Add generalized active space CI solver with OpenMP C kernels - #203

Would be nice to check the energy value you get here to the values from OpenMolcas for example (where GAS has been implemented previously). At the least, you should also make sure the energy of the GASCI calculation does not change, most of the tests in the file are just checking the RDM elements (and specifically trace of the RDM).

Would be nice to check the energy value you get here to the values from OpenMolcas for example (where GAS has been implemented previously). At the least, you should also make sure the energy of the GASCI calculation does not change, most of the tests in the file are just checking the RDM elements (and specifically trace of the RDM).

Thank you Matthew. I have addressed both points.

(1) I added fixed-orbital GASCI validation examples against OpenMolcas for six systems: NO+, NO, O2, O2+, O2-, and AlFe2O4+. The PySCF calculations use the same molecular definition and MO coefficients as the corresponding OpenMolcas calculations. I did not include the original OpenMolcas .h5 or .out output files in the repository; instead, I saved the required molecular data and MO coefficients as .npz files under data/, together with the corresponding .xyz geometries.

The PySCF GASCI energies are compared directly with the OpenMolcas results, and all tested roots agree within 1E-8 Eh. This tolerance is consistent with the fact that OpenMolcas prints the total energies to eight digits after the decimal point.

(2) I also added an explicit O2 triplet energy regression test to pyscf/mcscf/test/test_gasci.py. It checks the GASCI ground state energy against the fixed reference value -147.5163002382864 Eh. This should catch future changes in the GASCI energy, rather than only checking quantities such as RDM traces.

The updated GASCI test suite now passes all 13 tests.

The examples aren't actually run in any CI/CD pipeline, so they can drift undetected. Examples should not be the core validation, but just demonstration of how to use the code. I would prefer at least 1 (lightweight) test that validates against the OpenMolcas result, even if there is slightly looser tolerance (like 1e-6 or so). This should ideally be in addition to the o2 triplet test.

The examples aren't actually run in any CI/CD pipeline, so they can drift undetected. Examples should not be the core validation, but just demonstration of how to use the code. I would prefer at least 1 (lightweight) test that validates against the OpenMolcas result, even if there is slightly looser tolerance (like 1e-6 or so). This should ideally be in addition to the o2 triplet test.

That makes sense. I moved the core OpenMolcas validation into test_gasci.py as a CI-run regression test. The new test embeds the exact NO+ molecular definition and fixed MO coefficients directly in the test file, checks MO orthonormality, computes the lowest 30 states, filters the 10 singlet roots, and compares all of them against the OpenMolcas reference energies with atol=1e-7.

I also simplified the examples so that they serve as demonstrations rather than the main validation suite. They now contain one O2 multiroot example and one AlFe2O4+ single-root example, with representative output included as comments.

Should be fine, but down the line might be too tight. Could always loosen to 1e-7 (which is a value used in the mcscf module of pyscf).

Should be fine, but down the line might be too tight. Could always loosen to 1e-7 (which is a value used in the mcscf module of pyscf).

Agreed. I relaxed the O2 triplet energy regression tolerance from 1e-9 to 1e-7, with rtol=0. The new NO+/OpenMolcas multiroot regression also uses an absolute tolerance of 1e-7.

OK, sorry, I lied, one more thing before I'm done. There is no specific test here of contract_2e or contract_spin as such. If you had a tool to project CI vectors back and forth between the GAS Hilbert space and the FCI Hilbert space, this would be easy to implement even for complicated GAS setups (i.e., not just the GAS-as-CAS limiting case). The GAS Hilbert space is entirely within the FCI Hilbert space, so you can insert resolutions of the identity as you please,

$$ \langle I_{\textrm{GAS}}|\hat{O}|\Psi_{\textrm{GAS}}\rangle = \sum_{J,K} \langle I_{\textrm{GAS}}|J_{\textrm{FCI}}\rangle\langle J_{\textrm{FCI}}|\hat{O}|K_{\textrm{FCI}}\rangle\langle K_{\textrm{FCI}}|\Psi_{\textrm{GAS}}\rangle $$

which implies that if you

1. Map a GAS CI vector x (i.e., packed flat ndarray) into a FCI vector c (rectangular ndarray with many zeros)
2. Use the FCI contract_ function: oc = contract_o (c)
3. Project the operator-vector product oc back into the GAS space (necessarily discarding some nonzero coefficients corresponding to illegal determinants)

then the outcome should be the same as if you did the contract operation with the GAS function: ox = contract_o (x).

N.B. gas2fci/fci2gas converter functions for the CI vector would also be very useful for developing compatibility with the CSF solver.

OK, sorry, I lied, one more thing before I'm done. There is no specific test here of contract_2e or contract_spin as such. If you had a tool to project CI vectors back and forth between the GAS Hilbert space and the FCI Hilbert space, this would be easy to implement even for complicated GAS setups (i.e., not just the GAS-as-CAS limiting case). The GAS Hilbert space is entirely within the FCI Hilbert space, so you can insert resolutions of the identity as you please,

⟨ I GAS | O ^ | Ψ GAS ⟩ = ∑ J , K ⟨ I GAS | J FCI ⟩ ⟨ J FCI | O ^ | K FCI ⟩ ⟨ K FCI | Ψ GAS ⟩

which implies that if you

1. Map a GAS CI vector x (i.e., packed flat ndarray) into a FCI vector c (rectangular ndarray with many zeros)
2. Use the FCI contract_ function: oc = contract_o (c)
3. Project the operator-vector product oc back into the GAS space (necessarily discarding some nonzero coefficients corresponding to illegal determinants)

then the outcome should be the same as if you did the contract operation with the GAS function: ox = contract_o (x).

Hi Matt. Added reusable gas2fci / fci2gas converters and operator-level projection tests in 83e2ba7.

The tests compare the GAS contractions against the corresponding full-FCI contractions followed by projection, i.e. $O_{\mathrm{GAS}}x = P O_{\mathrm{FCI}} P^\dagger x$, for explicit spin-supergroup, supergroup, and RAS restrictions.

For the spin-complete supergroup and RAS cases, the spin-square tests also verify that the GAS space is invariant under $\hat{S}^2$, i.e. $\hat{S}^2_{\mathrm{FCI}} P^\dagger x = P^\dagger \hat{S}^2_{\mathrm{GAS}} x$.

Thanks for the suggestion — the converters should also be useful for future CSF-solver compatibility.