slater_basis parameter

The basis for the Slater interactions can be specified by slater_basis parameter. The format is slater_basis=[(basis, order), (basis, order),]. One needs to specify basis and order for each inequivalent shell as in slater_f and slater_uj parameters. The details of basis and order are described below.

• basis

  • 'cubic' : cubic harmonics (default)

  • 'spherical' : spherical harmonics in L-S basis

  • 'spherical_j' : spherical harmonics in J-Jz basis

• order

  • The order of basis is specified by index numbers.

  • The basis can be truncted if the number of indices is fewer than 2*l+1

  • Special word such as ‘eg’ and ‘t2g’ can be used.

  • If not specified, all bases are used in the default order

Examples

General examples:

slater_basis = [('spherical',), ('spherical', 4, 3, 2, 1, 0), ('spherical', 2, 1, 0)]

slater_basis = [('cubic',), ('cubic', 2, 4), ('cubic', 1, 0, 2)]
slater_basis = [('cubic',), ('cubic', 'eg'), ('cubic', 1, 0, 2)] # Equivalent

slater_basis = [('spherical_j',), ('spherical_j',), ('spherical',)]

slater_basis = 'cubic'  # Default
slater_basis = [('cubic',), ('cubic',), ('cubic',)] # Equivalent

More specific examples are shown below.

cubic harmonics for p orbitals

[model]
norb = 3
interaction = slater_uj
slater_uj = [(1, 4.0, 0.9)]
slater_basis = 'cubic'

Output of dcore_pre :

Generating U-matrix
slater_uj = [(1, 4.0, 0.9)]
slater_basis(basis) = ['cubic']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 1
    | F_2m = [4.  4.5]
    | basis/sp = ['x' 'y' 'z']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['x' 'y' 'z']
    | basis(dn) = ['x' 'y' 'z']

cubic harmonics for d orbitals

[model]
norb = 5
interaction = slater_uj
slater_uj = [(2, 4.0, 0.9)]
slater_basis = 'cubic'

Output of dcore_pre :

Generating U-matrix
slater_uj = [(2, 4.0, 0.9)]
slater_basis(basis) = ['cubic']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 2
    | F_2m = [4.         7.73006135 4.86993865]
    | basis/sp = ['xy' 'yz' 'z^2' 'xz' 'x^2-y^2']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['xy' 'yz' 'z^2' 'xz' 'x^2-y^2']
    | basis(dn) = ['xy' 'yz' 'z^2' 'xz' 'x^2-y^2']

cubic harmonics for f orbitals

[model]
norb = 7
interaction = slater_uj
slater_uj = [(3, 4.0, 0.9)]
slater_basis = 'cubic'

Output of dcore_pre :

Generating U-matrix
slater_uj = [(3, 4.0, 0.9)]
slater_basis(basis) = ['cubic']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 3
    | F_2m = [ 4.         10.72759974  7.1676259   5.3028777 ]
    | basis/sp = ['x(x^2-3y^2)' 'z(x^2-y^2)' 'xz^2' 'z^3' 'yz^2' 'xyz' 'y(3x^2-y^2)']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['x(x^2-3y^2)' 'z(x^2-y^2)' 'xz^2' 'z^3' 'yz^2' 'xyz' 'y(3x^2-y^2)']
    | basis(dn) = ['x(x^2-3y^2)' 'z(x^2-y^2)' 'xz^2' 'z^3' 'yz^2' 'xyz' 'y(3x^2-y^2)']

d-eg orbitals

[model]
norb = 2
interaction = slater_uj
slater_uj = [(2, 4.0, 0.9)]
slater_basis = [('cubic', 'eg'),]
# slater_basis = [('cubic', 2, 4),]  equivalent

Output of dcore_pre :

Generating U-matrix
slater_uj = [(2, 4.0, 0.9)]
slater_basis = [('cubic', 'eg')]
slater_basis(basis) = ['cubic']
slater_basis(order) = [['eg']]

Slater interactions
ish = 0
    | l = 2
    | F_2m = [4.         7.73006135 4.86993865]
    | basis/sp = ['xy' 'yz' 'z^2' 'xz' 'x^2-y^2']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['z^2' 'x^2-y^2']
    | basis(dn) = ['z^2' 'x^2-y^2']

d-t2g orbitals

[model]
norb = 3
interaction = slater_uj
slater_uj = [(2, 4.0, 0.9)]
slater_basis = [('cubic', 't2g'),]
# slater_basis = [('cubic', 0, 1, 3),]  equivalent

Output of dcore_pre :

Generating U-matrix
slater_uj = [(2, 4.0, 0.9)]
slater_basis = [('cubic', 't2g')]
slater_basis(basis) = ['cubic']
slater_basis(order) = [['t2g']]

Slater interactions
ish = 0
    | l = 2
    | F_2m = [4.         7.73006135 4.86993865]
    | basis/sp = ['xy' 'yz' 'z^2' 'xz' 'x^2-y^2']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['xy' 'yz' 'xz']
    | basis(dn) = ['xy' 'yz' 'xz']

d-(xy, x^2-y^2) orbitals

[model]
norb = 2
interaction = slater_uj
slater_uj = [(2, 4.0, 0.9)]
slater_basis = [('cubic', 0, 4),]

Output of dcore_pre :

Generating U-matrix
slater_uj = [(2, 4.0, 0.9)]
slater_basis = [('cubic', 0, 4)]
slater_basis(basis) = ['cubic']
slater_basis(order) = [[0, 4]]

Slater interactions
ish = 0
    | l = 2
    | F_2m = [4.         7.73006135 4.86993865]
    | basis/sp = ['xy' 'yz' 'z^2' 'xz' 'x^2-y^2']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['xy' 'x^2-y^2']
    | basis(dn) = ['xy' 'x^2-y^2']

spherical harmonics for p orbitals

[model]
norb = 3
interaction = slater_uj
slater_uj = [(1, 4.0, 0.9)]
slater_basis = 'spherical'

Output of dcore_pre :

Generating U-matrix
slater_uj = [(1, 4.0, 0.9)]
slater_basis(basis) = ['spherical']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 1
    | F_2m = [4.  4.5]
    | basis/sp = ['p-1' 'p+0' 'p+1']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['p-1' 'p+0' 'p+1']
    | basis(dn) = ['p-1' 'p+0' 'p+1']

spherical harmonics for d orbitals

[model]
norb = 5
interaction = slater_uj
slater_uj = [(2, 4.0, 0.9)]
slater_basis = 'spherical'

Output of dcore_pre :

Generating U-matrix
slater_uj = [(2, 4.0, 0.9)]
slater_basis(basis) = ['spherical']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 2
    | F_2m = [4.         7.73006135 4.86993865]
    | basis/sp = ['d-2' 'd-1' 'd+0' 'd+1' 'd+2']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['d-2' 'd-1' 'd+0' 'd+1' 'd+2']
    | basis(dn) = ['d-2' 'd-1' 'd+0' 'd+1' 'd+2']

spherical harmonics for f orbitals

[model]
norb = 7
interaction = slater_uj
slater_uj = [(3, 4.0, 0.9)]
slater_basis = 'spherical'

Output of dcore_pre :

Generating U-matrix
slater_uj = [(3, 4.0, 0.9)]
slater_basis(basis) = ['spherical']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 3
    | F_2m = [ 4.         10.72759974  7.1676259   5.3028777 ]
    | basis/sp = ['f-3' 'f-2' 'f-1' 'f+0' 'f+1' 'f+2' 'f+3']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['f-3' 'f-2' 'f-1' 'f+0' 'f+1' 'f+2' 'f+3']
    | basis(dn) = ['f-3' 'f-2' 'f-1' 'f+0' 'f+1' 'f+2' 'f+3']

j-jz basis for p orbitals

[model]
norb = 3
interaction = slater_uj
slater_uj = [(1, 4.0, 0.9)]
slater_basis = 'spherical_j'
spin_orbit = True

Output of dcore_pre :

Generating U-matrix
slater_uj = [(1, 4.0, 0.9)]
slater_basis(basis) = ['spherical_j']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 1
    | F_2m = [4.  4.5]
    | basis/sp = ['p-1' 'p+0' 'p+1']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['j1/2-1/2' 'j3/2-3/2' 'j3/2-1/2']
    | basis(dn) = ['j1/2+1/2' 'j3/2+3/2' 'j3/2+1/2']

j-jz basis for d orbitals

[model]
norb = 5
interaction = slater_uj
slater_uj = [(2, 4.0, 0.9)]
slater_basis = 'spherical_j'
spin_orbit = True

Output of dcore_pre :

Generating U-matrix
slater_uj = [(2, 4.0, 0.9)]
slater_basis(basis) = ['spherical_j']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 2
    | F_2m = [4.         7.73006135 4.86993865]
    | basis/sp = ['d-2' 'd-1' 'd+0' 'd+1' 'd+2']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['j3/2-3/2' 'j3/2-1/2' 'j5/2-5/2' 'j5/2-3/2' 'j5/2-1/2']
    | basis(dn) = ['j3/2+3/2' 'j3/2+1/2' 'j5/2+5/2' 'j5/2+3/2' 'j5/2+1/2']

j-jz basis for f orbitals

[model]
norb = 7
interaction = slater_uj
slater_uj = [(3, 4.0, 0.9)]
slater_basis = 'spherical_j'
spin_orbit = True

Output of dcore_pre :

Generating U-matrix
slater_uj = [(3, 4.0, 0.9)]
slater_basis(basis) = ['spherical_j']
slater_basis(order) = [None]

Slater interactions
ish = 0
    | l = 3
    | F_2m = [ 4.         10.72759974  7.1676259   5.3028777 ]
    | basis/sp = ['f-3' 'f-2' 'f-1' 'f+0' 'f+1' 'f+2' 'f+3']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['j5/2-5/2' 'j5/2-3/2' 'j5/2-1/2' 'j7/2-7/2' 'j7/2-5/2' 'j7/2-3/2' 'j7/2-1/2']
    | basis(dn) = ['j5/2+5/2' 'j5/2+3/2' 'j5/2+1/2' 'j7/2+7/2' 'j7/2+5/2' 'j7/2+3/2' 'j7/2+1/2']

j=5/2 for f orbitals

[model]
norb = 3
interaction = slater_uj
slater_uj = [(3, 4.0, 0.9)]
slater_basis = [('spherical_j', 0, 1, 2),]
spin_orbit = True

Output of dcore_pre :

Generating U-matrix
slater_uj = [(3, 4.0, 0.9)]
slater_basis = [('spherical_j', 0, 1, 2)]
slater_basis(basis) = ['spherical_j']
slater_basis(order) = [[0, 1, 2]]

Slater interactions
ish = 0
    | l = 3
    | F_2m = [ 4.         10.72759974  7.1676259   5.3028777 ]
    | basis/sp = ['f-3' 'f-2' 'f-1' 'f+0' 'f+1' 'f+2' 'f+3']
    |
    | in SO rep (after transformed, reordered, or truncated)
    | basis(up) = ['j5/2-5/2' 'j5/2-3/2' 'j5/2-1/2']
    | basis(dn) = ['j5/2+5/2' 'j5/2+3/2' 'j5/2+1/2']