Promotion order matters

[Vladimir_Konjkov]

Hello CASINO developers

I am faced with CASINO behavior that confused me much.

The wavefunction of a given CASSCF state is written as a sum of a configuration state functions CSF (for example linear combination of Slater determinants) each adapted to a total spin S.
Each CSF is constructed from a common set of orthonormal molecular orbitals, which are in turn expanded in basis functions.

For example if one perform CASSCF(3,8) for B atom in ORCA he will receive next set of Slater determinants with weights, combined in CSFs between the lines.

1 [2u000000] 0.963819759 <- ground state
------------------------------
2 [0u200000] -0.155310503
------------------------------
3 [0u020000] -0.155310503
------------------------------
4 [du00u000] -0.036515565
5 [ud00u000] 0.074901799
6 [uu00d000] -0.038386234
------------------------------
7 [d0u000u0] 0.021927936 <- only up electron promotes
8 [u0d000u0] -0.060742456
9 [u0u000d0] 0.038814521
------------------------------
10 [d00u0u00] 0.021927936 <- only up electron promotes
11 [u00d0u00] -0.060742456
12 [u00u0d00] 0.038814521

there are two determinants where only up electrons are promotes, and this can be done in two different ways for each determinant (CASSCF active space is starting from 2-nd orbital).

DET 7 1 PR 2 1 4 1
DET 7 1 PR 3 1 8 1
DET 10 1 PR 2 1 5 1
DET 10 1 PR 3 1 7 1

or

DET 7 1 PR 2 1 8 1
DET 7 1 PR 3 1 4 1
DET 10 1 PR 2 1 7 1
DET 10 1 PR 3 1 5 1

and these variants give significantly different VMC/DMC energies - the first is worse than the second, but if for the first variant, simultaneously change the sign of determinant weights the wave functions (and corresponding energies) become the same.

How is the permutation of the promotion order of electrons can change the sign of the resulting determinant?

Best Vladimir.

PS.
it's all a matter of Pauli exclusion principle - fermions WFN should be antisymmetric

[Pablo_Lopez_Rios]

Hi Vladimir,

In CASINO, the orbitals in each determinant are by default ordered sequentially, e.g., [123456], and excitations replace the specified orbital at its original position with the requested orbital, e.g., [123476]. The determinants are internally constructed strictly in this order. You need to ensure that the determinant coefficients that you provide correspond to the requested permutation, e.g., since by the properties of determinants [123476] = -[123467] you should apply a minus sign to the determinant coefficient.

This is all a matter of what the internal representation of the determinant is. Quantum chemistry codes tend to keep their indices sorted since excited determinants are generated internally, but CASINO goes with whatever the user provides.

Best,
Pablo