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FrancescoSegatta

Member

Registered: 2023-01-12

Posts: 3

Slater Determinant (printed with PRSD) sign convention

Dear Developers,

I would like to ask you a question related to the (global) sign convention of the linear combination (LC) of slater determinants (SD) that lies behind a given configuration (CSF) in a CASSCF/RASSCF calculation. I used PRSD to inspect them and found what is reported hereafter (for a RAS calculation of singlet states, with an active space of 3 occupied and 3 virtual orbitals and only single excitation allowed, and cleaning a bit the output format):

CSF #1:  222000   
SDs: + sqrt(1)  |222000|

CSF #2:  22ud00 
SDs: + sqrt(1/2)  |22ab00| - sqrt(1/2)  |22ba00|

CSF #3:  2u2d00
SDs:  - sqrt(1/2)  |2a2b00| + sqrt(1/2)  |2b2a00|

CSF #4:  u22d00
SDs:  + sqrt(1/2)  |a22b00| - sqrt(1/2)  |b22a00|

CSF #5:  22u0d0 
SDs:  + sqrt(1/2)  |22a0b0| - sqrt(1/2)  |22b0a0|

CSF #6:  22u00d 
SDs: + sqrt(1/2)  |22a00b| - sqrt(1/2)  |22b00a|

CSF #7:  2u200d 
SDs: - sqrt(1/2)  |2a200b| + sqrt(1/2)  |2b200a|

CSF #8:  u22d00   
SDs: + sqrt(1/2)  |a22b00| - sqrt(1/2) |b22a00|

CSF #9:  u220d0  
SDs: + sqrt(1/2) |a220b0| - sqrt(1/2) |b220a0|

CSF #10:  u2200d 
SDs: + sqrt(1/2)  |a2200b| - sqrt(1/2)  |b2200a|

In each of the CSF #2-#10 one expects the two terms of the LC to have opposite sign (computing singlet states). ok.
But what is the "global sign convention" for a given LC? What it seems from the printout is that, moving down the ladder of occupied MOs, the "global" sign of the LC changes, while moving up the ladder of the virtual MOs, the sign remains unchanged with respect to the sign defined for the CSF #2.

Since I do not see any physical reason for that change, what is the rationale behind such a sign convention? Is it related to how thinks are indexed?
I'm asking that because in other quantum chemistry software the convention seems different.

Best Regards,
Francesco

Ignacio

Administrator

From: Uppsala

Registered: 2015-11-03

Posts: 1,109

Re: Slater Determinant (printed with PRSD) sign convention

This is maybe too late, but I think this is just a matter of convention when defining the CSFs, and there are probably different conventions. This looks as if the contribution (factor) of a "2" orbital is +1 if the cumulative spin up to that point is integer and -1 if it's half-integer, but it might be some different rule. I guess it has to do with the default order of the spin-orbitals when defining a CSF, which may be different from the order when we specify a determinant as |2a2b00| or whatever. The phase of each CSF is to some extent arbitrary, and it could be different from other codes, although it should be consistent within a single code.

The sign convention seems to be stated here: https://doi.org/10.1002/jcc.21275 (Table 1). The step vectors 2 and 3 (which correspond to "d" and "2" orbitals) have a (-1)^b_k factor, where b_k is the 2S up to that point.