Excitation energies: UV/Vis, X-ray, CD, MCD
*******************************************

Ultraviolet-visible (UV/Vis) spectroscopy studies electronic
excitations of valence electrons, whereas X-ray spectroscopy studies
electronic excitations of core electrons. Excitation energies and
oscillator strengths are all available in ADF as applications of time-
dependent DFT (TD-DFT). Excitation energies can be calculated for
closed-shell as well as for open-shell molecules. It is also possible
to include spin-orbit coupling and to calculate core excitations
(X-ray absorption spectra). Circular dichroism (CD) is the
differential absorption of left- and right-handed circularly polarized
light. Starting from AMS2023, it is also possible to calculate
excitation energies and oscillator strengths using the GW-Bethe-
Salpeter equation (GW-BSE) formalism.

Tip:

  See also the ADF Advanced TD-DFT tutorial: TDDFT Study of 3
  different Dihydroxyanthraquinones


Excitation energies, UV/Vis spectra
===================================

You can calculate singlet-singlet and singlet-triplet excitation
energies for a closed-shell molecule by including the EXCITATIONS
block in the input file. See the next sections for settings of
technical parameters, the calculation of excitation energies for open-
shell molecules, inclusion of spin-orbit coupling, and the calculation
of CD spectra.

   EXCITATIONS
    EXACT
      {IRREP1 N1}
      {IRREP2 N2}
    END
    DAVIDSON
      {IRREP3 N3}
      {IRREP4 N4}
    END
    ALLOWED
    ONLYSING
    ONLYTRIP
    LOWEST nlowest
   End

Several options can be addressed with subkeys in the data block. This
functionality is based on TD-DFT and consequently has a different
theoretical foundation than the SCF techniques described elsewhere in
this User's Guide. Two possible ways are available to solve the
eigenvalue equation from which the excitation energies and oscillator
strengths are obtained, of which the iterative Davidson procedure is
the default. In this case, the program needs to know how many
excitation energies are needed per irrep, what accuracy is required,
and what type of excitation energies are required (singlet-singlet or
singlet-triplet). Suitable defaults have been defined for all of
these. Each of these points is discussed below.

"Exact diagonalization vs. iterative Davidson procedure"
   The most straightforward procedure is a direct diagonalization of
   the matrix from which the excitation energies and oscillator
   strengths are obtained. Since the matrix may become very large,
   this option is possible only for very small molecules. It can be
   activated by specifying the block EXACT as one of the subkeys in
   the Excitations data block. The default is the iterative Davidson
   method. A few of the lowest excitation energies and oscillator
   strengths are then found within an error tolerance. An advantage of
   the EXACT option is that additional information is produced, such
   as the Cauchy coefficients that determine the average dipole
   polarizability. The EXACT option cannot be used in unrestricted
   calculations.

"Singlet versus triplet"
   By default, the singlet-singlet and singlet-triplet excitation
   energies are both calculated. The singlets are handled first, then
   the corresponding triplet excitation energies. One can skip one of
   these two parts of the calculation by specifying either ONLYSING or
   ONLYTRIP as a subkey in the data block.

   In case of a calculation including spin-orbit coupling, one cannot
   separate the singlet-singlet and singlet-triplet excitations. The
   subkeys ONLYSING and ONLYTRIP are misused in this case to do a
   spin-restricted calculation, or a spin-polarized calculation,
   respectively. One should in fact only use the results of the spin-
   polarized calculation.

"Dipole-allowed versus general excitations."
   If you are interested in the optical absorption spectrum, you may
   not want to compute singlet-triplet excitation energies, nor
   singlet-singlet excitation energies which, by symmetry, have zero
   oscillator strengths. This subkey should not be used in case of
   spin-orbit coupling. The subkey ALLOWED tells ADF to treat only
   those irreducible representations for which the oscillator
   strengths will be nonzero. Of course, the oscillator strengths may
   still be negligibly small. The ALLOWED subkey automatically implies
   ONLYSING. The simplest, fastest, and recommended way to obtain
   information about the ten lowest dipole-allowed excitation energies
   would be:

   EXCITATIONS
    ALLOWED
    LOWEST 10
   END

"Which excitation energies and how many?"
   The user can specify how many excitation energies per irrep should
   be calculated. If no relevant input is provided, the program
   determines these numbers from the smallest differences between
   occupied and virtual Kohn-Sham orbital energies. By default, it
   looks at the 10 lowest orbital energy differences. This number can
   be modified, for example, by specifying inside the Excitation block
   key:

      LOWEST 30

   One should be aware that this procedure does not guarantee that the
   lowest 10 (or 30) excitation energies will actually be found, since
   the orbital energy difference approximation to the excitation
   energy is rather crude. However, if the program decides on the
   basis of this procedure to calculate 4 excitation energies in a
   certain irreducible representation, these 4 excitation energies are
   certainly the lowest in that particular irrep.

   The user has more control when the number of excitations per irrep
   is explicitly specified within the EXCITATIONS block key by the
   Davidson subkey:

      DAVIDSON
       E'' 5
       T1.u 2
      END

   The DAVIDSON sub block may contain any number of records and must
   end with a record END. In the subkey data block a list of irreps,
   followed by the number of requested excitation energies is
   specified. Note that the irrep name may not be identical to the
   usual ADF name. For example E'' is called EEE in ADF. The
   Excitation code will skip an irrep if the label is not recognized.
   For multidimensional irreps, only the first column is treated,
   because the other would produce identical output. This implies that
   the oscillator strengths for E-irreps have to be multiplied by 2
   and the oscillator strengths for T-irreps by 3. The ALLOWED subkey
   should not be used if irreps are specified with the Davidson block
   subkey, however, the subkey ONLYSING (or ONLYTRIP) can be used in
   this case.

   The EXACT sub-block, mentioned already above, can also be used to
   treat only a few irreps instead of all. The number of excitation
   energies does not have to be specified then.

Some early applications of the Excitation feature in ADF can be found
in Refs. [45] [46] [47] [48] [49] [50].

For closed-shell molecules the unrelaxed excited state electric dipole
moment is calculated, see Ref. [60]. In TDDFT-gradient calculations,
the relaxed excited state electric dipole moment is also calculated.
For the definition of unrelaxed and relaxed excited state densities,
see Ref. [59].


Tamm-Dancoff approximation
--------------------------

Excitation energies can be calculated using the Tamm-Dancoff
approximation (TDA) [1] if one includes, besides the EXCITATIONS block
key, the key TDA:

   TDA


GW-Bethe-Salpeter equation (GW-BSE)
-----------------------------------

Starting from AMS2023 it is possible to calculate excitation energies
using the GW-BSE method instead of TD-DFT. The implementation in ADF
is described in Ref. [61]. Currently, the BSE kernel is compatible
with all options inside the XC block and also with the TDA and spin-
orbit coupling. As GW-BSE uses the results of a GW calculation, GW-BSE
should also not be used in combination with solvent models, like
COSMO, or other environments. A GW-BSE calculation is performed by
including the BSE key in the EXCITATIONS block:

   EXCITATIONS
    ...
    BSE
   END

In a GW-Bethe-Salpeter Equation calculation a GW calculation is
performed to calculate single-particle energies and the screened
electron-electron interaction which are then used to calculate optical
excitation energies. In a hybrid-DFT calculation, KS-DFT single-
particle energies and the unscreened electron-electron interaction are
used instead to calculate optical excitation energies.

By default, the G_0W_0 variant of the GW approximation is used in a
BSE calculation. This can be changed in the GW block, see the GW
description. Almost all options in the GW block are compatible with a
BSE calculation. For open-shell systems, partially self-consistent GW
options are currently not supported.

The recommended numerical settings for a GW-BSE calculation are the
same as for a GW calculation, see the MBPT description. ADF uses
"Dependency bas=5e-3" and "RIHartreeFock DependencyThreshold=5e-3" in
case of (any variant of) GW and GW-BSE. This can be overwritten but it
is not recommended to do so. In case of convergence problems and/or
variational collapse it is recommended to switch on the
MODIFYEXCITATION key and restrict the particle-hole basis in which the
BSE is solved to particle-hole pairs with energy differences below for
instance 2 Hartree.

For instance, one could do the following:

   EXCITATIONS
    ...
    BSE
   END

   MODIFYEXCITATION
      UseOccVirtRange 0.0 2.0
   END

   GW
   ...
   END

   MBPT
   ...
   END


Full XC kernel
--------------

With XCFUN the full (non-ALDA) kernel can be evaluated, see the XCFUN
description. To use the non-ALDA kernel the keyword FULLKERNEL should
be put in the EXCITATIONS block. FULLKERNEL can be used with GGAs
(including hybrids and RS functionals) but not meta-GGAs or meta-
hybrids. For some of the LibXC functionals (mostly GGA's and hybrids)
it is also possible to use the Full (non-ALDA) kernel. FULLKERNEL
cannot be used in combination with excited state geometry
optimizations or other response properties. If symmetry is used, it is
recommended to check that SYMMETRY NOSYM gives the same results.

   XC
    ...
    XCFUN
   End
   EXCITATIONS
    ...
    FullKernel
   END

   XC
    LibXC ...
   End
   EXCITATIONS
    ...
    FullKernel
   END

Singlet-triplet excitations are now also possible with FULLKERNEL.


Plasmons in Molecules
---------------------

In Ref. [2] a scaling approach was used to identify plasmons in
molecules. To identify plasmons in general molecules, they proposed to
analyze the excitations by scaling the electron-electron interaction
in the response calculation. While single-particle excitations are
only slightly affected by this scaling, plasmonic excitations are
sensitive to variations in the scaling parameter \lambda. More
specifically, their squared excitation energies should increase
linearly with the scaling parameter \lambda. Scaling of Coulomb, XC,
and HF parts of the response kernel in TD-DFT can be performed using
the subkeywords SCALECOUL, SCALEXC, SCALEHF in the EXCITATIONS block.

   EXCITATIONS
      SCALECOUL scalecoul
      SCALEXC   scalexc
      SCALEHF   scalehf
   END

"SCALECOUL scalecoul"
   Scaling of Coulomb kernel with scale parameter scalecoul.

"SCALEXC scalexc"
   Scaling of the XC-kernel (excluding a possible HF-part) with scale
   parameter scalexc.

"SCALEHF scalehf"
   Scaling of the HF part of the kernel with scale parameter scalehf.


Transition dipole moments between excited states
------------------------------------------------

In Ref. [60] transition dipole moments (electric dipole approximation)
between different excited states are calculated. In ADF these
transition dipole moments can be calculated if the subkey ESESTDM is
included. Spin-orbit coupling can be used. Only implemented for
closed-shell molecules.

   EXCITATIONS
      ESESTDM
   END

"ESESTDM"
   Compute transition dipole moments between excited states.


Accuracy and other technical parameters
---------------------------------------

For details regarding the (near linear scaling and parallelized)
implementation, please check Refs. [3] [4].

A summary of technical parameters with their defaults is:

   EXCITATIONS
    VECTORS 40
    TOLERANCE 1e-6
    ORTHONORMALITY 1e-8
    ITERATIONS 200
    KFWRITE 3
   END

"VECTORS vectors"
   The maximum number of trial vectors in the Davidson algorithm for
   which space is allocated. If this number is small less memory will
   be needed, but the trial vector space is smaller and has to be
   collapsed more often, at the expense of CPU time. The default is
   usually adequate.

"TOLERANCE tolerance"
   Specifies the error tolerance in *the square* of the excitation
   energies in Hartree units. The default is probably acceptable but
   we recommend that you verify the results against a stricter default
   (e.g. 1e-8) for at least a few cases. In case of a BSE calculation,
   the default is 1e-5.

"ORTHONORMALITY orthonormality"
   The Davidson algorithm orthonormalizes its trial vectors.
   Increasing the default orthonormality criterion increases the CPU
   time somewhat, but is another useful check on the reliability of
   the results.

"ITERATIONS iterations"
   The maximum number of attempts within which the Davidson algorithm
   has to converge. The default appears to be adequate in most cases.

"KFWRITE kfwrite"
   If kfwrite is 0 then do not write contributions, transition
   densities, and restart vectors to adf.rkf, since this can lead to a
   huge adf.rkf, especially if many excitations are calculated.
   Default value kfwrite is 3, which means that contributions,
   transition densities, and restart vectors are written to adf.rkf.


Excitation energies for open-shell systems
==========================================

Excitation energies can be obtained for open-shell systems in a spin-
unrestricted TDDFT calculation [5]. To perform an open-shell TD-DFT
calculation (unrestricted TDDFT) one just needs to do an unrestricted
SCF calculation and use the EXCITATIONS keyword. For ROKS TDA-DFT [10]
[62] in addition one needs to include the subkey ROSCF of the key SCF
and the key TDA (Tamm-Dancoff approximation). Presently the excitation
energies can only be found with Davidson's procedure. In case of spin-
orbit coupling, see the section on spin-orbit coupled excitation
energies open-shell molecule.

The printed symmetry in the output in TDDFT calculations is actually
the symmetry of transition density. For closed-shell systems, the
symmetry of the excited state is the same as the symmetry of the
transition density, while for open-shell systems, the symmetry of the
excited states is the direct product between the symmetry of the
transition density and the ground state symmetry. Note that the ground
state symmetry of an open-shell molecule is not necessarily A1.

For degenerate representations such as the 2-dimensional
E-representations or the 3-dimensional T-representations, the
occupation should be either fully occupied or zero. For example, for
an orbital in an E-representation the \alpha and \beta occupation
number should be either 2 or 0. The \alpha occupation number can of
course be different from the \beta occupation number.


Unrestricted TDDFT
------------------

In case of unrestricted TDDFT the spin of the excited state is not
well defined. As for the spin-state in unrestricted TDDFT, the general
rule is that if the excited state mainly results from transitions from
the singly occupied orbitals to virtual orbitals or from fully
occupied orbitals to the singly occupied orbitals, the spin state of
the excited state should roughly be the same as that of the ground
state. However, if the excited state mainly comes from transitions
from fully occupied orbitals to virtual orbitals, the spin state of
the excited state is usually a mixture since TDDFT can only deal with
single excitations within the adiabatic approximation for the XC
kernel [7]. Sometimes we simply assume that the spin state of this
kind of excited state is the same as that of the ground state [5]. In
the MO → MO transitions part for the excitations of the output file,
the spin of each molecular orbital is also specified to help assign
the spin state of the excited states. The transitions are always from
\alpha spin-orbital to \alpha spin-orbital or from \beta spin-orbital
to \beta spin-orbital.

Excited state gradients can be calculated.


Unrestricted TDDFT: spin-flip excitations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Spin-flip excitation energies [8] [9] can only be calculated with
Tamm-Dancoff approximation (TDA) [1] and Davidson's method. This
unrestricted TDDFT method cannot be used in case of spin-orbit
coupling. Excited state gradients can be calculated.

To calculate spin-flip excitation energies, one must specify two keys
SFTDDFT and TDA:

   SFTDDFT
   TDA
   Unrestricted
   SpinPolarization spinpolarization
   EXCITATIONS
      {...}
   End

For open-shell molecules, a spin-flip transition can result in a
transition to the ground state with a different S_z value, while the
symmetry of the transition density is A1. The excitation energy of
this transition should be zero and this can be used to test the
reliability of spin-flip TDDFT.

The symmetry of the excited states can be determined in the same way
as that in spin-unrestricted TDDFT calculations. As for the spin
state, similar to that in the spin-unrestricted TDDFT calculations,
some states may be more or less pure spin states, others may just be
mixtures. The users can interpret the excited state through the
transitions that contribute to this state. Note that the transitions
are always from \alpha spin-orbital to \beta spin-orbital in spin-flip
calculations, or from \beta spin-orbital to \alpha spin-orbital.


ROKS TDA-DFT
------------

In case of ROKS-TDA-DFT [10] [62] the spin of the excited state is
well defined. The default is that the spin of the excited state is the
same as that of the ground state (SS, S_{excited} = S_{ground state}).
In spin-flip up ROKS TDA-DFT calculations the spin of the excitation
is 1 higher than the spin of the ground state (SU, S_{excited} =
S_{ground state} + 1). In spin-flip down ROKS TDA-DFT calculations
(implemented in ADF2026) the spin of the excitation is 1 lower than
the spin of the ground state (SD, S_{excited} = S_{ground state} - 1).
In this case the spin of the ground state should be at least 1. This
ROKS TDA-DFT method can be used in case of spin-orbit coupling if one
includes spin-orbit coupling perturbatively, see spin-orbit coupled
excitation energies open-shell molecule.

Excited state gradients cannot be calculated with ROKS-TDA-DFT.

The ROKS method implemented in ADF needs integer occupation numbers,
and a positive spin polarization, see ROKS for high-spin open-shell
molecules. It requires the subkey ROSCF of the key SCF (and the
keyword UNRESTRICTED, and the key SPINPOLARIZATION), and the key TDA
(Tamm-Dancoff approximation), thus add:

   TDA
   Scf
      ROSCF
      End
   End
   Unrestricted
   SpinPolarization spinpolarization
   EXCITATIONS
      {...}
   End

One can do spin-conserved (same-spin, SS), spin-flip up (SU), or spin-
flip down (SD) ROKS TDA-DFT calculations, or do combinations of them.
One can specify the corresponding types with SSType, SUType, and
SDType. If spin-flip down (SD) is included one should use an Abelian
point-group symmetry (symmetry NOSYM can always be used).

Remark: with the subkey Excitations%Lowest one can specify the number
of wanted excited states. For example, if one calculates SS-SU and 2
numbers are provided, the first number is the wanted number of excited
states with same spin (SS) and the second number the wanted number of
excited states with higher spin (SU).

   Excitations
      RoksTddftType
         Calculate [SD | SS | SU | SD-SS | SS-SU | SD-SS-SU]
         SDType [SD-TDA | SD-CIS | QSD-TDA]
         SSType [R-TDA | S-TDA | X-TDA]
         SUType [SF-TDA]
      End
   End

"Excitations"
   "RoksTddftType"
      Type:
         Block

      Description:
         Excitation energies from RoksTddft: UV/Vis

      "Calculate"
         Type:
            Multiple Choice

         Default value:
            SS

         Options:
            [SD, SS, SU, SD-SS, SS-SU, SD-SS-SU]

         GUI name:
            RoksTddft Calculate

         Description:
            Specifies the types of RoksTddft methods to target several
            target spins: spin-down (SD), same-spin (SS), and/or spin-
            up (SU).

      "SDType"
         Type:
            Multiple Choice

         Default value:
            QSD-TDA

         Options:
            [SD-TDA, SD-CIS, QSD-TDA]

         GUI name:
            RoksTddft SD-Type

         Description:
            Specifies the flavor to be used in the case of spin-down
            (SD) RoksTddft

      "SSType"
         Type:
            Multiple Choice

         Default value:
            S-TDA

         Options:
            [R-TDA, S-TDA, X-TDA]

         GUI name:
            RoksTddft SS-Type

         Description:
            Specifies the flavor to be used in the case of same-spin
            (SS) RoksTddft

      "SUType"
         Type:
            Multiple Choice

         Default value:
            SF-TDA

         Options:
            [SF-TDA]

         GUI name:
            RoksTddft SU-Type

         Description:
            Specifies the flavor to be used in the case of spin-up
            (SU) RoksTddft


ROKS TDA-DFT: spin-conserved excitations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In spin-conserved ROKS-TDA-DFT calculations (same spin, SS) the spin
of the excitation is the same as the spin of the ground state. For
example, if the ground state is a doublet, then the excited states in
spin-conserved ROKS TDA-DFT will also be doublet states. There are
several flavors (SSType) of spin-conserved ROKS TDA-DFT [10] [30],
namely R-TDA [7], S-TDA [34], and X-TDA [35]. R-TDA is the original
restricted open-shell TD-DFT method within the TDA by Rinkevicius and
co-workers. S-TDA stands for spin-adapted TD-DFT within TDA in which
also triplet coupled closed-to-virtual single excitations are
incorporated. S-TDA is the default method in ADF. When one applies RPA
corrections to the frequency-independent kernel in S-TDA, one obtains
the X-TDA method. Li and Liu have shown that X-TDA is just as good as
S-TDA but due to simplicity, X-TDA may be more efficient. Note that
for X-TDA the calculation of HF exchange integrals is needed, thus for
LDA and GGA functionals one needs to include the key HARTREEFOCK.


ROKS TDA-DFT: spin-flip up excitations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In spin-flip up ROKS TDA-DFT (SFU-TDA) calculations the spin of the
excitation is 1 higher than the spin of the ground state. For example,
if the ground state is a doublet, then the excited states in spin-flip
up ROKS TDA-DFT will be quartet states. Excited state gradients cannot
be calculated with ROKS-TDA-DFT. One flavor (SUType) of spin-flip up
ROKS TDA-DFT is implemented, namely SF-TDA [10].


ROKS TDA-DFT: spin-flip down excitations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In spin-flip down ROKS TDA-DFT (SFD-TDA) calculations the spin of the
excitation is 1 lower than the spin of the ground state. For example
if the ground state is a triplet, then the excited states in spin-flip
down ROKS TDA-DFT will be singlet states. Excited state gradients
cannot be calculated with ROKS-TDA-DFT. There are several flavors
(SDType) of spin-flip down ROKS TDA-DFT, namely SD-TDA, SD-CIS, and
QSD-TDA, all described in Ref. [62]. Two fully spin-adapted ROKS-SFD-
TDA methods were described in this paper that either stem from
configuration interaction with single excitations (SD-CIS, in the
paper called SF-CIS) or from the equation-of-motion ansatz (SD-TDA, or
in the paper called SF-TDA). It was shown in the paper that the spin-
adaptation in SF-CIS (SD-CIS) and SF-TDA (SD-TDA) gives rise to
artificial double counting of correlation effects by incorporating
double excitations. When discarding this double counting, one ends up
with an excited state that is partly spin-adapted (only in the open-
to-open configurations). In the paper this method was called quasi-
spin-adapted SF-TDA (QSD-TDA, or in the paper called Q-SF-TDA) and was
shown in the paper to be a stable and efficient method that performs
similarly to spin-unrestricted SFD-TD-DFT.


Spin-flip excitations: ALDA kernel
----------------------------------

In spin-flip unrestricted TDDFT and spin-flip ROKS-TDA, the XC kernel
can be calculated directly from the XC potential. To use the LDA
potential for the XC kernel, which roughly corresponds to the ALDA in
ordinary TDDFT, one must specify the key

   FORCEALDA

anywhere in the input file. Only calculations using the LDA potential
in the SCF are fully tested. Using other GGA potentials in the SCF and
using the FORCEALDA key at the same time may introduce unreasonable
results, while using LB94 or SAOP potential in the SCF without the
FORCEALDA key may give unstable results. Unstable results have been
reported for the PW91 functional. In AMS2025, the stability of the
results without using the FORCEALDA key have been improved by treating
small spin-polarization densities in the same way as when FORCEALDA is
used.


Select (core) excitation energies, X-ray absorption
===================================================

Two methods can be used to reduce the computational costs of, for
example, core excitation energies, or some other high-lying excitation
energy. In the state-selective method scheme a guess vector for the
orbital transition has to be provided. An overlap criterion is used to
follow the desired eigenvector. In this scheme the one-electron
excited state configuration space remains complete, see Ref [12]. In
the second scheme, the range of excitations that are calculated is
modified, which means that the one-electron excited state
configuration space is reduced to the interesting part, see Ref. [11].
The calculated excited states are more accurate with the state
selective method if convergence is reached, however, the second scheme
is more robust, and it is easier to find convergence.

These selection methods can, for example, also be used in case one
calculates excitation energies as Kohn-Sham orbital energy
differences, see subkey SINGLEORBTRANS of the key EXCITATIONS.

For X-ray spectra it can be important to include quadrupole
intensities.


State selective optimization excitation energies
------------------------------------------------

The state selective method (key SELECTEXCITATION) can be used to
reduce the computational costs of, for example, core excitation
energies. In this scheme a guess vector for the orbital transition has
to be provided. It should be used in combination with the Davidson
method to calculate excitation energies. An overlap criterion is used
to follow the wanted eigenvector. This method for state selective
optimization of excitation energies is based on the method by
Kovyrshin and Neugebauer, see Ref. [12]. This key can also be used in
case of spin-orbit coupling. The use of the key SELECTEXCITATION is
similar to the use of the key MODIFYEXCITATION. However, the key
SELECTEXCITATION cannot be used in combination with the key
MODIFYEXCITATION. In the state selective method (key SELECTEXCITATION)
the one-electron excited state configuration space remains complete,
whereas it is reduced in the scheme with the MODIFYEXCITATION key.

The starting guess vector(s) for the excitation energies can be
selected, for example, by selecting 1 occupied orbital and 1 virtual
orbital.

   SELECTEXCITATION
      OscStrength oscstrength
      UseOccVirtRange elowoccvirt ehighoccvirt
      UseOccVirtNumbers nrlowoccvirt nrhighoccvirt
      UseOccRange elowocc ehighocc
      UseVirtRange elowvirt ehighvirt
      UseOccupied
         irrep orbitalnumbers
         irrep orbitalnumbers
         ...
      SubEnd
      UseVirtuaL
         irrep orbitalnumbers
         irrep orbitalnumbers
         ...
      SubEnd
      UseScaledZORA
   end

"OscStrength oscstrength"
   Use only pairs of an occupied and virtual orbital as guess vectors,
   for which the oscillator strength of the single-orbital transition
   is larger than oscstrength.

"UseOccVirtRange elowoccvirt ehighoccvirt"
   Use only pairs of an occupied and virtual orbital as guess vectors,
   for which the orbital energy difference is between elowoccvirt and
   ehighoccvirt (in hartree). If one is interested in the lowest
   excitation energies, use for elowoccvirt a value smaller than the
   HOMO-LUMO gap, and for ehighoccvirt a value larger than the energy
   range one is interested in.

"UseOccVirtNumbers nrlowoccvirt nrhighoccvirt"
   Use only pairs of an occupied and virtual orbital as guess vectors,
   for which in the sorted list of the orbital energy differences, the
   number of the single-orbital transition is between nrlowoccvirt and
   nrhighoccvirt.

"UseOccRange elowocc ehighocc"
   Use only occupied orbitals in the guess vectors which have orbital
   energies between elowocc and ehighocc (in hartree).

"UseVirtRange elowvirt ehighvirt"
   Use only virtual orbitals in the guess vectors which have orbital
   energies between elowvirt and ehighvirt (in hartree).

"UseOccupied"
   Use only the occupied orbitals in the guess vectors which are
   specified.

"UseVirtual"
   Use only the virtual orbitals in the guess vectors which are
   specified.

"irrep"
   The name of one of the irreducible representations (not a
   subspecies) of the point group of the system. See the Appendix for
   the irrep names as they are used in ADF.

"orbitalnumbers"
   A series of one or more numbers: include all numbers of the
   orbitals in the guess vectors that are to be used. In an
   unrestricted calculation the same numbers are used for the
   spin-\alpha orbitals and the spin-\beta orbitals.


Modify range of excitation energies
-----------------------------------

The key MODIFYEXCITATION can be used to reduce the computational costs
of, for example, core excitation energies. This key can also be used
in case of spin-orbit coupling. The use of the key MODIFYEXCITATION is
similar to the use of the key SELECTEXCITATION. However, the key
MODIFYEXCITATION cannot be used in combination with the key
SELECTEXCITATION. In the state selective method (key SELECTEXCITATION)
the one-electron excited state configuration space remains complete,
whereas it is (effectively) reduced in the scheme with the
MODIFYEXCITATION key.

One possibility is to allow only selected occupied orbitals and/or
selected virtual orbitals in the TDDFT calculations. In this scheme
the complete one-electron excited state configuration space is reduced
to the subspace where only the core electrons are excited, see Stener
et al. [11]. In the actual implementation this is done by artificially
changing the orbital energies of the uninteresting occupied orbitals
to a large negative value (default -1d6 Hartree), and by artificially
changing the orbital energies of the uninteresting virtual orbitals to
a large positive value (default 1d6).

In ADF2010 an extra possibility is added with the subkey
UseOccVirtRange, which restricts the space of excitation energies, by
allowing only pairs of occupied and virtual orbitals, for which the
difference in orbital energy is between a certain range.

   MODIFYEXCITATION
      OscStrength oscstrength
      UseOccVirtRange elowoccvirt ehighoccvirt
      UseOccVirtNumbers nrlowoccvirt nrhighoccvirt
      UseOccRange elowocc ehighocc
      UseVirtRange elowvirt ehighvirt
      UseOccupied
         irrep orbitalnumbers
         irrep orbitalnumbers
         ...
      SubEnd
      UseVirtuaL
         irrep orbitalnumbers
         irrep orbitalnumbers
         ...
      SubEnd
      SetOccEnergy esetocc
      SetLargeEnergy epsbig
      UseScaledZORA
   end

"OscStrength oscstrength"
   Use only pairs of an occupied and virtual orbital as guess vectors,
   for which the oscillator strength of the single-orbital transition
   is larger than oscstrength.

"UseOccVirtRange elowoccvirt ehighoccvirt"
   Use only pairs of an occupied and virtual orbital, for which the
   orbital energy difference is between elowoccvirt and ehighoccvirt
   (in hartree).

"UseOccVirtNumbers nrlowoccvirt nrhighoccvirt"
   Use only pairs of an occupied and virtual orbital as guess vectors,
   for which in the sorted list of the orbital energy differences, the
   number of the single-orbital transition is between nrlowoccvirt and
   nrhighoccvirt.

"UseOccRange elowocc ehighocc"
   Use only occupied orbitals which have orbital energies between
   elowocc and ehighocc (in Hartree).

"UseVirtRange elowvirt ehighvirt"
   Use only virtual orbitals which have orbital energies between
   elowvirt and ehighvirt (in Hartree).

"UseOccupied"
   Use only the occupied orbitals which are specified.

"UseVirtual"
   Use only the virtual orbitals which are specified.

"irrep"
   The name of one of the irreducible representations (not a
   subspecies) of the point group of the system. See the Appendix for
   the irrep names as they are used in ADF.

"orbitalnumbers"
   A series of one or more numbers: include all numbers of the
   orbitals that are to be used. In an unrestricted calculation the
   same numbers are used for the spin-\alpha orbitals and the
   spin-\beta orbitals.

"SetOccEnergy esetocc"
   All occupied orbitals that are used will change their orbital
   energy to esetocc. In practice only useful if one has selected one
   occupied orbital energy, and one wants to change this to another
   value. Default: the orbital energies of the occupied orbitals that
   are used are not changed.

"SetLargeEnergy epsbig"
   The orbital energies of the uninteresting occupied orbitals are
   changed to -epsbig Hartree, and the orbital energies of the
   uninteresting virtual orbitals are changed to epsbig Hartree
   (Default: epsbig = 1d6 Hartree).

"UseScaledZORA"
   Use everywhere the scaled ZORA orbital energies instead of the ZORA
   orbital energies in the TDDFT equations. This can improve deep core
   excitation energies. Only valid if ZORA is used. Default: use the
   unscaled ZORA orbital energies.


Excitations as orbital energy differences
-----------------------------------------

Instead of the relatively expensive TDDFT calculation of excitation
energies, sometimes just calculating Kohn-Sham orbital energy
differences may already be useful. The subkeyword SINGLEORBTRANS in
the block key EXCITATIONS will calculate excitation energies as Kohn-
Sham orbital energy differences. The subkeyword SINGLEORBTRANS in the
block key EXCITATIONS replaces the obsolete key KSSPECTRUM. For a
given excitation from an occupied orbital to a virtual orbital the
oscillator strength is calculated from the dipole transition moment
between this occupied orbital and this virtual orbital. This method is
best suited if LDA or a GGA is used in the SCF.

Especially useful for core excitation energy calculations. If
SINGLEORBTRANS is used, it is possible to use fractional occupation
numbers in the SCF, as is used in the DFT transition state (DFT-TS)
scheme, see, for example, Ref. [15]. Note: for fractional occupation
numbers, typically an orbital is treated in the excitation calculation
as if it is fully occupied if the occupation number is 1.5 or more,
and it is treated as if it is fully unoccupied if the occupation
number is 0.5 or less. In the transition state procedure for core
excitations half an electron is moved from an initial core orbital to
a final virtual orbital and the SCF KS eigenvalue difference is taken
as excitation energy. This is the original Slater formulation and can
be justified by doing a Taylor expansion of total energy with respect
to occupation numbers. It can be shown that errors arise from third
order terms, which are small. This, however, is not very practical: to
converge SCF by putting half an electron in virtual orbitals is very
hard, usually it works only for the lowest virtuals. In order to avoid
this problem one neglects the half an electron in virtual, this scheme
(only half an electron removed from occupied orbital) is called
Transition Potential (TP) and is widely employed in XAS from K-edges.

   EXCITATIONS
    SingleOrbTrans
   END

"SingleOrbTrans"
   keyword to use only orbital energy differences

The value for the subkeyword KFWRITE in the EXCITATIONS block key is
set by default to 0 in case of SINGLEORBTRANS. If kfwrite is 0 then do
not write contributions, transition densities, and restart vectors to
adf.rkf, since this can lead to a huge adf.rkf, especially if many
excitations are calculated.


Quadrupole intensities in X-ray spectroscopy
--------------------------------------------

For electronic excitations in the ultraviolet and visible range of the
electromagnetic spectrum, the intensities are usually calculated
within the dipole approximation, which assumes that the oscillating
electric field is constant over the length scale of the transition.
For the short wavelengths used in hard X-ray spectroscopy, the dipole
approximation may not be adequate. In particular, for metal K-edge
X-ray absorption spectroscopy (XAS), it becomes necessary to include
higher-order contributions. An origin-independent calculation of
quadrupole intensities in XAS was implemented in ADF by Bernadotte et
al., see Ref. [16]. These quadrupole intensities may also be important
for the calculation of X-ray emission spectroscopy, see section on
XES. Cannot be used in combination with spin-orbit coupling.

Usage

   EXCITATIONS
     ...
     XAS
     {ALLXASMOMENTS}
     {ALLXASQUADRUPOLE}
   END

"XAS"
   Use XAS within the EXCITATIONS block to initiate the calculation of
   the higher-order multipole moment integrals and the calculation of
   the quadrupole oscillator strengths. This will only print the total
   oscillator strength and the excitation energy.

"ALLXASMOMENTS"
   Use ALLXASMOMENTS within the EXCITATIONS block in combination with
   the XAS keyword. This will print out all the individual transition
   moments used within the calculation of the total oscillator
   strength.

"ALLXASQUADRUPOLE"
   Use ALLXASQUADRUPOLE within the EXCITATIONS block in combination
   with the XAS keyword. This will print out the individual oscillator
   strength components to the total oscillator strength.

The MODIFYEXCITATION or SELECTEXCITATION keyword could be used to
select a core orbital.


XES: X-ray emission spectroscopy
================================

For the calculation of XES (X-ray emission spectroscopy) spectra the
approach of Ref. [17] is implemented in the ADF program by Atkins et
al., see Ref. [18]. This is a frozen orbital, one-electron ΔDFT
approach which uses orbital energy differences between occupied
orbitals to model the X-ray emission energies. Even though it is the
simplest possible approximation for the calculation of XES spectra, it
has been shown to work well for V2C-XES (valence-to-core X-ray
emission spectroscopy) spectra of transition metal complexes.

The XES keyword initiates the calculation of X-ray emission energies
to a core orbital. In addition to dipole oscillator strengths this
keyword also triggers the calculation of the higher-order moment
integrals and the calculation of the quadrupole oscillator strengths.
By default, it calculates the emission to the first orbital in the
first symmetry, often the deepest core orbital, and only prints the
total oscillator strength and the excitation energy. Cannot be used in
combination with spin-orbit coupling.

   XES
      {COREHOLE irrep number}
      {ALLXESMOMENTS}
      {ALLXESQUADRUPOLE}
   END

"COREHOLE irrep number"
   This allows the selection of the acceptor orbital for the
   calculation of the emission oscillator strengths. For example
   'COREHOLE A1 2' calculates oscillator strengths to the orbital 2 in
   irrep A1.

"ALLXESMOMENTS"
   Use ALLXESMOMENTS within the XES block. This will print out all the
   individual transition moments used within the calculation of the
   total oscillator strength.

"ALLXESQUADRUPOLE"
   Use ALLXESQUADRUPOLE within the XES block. This will print out the
   individual oscillator strength components to the total oscillator
   strength.


Excitation energies and spin-orbit coupling
===========================================

Spin-orbit coupling can be included in the TDDFT calculation of
excitation energies. Two methods can be used in ADF. The first one
includes spin-orbit coupling as a perturbation to a scalar
relativistic calculation of excitation energies, in which spin-orbit
coupling matrix elements (SOCMEs) between excited states are
calculated. The second one includes spin-orbit coupling self-
consistently in the ground state calculation. If spin-orbit coupling
is large, the second one is more accurate for closed-shell ground
states, but is also more time-consuming. For open-shell ground states
the perturbative ROKS-TDA-SOC method is to be preferred over the
highly approximate method which includes spin-orbit coupling self-
consistently.

The results of these spin-orbit coupled TDDFT calculations include the
calculation of the zero field splitting (ZFS) of triplet excited
states in case of a closed-shell ground state and the calculation of
radiative rate constants, which could be used to calculate radiative
phosphorescence lifetimes.


Perturbative inclusion of spin-orbit coupling
---------------------------------------------

   SOPERT
     {NCALC ncalc}
     {ESHIFT eshift}
     {GSCorr Yes/No}
   End
   Relativity
     Level Scalar
     Formalism ZORA
   End
   EXCITATIONS
   END

The perturbative method, which is described in Ref. [19], is an
approximate time-dependent density-functional theory (TDDFT) formalism
to deal with the influence of spin-orbit coupling on the excitation
energies for closed-shell systems. In this formalism scalar
relativistic TDDFT calculations are first performed to determine the
lowest single-group excited states and the spin-orbit coupling
operator is applied to these single-group excited states to obtain the
excitation energies with spin-orbit coupling effects included. The
computational effort of the present method is much smaller than that
of the two-component TDDFT formalism. The compositions of the double-
group excited states in terms of single-group singlet and triplet
excited states are obtained automatically from the calculations. In
Ref. [19] it was shown that the present formalism provides reasonable
excitation energies for transitions not involving 5p and 6p orbitals.
For transitions involving 5p orbitals, one can still obtain acceptable
results for excitations with a small truncation error, while the
formalism will fail for transitions involving 6p orbitals, especially
6p1/2 spinors. In AMS2025 a bug has been fixed such that one can use
symmetry C(2H).

Although this method is not completely correctly implemented for
(meta-)hybrids or Hartree-Fock, it still gives reasonable excitation
energies, and can thus be useful also in that case.

"NCALC=ncalc"
   Number of spin-orbit coupled excitation energies to be calculated.
   Default (and maximum) value: 4 times the number of scalar
   relativistic singlet-singlet excitations.

"ESHIFT=eshift"
   The actually calculated eigenvalues are calculated up to the
   maximum singlet-singlet or singlet-triplet scalar relativistic
   excitation energy plus eshift (in Hartree). Default value: 0.2
   Hartree.

"GSCORR Yes/No"
   If Yes, the singlet ground state is included, which allows spin-
   orbit coupling to affect the ground-state energy as well. In that
   case, the spin-orbit matrix is built in the basis of the singlet
   and triplet excited states together with the ground state. The
   default is Yes.

Some extra information about the spin-orbit matrix is written to the
output if one includes

   SOPERT
     {NCALC ncalc}
     {ESHIFT eshift}
     {GSCorr Yes/No}
   End
   PRINT SOMATRIX

If one includes PRINT SOMATRIX the spin-orbit matrix (in Hartree) on
basis of singlet and triplet excited states will be printed. Note that
a triplet has three sublevels. The numbering of the basis of singlets
and triplets is given above the spin-orbit matrix. The spin-orbit
matrix has a real and imaginary part. On the diagonal the singlet or
triplet energies is added. This spin-orbit matrix is the one that is
diagonalized to get the spin-orbit coupled excitation energies. For
example, for the effect of spin-orbit coupling between T1 and S1 one
finds 3 complex values, between the 3 sublevels of T1 and the one
level of S1. For a magnitude one could take the square root of the
(real part squared + imaginary part squared).

Starting from ADF2018 these spin-orbit coupling matrix elements
(SOCMEs) are printed in an easier format in the output. The SOCME that
is printed is calculated as a root mean square: square root of (the
sum of squares of spin-orbit coupling matrix elements of all sublevels
of the uncoupled states).


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

The perturbative ROKS-TDA-SOC method described in Refs. [10] [63], is
an approximate time-dependent density-functional theory (TDDFT)
formalism to deal with the influence of spin-orbit coupling effect on
the excitation energies for open-shell systems. In ROKS-TDA-SOC spin-
orbit coupling (SOC) is included perturbatively on top of a restricted
open-shell Kohn–Sham (ROKS) time-dependent density functional theory,
in which the Tamm–Dancoff approximation (TDA) is used. Thus scalar
relativistic excited states are calculated with TDA-DFT, after which
the Hamiltonian that includes SOC is diagonalized in the basis of
these states.

The ROKS method implemented in ADF needs integer occupation numbers,
and a positive spin polarization, see ROKS for high-spin open-shell
molecules. It requires the subkey ROSCF of the key SCF (and the
keyword UNRESTRICTED, and the key SPINPOLARIZATION), and the key TDA
(Tamm-Dancoff approximation). In order to perform ROKS-TDA-SOC
calculations in addition one needs to include the block keyword
SOPERT. Symmetry NOSYM should be used. One needs to specify (with
Excitations%RoksTddftType%Calculate) which scalar relativistic ROKS-
TDA excitations to include: those with the same spin as the ground
state (SS, S_{excited} = S_{ground state}),  those with a spin 1
higher than the ground state (SU, S_{excited} = S_{ground state} + 1),
and/or those with a spin 1 lower than the ground state (SD,
S_{excited} = S_{ground state} - 1). The possibility to include spin-
flip down (SD) excitations in ROKS-TDA-SOC, see Ref. [63], is
implemented in ADF2026.

   UnRestricted
   SpinPolarization spinpolarization
   Symmetry NOSYM
   Scf
      ROSCF
      End
   End
   TDA
   SOPERT
   End
   Relativity
     Level Scalar
   End
   EXCITATIONS
      RoksTddftType
         Calculate [SD | SS | SU | SD-SS | SS-SU | SD-SS-SU]
      End
   END

Remark: with the subkey Excitations%Lowest one can specify the number
of wanted excited states. For example, if one calculates SD-SS-SU and
3 numbers are provided, the first number is the wanted number of
excited states with lower spin (SD), the second number the wanted
number of excited states with same spin (SS), and the third number the
wanted number of excited states with higher spin (SU). For open-shell
excitations typically the excited states with higher spin than the
ground state are higher in energy than those of the excited states
with the same spin as the ground state, and excited states with lower
spin than the ground state may be lowest.


Self-consistent spin-orbit coupling
-----------------------------------

   Relativity
     Level Spin-Orbit
     Formalism ZORA
   End
   EXCITATIONS
     {ALSORESTRICTED}
   END

Starting from the ADF2006.01 version, relativistic TDDFT including
spin-orbit coupling has been implemented for closed-shell molecules
with full use of double-group symmetry [20]. This relativistic time-
dependent density-functional theory (TDDFT) is based on the two-
component zeroth-order regular approximation (ZORA) and a noncollinear
exchange-correlation (XC) functional. This two-component TDDFT
formalism has the correct non-relativistic limit and affords the
correct threefold degeneracy of triplet excitations. In AMS2025 a bug
has been fixed such that one can use symmetry C(2H).

Starting from AMS2023 spin-orbit coupling can be included in the
calculation of CD spectra [32], see section on CD spectra.

In case of a calculation including spin-orbit coupling one cannot
separate the singlet-singlet and singlet-triplet excitations. By
default, the spin-polarized excitation energies are calculated (the
noncollinear scheme is used for the spin-dependent exchange-
correlation kernel). The subkey ALSORESTRICTED can be used to include
also excitation energies in which a spin-restricted exchange-
correlation kernel is used. One should in fact only use the results of
the spin-polarized calculation, which is based on the noncollinear
exchange-correlation (XC) functional. For the same reason, the ALLOWED
subkey should not be used if spin-orbit coupling is included.

To perform a spin-orbit coupled TDDFT calculation one only needs to do
a spin-orbit coupled SCF calculation and use the EXCITATIONS keyword.
The molecule needs to be closed-shell, and should be calculated spin-
restricted. Thus do not use the UNRESTRICTED, COLLINEAR, or
NONCOLLINEAR keyword. See, however, also next section.

The contribution to the double group excited states in terms of
singlet and triplet single group excited states can be estimated
through the inner product of the transition density matrix obtained
from two-component and scalar relativistic TDDFT calculations to
better understand the double group excited states [21]. In order to
obtain this analysis, one needs to perform a scalar relativistic TDDFT
calculation of excitation energies on the closed-shell molecule first,
and use the resulting adf.rkf as a fragment in the spin-orbit coupled
TDDFT calculation of excitation energies, including the keyword
STCONTRIB (Singlet and Triplet CONTRIButions):

   STCONTRIB

This STCONTRIB analysis is not performed for (meta-)hybrids, unless
one uses the Tamm-Dancoff approximation (TDA) approximation, but then
it may also fail. If one wants a similar analysis for (meta-)hybrids,
one may consider the perturbative inclusion of spin-orbit coupling in
the calculation of excitation energies.

Note that if hybrids are used, the dependency key is automatically
set, and this may effectively reduce the number of excitations, which
may give problems in the STCONTRIB analysis. A workaround for these
problems is to first calculate the scalar relativistic fragment
without the EXCITATIONS keyword. Use the adf.rkf of this calculation
as fragment in a scalar relativistic calculation with the EXCITATIONS
keyword. Use the adf.rkf of the second calculation as fragment in the
spin-orbit coupled calculation, including the STCONTRIB keyword.


Highly approximate spin-orbit coupled excitation energies for open-shell molecule
---------------------------------------------------------------------------------

Excitation energies can be obtained for open-shell systems in a spin-
unrestricted TDDFT calculation including spin-orbit coupling. This
approximate method uses a single determinant for the open-shell ground
state. The Tamm-Dancoff approximation (TDA) is needed and symmetry
NOSYM should be used. Best is to use the noncollinear approximation.
For analysis it is advised to calculate the molecule also with the
scalar relativistic spin-restricted method and use it as fragment in
the spin-orbit coupled calculation. This will make it easier to
identify the excitations.

   Unrestricted
   Symmetry NOSYM
   Relativity
     Level Spin-Orbit
     Formalism ZORA
     SpinOrbitMagnetization NONCOLLINEAR
   End
   TDA
   Excitations
   End

Note that this approximate method for open-shell molecules is not able
to show the subtle effects of spin-orbit coupling. Some of the reasons
are the approximate nature of the XC functionals for open-shell
molecules, the single determinant that is used for the open-shell
ground state, and that only single excitations are included in the
excitation. If one does not include spin-orbit coupling the spin-
unrestricted TDDFT approach introduces spin-contamination such that
the result does not represent transitions between pure spin states.
Inclusion of spin-orbit coupling will not simplify this. However, if
spin-orbit coupling is large, then this method may help to identify
excitations.

Note that the approximations made in this approximate method are much
worse than for spin-orbit coupled TDDFT for closed-shell systems. In
that case one can get a reasonable description of the subtle effects
of spin-orbit coupling, for example, for the zero-field splitting of a
triplet excited state.


CV(n)-DFT: Constricted Variational DFT
======================================

In the constricted nth order variational density functional method
(CV(n)-DFT) [22] [6] [23] the occupied excited state orbitals are
allowed to relax in response to the change of both the Coulomb and
exchange-correlation potential in going from the ground state to the
excited state. This theory is not time-dependent nor is it based on
response theory. It is instead variational in nature and has been
termed constricted variational DFT or CV(n)-DFT.

Due to bugs in older versions it is important to use ADF2016.105 or
later. In ADF2017 the relaxation density is a bit differently
calculated than in ADF2016, which will slightly modify the results
compared to ADF2016. In ADF2017 singlet-triplet excitations are added
[24]. Different working equations have been implemented in different
modified versions of ADF. In ADF2016 and ADF2017 all CV-DFT excitation
energies are calculated consistently, this is why some energies may
differ from previously published values.

CV(n)-DFT requires an all-electron basis set.

The CVNDFT block key regulates the execution of the CV(n)-DFT code,
which calculates singlet or triplet electronic excitations for closed-
shell molecules. Note that one has to choose either singlet-singlet
excitations (subkey ONLYSING in EXCITATIONS) or singlet-triplet
excitations (subkey ONLYTRIP in EXCITATIONS), in case of CV(n)-DFT,
one cannot calculate them both in one run. The parameter n in (n)
describes the order of this theory. There are two limiting cases
implemented in the CV(n)-DFT code: n=2 and n= \infty. Since CV(n)-DFT
coincides in the second order with TDDFT, the CV(n)-DFT code is
designed as an extension of the EXCITATIONS module. Therefore, the
EXCITATIONS block has to be present in the input file together with
CVNDFT. CVNDFT takes all the information about the number of
excitations and their symmetry from the EXCITATIONS block. Moreover,
the TDDFT transition density vectors are used as an initial guess for
the CV(\infty)-DFT calculations.

There are a few approximations within the CV(\infty)-DFT approach,
therefore there are a few corresponding sub-blocks in the CVNDFT
block. In the simplest case the TDDFT transition density U-vector is
substituted into the infinite order CV(\infty)-DFT excitation energy.
This approximation corresponds to the CV_DFT sub-block, so the input
fragment would look like this:

   CVNDFT
     CV_DFT &
     SUBEND
   END
   EXCITATIONS
     ONLYSING|ONLYTRIP
     ...
   END

In general, the U-vector has to be optimized with respect to the
infinite order CV(\infty)-DFT excitation energy. It is accomplished
iteratively in the SCF-CV(\infty)-DFT method [25], which is invoked
when the SCF_CV_DFT sub-block is present. Therefore, there are input
parameters that control the SCF procedure, i.e. the total number of
iterations and tolerance. The corresponding input fragment would look
like this:

   CVNDFT
     ITERATION iter
     TOLERANCE tol
     SCF_CV_DFT &
       DAMPING mix
     SUBEND
   END
   EXCITATIONS
     ONLYSING|ONLYTRIP
     ...
   END

"iter"
   iter is the maximum number of iterations. Default 50.

"tol"
   tol is the convergence criterion, i.e. the SCF-CV(\infty)-DFT
   procedure stops when the given accuracy is achieved. Default 1e-4.

"mix"
   mix is the relative weight of the new U-vector that is added to the
   one from the previous iteration (default value is mix=0.2).

Another feature of CV(\infty)-DFT is a possibility to optimize (or
relax) the molecular orbitals for the particular excitation. This is
accomplished in the R-CV(\infty)-DFT method, see for an application
Ref. [26]. In the last case, the transition U-vector is frozen, while
the orbitals are relaxed. For example, the input fragment which
employs the U-vector from TDDFT would look like this:

   CVNDFT
     ITERATION iter
     TOLERANCE tol
     R_CV_DFT &
       RELAXALPHA start_a_iter
       RELAXBETA  start_b_iter
       DAMPORBRELAX mix_relax
     SUBEND
   END
   EXCITATIONS
     ONLYSING|ONLYTRIP
     ...
   END

"start_a_iter"
   start_a_iter is the SCF cycle number at which the relaxation of
   \alpha orbitals starts. Default value 1.

"start_b_iter"
   start_b_iter is the SCF cycle number at which the relaxation of
   \beta orbitals starts. Default value 1.

   Since there is no optimization of the U-vector in R-CV(\infty)-DFT,
   the relaxation of orbitals (either \alpha or \beta or both) should
   be turned on at the very first iteration.

"mix_relax"
   The mix_relax parameter defines the relative weight of the new
   relaxation vector that is added to the one from the previous
   iteration (the default value is mix_relax=0.2).

Note that all three methods can be used in the input file. The
corresponding input fragment would look like this:

   CVNDFT
     ITERATION iter
     TOLERANCE tol
     CV_DFT &
     SUBEND
     SCF_CV_DFT &
       DAMPING mix
     SUBEND
     R_CV_DFT &
       RELAXALPHA start_a_iter
       RELAXBETA  start_b_iter
       DAMPORBRELAX mix_relax
     SUBEND
   END
   EXCITATIONS
     ONLYSING|ONLYTRIP
     ...
   END

In this case the U-vector will be optimized with respect to the
infinite order CV(\infty)-DFT excitation energy by the SCF-
CV(\infty)-DFT code and then supplied to the R-CV(\infty)-DFT code.

CVNDFT prints in the output the excitation energies as well as the
maximum value of the lambda parameter \lambda. This number corresponds
to the largest singular value of the singular value decomposition of
the U matrix/vector (i.e. the U-vector can be considered as a matrix
with *nvir* rows and *nocc* columns, where *nvir* is the number of
virtual orbitals and *nocc* is the number of occupied orbitals). If
this number is close to \pi / 2 (i.e. \lambda_{max} \approx 1.57),
then the corresponding electronic excitation can be approximated by a
single natural transition orbital (NTO) to another single NTO
transition. This type of transitions is obtained in \Delta SCF.
Therefore, this information can be used, for example, to compare the
CV(\infty)-DFT results to the \Delta SCF excitation energies.


HDA, TD-DFT+TB, TDDFT-aas, sTDDFT
=================================

There are several approximations that can be made to speed up the TD-
DFT excitation calculations that will be discussed here: HDA, TD-
DFT+TB, TDA-DFT+TB, TDDFT-aas, sTDA, and sTDDFT. An alternative to
these approximations is PolTDDFT, which is discussed in a different
part of the ADF manual.


HDA: Hybrid Diagonal Approximation
----------------------------------

This method is only relevant if a (meta-)hybrid is used in the SCF.
The hybrid diagonal approximation (HDA) [58] is based on utilizing the
hybrid exchange only for the diagonal terms in the response equations
to calculate excitations. This allows one to limit the computational
cost of the TD-DFT simulation while keeping basically the same
accuracy as in the full TD-DFT scheme using hybrid xc-functionals. It
is furthermore not necessary to correct all the diagonal terms with
hybrid exchange. A cutoff parameter can be chosen in the input in
order to reduce the number of diagonal terms that have to be
calculated, which can speed up the HDA calculations.

   EXCITATIONS
    HDA
    HDA_CutOff hda_cutoff
   End

"HDA"
   Activate the hybrid diagonal approximation (HDA).

"hda_cutoff"
   Cutoff on difference in energy between virtual and occupied
   orbitals eps_virt-eps_occ, in order to reduce the number of
   diagonal terms that are corrected. Default is a huge value, which
   means that all diagonal terms are corrected.

This method cannot be used in excited state geometry optimizations.
For spin-orbit coupled HDA excitations one needs to use symmetry
NOSYM:

   relativity
    level spin-orbit
   end
   Symmetry NOSYM
   EXCITATIONS
    HDA
    HDA_CutOff hda_cutoff
   End


TD-DFT+TB, TDA-DFT+TB
---------------------

The basic idea of TD-DFT+TB [27] is to use the molecular orbitals from
a DFT ground state calculation as input to an excited state
calculation with TD-DFTB coupling matrices. TB means tight binding. In
ADF2023 TDA-DFT+TB is implemented, which is similar to TD-DFT+TB, but
now using TDA. Starting from ADF2023 one can also use TD-DFT+TB and
TDA-DFT+TB for open-shell molecules. If many excitations are required,
this method will speed up the calculation drastically in comparison to
the standard time needed for TDDFT calculations of excitation
energies. This method is best suited if a (meta-)GGA or LDA is used in
the SCF. Since ADF2023 the TD-DFT+TB method can be used in excited
state geometry optimizations in case of closed-shell singlet-singlet
and closed-shell singlet-triplet excitations, see Ref. [13].

For TD-DFT+TB use

   SYMMETRY NOSYM
   EXCITATIONS
     TD-DFTB
   END

For TDA-DFT+TB use

   SYMMETRY NOSYM
   EXCITATIONS
     TDA-DFTB
   END

One can use this, for example, in combination with the
MODIFYEXCITATION key to use only single-orbital transitions that have
a minimal oscillator strength:

   SYMMETRY NOSYM
   EXCITATIONS
     TD-DFTB
     ONLYSING
   END
   MODIFYEXCITATION
     OscStrength 0.001
   End

In TD-DFT+TB reducing the space of single-orbital transitions with the
key MODIFYEXCITATION will reduce the memory needed in ADF. Starting
from ADF2016.102 for TD-DFT+TB only the lowest (in energy) 100000
single orbital transitions are included by default. One can override
this by using a different value for the subkey UseOccVirtNumbers of
the key MODIFYEXCITATION.

   MODIFYEXCITATION
     UseOccVirtNumbers 1 100000
   END

The value for the subkeyword KFWRITE in the EXCITATIONS block key is
set by default to 0 in case of TD-DFT+TB. If kfwrite is 0 then do not
write contributions, transition densities, and restart vectors to
adf.rkf, since this can lead to a huge adf.rkf, especially if many
excitations are calculated.


TDDFT-aas
---------

TDDFT-aas [14] (TDDFT-approximate auxiliary s function) resembles the
TDDFT-as method and the TDDFT+TB method: when solving the K coupling
integrals in the excited state, an approximation is made instead of
calculating the exact two-electron integrals. TDDFT-aas utilizes the
\gamma_{AB} function from the TDDFT-as method to calculate the
coupling matrix K:

   K_{i a, j b}=\sum_{A B} q_{i a, A} \gamma_{A B} q_{j b, B} \\
   \gamma\left(R_{A B}\right)=\frac{\operatorname{erf}\left(R_{A B}
   \sqrt{\alpha / 2}\right)}{R_{A B}}

In ADF, TDDFT-aas can be calculated for singlet excited states only,
symmetry NOSYM is required. This method is best suited if a (meta-)GGA
or LDA is used in the SCF. The TDDFT-aas method can be used in excited
state geometry optimizations in case of closed-shell singlet-singlet
excitations, see Ref. [14].

For TDDFT-aas use

   SYMMETRY NOSYM
   EXCITATIONS
     ONLYSING
     AAS
     {AASALPHA aasalpha}
   END

"aasalpha"
   The alpha value for the TDDFT-aas method. Default alpha=0.2.


sTDA, sTDDFT
------------

The simplified Tamm-Dancoff approach (sTDA) [28] and simplified time-
dependent DFT approach (sTDDFT) [29] by Grimme et al. are implemented
in ADF. In these approaches the time-dependent DFT equations are
simplified by the evaluation of the two-electron integrals as short-
range damped Coulomb interactions between (transition) charge density
monopoles and a truncation of the single excitation expansion space.
These methods are best suited if a (meta-)hybrid or a range-separated-
hybrid is used in the SCF. These methods will speed up the calculation
drastically in comparison to the standard time needed for TDA or TDDFT
calculations of excitation energies for hybrids. Starting from ADF2023
one can use sTDA and sTDDFT for open-shell molecules. These methods
cannot be used in excited state geometry optimizations.

An example application of sTDDFT is given in the TDDFT advanced
tutorial

For sTDA use:

   SYMMETRY NOSYM
   EXCITATIONS
    sTDA
   END

For sTDDFT use:

   SYMMETRY NOSYM
   EXCITATIONS
    sTDDFT
   END

For global hybrids ADF will use the parameters \alpha and \beta that
depend on the amount of exact exchange a_x as:

   \alpha = \alpha_1 + \alpha_2 a_x \\ \beta = \beta_1 + \beta_2 a_x

where \alpha_1 = 1.42, \alpha_2 = 0.48, \beta_1 = 0.2, \beta_2 = 1.83
are used in ADF. These values are fitted parameters by Grimme [28].
For range-separated hybrids one should set the parameters \alpha,
\beta and a_x in the input of ADF with the keyword MODIFYEXCITATION.
See Ref. [31] for parameters that are used by Risthaus et al. for a
few range-separated functionals. However, the \alpha and \beta
parameters are mixed up in Ref. [31]. Thus use:

 
| **Functional**                                                                                                                                                                                           | a_x                                                                                                  | \alpha                                                                                               | \beta                                                                                                |
 
| CAM-B3LYP                                                                                                                                                                                                | 0.38                                                                                                 | 0.90                                                                                                 | 1.86                                                                                                 |
 
| WB97                                                                                                                                                                                                     | 0.61                                                                                                 | 4.41                                                                                                 | 8.00                                                                                                 |
 
| WB97X                                                                                                                                                                                                    | 0.56                                                                                                 | 4.58                                                                                                 | 8.00                                                                                                 |
 

For example, for CAM-B3LYP use

   MODIFYEXCITATION
     GrimmeAlpha 0.9
     GrimmeBeta 1.86
     GrimmeAex 0.38
   END

If one uses the keyword MODIFYEXCITATION one may also set more
parameters that are used in the sTDA and sTDDFT approach.

   MODIFYEXCITATION
     GrimmeAlpha Alpha
     GrimmeBeta Beta
     GrimmeAex Aex
     GrimmeDEmax DEmax
     GrimmeTPmin TPmin
     GrimmePertC|NoGrimmePertC
   END

"GrimmeAlpha Alpha"
   To set the \alpha parameters, should be used with GrimmeBeta and
   GrimmeAex.

"GrimmeBeta Beta"
   To set the \beta parameters, should be used with GrimmeAlpha and
   GrimmeAex.

"GrimmeAex Aex"
   To set a_x that is used in the sTDA or sTDDFT part of the
   calculation, should be used with GrimmeAlpha and GrimmeBeta.

"GrimmeDEmax DEmax"
   Single orbital transitions that have an orbital energy difference
   less than DEmax (in Hartree) are included. Default value DEmax =
   0.4 Hartree. See also the meaning of E_{max} in Grimme [28].

"GrimmeTPmin TPmin"
   Single orbital transitions that would have a cumulative
   perturbative energy contribution larger than TPmin are included.
   Default value TPmin = 10^{-4}. See also the meaning of t_p in
   Grimme [28].

"GrimmePertC|NoGrimmePertC"
   In case of GrimmePertC, which is the default, if the cumulative
   perturbative energy contribution for a single orbital transition is
   smaller than TPmin, this contribution is used to modify the
   diagonal value of a matrix that is used in the sTDA or the sTDDFT
   method. If NoGrimmePertC is included such contributions are
   neglected. See Grimme [28] for more details on the truncation of
   the single excitation space.

Starting from ADF2016.102 for sTDA or sTDDFT only the lowest (in
energy) 10000 single orbital transitions are included by default. One
can override this by using a different value for the subkey
UseOccVirtNumbers of the key MODIFYEXCITATION.

   MODIFYEXCITATION
    UseOccVirtNumbers 1 10000
   END

The value for the subkeyword KFWRITE in the EXCITATIONS block key is
set by default to 0 in case of sTDA or sTDDFT. If kfwrite is 0 then do
not write contributions, transition densities, and restart vectors to
adf.rkf, since this can lead to a huge adf.rkf, especially if many
excitations are calculated.


CD spectra
==========

Circular dichroism (CD) is the differential absorption of left- and
right-handed circularly polarized light. Starting from ADF2010
Hartree-Fock and hybrids can also be used to calculate CD spectra.

Starting from AMS2023, spin-orbit coupling can be included in the
calculation of CD spectra, see Ref. [32], which allows one, for
example, to calculate circularly polarized luminescence (CPL) for
spin-forbidden transitions such as T1-S0 emission, and spin-forbidden
electronic circular dichroism. These are spin-forbidden if one would
not include spin-orbit coupling. Phosphorescence dissymmetry factors
are calculated.

   EXCITATIONS
    CDSPECTRUM
    ANALYTICAL
    VELOCITY
   End

"CDSPECTRUM"
   If the subkey *CDSPECTRUM* is included in the key EXCITATIONS, the
   rotatory strengths for the calculated excitations are calculated,
   in order to simulate Circular Dichroism (CD) spectra [36] [37].
   Interesting for chiral molecules.

"Velocity"
   If the subkey *VELOCITY* is included, ADF calculates the dipole-
   velocity representation of the oscillator strength. If applicable
   (use of subkey *CDSPECTRUM*) the dipole-velocity representation of
   the rotatory strength is calculated. Default the dipole-length
   representation of the oscillator strength and rotatory strength is
   calculated. Remark: in case of (range separated) (meta-) hybrids
   and TDA the dipole-length and velocity representation of the
   oscillator strength are not the same (not even in the basis set
   limit), see Ref. [33].


MCD
===

MCD or magnetic circular dichroism is the differential absorption of
left and right circularly polarized light in the presence of a
magnetic field. MCD intensity is usually described in terms of
different contributions called A, B and C terms, see Refs. [38] [39].
A further parameter D is often discussed in MCD studies. D is
proportional to the intensity of an absorption band and is closely
related to the oscillator strength. A and B terms for closed and open-
shell molecules and C terms of open-shell molecules induced by spin-
orbit coupling can be calculated. Starting from ADF2010, C terms
related to spatially degenerate states, i.e. breaking of degeneracies
can be calculated. The A term is only calculated in case the molecule
has symmetry and then only for excited states that belong to multi-
dimensional irreps.

For MCD calculations for molecules that have C(2) or D(2) symmetry use
SYMMETRY NOSYM.

**Input options**

   EXCITATIONS
      MCD {options}
      ONLYSINGLET
      {SELECT transition number}
      {DTENSOR {Dxx Dxy Dyy Dxz Dyz|D E/D}}
   End

   ALLPOINTS
   {Relativity
      Level Scalar
      Formalism ZORA
    End}
   {SOMCD}
   {ZFS}

"MCD"
   If the subkey MCD is included in the key EXCITATIONS the MCD
   parameters of some or all of the excitations considered in the
   TDDFT procedure are calculated [51] [52] [53] [41]. This subkey
   should not be used with spin-orbit coupling (but, see below).
   Several other keywords could be important.

   ALLPOINTS: required for an MCD calculation.

   ONLYSINGLET: this keyword should be used in combination with a MCD
   calculation.

   RELATIVITY: Scalar Relativistic ZORA: required for a calculation of
   temperature-dependent C terms. In this case the keyword SOMCD must
   also be added as a key by itself, and the calculation must be
   unrestricted. If only A and B terms are calculated then ZORA is not
   needed but can be included if desired.

   ZFS: If the ZFS keyword and MCD with SOMCD are also included then
   the influence of the calculated zero-field splitting (ZFS) on the
   temperature-dependent MCD is evaluated. The MCD in the presence of
   ZFS is described as anisotropic in the output because the Zeeman
   splitting becomes orientation dependent in the presence of ZFS.

   In ADF2010 the temperature-dependent MCD due to the breaking of
   degeneracies of excited states by spin-orbit coupling can be
   calculated. Although all temperature-dependent MCD is typically
   called "C terms", the parameters associated with the MCD are
   labeled "CE" to distinguish them from the MCD due to mixing between
   states caused by spin-orbit coupling that is labeled "C". The CE
   terms have a derivative shape like A terms. They have the same
   temperature-dependence as normal C terms. If they are present, CE
   terms are calculated automatically along with C terms if the
   keyword SOMCD is included in the input.

"MCD {options}"
   Options include NMCDTERM, NMIX, DCUTOFF, MCDOUT, CGOUT, NANAL,
   NANAL2, FULLOMEGA, NOAB, NODIRECT, NOCG, CONVCG, ITERCG, ITER2CG,
   BMIN, BMAX, TMIN, TMAX and NTEMP.

   "NMCDTERM=nmcdterm"
      Number of excitations for which MCD parameters are to be
      calculated. The nmcdterm lowest energy excitations are treated.
      The default is the number of transitions considered in the TDDFT
      calculation.

   "NMIX=nmix"
      Number of transitions allowed to mix in a SOS calculation.
      Default is the number of transitions considered in the TDDFT
      calculation.

   "DCUTOFF=dcutoff"
      MCD parameters will only be calculated for transitions with
      sufficient intensity. Each Cartesian component of each
      transition is considered separately. If the dipole strength D of
      that component is below dcutoff then the MCD is not calculated.
      The default is 1.0e-6.

   "MCDOUT=mcdout"
      Number that determines the amount of output to be printed about
      the MCD calculation. Higher means more output. Possible values
      are 0, (orientationally averaged and Cartesian components of MCD
      parameters only) 1 (as for 0 but with the addition of a short
      analysis) or 2 (as for 1 but with the addition of a lengthy
      analysis). Theoretical analyses of MCD parameters are presented
      in several places including Refs. [38] [39] [52] [53] [41]. The
      default for MCDOUT is 0.

   "CGOUT=cgout"
      The perturbed transition densities used to evaluate the B and C
      term parameters can be obtained through an iterative conjugate-
      gradient procedure. Convergence information of the conjugate-
      gradient algorithm is printed every cgout iterations. Default is
      10.

   "NANAL=nanal, NANAL2=nanal2"
      If MCDOUT is set to 2, a detailed analysis of the B and/or C
      term parameters in terms of which states mix and how much MCD
      each mixing causes, is presented. The parameters NANAL and
      NANAL2 determine how many contributions are included in the
      analyses. Defaults are 10 for NANAL and 5 for NANAL2.

   "FULLOMEGA"
      A standard TDDFT calculation involves the solution of an
      eigenvalue equation to obtain the excitation energies and
      transition densities of interest. ADF can solve this eigenvalue
      equation two ways: through diagonalization of the full Omega
      matrix or through the Davidson procedure where Omega is never
      explicitly constructed. Construction of the complete Omega
      matrix is generally only feasible for smaller problems. The
      matrix Omega appears again in the equations solved to obtain
      MCD. Here again Omega can be built or only the products of Omega
      with a vector can be used as is the case in the Davidson
      procedure. The default is to not construct Omega. If the keyword
      FULLOMEGA is included then Omega is constructed. Note that the
      choice of FULLOMEGA is completely independent of whether EXACT
      or DAVIDSON is chosen in the earlier TDDFT calculation.

   "NOAB"
      If this keyword is included then A and B terms are not
      calculated. NOAB only makes sense if SOMCD is included in the
      input otherwise no MCD will be calculated at all.

   "NODIRECT"
      The perturbed transition density needed to evaluate B and C term
      parameters is obtained through the solution of a large system of
      equations. This system of equations is solved in two ways:
      through a sum-over-states (SOS) type approach where the solution
      is expanded in a known set of transition densities or through
      the direct solution of the system of equations by the conjugate
      gradient procedure. The SOS method is much faster but also less
      accurate, particularly for larger systems. By default, MCD
      parameters are evaluated through both approaches. If the
      NODIRECT keyword is included then only the SOS calculation is
      performed.

   "NOCG"
      The conjugate gradient procedure is first used in combination
      with a preconditioner that generally speeds up convergence
      significantly. If no solution is found in a reasonable number of
      iterations then the procedure is restarted without the
      preconditioner. If the NOCG keyword is included then the
      preconditioner is never used.

   "CONVCG"
      Convergence criterion for the CG iterative methods. The default
      value of 0.01 is probably good enough for most applications.
      This choice seems to produce B and C terms that are converged to
      3 significant figures. Except for small systems, it is not
      recommended that CONVCG be set to a much smaller number as this
      will probably cause a large number of convergence failures.

   "ITERCG=itercg"
      Number of iterations before failure in the first
      (preconditioned) CG solver. This solver either succeeds quickly
      or not at all so the default value is 30.

   "ITER2CG=iterc2g"
      Number of iterations before failure in the B of C term parameter
      calculation of the unconditioned CG solver. This solver is often
      slow so the default value is 200.

   "BMIN=bmin, BMAX=bmax, NBFIELD=nbfield, TMIN=tmin, TMAX=tmax,
   NTEMP=ntemp"
      Temperature dependent MCD intensity often varies nonlinearly
      with T and B when T is small and/or B is large. It may therefore
      be of interest to evaluate the MCD intensity over a range of
      temperatures and/or magnetic fields. This can be achieved
      through the use of the BMIN, BMAX, NBFIELD, TMIN, TMAX and NTEMP
      keywords. The MIN and MAX keywords give the maximum values of B
      or T. NBFIELD and NTEMP indicate how many values are to be
      considered. Note that magnetic fields are assumed to be given in
      Tesla and temperatures in Kelvin. For example, BMIN=1, BMAX=5,
      NBFIELD=5 means that fields of 1,2,3,4 and 5 T will be
      considered. Defaults are BMIN=BMAX=1, TMIN=TMAX=5 and
      NBFIELD=NTEMP=1.

"SELECT nselect1 nselect2 nselect3..."
   Rather than selecting the first nmcdterm transitions for
   consideration individual transitions can be selected through the
   SELECT keyword. The transitions of interest are listed after the
   SELECT keyword. Note that the numbering follows that given in the
   summary table at the end of the TDDFT calculation. To consider a
   degenerate transition only the first component need be included.
   Note that it makes no sense to use both the SELECT and NMCDTERM
   keywords together.

"DTENSOR Dxx Dxy Dyy Dxz Dyz" "DTENSOR D E/D"
   As noted earlier, if the ZFS keyword is included with MCD and SOMCD
   then the influence of zero-field splitting on temperature-dependent
   MCD will be evaluated. As an alternative to the ZFS keyword the
   D-tensor parameters can be entered directly through the DTENSOR
   keyword in the EXCITATIONS block. Two input formats are possible.
   Five real numbers Dxx Dxy Dyy Dxz Dyz can be entered. These five
   numbers are sufficient to define the traceless tensor D.
   Alternatively, the two parameters D and E/D can be entered. In this
   case the coordinate system chosen to define the molecular geometry
   must be the principal axis system of the D-tensor. D, Dxx, Dxy,
   Dyy, Dxz and Dyz should be given in wavenumbers (cm-1).

**Notes**

If an MCD calculation is run, the transition densities obtained in the
TDDFT calculation are saved to adf.rkf. For large molecules this can
result in a very large adf.rkf file.

An MCD calculation relies on the excitation energies and, in
particular, the transition densities that result from the preceding
TDDFT calculation. If the results of the TDDFT calculation are poor
then it is likely that the results of the MCD calculation will be
poor. It therefore should be kept in mind that most TDDFT calculations
will make use of the Davidson method for finding the eigenvalues and
eigenvectors of the TDDFT equation. The Davidson approach involves
some approximations that can lead to some variation in results with
the applied parameters. The most important example of this is the fact
that the results vary depending on how many eigenvalue/eigenvector
pairs are calculated, ie how many transitions are selected through the
LOWEST keyword. The variation is small for the eigenvalues (excitation
energies) but can be significant for the eigenvectors (transition
densities). A variation in the transition densities leads to variation
in the transition dipoles which can significantly impact calculated
MCD parameters. The moral of this story is that when calculating MCD
parameters it is best to choose one value of LOWEST and stick with it.

The most time-consuming part of an MCD calculation is the solution of
the system of equations through the conjugate-gradient solver. The
solver can fail so be aware of warnings concerning convergence in the
output. A few hints to improve convergence are: a) choose a value of
LOWEST that is at least double the number of transitions for which you
desire MCD parameters. This helps to improve the SOS calculation which
provides an initial guess for the conjugate gradient solver. The
solver is sensitive to the initial guess so changing LOWEST by a small
amount may help (or hinder) convergence significantly. Keep the
previous note in mind when playing with LOWEST however. b) The
preconditioned conjugate gradient solver is usually fast but does not
converge monotonically to the correct answer. The unpreconditioned
solver is much slower but tends to converge monotonically. If the
preconditioned solver fails but leaves a fairly well converged result
for the unpreconditioned solver the latter usually converges quickly.
If the preconditioned solver does not leave a fairly well converged
result it may be worth changing the number of iterations it uses since
a few iterations earlier or later may provide a much better converged
answer. c) The SELECT keyword can be used to work on the remaining
transitions for which converged results have not been obtained.

All MCD parameters are presented in au. To convert A and C terms to
the alternative unit D^2 (Debye squared) the value in au should be
multiplied by 6.46044. To convert the B term to the alternative unit
of D^2 /cm-1 the value in au should be multiplied by 2.94359e-05.

The A, B and C terms are defined through the equation suggested by
Stephens (equation 1 in [41] and also see refs. [38] [39] [40]). This
equation assumes that MCD intensity varies linearly with applied
magnetic field and that the temperature-dependent component varies
linearly with temperature as 1/T. For the most part, these assumptions
are reasonable. An exception is that the temperature-dependent part
varies from linearity when T is very small. To allow for this
situation a temperature and magnetic field dependent multiplicative
constant (chi(B,T)) is evaluated whenever temperature-dependent MCD
parameters are considered. This constant includes all magnetic field
and temperature dependence of the temperature-dependent MCD. Thus
chi(B,T)*C can be used in place of B*C/kT in equation 1 of [41] when
MCD spectra are to be simulated. Note that, since the g-factor for all
states is here approximated by 2.0, chi applies to all transitions.


Analysis
========

By default, the ADF output will show the excitation energies,
oscillator strengths from ground state to excited state, and
transition dipole moments from ground state to excited state. Also, an
analysis in terms of single orbital transitions (occupied to virtual)
is given, and their contributions to the transition dipole moment. For
analysis reasons, one might be interested in a pure single orbital
transition. Such information can be obtained in case one calculates
excitation energies as Kohn-Sham orbital energy differences, see
subkey SINGLEORBTRANS of the key EXCITATIONS.


NTO: Natural Transition Orbitals
--------------------------------

Natural Transition Orbitals [42] come from a transformation of the
transition density matrix. They are the closest you can get to
visualizing an excitation as a one-electron excitation from 1 orbital
to another. They provide insight into the localization of excitations.
Calculation of Natural Transition Orbitals (NTOs) will be performed if
the subkey NTO is included in the key EXCITATIONS.

   EXCITATIONS
     ...
     NTO
   End

For (meta-)hybrids and range-separated functionals NTOs will be
calculated only if one uses the Tamm-Dancoff approximation (TDA).

NTOs can be visualized in AMSspectra as shown in the GUI tutorial .


SFO analysis
------------

In ADF2018 an SFO analysis of the excitation is implemented. This
includes an analysis of the transition dipole moment. Especially
useful in a fragment calculation, where one only has a few fragments.
Not implemented for spin-orbit coupled excitations.

   EXCITATIONS
     ...
     SFOANALYSIS
     N_SFO NMAXPRINTED
   End

"SFOANALYSIS"
   If SFOANALYSIS is included, the SFO analysis of the excitation will
   be calculated.

"N_SFO NMAXPRINTED"
   NMAXPRINTED is the maximum number of printed contributions. Default
   value for NMAXPRINTED is 40.

The SFO - SFO dipole matrix elements can be printed in the output
with:

   PRINT DIPOLEMAT


Charge-transfer descriptors
---------------------------

Two methods for calculating charge-transfer descriptors are
implemented (also working for spin-orbit coupled excitations).

The first method is the charge-transfer diagnostic overlap quantity
LAMBDA, developed by Peach, Tozer, et al. [43] and a hole-electron
distance R_HE, see Guido, Adamo, et al. [44].

The second method is the charge transfer descriptor CT, hole-electron
distance R_HE, and some other descriptors, developed by Plasser,
Lischka, et al. [54] [55] [56] [57]. For this method, a fragment
calculation is required. Using a similar method, a charge transfer
descriptor CT_AT was defined that uses an atomic distance criterion,
which will also be calculated in case of atomic fragments.

   EXCITATIONS
     ...
     DESCRIPTORS
     {Descriptors_CT_AT_Rab Rab}
   END

"DESCRIPTORS"
   If DESCRIPTORS is included, the charge transfer descriptors are
   calculated.

"Descriptors_CT_AT_Rab Rab"
   Rab is the atomic distance criterion used for the calculation of
   CT_AT (in Angstrom). Default value for Rab is 2 Angstrom.


ROSE or FOCDFT localized orbitals
---------------------------------

The localized orbitals that ROSE calculates or FOCDFT calculates can
be used in time-dependent DFT calculations using ADF, see section on
exciton transfer integrals with ROSE or FOCDFT. These methods are
especially useful if one wants to analyze the local excitation (LE) or
charge transfer (CT) character of an excitation, and allow one to
study diabatic states that are pure local excitations or pure charge
transfer excitations and calculate possible electronic couplings
between them.

[1] S. Hirata and M. Head-Gordon, *Time-dependent density functional
    theory within the Tamm-Dancoff approximation*, Chemical Physics
    Letters 314, 291 (1999)

[2] S. Bernadotte, F. Evers, and C.R. Jacob, *Plasmons in Molecules*,
    Journal of Physical Chemistry C 117, 1863 (2013)

[3] S.J.A. van Gisbergen, J.G. Snijders and E.J. Baerends,
    *Implementation of time-dependent density functional response
    equations*, Computer Physics Communications 118, 119 (1999)

[4] S.J.A. van Gisbergen, C. Fonseca Guerra and E.J. Baerends,
    *Towards excitation energies and (hyper)polarizability
    calculations of large molecules. Application of parallelization
    and linear scaling techniques to time-dependent density functional
    response theory*, Journal of Computational Chemistry 21, 1511
    (2000)

[5] F. Wang and T. Ziegler, *Excitation energies of some d1 systems
    calculated using time-dependent density functional theory: an
    implementation of open-shell TDDFT theory for doublet-doublet
    excitations*, Molecular Physics 102, 2585 (2004)

[6] M. Krykunov and T. Ziegler, *Self-consistent Formulation of
    Constricted Variational Density Functional Theory with Orbital
    Relaxation. Implementation and Applications*, Journal of Chemical
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