Molcas Forum

wingsofephemera

Member

Registered: 2019-10-11

Posts: 10

[SOLVED] Symmetries in RASSI

Hi all, I've been trying to work around symmetry as in RASSI module.
I'll use what's outputted from dens2file (https://molcas.gitlab.io/OpenMolcas/dox … urce.xhtml) as an example,
if

ao_mode == true

, then the transition density matrices outputted are stored in symmetry, say the initial state and final state have the same symmetry,
then TDMs are in diagonal symmetry blocks, this is easily understandable, since if we are expanding the TDMs into a full nbst*nbst sized matrix, all the other matrix elements are zeroes except ones that use the AO basis sets in the same symmetry.

But I guess my confusion is when the initial and final state have different symmetries (but the span the same space), then the TDMs are in what's called non-diagonal symmetry block, so how exactly this non-diagonal symmetry blocks are formed?

Say, if I want to expand the non-diagonal symmetry blocked matrix into a full nbst*nbst sized matrix, where are the non-zero (or the zero) matrix elements located?

I've been trying to decipher what's in mktdzz (https://molcas.gitlab.io/OpenMolcas/dox … urce.xhtml) but hardly made any progress.

Please anybody would provide some kind of pictorial representation, such as this one

to illustrate what a non-diagonal symmetry block is? Presumably using different irreps of the initial and final states?

Thanks very much in advance!

Ignacio

Administrator

From: Uppsala

Registered: 2015-11-03

Posts: 1,079

Re: [SOLVED] Symmetries in RASSI

If I remember correctly, the non-zero blocks are those whose product is the same as the product of the two state symmetries. For example, for C2h we have the following multiplication table:

   || Ag | Bg | Au | Bu |
=========================
Ag || Ag | Bg | Au | Bu |
-------------------------
Bg || Bg | Ag | Bu | Au |
-------------------------
Au || Au | Bu | Ag | Bg |
-------------------------
Bu || Bu | Au | Bg | Ag |

If the two states have the same symmetry, their product is Ag, and the only non-zero blocks are the diagonal blocks.

   || Ag | Bg | Au | Bu |
=========================
Ag || Ag |    |    |    |
-------------------------
Bg ||    | Ag |    |    |
-------------------------
Au ||    |    | Ag |    |
-------------------------
Bu ||    |    |    | Ag |

If the two states are e.g. Bg and Au, their product is Bu, and the only non-zero blocks are Ag×Bu, Bg×Au, Au×Bg and Bu×Ag.

   || Ag | Bg | Au | Bu |
=========================
Ag ||    |    |    | Bu |
-------------------------
Bg ||    |    | Bu |    |
-------------------------
Au ||    | Bu |    |    |
-------------------------
Bu || Bu |    |    |    |

wingsofephemera

Member

Registered: 2019-10-11

Posts: 10

Re: [SOLVED] Symmetries in RASSI

If I remember correctly, the non-zero blocks are those whose product is the same as the product of the two state symmetries. For example, for C2h we have the following multiplication table:

   || Ag | Bg | Au | Bu |
=========================
Ag || Ag | Bg | Au | Bu |
-------------------------
Bg || Bg | Ag | Bu | Au |
-------------------------
Au || Au | Bu | Ag | Bg |
-------------------------
Bu || Bu | Au | Bg | Ag |

If the two states have the same symmetry, their product is Ag, and the only non-zero blocks are the diagonal blocks.

   || Ag | Bg | Au | Bu |
=========================
Ag || Ag |    |    |    |
-------------------------
Bg ||    | Ag |    |    |
-------------------------
Au ||    |    | Ag |    |
-------------------------
Bu ||    |    |    | Ag |

If the two states are e.g. Bg and Au, their product is Bu, and the only non-zero blocks are Ag×Bu, Bg×Au, Au×Bg and Bu×Ag.

   || Ag | Bg | Au | Bu |
=========================
Ag ||    |    |    | Bu |
-------------------------
Bg ||    |    | Bu |    |
-------------------------
Au ||    | Bu |    |    |
-------------------------
Bu || Bu |    |    |    |

This makes a lot more sense, thanks Ignacio!