Orbital printing format in DIRAC

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Vsevolod Dergachev

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Oct 11, 2021, 4:49:08 PMOct 11
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Dear DIRAC Community,

Could you please help me understand the orbital printing format in DIRAC activated by **ANALYZE .PRIVEC option? Specifically, I don't fully understand why there are four columns for orbital coefficients. My guess these correspond to spin up/ spin down and real/imaginary parts of the large component. Could you tell me please if I am right or wrong and what the order of the columns?

As a test, I run single point DHF for a hydrogen molecule with (D2h) and without symmetry. Using symmetry, most of the orbital coefficients are printed to a single column, whereas without symmetry - to all four.

I have attached two output files.

Thank you very much,
Seva
hydrogen_D2h.out
hydrogen_no_sym.out

Visscher, L.

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Oct 12, 2021, 2:33:51 AMOct 12
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Dear Seva,

DIRAC uses a quaternion transformation to reduce the size of the Fock matrix that is to be handled. As a consequence orbitals are obtained in quaternion format, which consists of one real and three imaginary numbers. They can be related to compolex numbers and spin, however. 
Doing this, when symmetry is not used the columns are: 

first column: real part of alpha coefficients
second column: imaginary part of alpha coefficients
third column: minus real part of beta coefficients
fourth column: imaginary part of beta coefficients

Note that all orbitals have non-zero alpha and beta parts. Note also the minus sign in the relation for the beta-real coefficient.

If symmetry is used it is (for the so-called real point groups, D2h, D2, C2v or higher) possible to apply a phase transformation to the symmetry-adapted basis functions and write all coefficients as real numbers. 
This does not mean orbitals are spin-pure, spin-orbit coupling will still lead to orbitals with mixed alpha-beta character but which part is alpha and which part is beta can be predicted by group theory.

best regards,

Luuk

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<hydrogen_D2h.out><hydrogen_no_sym.out>

Trond SAUE

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Oct 12, 2021, 2:54:00 AMOct 12
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Dear Seva,

the two first columns are the alpha-part and the final two the beta-component of the spinor.

The reason why the D2h output looks nicer than the C1 output is because of symmetry.

A bit up in the output you find the following

                                 Spinor structure
                                 ----------------


   * Fermion irrep no.: 1            * Fermion irrep no.: 2

      La  |  Ag (1)  B1g(2)  |                La  |  Au (1)  B1u(2)  |
      Sa  |  Au (1)  B1u(2)  |                Sa  |  Ag (1)  B1g(2)  |
      Lb  |  B2g(3)  B3g(4)  |                Lb  |  B2u(3)  B3u(4)  |
      Sb  |  B2u(3)  B3u(4)  |                Sb  |  B2g(3)  B3g(4)  |

The point is that the complete spinor transforms according to fermion irreps. These are the extra irreps, spanned by half-integer spin functions, that are added when you go from single to double point groups. However, each component of the spinor is a complex function and its real and imaginary parts span boson irreps, so the irreps you find in a "normal" character table. This realization is the basis of the quaternion symmetry scheme of DIRAC

http://dx.doi.org/10.1063/1.479958

When there is enough symmetry DIRAC will fix the irreps associated with each real and imaginary part of spinors, as you see above. This already gives more neat outputs, as you have observed, but is exploited to give significant reductions in computing time and storage. When there is no symmetry you see

                                 Spinor structure
                                 ----------------


   * Fermion irrep no.: 1
      La  |  A  (1)  A  (1)  |
      Sa  |  A  (1)  A  (1)  |
      Lb  |  A  (1)  A  (1)  |
      Sb  |  A  (1)  A  (1)  |

so now contributions to coefficients come all over the place.

You should also note that when you have symmetry DIRAC will form symmetry-adapted combinations of basis functions. These are indicated in the output

  Symmetry Orbitals
  -----------------

  Number of orbitals in each symmetry:          161   100   100    54   161   100   100    54
  Number of large orbitals in each symmetry:     54    30    30    16    54    30    30    16
  Number of small orbitals in each symmetry:    107    70    70    38   107    70    70    38

* Large component functions

  Symmetry  Ag ( 1)

       10 functions:    H  s   1+2
        5 functions:    H  pz  1-2
        4 functions:    H  dxx 1+2
        4 functions:    H  dyy 1+2
        4 functions:    H  dzz 1+2
        3 functions:    H  fxxz1-2
        3 functions:    H  fyyz1-2
        3 functions:    H  fzzz1-2

....

and the vector print is based on these. If you want the vectors printed without symmetry combinations you can use this keyword

http://www.diracprogram.org/doc/release-21/manual/analyze/privec.html#aolab

All the best,

    Trond


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Trond Saue
Laboratoire de Chimie et Physique Quantiques
UMR 5626 CNRS --- Université Toulouse III-Paul Sabatier
118 route de Narbonne, F-31062 Toulouse, France

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Book: Principles and Practices of Molecular Properties: Theory, Modeling, and Simulations

Vsevolod Dergachev

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Oct 12, 2021, 12:16:28 PMOct 12
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Dear Trond,

Thank you very much for such detailed answer!

Sincerely Yours,
Seva

пн, 11 окт. 2021 г. в 23:54, Trond SAUE <trond...@irsamc.ups-tlse.fr>:
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