double_factorized_2body

double_factorized_2body(two_body_tensor, tol, max_vecs=None, cholesky=True)

Double-factorized decomposition of a real two-body tensor.

Represents $h_{pqrs} = \sum_t \sum_{kl} Z^{(t)}_{kl} U^{(t)}_{pk} U^{(t)}_{qk} U^{(t)}_{rl} U^{(t)}_{sl}$, returning a list of $(Z, U)$ terms where each $Z$ is a real symmetric diagonal Coulomb matrix and each $U$ is a unitary orbital rotation.

When cholesky is True (the default behavior) the outer factorization uses a modified Cholesky decomposition; otherwise it uses a truncated eigendecomposition.

Parameters

• two_body_tensor – the real two-body tensor of shape (norb, norb, norb, norb), in chemist ordering $(pq|rs)$ (i.e. the index pairs $(p,q)$ and $(r,s)$ each label a charge density, as in the reconstruction formula above).
• tol – the tolerance controlling the truncation error of the decomposition.
• max_vecs – the maximum number of terms to retain. Defaults to norb * (norb + 1) / 2.
• cholesky – whether to use the modified Cholesky decomposition (True) or a truncated eigendecomposition (False) for the outer factorization.

Returns

A list of $(Z, U)$ 2-tuples, where Z is the diagonal Coulomb matrix and U is the orbital rotation.