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:mod:`symm` -- Point group symmetry and spin symmetry

.. automodule:: pyscf.symm

PySCF supports D2h symmetry and linear molecule symmetry (Dooh and Coov). For D2h, the direct production of representations are

D2h A1g B1g B2g B3g A1u B1u B2u B3u
A1g A1g              
B1g B1g A1g            
B2g B2g B3g A1g          
B3g B3g B2g B1g A1g        
A1u A1u B1u B2u B3u A1g      
B1u B1u A1u B3u B2u B1g A1g    
B2u B2u B3u A1u B1u B2g B3g A1g  
B3u B3u B2u B1u A1u B3g B2g B1g A1g

The multiplication table for XOR operator is

XOR 000 001 010 011 100 101 110 111
000 000              
001 001 000            
010 010 011 000          
011 011 010 001 000        
100 100 101 110 111 000      
101 101 100 111 110 001 000    
110 110 111 100 101 010 011 000  
111 111 110 101 100 011 010 001 000

Comparing the two table, we notice that the two tables can be changed to each other with the mapping

D2h XOR ID
A1g 000 0
B1g 001 1
B2g 010 2
B3g 011 3
A1u 100 4
B1u 101 5
B2u 110 6
B3u 111 7

The XOR operator and the D2h subgroups have the similar relationships. We therefore use the XOR operator ID to assign the irreps (see :file:`pyscf/symm/param.py`).

C2h XOR ID C2v XOR ID D2 XOR ID
Ag 00 0 A1 00 0 A1 00 0
Bg 01 1 A2 01 1 B1 01 1
Au 10 2 B1 10 2 B2 10 2
Bu 11 3 B2 11 3 B3 11 3
Cs XOR ID Ci XOR ID C2 XOR ID
A' 0 0 Ag 0 0 A 0 0
B" 1 1 Au 1 1 B 1 1

To easily get the relationship between the linear molecule symmetry and D2h/C2v, the ID for irreducible representations of linear molecule symmetry are chosen as (see :file:`pyscf/symm/basis.py`)

D_{\infty h} ID D_{2h} ID
A1g 0 Ag 0
A2g 1 B1g 1
A1u 5 B1u 5
A2u 4 Au 4
E1gx 2 B2g 2
E1gy 3 B3g 3
E1uy 6 B2u 6
E1ux 7 B3u 7
E2gx 10 Ag 0
E2gy 11 B1g 1
E2ux 15 B1u 5
E2uy 14 Au 4
E3gx 12 B2g 2
E3gy 13 B3g 3
E3uy 16 B2u 6
E3ux 17 B3u 7
E4gx 20 Ag 0
E4gy 21 B1g 1
E4ux 25 B1u 5
E4uy 24 Au 4
E5gx 22 B2g 2
E5gy 23 B3g 3
E5uy 26 B2u 6
E5ux 27 B3u 7

and

C_{\infty v} ID C_{2v} ID
A1 0 A1 0
A2 1 A2 1
E1x 2 B1 2
E1y 3 B2 3
E2x 10 A1 0
E2y 11 A2 1
E3x 12 B1 2
E3y 13 B2 3
E4x 20 A1 0
E4y 21 A2 1
E5x 22 B1 2
E5y 23 B2 3

So that, the subduction from linear molecule symmetry to D2h/C2v can be achieved by the modular operation %10.

In many output messages, the irreducible representations are labeld with the IDs instead of the irreps' symbols. We can use :func:`symm.addons.irrep_id2name` to convert the ID to irreps' symbol, e.g.:

>>> from pyscf import symm
>>> [symm.irrep_id2name('Dooh', x) for x in [7, 6, 0, 10, 11, 0, 5, 3, 2, 5, 15, 14]]
['E1ux', 'E1uy', 'A1g', 'E2gx', 'E2gy', 'A1g', 'A1u', 'E1gy', 'E1gx', 'A1u', 'E2ux', 'E2uy']

Enabling symmetry in other module

  • SCF

    To control the HF determinant symmetry, one can assign occupancy for particular irreps, e.g.

.. literalinclude:: /../examples/scf/13-symmetry.py

  • FCI

    FCI wavefunction symmetry can be controlled by initial guess. Function :func:`fci.addons.symm_initguess` can generate the FCI initial guess with the right symmetry.

  • MCSCF

    The symmetry of active space in the CASCI/CASSCF calculations can controlled

.. literalinclude:: /../examples/mcscf/21-active_space_symmetry.py

  • MP2 and CCSD

    Point group symmetry are not supported in CCSD, MP2.

Examples

  • :source:`examples/symm/30-guess_symmetry.py`
  • :source:`examples/symm/31-symmetrize_space.py`
  • :source:`examples/symm/32-symmetrize_natural_orbital.py`
  • :source:`examples/symm/33-lz_adaption.py`

Program reference

geom

.. automodule:: pyscf.symm.geom
   :members:

Symmtry adapted basis

.. automodule:: pyscf.symm.basis
   :members:

addons

.. automodule:: pyscf.symm.addons
   :members:

Clebsch Gordon coefficients

.. automodule:: pyscf.symm.cg
   :members:

Parameters

.. automodule:: pyscf.symm.param
   :members:

Spherical harmonics

.. automodule:: pyscf.symm.sph
   :members:

Wigner rotation Dmatrix

.. automodule:: pyscf.symm.Dmatrix
   :members: