S-Erik
commented
3 months ago Implementing the suggested change seems to cause problems with the AngularMomentum overlap setter function:
@overlap.setter
def overlap(self, overlap: np.ndarray | None) -> None:
self._overlap = overlap
if overlap is not None:
norb = self.num_spatial_orbitals
delta = np.eye(2 * norb)
delta[:norb, :norb] -= overlap.T @ overlap
delta[norb:, norb:] -= overlap @ overlap.T
summed = np.einsum("ij->", np.abs(delta))
if not np.isclose(summed, 0.0, atol=1e-6):
LOGGER.warning(
"The provided alpha-beta overlap matrix is NOT unitary! This can happen when "
"the alpha- and beta-spin orbitals do not span the same space. To provide an "
"example of what this means, consider an active space chosen from unrestricted-"
"spin orbitals. Computing <S^2> within this active space may not result in the "
"same <S^2> value as obtained on the single-reference starting point. More "
"importantly, this implies that the inactive subspace will account for the "
"difference between these two <S^2> values, possibly resulting in significant "
"spin contamination in both subspaces. You should verify whether this is "
"intentional/acceptable or whether your choice of active space can be improved."
" As a reference, here is the summed-absolute deviation of `S^T @ S` from the "
"identity: %s",
str(summed),
)This function checks if the overlap matrix is unitary, which was added in qiskit_nature verion 0.7.2, see #1292. This is done by using a helper array delta. Since the suggested change turns the overlap in a numpy array of dtype complex while delta is of dtype float, this results in the error
UFuncTypeError: Cannot cast ufunc 'subtract' output from dtype('complex128') to dtype('float64') with casting rule 'same_kind'.
What should happen?
Change
return coeff_a.T @ overlap @ coeff_bto
return (coeff_a.conj().T @ overlap @ coeff_b).realin get_overlap_ab_from_qcschema since the overlap matrix is real and casting to real values does not change the matrix.
mrossinek
Environment
What is happening?
Summary
The
get_overlap_ab_from_qcschemafunction calculates the molecular orbital (MO) overlap matrix from the atomic orbital (AO) overlap matrix and the MO coefficients. The function returnswhere
overlapis the AO overlap matrix, andcoeff_aandcoeff_bare the MO coefficients of the $\alpha$ (up)- and $\beta$ (down)-spin MOs. For real coefficientscoeff_aandcoeff_bthis is correct. For complex coefficientscoeff_aandcoeff_bthe following should be correct (and is also correct for real coefficients):Mathematical Background
The
get_overlap_ab_from_qcschemafunction calculates the overlap matrix $$O{ij}^\mathrm{MO}=c^T{pi}\cdot O{pq}^\mathrm{AO}\cdot c{qj},$$ where we abuse notation by mixing matrix multiplication and index notation. $\cdot$ denotes matrix multiplication, $O{ij}^\mathrm{MO}$($O{pq}^\mathrm{AO}$) are the MO(AO) overlap matrix entries, and $c{pi}$ are the MO coefficients. Further, $i$ and $j$ denote different MOs while $p$ and $q$ denote different AOs. This can be written with sums instead of matrix multiplication as follows $$O{ij}^\mathrm{MO}=\sum{p,q}c{pi}c{qj}O{pq}^\mathrm{AO},$$But on the other hand $O{ij}^\mathrm{MO}$ should be defined as $$O{ij}^\mathrm{MO}=\braket{i|j}.$$ We can now expand the MOs in terms of AOs: $$\ket{j}=\sumq c{qj}\ket{q}$$ with $c_{qj}=\braket{q|j}$. This implies $$\bra{j}=\sumq c^\ast{qj}\bra{q}.$$ With this we find $$O{ij}^\mathrm{MO}=\braket{i|j}=\sum{p,q}c^\ast{pi}c{qj}\braket{p|q}\sum{p,q}c^\ast{pi}c{qj}O{pq}^\mathrm{AO},$$ where we used $O{pq}^\mathrm{AO}=\braket{p|q}$. Writing the above equation using matrix multiplication we find $$O{ij}^\mathrm{MO}=c^\dagger{pi}\cdot O{pq}^\mathrm{AO}\cdot c{qj},$$ where $c^\dagger{pi}=(c^T_{pi})^\ast$. From this it is obvious to me that
should be the correct implementation.
How can we reproduce the issue?
I do not think a minimal working example is necessary here.
What should happen?
Change
to
in
get_overlap_ab_from_qcschema.Any suggestions?
No response