ffsim.transition_rdms¶

ffsim.transition_rdms(bra, ket, norb, nelec, *, rank=1, spin_summed=False, reorder=True)[source]¶

Return the transition reduced density matrices between two state vectors.

Transition RDMs use the same operators and index conventions as rdms(), with each expectation value replaced by a matrix element between bra and ket. For example:

rdm1[p, q] = ⟨bra| p+ q |ket⟩
rdm2[p, q, r, s] = ⟨bra| p+ r+ s q |ket⟩  (reorder=True)
rdm2[p, q, r, s] = ⟨bra| p+ q r+ s |ket⟩  (reorder=False)

Both inputs contain ket-state amplitudes; this function conjugates bra. The states must have the same numbers of alpha and beta electrons. They need not be normalized or orthogonal, and the result is not divided by their overlap. Passing the same state for both arguments gives its ordinary RDMs, but rdms() computes those more efficiently.

Note

Ranks 1 through 4 are supported, with the same memory requirements per real tensor family as rdms(). Spin-resolved calculations at ranks 3 and 4 embed both states in the spinless representation.

Parameters:

bra (ndarray) – The state vector on the left of the matrix element.
ket (ndarray) – The state vector on the right of the matrix element.
norb (int) – The number of spatial orbitals.
nelec (tuple[int, int]) – The number of alpha and beta electrons in each state.
rank (int) – The highest rank of reduced density matrix to return, from 1 to 4.
spin_summed (bool) – Whether to return the “spin-summed” RDMs.
reorder (bool) – Whether to reorder the indices of the reduced density matrix.

Return type:

ndarray | tuple[ndarray, ...]

Returns:

All transition RDMs up to and including rank, in increasing order of rank. For rank=1, the 1-RDM is returned directly; otherwise a tuple is returned. All tensors have complex dtype.

Spin-summed rank-k tensors have shape (norb,) * (2*k). Spin-resolved tensors have shape (k+1,) + (norb,) * (2*k), with the same spin blocks as rdms(): (a, b), (aa, ab, bb), (aaa, aab, abb, bbb), and (aaaa, aaab, aabb, abbb, bbbb).