I think you may have misread the section - the document says:
When noise mitigation is enabled, even though the result does not fall within chemical accuracy (defined as being within 0.0016 Hartree of the exact result), it is fairly close to the exact solution.
So, the VQE result is not within chemical accuracy, but it is fairly close to chemical accuracy.
Computing the exchange-correlation functional to sufficiently high accuracy is QMA-hard, where QMA is the quantum version of NP. In particular, this means that it is will all likelihood hard even for a quantum computer.
Something to keep in mind is that integrals often hold two forms of symmetries - not only the symmetries enumerated in that table between the indices, but also the spins of the orbitals. I think this is explained well in Microsoft's documents. For example, if the input file notes $a^\dagger_0 a^\dagger_1 a_1 a_0$, we need to consider the potential ...
There are several startups that have formed around QC-assisted drug discovery. The ones listed below have resources on their websites that you might find helpful.
For a general understanding of molecular simulation, Simulating Molecules using VQE from the Qiskit Textbook is a great resource if you're generally familiar with QC (...