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Density Inputs / Potential Outputs Should be Consistent and Flexible for RKS and UKS #79

@wavefunction91

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@wavefunction91

As of #78, UKS adopts a canonical Pauli definition of the density input / VXC output. RKS currently takes $D_\alpha$, which is not consistent with the Pauli definition. Looking forward to GKS, the Pauli implementation is superior as all input densities are Hermitian (as is not the case in the spin-separated convention), while one can go either way for RKS/UKS. We should also support the latter to allow for simple integration with spin-separated codes (which are the norm).

  1. RKS needs to (optionally) accept $D_s$ as input - this could be resolved by a strong type template parameter toggling the expected input. Luckily, the potential remains the same for either spin-separated or Pauli definitions of the density / potential.
  2. UKS needs to (optionally) accept $D_\alpha$ / $D_\beta$ as input and return $V_\alpha$ / $V_\beta$ as output. This does change the expected output by $\pm$ factors when this option would be enabled

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