ROKS-TDA-SOC
************

ADF Spectroscopy UV/Vis Excitations ROKS-TDA-SOC Spin-orbit coupling
Unrestricted

This tutorial requires AMS2026 or later.

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 by means of time-dependent density
funcitonal theory (TD-DFT) using a restricted open-shell Kohn–Sham
(ROKS) framework, as well as applying the Tamm–Dancoff approximation
(TDA). Next, in the basis of these ROKS-TDA states, SOC is included,
which in total defines the ROKS-TDA-SOC method.

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 not deduced
here, but 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 [image: SymmTool][image] button: symmetry should be ATOM
   Use the Main [image: ADFPanel][image] 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
 [image][image]

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**
 [image][image]

Results can be viewed with AMSspecta.

   Start the AMSspectra module by clicking **SCM → Spectra**
 [image][image]

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 (5d^96s^2) 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.


ZnCH_3
======

ZnCH_3 was investigated in Ref. [1] which has a ^2A_1 (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 ZnCH_3 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 [image: SymmTool][image] 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)
 [image]

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 [image: ADFPanel][image] 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**
 [image][image]

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**
 [image][image]

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 X_1 0^+
ground state, and low X_2 1, a 2, and b 0^+ excited states. The
energetic difference between X_1 0^+ and X_2 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 [image: SymmTool][image] button: symmetry should be C(lin)
   Use the Main [image: ADFPanel][image] 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**
 [image]

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**
 [image][image]

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 X_1
0^+ ground state (no degeneracy, at 0 eV), X_2 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.


OsCl_6^-
========

The octahedral OsCl_6^- 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 5d^3 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 [image: SymmTool][image] button: symmetry should be O(h)
   Use the Main [image: ADFPanel][image] 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**
 [image]

   Use the panel bar **Model →  Solvation**
   Select **Solvation method → COSMO**
   Select **COSMO Solvent → Water**
 [image]

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**
 [image][image]

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] 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] 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)
