# pyscf.symm package¶

## pyscf.symm.Dmatrix module¶

Wigner rotation D-matrix for real spherical harmonics

pyscf.symm.Dmatrix.Dmatrix(l, alpha, beta, gamma, reorder_p=False)[source]

Wigner rotation D-matrix

D_{mm’} = <lm|R(alpha,beta,gamma)|lm’> alpha, beta, gamma are Euler angles (in z-y-z convention)

Kwargs:

reorder_p (bool): Whether to put the p functions in the (x,y,z) order.

pyscf.symm.Dmatrix.dmatrix(l, beta, reorder_p=False)[source]

Wigner small-d matrix (in z-y-z convention)

pyscf.symm.Dmatrix.get_euler_angles(c1, c2)[source]

Find the three Euler angles (alpha, beta, gamma in z-y-z convention) that rotates coordinates c1 to coordinates c2.

yp = numpy.einsum(‘j,kj->k’, c1[1], geom.rotation_mat(c1[2], beta)) tmp = numpy.einsum(‘ij,kj->ik’, c1 , geom.rotation_mat(c1[2], alpha)) tmp = numpy.einsum(‘ij,kj->ik’, tmp, geom.rotation_mat(yp , beta )) c2 = numpy.einsum(‘ij,kj->ik’, tmp, geom.rotation_mat(c2[2], gamma))

(For backward compatibility) if c1 and c2 are two points in the real space, the Euler angles define the rotation transforms the old coordinates to the new coordinates (new_x, new_y, new_z) in which c1 is identical to c2.

tmp = numpy.einsum(‘j,kj->k’, c1 , geom.rotation_mat((0,0,1), gamma)) tmp = numpy.einsum(‘j,kj->k’, tmp, geom.rotation_mat((0,1,0), beta) ) c2 = numpy.einsum(‘j,kj->k’, tmp, geom.rotation_mat((0,0,1), alpha))

Solve eigenvalue problem based on the symmetry information for basis. See also pyscf/lib/linalg_helper.py `eigh_by_blocks()`

Examples:

```>>> from pyscf import gto, symm
>>> mol = gto.M(atom='H 0 0 0; H 0 0 1', basis='ccpvdz', symmetry=True)
>>> c = numpy.hstack(mol.symm_orb)
>>> vnuc_so = reduce(numpy.dot, (c.T, mol.intor('int1e_nuc_sph'), c))
>>> orbsym = symm.label_orb_symm(mol, mol.irrep_name, mol.symm_orb, c)
>>> symm.eigh(vnuc_so, orbsym)
(array([-4.50766885, -1.80666351, -1.7808565 , -1.7808565 , -1.74189134,
-0.98998583, -0.98998583, -0.40322226, -0.30242374, -0.07608981]),
...)
```

Find the list of MO of that thransform like particular irrep

Args:
moso2D float array

Overlap matrix of symmetry-adapted AO and MO, it can be obtained by reduce(numpy.dot, (csym.T.conj(), s, mo)), where csym taken from mol.symm_orb, and s is the AO overlap matrix

ovlpso2D float array

Overlap matrix between symmetry-adapted AO, it can be obtained by reduce(numpy.dot, (csym.T.conj(), s, csym))

Kwargs:
thrfloat

Returns:

1D bool array to select symmetry adapted MO

Convert the internal irrep ID to irrep symbol

Args:
gpnamestr

The point group symbol

irrep_idint

See IRREP_ID_TABLE in pyscf/symm/param.py

Returns:

Irrep sybmol, str

Convert the irrep symbol to internal irrep ID

Args:
gpnamestr

The point group symbol

symbstr

Irrep symbol

Returns:

Irrep ID, int

pyscf.symm.addons.label_orb_symm(mol, irrep_name, symm_orb, mo, s=None, check=True, tol=1e-09)[source]

Label the symmetry of given orbitals

irrep_name can be either the symbol or the ID of the irreducible representation. If the ID is provided, it returns the numeric code associated with XOR operator, see `symm.param.IRREP_ID_TABLE()`

Args:

mol : an instance of `Mole`

irrep_namelist of str or int

A list of irrep ID or name, it can be either mol.irrep_id or mol.irrep_name. It can affect the return “label”.

symm_orblist of 2d array

mo2d array

the orbitals to label

Returns:

list of symbols or integers to represent the irreps for the given orbitals

Examples:

```>>> from pyscf import gto, scf, symm
>>> mol = gto.M(atom='H 0 0 0; H 0 0 1', basis='ccpvdz',verbose=0, symmetry=1)
>>> mf = scf.RHF(mol)
>>> mf.kernel()
>>> symm.label_orb_symm(mol, mol.irrep_name, mol.symm_orb, mf.mo_coeff)
['Ag', 'B1u', 'Ag', 'B1u', 'B2u', 'B3u', 'Ag', 'B2g', 'B3g', 'B1u']
>>> symm.label_orb_symm(mol, mol.irrep_id, mol.symm_orb, mf.mo_coeff)
[0, 5, 0, 5, 6, 7, 0, 2, 3, 5]
```

Pick orbitals to form a determinant which has the right symmetry. If solution is not found, return []

std_symb(‘d2h’) returns D2h; std_symb(‘D2H’) returns D2h

pyscf.symm.addons.symmetrize_multidim(mol, mo, s=None, check=True, tol=1e-10, keep_phase=True)[source]

Symmetrize orbitals with respect to multidimensional irreps.

Make coefficients of partner functions of multidimensional irreps to be the same. The functions uses the convention of the libmsym interface, that introduces underscores to the labels of multidimensional irreps partners.

Args:
molan instance of `Mole`

Symmetry-adapted basis with multidimensional irreps should be generated by libmsym

mo2D float array

The orbital space to symmetrize

Kwargs:
s2D float array

Overlap matrix. If not given, overlap is computed with the input mol.

checkbool

Whether to check orthogonality of input orbitals and try to fix it

tolfloat

Orthogonality tolerance

keep_phasebool

Whether to keep original orbital phases, rather then make them coherent with the first partner

Returns:

2D orbital coefficients

Symmetrize the given orbitals.

This function is different to the `symmetrize_space()`: In this function, each orbital is symmetrized by removing non-symmetric components. `symmetrize_space()` symmetrizes the entire space by mixing different orbitals.

Note this function might return non-orthorgonal orbitals. Call `symmetrize_space()` to find the symmetrized orbitals that are close to the given orbitals.

Args:
mo2D float array

The orbital space to symmetrize

Kwargs:
orbsyminteger list

Irrep id for each orbital. If not given, the irreps are guessed by calling `label_orb_symm()`.

s2D float array

Overlap matrix. If given, use this overlap than the the overlap of the input mol.

Returns:

2D orbital coefficients

Examples:

```>>> from pyscf import gto, symm, scf
>>> mol = gto.M(atom = 'C  0  0  0; H  1  1  1; H -1 -1  1; H  1 -1 -1; H -1  1 -1',
...             basis = 'sto3g')
>>> mf = scf.RHF(mol).run()
>>> mol.build(0, 0, symmetry='D2')
>>> mo = symm.symmetrize_orb(mol, mf.mo_coeff)
>>> print(symm.label_orb_symm(mol, mol.irrep_name, mol.symm_orb, mo))
['A', 'A', 'B1', 'B2', 'B3', 'A', 'B1', 'B2', 'B3']
```
pyscf.symm.addons.symmetrize_space(mol, mo, s=None, check=True, tol=1e-09, clean=False)[source]

Symmetrize the given orbital space.

This function is different to the `symmetrize_orb()`: In this function, the given orbitals are mixed to reveal the symmtery; `symmetrize_orb()` projects out non-symmetric components for each orbital.

Args:

mol : an instance of `Mole`

mo2D float array

The orbital space to symmetrize

Kwargs:
s2D float array

Overlap matrix. If not given, overlap is computed with the input mol.

checkbool

Whether to check orthogonality of input orbitals and try to fix it

tolfloat

Orthogonality tolerance

cleanbool

Whether to zero out symmetry forbidden orbital coefficients

Returns:

2D orbital coefficients

Examples:

```>>> from pyscf import gto, symm, scf
>>> mol = gto.M(atom = 'C  0  0  0; H  1  1  1; H -1 -1  1; H  1 -1 -1; H -1  1 -1',
...             basis = 'sto3g')
>>> mf = scf.RHF(mol).run()
>>> mol.build(0, 0, symmetry='D2')
>>> mo = symm.symmetrize_space(mol, mf.mo_coeff)
>>> print(symm.label_orb_symm(mol, mol.irrep_name, mol.symm_orb, mo))
['A', 'A', 'A', 'B1', 'B1', 'B2', 'B2', 'B3', 'B3']
```

## pyscf.symm.basis module¶

pyscf.symm.basis.linearmole_irrep_id2symb(gpname, irrep_id)[source]
pyscf.symm.basis.linearmole_irrep_symb2id(gpname, symb)[source]
pyscf.symm.basis.linearmole_symm_descent(gpname, irrep_id)[source]

Map irreps to D2h or C2v

pyscf.symm.basis.so3_irrep_id2symb(irrep_id)[source]
pyscf.symm.basis.so3_irrep_symb2id(symb)[source]
pyscf.symm.basis.symmetrize_matrix(mat, so)[source]
pyscf.symm.basis.tot_parity_odd(op, l, m)[source]

## pyscf.symm.cg module¶

pyscf.symm.cg.cg_spin(l, jdouble, mjdouble, spin)[source]

Clebsch Gordon coefficient of <l,m,1/2,spin|j,mj>

## pyscf.symm.geom module¶

exception pyscf.symm.geom.RotationAxisNotFound[source]
class pyscf.symm.geom.SymmSys(atoms, basis=None)[source]

Bases: `object`

property atom_coords
cartesian_tensor(n)[source]
has_icenter()[source]
has_improper_rotation(axis, n)[source]
has_mirror(perp_vec)[source]
has_rotation(axis, n)[source]
search_c2x(zaxis, n)[source]

C2 axis which is perpendicular to z-axis

search_c_highest(zaxis=None)[source]
search_mirrorx(zaxis, n)[source]

mirror which is parallel to z-axis

search_possible_rotations(zaxis=None)[source]

If zaxis is given, the rotation axis is parallel to zaxis

symmetric_for(op)[source]
pyscf.symm.geom.alias_axes(axes, ref)[source]

Rename axes, make it as close as possible to the ref axes

pyscf.symm.geom.argsort_coords(coords, decimals=None)[source]
pyscf.symm.geom.as_subgroup(topgroup, axes, subgroup=None)[source]
pyscf.symm.geom.check_given_symm(gpname, atoms, basis=None)

Check whether the declared symmetry (gpname) exists in the system

If basis is specified, this function checks also the basis functions have the required symmetry.

Args:
gpname: str

point group name

atoms: list

[[symbol, [x, y, z]], [symbol, [x, y, z]], …]

pyscf.symm.geom.check_symm(gpname, atoms, basis=None)[source]

Check whether the declared symmetry (gpname) exists in the system

If basis is specified, this function checks also the basis functions have the required symmetry.

Args:
gpname: str

point group name

atoms: list

[[symbol, [x, y, z]], [symbol, [x, y, z]], …]

pyscf.symm.geom.closest_axes(axes, ref)[source]
pyscf.symm.geom.detect_symm(atoms, basis=None, verbose=2)[source]

Detect the point group symmetry for given molecule.

Return group name, charge center, and nex_axis (three rows for x,y,z)

pyscf.symm.geom.get_subgroup(gpname, axes)[source]
pyscf.symm.geom.householder(vec)[source]
pyscf.symm.geom.parallel_vectors(v1, v2, tol=1e-05)[source]
pyscf.symm.geom.rotation_mat(vec, theta)[source]

rotate angle theta along vec new(x,y,z) = R * old(x,y,z)

pyscf.symm.geom.shift_atom(atoms, orig, axis)[source]
pyscf.symm.geom.sort_coords(coords, decimals=None)[source]
pyscf.symm.geom.subgroup(gpname, axes)
pyscf.symm.geom.symm_identical_atoms(gpname, atoms)[source]

Symmetry identical atoms

pyscf.symm.geom.symm_ops(gpname, axes=None)[source]

## pyscf.symm.param module¶

pyscf.symm.param.SPHERIC_GTO_PARITY_ODD = (((0, 0, 0),), ((1, 0, 0), (0, 1, 0), (0, 0, 1)), ((1, 1, 0), (0, 1, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0)), ((0, 1, 0), (1, 1, 1), (0, 1, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0)), ((1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0)), ((0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0)), ((1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0)), ((0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0)), ((1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0)), ((0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0)), ((1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0)), ((0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0)), ((1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0)), ((0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0)), ((1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0), (1, 0, 1), (0, 0, 0)), ((0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (1, 1, 1), (0, 1, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1), (1, 0, 0)))
def parity_lm(c, l):

n = 0 for lx in reversed(range(0, l+1)):

for ly in reversed(range(0, l+1 - lx)):

lz = l - lx - ly if c[n] != 0:

return (lx%2, ly%2, lz%2)

n += 1

[[parity_lm(c, l) for c in gto.cart2sph(l).T] for l in range(16)]

## pyscf.symm.sph module¶

Spherical harmonics

pyscf.symm.sph.cart2spinor(l)[source]

Cartesian to spinor for angular moment l

pyscf.symm.sph.multipoles(r, lmax, reorder_dipole=True)[source]

Compute all multipoles upto lmax

Kwargs:
reorder_pbool

sort dipole to the order (x,y,z)

pyscf.symm.sph.real2spinor(l, reorder_p=True)
pyscf.symm.sph.real2spinor_whole(mol)

Transformation matrix that transforms real-spherical GTOs to spinor GTOs for all basis functions

Examples:

```>>> from pyscf import gto
>>> from pyscf.symm import sph
>>> mol = gto.M(atom='H 0 0 0; F 0 0 1', basis='ccpvtz')
>>> ca, cb = sph.sph2spinor_coeff(mol)
>>> s0 = mol.intor('int1e_ovlp_spinor')
>>> s1 = ca.conj().T.dot(mol.intor('int1e_ovlp_sph')).dot(ca)
>>> s1+= cb.conj().T.dot(mol.intor('int1e_ovlp_sph')).dot(cb)
>>> print(abs(s1-s0).max())
>>> 6.66133814775e-16
```
pyscf.symm.sph.real_sph_vec(r, lmax, reorder_p=False)[source]

Computes (all) real spherical harmonics up to the angular momentum lmax

pyscf.symm.sph.sph2spinor(l, reorder_p=True)[source]
pyscf.symm.sph.sph2spinor_coeff(mol)[source]

Transformation matrix that transforms real-spherical GTOs to spinor GTOs for all basis functions

Examples:

```>>> from pyscf import gto
>>> from pyscf.symm import sph
>>> mol = gto.M(atom='H 0 0 0; F 0 0 1', basis='ccpvtz')
>>> ca, cb = sph.sph2spinor_coeff(mol)
>>> s0 = mol.intor('int1e_ovlp_spinor')
>>> s1 = ca.conj().T.dot(mol.intor('int1e_ovlp_sph')).dot(ca)
>>> s1+= cb.conj().T.dot(mol.intor('int1e_ovlp_sph')).dot(cb)
>>> print(abs(s1-s0).max())
>>> 6.66133814775e-16
```
pyscf.symm.sph.sph_pure2real(l, reorder_p=True)[source]

Transformation matrix: from the pure spherical harmonic functions Y_m to the real spherical harmonic functions O_m:

```O_m = \sum Y_m' * U(m',m)
```

Y(-1) = 1/sqrt(2){-iO(-1) + O(1)}; Y(1) = 1/sqrt(2){-iO(-1) - O(1)} Y(-2) = 1/sqrt(2){-iO(-2) + O(2)}; Y(2) = 1/sqrt(2){iO(-2) + O(2)} O(-1) = i/sqrt(2){Y(-1) + Y(1)}; O(1) = 1/sqrt(2){Y(-1) - Y(1)} O(-2) = i/sqrt(2){Y(-2) - Y(2)}; O(2) = 1/sqrt(2){Y(-2) + Y(2)}

Kwargs:

reorder_p (bool): Whether the p functions are in the (x,y,z) order.

Returns:

2D array U_{complex,real}

pyscf.symm.sph.sph_real2pure(l, reorder_p=True)[source]

Transformation matrix: from real spherical harmonic functions to the pure spherical harmonic functions.

Kwargs:

reorder_p (bool): Whether the real p functions are in the (x,y,z) order.

## Module contents¶

This module offers the functions to detect point group symmetry, basis symmetriziation, Clebsch-Gordon coefficients. This module works as a plugin of PySCF package. Symmetry is not hard coded in each method.