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.

/scm-uploads/doc/ADF/_images/Exciton_Hopping.png

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. See for a complete description the FDE part of the ADF manual, and 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 can not 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 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 be not 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 case of ROSE or FOCDFT TD-DFT excitation calculations, best is to use an XC-functional that can calculate CT excitations reasonably accurate, like, for example, range separated hybrids.

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

\[\begin{split}{\rm A}(S_n) + {\rm B}(S_0) \longleftrightarrow {\rm A}(S_0) + {\rm B}(S_m) \\ {\rm A}(T_n) + {\rm B}(S_0) \longleftrightarrow {\rm A}(S_0) + {\rm B}(T_m)\end{split}\]

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

One of the possibilities using the key ExcitonTransfer 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 part of the 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 that 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 not so 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 a 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
/scm-uploads/doc/ADF/_images/Exciton_CT.png

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).

\[\begin{split}{\rm A}(S_n) + {\rm B}(S_0) \longleftrightarrow {\rm A}^{+} + {\rm B}^{-} (CT_k) \longleftrightarrow {\rm A}^{-} + {\rm B}^{+} (CT_l) \longleftrightarrow {\rm A}(S_0) + {\rm B}(S_m) \\ {\rm A}(T_n) + {\rm B}(S_0) \longleftrightarrow {\rm A}^{+} + {\rm B}^{-} (CT_k) \longleftrightarrow {\rm A}^{-} + {\rm B}^{+} (CT_l) \longleftrightarrow {\rm A}(S_0) + {\rm B}(T_m)\end{split}\]

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
   FilteredCouplings
      MaxEnergy float
      MaxEnergyDiff float
      MinCoupling float
      MinEnergy float
   End
   FullRun Yes/No
   LocalCouplingsOnly Yes/No
   Localize [OccupiedOnly | OccupiedAndVirtual]
   Output [AllCouplings | FilteredCouplings | AllAndFilteredCouplings]
   SecondOrder Yes/No
End
ExcitonTransfer
Type:

Block

Description:

Block key for exciton transfer integrals with ROSE or FOCDFT.

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.

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

References