Exciton transfer integrals
**************************

ADF can provide input parameters, such as exciton transfer integrals,
that are needed in approximate methods that model exciton transport
properties, like Förster and Marcus theory.

   [image]

Example electron configuration for exciton hopping from A to B or
back, where A and B are nearby molecules, or different parts of a
complex AB. The hopping can occur from the diabatic state where A is
in an excited state and B is in the ground state to the diabatic state
where A is in the ground state and B is in an excited state (or back).

   {\rm A}^{*} + {\rm B} \longleftrightarrow {\rm A} + {\rm B}^{*}

In theoretical models of exciton hopping, the whole system is divided
into fragments, in which an exciton is localized on a fragment
(diabatic state), and can hop from one fragment to another. The models
require accurate values of excitonic couplings for exciton transfer
(also referred to as exciton transfer integrals) and exciton site
energies (energy of an exciton when it is localized at a particular
molecule) as a function of the geometric conformation of adjacent
molecules. Exciton transfer integrals can be calculated from the
energetic splitting of excitation energies in a system consisting of
two adjacent molecules, also called "Davydov splitting". ADF can also
calculate exciton transfer integrals by the use of its unique fragment
approach. ADF allows one to use molecular orbitals on individual
molecules as a basis set in calculations on a system composed of two
or more molecules. Also, ADF's methods are applicable in cases where
an orbital on one molecule couples with two or more orbitals on
another molecule.


Exciton transfer integrals with FDE
===================================

The linear-response subsystem TDDFT code couples the monomer
excitations to obtain the excited states of the total system (often
denoted as coupled frozen density embedding, FDEc). It can be used,
for example, to calculate singlet energy transfer integrals. For a
complete description, see the FDE part of the ADF manual, and in
particular the part about subsystem TDDFT.


Exciton transfer integrals with ROSE or FOCDFT
==============================================

The methods used for calculating exciton transfer integrals with ROSE
and FOCDFT are very similar. However, the ROSE localized orbitals and
FOCDFT localized orbitals may be quite different. Note that the
methods used here can also be used solely for analysis of the
character of the excited states, especially if Excitations%descriptors
is included.


ROSE localized orbitals
-----------------------

The ROSE localized orbitals can be used to calculate exciton transfer
integrals. ROSE is a method that calculates a set of re-canonicalized
intrinsic localized molecular orbitals (RILMOs), see also Ref. [1].
While almost fully localized on individual fragments, RILMOs are
delocalized within a fragment and can be interpreted as perturbed
fragment orbitals. The ROSE localized occupied orbitals and the
localized valence virtual orbitals are not very basis set dependent.
The rest of the ROSE virtuals, the so-called hard virtuals, are
(typically) more basis-set dependent. The occupied RILMOs span the
occupied space of the supermolecular calculation exactly. As ROSE
truncates the virtual space, one may need to vary the value of the
Additional_virtuals_cutoff key to check convergence of results that
depend on electronically excited states.

In combination with the EXCITATIONS and EXCITONTRANSFER key one can
use the localized orbitals (RILMOs) that ROSE makes to calculate
localized excitations on fragments (quasi-diabatic states) and
calculate electronic couplings between them. The Tamm–Dancoff
approximation (TDA) is required. Symmetry cannot be used. All-electron
basis sets should be used.


FOCDFT localized orbitals
-------------------------

FOCDFT can be used to calculate exciton transfer integrals. FOCDFT is
a fragment orbital based method for constraining (integer) charges and
spins on fragments. It uses the (localized) Lowdins that ADF makes to
constrain charges and spins on fragments. In practice this method
works best if there are no covalent bonds or if one does not want
covalent bonds between the fragments. The FOCDFT localized occupied
orbitals are not very basis-set dependent. The FOCDFT localized
virtual orbitals, on the other hand, are much more basis-set
dependent. The occupied FOCDFT orbitals span a different space than
the occupied orbitals coming from a supermolecular calculation,
because of the constraints used in FOCDFT. If one has a neutral
complex and the constraints are that there should be zero charge and
spin on all the fragments, this difference may not be so large.
However, FOCDFT allows one also to constrain a charge on one of the
fragments, whereas a calculation on the whole complex with the same
charge would most likely give a much more delocalized charge. FOCDFT
does not truncate the virtual space.

In combination with the EXCITATIONS and EXCITONTRANSFER key one can
use the localized orbitals that FOCDFT makes to calculate localized
excitations on fragments (diabatic states) and calculate electronic
couplings between them. The Tamm–Dancoff approximation (TDA) is
required. Symmetry can be used. Frozen core basis sets are allowed.


Exciton transfer integrals
--------------------------

Because charge-transfer (CT) excitations are calculated in ROSE or
FOCDFT TD-DFT excitation calculations, it is best to use an XC
functional that can calculate CT excitations reasonably accurately,
such as range-separated hybrids.

One can study, for example, singlet energy transfer (SET) or triplet
energy transfer (TET) with ROSE or FOCDFT.

   {\rm A}(S_n) + {\rm B}(S_0) \longleftrightarrow {\rm A}(S_0) + {\rm
   B}(S_m)  \\ {\rm A}(T_n) + {\rm B}(S_0) \longleftrightarrow {\rm
   A}(S_0) + {\rm B}(T_m)

including couplings to charge separated states, since CT states are
calculated in this method.

One possibility using the ExcitonTransfer key is to make purely local
excitations (LEs) and purely charge-transfer (CT) excitations, if one
sets the subkey ExcitonTransfer%Localize to OccupiedAndVirtual. The
calculated LEs are then linear combinations of single-orbital
transitions from occupied orbitals on one fragment to virtual orbitals
on the same fragment. The calculated CT excitations are linear
combinations of single-orbital transitions from occupied orbitals on
one fragment to virtual orbitals on a different fragment. This is how
excitations were analyzed in Ref. [1] using ROSE localized orbitals.
The ROSE localized occupied orbitals and localized 'hard virtual'
orbitals are not very basis-set dependent. In many cases excitations
from occupied to the 'hard virtual' orbitals are the more interesting
ones.

One can calculate an effective excitonic coupling between local
diabatic states that include intermediate CT states by means of
second-order perturbation corrections. This is only possible if the
Localize keyword is set to OccupiedAndVirtual, the key SecondOrder is
included, and the key Output is set to FilteredCouplings or
AllAndFilteredCouplings. In case of a direct coupling between two
local diabatic states, a second-order correction (involving a sum over
diabatic CT states) is then added.

Part of the Engine ADF input in case of ROSE could look like:

   Basis
     ...
     Core None
   End
   ROSE
     FragmentFile Frag1.results/adf.rkf
     FragmentFile Frag2.results/adf.rkf
   End
   TDA
   Excitations
     descriptors
   End
   ExcitonTransfer
     Localize    OccupiedAndVirtual
     SecondOrder True
     Output      AllAndFilteredCouplings
   End

This method, which in many cases is adequate for ROSE, is not the
recommended way if one uses FOCDFT localized orbitals, since all
FOCDFT localized virtual orbitals are rather basis set dependent. In
order to make LEs and CT excitations that are less basis-set
dependent, a method has been implemented that combines the LEs from
one fragment and CT excitations coming from the same fragment. This
method will be used if one sets the subkey ExcitonTransfer%Localize to
OccupiedOnly. The calculated excitations are, in this case, linear
combinations of single-orbital transitions from occupied orbitals on
one fragment to virtual orbitals on all fragments. Of course, this
method can also be used for ROSE localized orbitals. One needs to use
the Excitations%descriptors key to see whether an excitation is mostly
an LE or a CT excitation.

Part of the Engine ADF part of the input in case of FOCDFT could look
like:

   Fragments
     Frag1 Frag1.results/adf.rkf
     Frag2 Frag2.results/adf.rkf
   End
   FOCDFT
   End
   TDA
   Excitations
     descriptors
   End
   ExcitonTransfer
     Localize OccupiedOnly
   End

   [image]

Example electron configuration for charge or exciton hopping from
molecule A to molecule B or back. The hopping can occur from the
diabatic state where A is in an excited state and B is in the ground
state to the diabatic state where A is in the ground state and B is in
an excited state or to a diabatic state where both A and B are charged
(CT state).

   {\rm A}(S_n) + {\rm B}(S_0) \longleftrightarrow {\rm A}^{+} + {\rm
   B}^{-} (CT_k) \longleftrightarrow {\rm A}^{-} + {\rm B}^{+} (CT_l)
   \longleftrightarrow {\rm A}(S_0) + {\rm B}(S_m) \\ {\rm A}(T_n) +
   {\rm B}(S_0) \longleftrightarrow {\rm A}^{+} + {\rm B}^{-} (CT_k)
   \longleftrightarrow {\rm A}^{-} + {\rm B}^{+} (CT_l)
   \longleftrightarrow {\rm A}(S_0) + {\rm B}(T_m)

With ROSE or FOCDFT, oscillator strengths can be calculated for the
absorption of light to localized diabatic states. Thus, with ROSE or
FOCDFT it is possible to investigate, for example, the absorption
process where A is locally excited, which is followed by a charge-
separation process (like in photosynthesis):

   {\rm A} + {\rm B} + h \nu \longrightarrow {\rm A}^{*} + {\rm B}
   \longrightarrow {\rm A}^{+} + {\rm B}^{-}

One could also study a reverse process (like in OLED devices):

   {\rm A}^{+} + {\rm B}^{-} \longrightarrow  {\rm A}^{*} + {\rm B}
   \longrightarrow {\rm A} + {\rm B} + h \nu

In order to calculate couplings from LEs from one fragment and CT
excitations coming from the same fragment, it is best to set the
subkey ExcitonTransfer%Localize to OccupiedAndVirtual.


ExcitonTransfer key
-------------------

   ExcitonTransfer
      CoupleFinalStates Yes/No
      FilteredCouplings
         MaxEnergy float
         MaxEnergyDiff float
         MinCoupling float
         MinEnergy float
      End
      FullRun Yes/No
      LocalCouplingsOnly Yes/No
      Localize [OccupiedOnly | OccupiedAndVirtual]
      Lowest integer_list
      LowestCT integer_list
      Output [AllCouplings | FilteredCouplings | AllAndFilteredCouplings]
      SecondOrder Yes/No
   End

"ExcitonTransfer"
   Type:
      Block

   Description:
      Block key for exciton transfer integrals with ROSE or FOCDFT.

   "CoupleFinalStates"
      Type:
         Bool

      Default value:
         No

      Description:
         Calculate delocalized excited states in the basis of the
         localized excited states that are calculated, thus couple the
         localized excited states.

   "FilteredCouplings"
      Type:
         Block

      Description:
         Details on filter used for electronic couplings in the output

      "MaxEnergy"
         Type:
            Float

         Default value:
            10.0

         Unit:
            eV

         Description:
            Max. energy (in eV) of diabatic states

      "MaxEnergyDiff"
         Type:
            Float

         Default value:
            0.5

         Unit:
            eV

         Description:
            Max. energy difference (in eV) between diabatic states

      "MinCoupling"
         Type:
            Float

         Default value:
            0.1

         Description:
            Min. coupling value (in meV) that is printed

      "MinEnergy"
         Type:
            Float

         Default value:
            0.0

         Unit:
            eV

         Description:
            Min. energy (in eV) of diabatic states

   "FullRun"
      Type:
         Bool

      Default value:
         No

      Description:
         Include run without restriction of localization of occupied
         and/or virtual orbitals.

   "LocalCouplingsOnly"
      Type:
         Bool

      Default value:
         No

      Description:
         Only account for couplings between local diabatic states

   "Localize"
      Type:
         Multiple Choice

      Default value:
         OccupiedOnly

      Options:
         [OccupiedOnly, OccupiedAndVirtual]

      Description:
         Localize OccupiedAndVirtual means that separately purely
         localized excitations and purely charge-transfer excitations
         are calculated. Localize OccupiedOnly means that an
         excitation may have local and charge-transfer character, but
         the excitation only has contributions from occupied orbitals
         on one fragment. Only relevant in case block key FOCDFT is
         used or ROSE orbitals are used.

   "Lowest"
      Type:
         Integer List

      GUI name:
         Number of excitations

      Description:
         Number of lowest excitations to compute for each fragment.
         These are purely local excitations if the Localize
         OccupiedAndVirtual key is specified.

   "LowestCT"
      Type:
         Integer List

      GUI name:
         Number of charge-transfer excitations

      Description:
         Number of lowest charge-transfer excitations to compute for
         each fragment. Only relevant if the Localize
         OccupiedAndVirtual key is also specified.

   "Output"
      Type:
         Multiple Choice

      Default value:
         AllCouplings

      Options:
         [AllCouplings, FilteredCouplings, AllAndFilteredCouplings]

      Description:
         Amount of output

   "SecondOrder"
      Type:
         Bool

      Default value:
         No

      Description:
         Include 2nd-order correction to electronic couplings

[1] S. Sen, B. Senjean, L. Visscher, *Characterization of excited
    states in time-dependent density functional theory using localized
    molecular orbitals*, Journal of Chemical Physics 158, 054115
    (2023)
