ROKS-TDA-SOC

The perturbative method ROKS-TDA-SOC method, described in Refs. [1] [2], is a method to deal with the influence of spin-orbit coupling (SOC) on the excitation energies for open-shell systems. In this method first the scalar relativistic excited states with well defined spin are calculated with restricted open-shell Kohn–Sham (ROKS) and the Tamm–Dancoff approximation (TDA). Next in the basis of these states SOC is included.

In this Tutorial ROKS-TDA-SOC is used for a few systems as described in Refs. [1] [2].

Remark: All systems calculated in the Tutorial contain some symmetry. These symmetries may have multi-dimensional irreps, in which case one can expect degeneracies in the excited state energies. In the ROKS-TDA-SOC implementation in ADF, however, symmetry can not be used. To preserve degeneracy also in calculations in which no symmetry is used it is required that all subspecies of an irrep for a given excited state are calculated (included). This means that one should choose the number of wanted excitations wisely (which is done is this Tutorial). One may have to iterate over the number of wanted excited states, if one does not observe the expected degeneracies. One may also first do ROKS-TDA calculations, thus without perturbative SOC, in which one can use higher symmetries. Note that numerical issues may also lift degeneracies. The symmetry labels labels are taken over from Refs. [1] [2].

Second remark: The choice of the XC functional will not be discussed here, see Refs. [1] [2] for more details. Here we pick one of the XC-functionals that is used in these papers for a certain system. The calculations will take much more time if one uses a hybrid functional instead of an LDA or GGA.

Au

The gold atom was investigated in Ref. [1] which has a ²S_1/2 (doublet) ground state. The spin-orbit splitting between the lowest ²D_3/2 and ²D_5/2 excited states was calculated. LDA is used with a QZ4P basis set.

Start AMSinput

Add an Au atom at the origin

Symmetrize the system by clicking the button: symmetry should be ATOM

Use the Main panel

Tick the Unrestricted check box

Set the Spin Polarization to 1.0

Select Basis Set → QZ4P

Select Frozen Core → None

Select Numerical Quality → Good

Use the panel bar Model → Spin and Occupation

Tick the Use ROSCF check box

8 doublet (same spin as ground state, SS) and 15 quartet (spin up, SU) excited states are taken as basis for the ROKS-TDA-SOC calculation.

Use the panel bar Properties → Excitations (UV/Vis), CD

Type of excitation → Spin-Orbit (Perturbative)

Check the TDA checkbox

Check the ROKS TDA checkbox

Enter 8 15 for the ‘Number of excitations`

Use the panel bar Details → ROKS, sTDA, sTDDFT

Select RoksTddft Calculate → SS-SU

Save and Run

Results can be viewed with AMSspecta.

Start the AMSspectra module by clicking SCM → Spectra

The scalar relativistic ²D excited state (2 times 5 is 10-fold degenerate, at 1.32 eV) is split by SOC into a ²D_5/2 excited state (6-fold degeneracy, at 0.72 eV) and a ²D_3/2 excited state (4-fold degeneracy, at 2.23 eV). The SOC-splitting \(\Delta\) = 2.23 - 0.72 = 1.51 eV.

Remark: The basis set convergence of (the spin-orbit splitting of) the ²D (5d⁹6s²) excited state is fast, since one already needs a good description of the 5d- and 6s-orbital in the ground state. For example, using a TZ2P basis set would also give reasonable numbers.

ZnCH₃

ZnCH₃ was investigated in Ref. [1] which has a ²A₁ (doublet) ground state and a lowest ²E excited state. The SOC splitting of ²E state into a ²E_1/2 and a ²E_3/2 state was calculated. In the ground state as well as in the lowest excited state the unpaired electron is mostly located at the Zn atom. The hybrid functional BHandH is used with a QZ4P basis set on the Zn atom and a TZ2P basis set on the other atoms.

The geometry of ZnCH₃ was optimized using B3LYP with a TZ2P all electron basis set:

C       0.00000000       0.00000000       0.00000000
Zn      0.00000000       0.00000000       2.05000000
H      -0.51875000      -0.89850136      -0.33000000
H      -0.51875000       0.89850136      -0.33000000
H       1.03750000       0.00000000      -0.33000000

Start AMSinput

Copy-paste the coordinates into AMSinput

Symmetrize the system by clicking the button: symmetry should be C(3v)

Panel bar Model → Regions

Select the Zn atom

Click the “+” button to add a new region (which contains the Zn atom)

This region will be used to define a larger basis set for the Zn atom than for the rest of the molecule.

Use the Main panel

Tick the Unrestricted check box

Set the Spin Polarization to 1.0

Select XC Functional → Hybrid → BHandH

Select Basis Set → TZ2P

Select Frozen Core → None

Select Numerical Quality → Good

Use the panel bar Details → Numerical Quality

Select for ‘Region_1’ Quality → QZ4P

5 doublet (same spin as ground state, SS) and 2 quartet (spin up, SU) excited states are taken as basis for the ROKS-TDA-SOC calculation.

Use the panel bar Model → Spin and Occupation

Tick the Use ROSCF check box

Use the panel bar Properties → Excitations (UV/Vis), CD

Type of excitation → Spin-Orbit (Perturbative)

Check the TDA checkbox

Check the ROKS TDA checkbox

Enter 5 2 for the ‘Number of excitations`

Use the panel bar Details → ROKS, sTDA, sTDDFT

Select RoksTddft Calculate → SS-SU

Save and Run

After the calculation finishes click SCM → Spectra

The scalar relativistic ²E excited state (2 times 2 is 4-fold degenerate, at 2.93 eV) is split by SOC into a ²E_1/2 excited state (2-fold degeneracy, at 2.91 eV) and a ²E_3/2 excited state (2-fold degeneracy, at 2.95 eV). The SOC-splitting \(\Delta\) = 2.95 - 2.91 = 0.04 eV.

Remark: The basis set convergence of (the small spin-orbit splitting of) the ²E excited state is slow, since this is more like a Rydberg state. Using a TZ2P basis set on the Zn atom would give different numbers.

TeSe

The diatomic SeTe was investigated in Ref. [2] which has a X \(³\Sigma^{-}\) (triplet) ground state and low a \(¹\Delta\) and b \(¹\Sigma^{+}\) (singlet) excited states. SOC splits these states into a X₁ 0⁺ ground state, and low X₂ 1, a 2, and b 0⁺ excited states. The energetic difference between X₁ 0⁺ and X₂ 1 is usually called the zero-field splitting (ZFS). PBE is used with a TZ2P basis set.

Start AMSinput

Add a Se atom and a Te atom with a bond length of 2.359

Symmetrize the system by clicking the button: symmetry should be C(lin)

Use the Main panel

Tick the Unrestricted check box

Set the Spin Polarization to 2.0

Select XC Functional → GGA → PBE

Select Basis Set → TZ2P

Select Frozen Core → None

Select Numerical Quality → Good

14 singlet (spin down, SD), 13 triplet (same spin as ground state, SS) and 3 quintet (spin up, SU) excited states are taken as basis for the ROKS-TDA-SOC calculation.

Use the panel bar Model → Spin and Occupation

Tick the Use ROSCF check box

Use the panel bar Properties → Excitations (UV/Vis), CD

Type of excitation → Spin-Orbit (Perturbative)

Check the TDA checkbox

Check the ROKS TDA checkbox

Enter 14 13 3 for the ‘Number of excitations`

Use the panel bar Details → ROKS, sTDA, sTDDFT

Select RoksTddft Calculate → SD-SS-SU

Save and Run

After the calculation finishes click SCM → Spectra

The scalar relativistic X \(³\Sigma^{-}\) (3-fold degeneracy, at 0 eV) ground state, the a \(¹\Delta\) (2-fold degeneracy, at 0.36 eV) and b \(¹\Sigma^{+}\) (no degeneracy, at 0.76 eV) are split by SOC into a X₁ 0⁺ ground state (no degeneracy, at 0 eV), X₂ 1 (2-fold degeneracy, at 0.16 eV), a 2 (2-fold degeneracy, ata 0.53 eV), and b 0⁺ (no degenaracy, at 1.08 eV) excited states. The ZFS is thus calculated to be 0.16 eV.

OsCl₆^-

The octahedral OsCl₆^- was investigated in Ref. [2] which has a ⁴A_2g (quartet) ground state, and low ²E_g, ²T_1g, and ²T_2g (doublet) excited states, all having a high Os 5d³ character. The states are split by SOC into the 0 \(\Gamma_{8g}\), 1 \(\Gamma_{8g}\), 2 \(\Gamma_{8g}\), 1 \(\Gamma_{6g}\), 1 \(\Gamma_{7g}\), and 3 \(\Gamma_{8g}\) states.

PBE is used with a TZ2P basis set. Like was done in Ref. [2] the conductor-like screening model (COSMO) was used to suppress numerical instabilities due to highly diffuse orbitals in the case of negatively charged complexes. The geometry is taken from Ref. [2].

Os   0.00   0.00   0.00
Cl   2.28   0.00   0.00
Cl   0.00   2.28   0.00
Cl   0.00   0.00   2.28
Cl  -2.28   0.00   0.00
Cl   0.00  -2.28   0.00
Cl   0.00   0.00  -2.28

Start AMSinput

Copy-paste the coordinates into AMSinput

Symmetrize the system by clicking the button: symmetry should be O(h)

Use the Main panel

Set the Total Charge to -1.0

Tick the Unrestricted check box

Set the Spin Polarization to 3.0

Select XC Functional → GGA → PBE

Select Basis Set → TZ2P

Select Frozen Core → None

Select Numerical Quality → Good

Use the panel bar Model → Solvation

Select Solvation method → COSMO

Select COSMO Solvent → Water

8 doublet (spin down, SD) excited states are taken as basis for the ROKS-TDA-SOC calculation.

Use the panel bar Model → Spin and Occupation

Tick the Use ROSCF check box

Use the panel bar Properties → Excitations (UV/Vis), CD

Type of excitation → Spin-Orbit (Perturbative)

Check the TDA checkbox

Check the ROKS TDA checkbox

Enter 8 for the ‘Number of excitations`

Use the panel bar Details → ROKS, sTDA, sTDDFT

Select RoksTddft Calculate → SD

Save and Run

After the calculation finishes click SCM → Spectra

The scalar relativistic ⁴A_2g (4-fold degeneracy, at 0 eV) ground state, and ²E_g (2 times 2 is 4-fold degenerate, at 0.47 eV), ²T_1g (2 times 3 is 6-fold degenerate, at 0.50 eV), and ²T_2g (2 times 3 is 6-fold degenerate, at 0.80 eV) excited states are split by SOC into a 0 \(\Gamma_{8g}\) ground state (4-fold degeneracy, at 0 eV), and 1 \(\Gamma_{8g}\) (4-fold degeneracy, at 0.44 eV), 2 \(\Gamma_{8g}\) (4-fold degeneracy, at 0.61 eV), 1 \(\Gamma_{6g}\) (2-fold degeneracy, at 0.62 eV), 1 \(\Gamma_{7g}\) (2-fold degeneracy, at 0.91 eV), and 3 \(\Gamma_{8g}\) (4-fold degeneracy, at 1.20 eV) excited states.

[1] (1,2,3,4,5,6)

C.S. Chibueze and L. Visscher, Restricted open-shell time-dependent density functional theory with perturbative spin–orbit coupling, Journal of Chemical Physics 161, 094112 (2024)

[2] (1,2,3,4,5,6,7,8)

C.S. Chibueze and L. Visscher, Restricted open-shell time-dependent density functional theory with perturbative spin–orbit coupling: Inclusion of spin-flip-down states, Journal of Chemical Physics 164, 024101 (2026)