Cross-project integration: DF-MP2

The canonical integration demo is examples/quantum_chemistry/df_mp2_synthetic.py. It computes a density-fitted MP2 correlation energy on a small synthetic system, with each stage handled by a different library in the suite:

Stage	Library	API used
Random occupied / virtual coefficients (demo only)	trnrand	normal()
Cholesky of the DF metric $J$	trnsolver	cholesky()
Half-transform $\mu\nu P \to ia P$ via GEMM	trnblas	gemm(), batched_gemm()
Pair energy contraction $B_{iaP} B_{jbP}$	trntensor	einsum("ap,bp->ab", ...)
FLOPs estimate	trntensor	estimate_flops()

The same pattern could substitute trnsparse for the DF tensor when shell screening is enabled (not shown in the minimal demo — see trnsparse/examples/sparse_fock.py).

Run it

python examples/quantum_chemistry/df_mp2_synthetic.py --demo

Output reports per-stage wall time and a FLOPs estimate from the contraction planner.

Why this example

DF-MP2 is the natural proving ground for the suite: it exercises dense GEMM (trnblas), small linear algebra (trnsolver), and tensor contractions with an FLOPs budget (trntensor). On Trainium, this is the path toward NKI-backed post-Hartree-Fock correlation methods.