Integration of EG-XC into Mess

[ESEberhard]

Fix Ups

• Implement transformations between spherical and cartesian primitive gaussians #19

Runtime / Memory Related Issues

• Implement Density Fitting for ERI-tensors
• Only use double precision where it is needed

Extending the SCF submodule

• The current SCF implementation seems limited to hatree fock calculations, which should be extended to DFT. I consider the work by Lehtola et al. a great guide to the numerics of DFT methods in finite basis sets.
• More sophisticated initial guesses required e.g. minao see pyscf
  • Should be implemented in a runtime-aware way
• Add convergence acceleration methods like Direct Inversion of the Iterative Subspace (DIIS) see psi4 explanation
  • (Unit) test for a larger molecule with a larger basis (this is typically when these methods like DIIS are required to avoid energy oscillations along the solver trajectory).

Programming Interface

• Common interface for optimizers (SCF and direct minimization)
• Interface for learnable XC-Functionals, including learnable parameters and nuclei positions

    class XCEnergyProtocol(Protocol):
      def __call__(
          self, 
          params: NnParams,
          density_matrix: jax.Array,
          nuclei_positions: jax.Array,
          ao_value_grid: jax.Array
      ) -> jax.Array:
          ...

  the explicit dependence of $E_{XC}$ on the denstity matrix $P$ is required to obtain the exchange correlation potential $V_{XC} = \frac{\partial E_{XC}}{\partial P}$

[hatemhelal]

Thank you for preparing this and these all make sense at a high-level to me.

The current SCF implementation seems limited to hatree fock calculations

The current SCF implementation was an initial prototype so is quite limited in functionality. At the time I thought it would be nice to compare SCF to direct minimisation but to do this correctly I think it would need DIIS as well as extending direct minimisation that aren't currently available in the minimise interface. Let us know if that sounds interesting and we could coordinate.