I am looking for a good approximation ... there are many out there but ideally it should be

1. Ideally, [$]C^\infty[$] smooth (a tempered distribution).

2. Always positive

3. Preserves/compatible with monotone schemes

4, Works with the PDE for gamma sensitivity

5. A bunch of deltas out there that don't cut the mustard.

6, No handwaving and well-known fixes.

The derivative of Fermi Dirac function looks not bad but I can't see the full consequences. An approximation of an approximation makes things worse?