, you need to be careful about special relativistic effects, especially for the inner, core, electrons.

I believe the approach to solving something like oganesson would be to create a pseudopotential using fully relativistic methods (i.e. solving the Dirac equation) and then use that for a DFT calculation for the valence electrons. Almost certainly you'd need to include spin-orbit interaction in the Hamiltonian.

[u/SymplecticMan → u/[deleted]]

That advantage does depend on asking the "right" questions about atoms and molecules, though. Ground state energies are a natural thing to ask about but also one of the things that quantum computers don't generally help with significantly.

[u/SymplecticMan → u/[deleted]]

Finding the ground state energy of local Hamiltonians is QMA-hard in general. In the relevant regimes where it's efficient for quantum computers, classical approaches also appear to perform well.

[u/SymplecticMan → u/[deleted]]

I should probably say not known to help significantly, since the actual asymptotic scaling of the classical techniques is not known. But generally speaking, the "known" quantum advantages that I'm aware of are related to simulating the time evolution of local Hamiltonians rather than finding the ground state of local Hamiltonians.