Charge transfer integrals (transport properties)
************************************************

ADF can provide input parameters, such as charge transfer integrals,
that are needed in approximate methods that model charge transport
properties. ADF also supports direct calculation of these transfer
integrals through its unique fragment approach.

   [image]

Example electron configuration for charge hopping between A and B,
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 charged
and B is in the ground state to the diabatic state where A is in the
ground state and B is charged (or back).

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

In theoretical models of charge transport in organic materials, see
Refs. [6] [7] [8], the whole system is divided into fragments, in
which an electron or hole is localized on one fragment and can hop to
another. In the tight-binding approximation used in these models, the
electron or hole is represented by a single orbital, and it is assumed
that only the nearest neighboring fragments can couple. The models
require accurate values of electronic couplings for charge transfer
and site energies as a function of the geometric conformation of
adjacent molecules. Electronic couplings for charge transfer are also
referred to as charge transfer integrals or hopping matrix elements. A
site energy is the energy associated with a charge localized on a
particular molecule. Charge transfer integrals for hole transport can
be calculated from the energetic splitting of the two highest-occupied
molecular orbitals (HOMO and HOMO-1) in a system consisting of two
adjacent molecules, also called the Energy Splitting In Dimer (ESID)
method. For electron transport, these can be calculated from the two
lowest-unoccupied orbitals (LUMO and LUMO+1) in this ESID method. ADF
can also calculate transfer integrals directly using its fragment
approach; see Refs. [7] [8]. 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. The charge transfer integrals
obtained in this way differ significantly from values estimated from
the energy splitting between the highest-occupied molecular orbitals
in a dimer. The difference is due to the nonzero spatial overlap
between the molecular orbitals on adjacent molecules. ADF's methods
can also treat cases where an orbital on one molecule couples with two
or more orbitals on another molecule.


Charge transfer integrals with the TRANSFERINTEGRALS key
========================================================

In this method, the matrix elements of the molecular Kohn-Sham
Hamiltonian H^{KS} in the basis of fragment orbitals are used to
calculate site energies and charge transfer integrals. Likewise, the
overlap integrals between fragment orbitals are calculated. No
explicit electrons are removed or added in this method. For electron
mobility calculations, the fragment LUMOs are considered. For hole
mobility calculations, the fragment HOMOs are considered.

To calculate the charge transfer integrals, spatial overlap integrals,
and site energies, include the key TRANSFERINTEGRALS in the input for
ADF. Symmetry NOSYM should be used. The molecular system typically
should be built from two fragments. In the fragment calculation, full
symmetry can be used.

   TRANSFERINTEGRALS
   Symmetry NOSYM
   Fragments
    frag1 frag1.results/adf.rkf
    frag2 frag2.results/adf.rkf
   End

By default, integrals are calculated only for the HOMO (LUMO) of the
fragments, and possibly HOMO-1, HOMO-2 (LUMO+1, LUMO+2) if the energy
levels of those fragment orbitals are close to the HOMO (LUMO) of that
fragment. To calculate the matrix elements and overlap integrals based
on all fragment orbitals, one can use the key:

   PRINT FMATSFO

The method described here for calculating charge transfer integrals is
more approximate than the FDE-based method described next. The main
difference is in how effects of a localized charge are treated.

If two fragments are used, the electronic coupling V, also known as
the effective (generalized) transfer integral J_eff, for hole transfer
or electron transfer is calculated as V = (J-S(e1+e2)/2)/(1-S^2). Here
e1 and e2 are the site energies of fragment 1 and 2, respectively. J
is the charge transfer integral, and S the overlap integral.

   e_1 & = \langle \phi_1 | H^{KS} | \phi_1 \rangle \\ e_2 & = \langle
   \phi_2 | H^{KS} | \phi_2 \rangle \\ J   & = \langle \phi_1 | H^{KS}
   | \phi_2 \rangle \\ S   & = \langle \phi_1 | \phi_2 \rangle \\ V
   & = \frac{J-S(e1+e2)/2}{1-S^2}

In electron mobility calculations, \phi_1 is the LUMO of fragment 1
and \phi_2 is the LUMO of fragment 2. In hole mobility calculations,
\phi_1 is the HOMO of fragment 1 and \phi_2 is the HOMO of fragment 2.
The electronic coupling between donor HOMO and acceptor LUMO, and vice
versa, are also calculated, which represents the probability of a
charge recombination process.

If there is (near) degeneracy in the fragment HOMO and/or LUMO,
multiple electronic couplings V_i are printed. A total electronic
coupling is calculated as

   V_{tot} = \sqrt{\sum_{deg} {V_i}^2}

   TIDegeneracyThreshold float

"TIDegeneracyThreshold"
   Type:
      Float

   Default value:
      0.1

   Unit:
      eV

   Description:
      If the orbital energy of the fragment MO is within this
      threshold with fragment HOMO or LUMO energy, then this fragment
      MO is included in the calculation of the transfer integrals.
      Relevant in case there is (near) degeneracy.


Charge transfer integrals with FDE
==================================

**Overview**

The ELECTRONTRANSFER key block invokes the calculation of Hamiltonian
(site energies and couplings) and overlap matrix elements with FDE-
derived localized states. Two FDE calculations are (not strictly)
needed before running the ELECTRONTRANSFER calculation. The calculated
matrix elements are theoretically similar to the ones obtained with
the TRANSFERINTEGRALS keyword. A review casts ELECTRONTRANSFER in the
state-of-the-art of methods for computing charge transfer couplings,
and provides a step-by-step guide for computing such couplings with
ADF [1].

**Features**

* Effects of orbital relaxation due to localized charges, Refs. [9]
  [2].

* Effects of polarization due to molecules in the environment, Ref.
  [2].

* ELECTRONTRANSFER has linear scaling with the number of fragments
  when the system is composed of more than one fragment.

* The code can tackle hole transfer, electron transfer, charge
  separation and charge recombination processes [3].

* Can compute transfer integrals from diabats obtained with the
  Constrained DFT method (i.e., invoked by using the experimental CDFT
  keyword of ADF). This includes the calculation of couplings between
  diabats made of a single fragment.

* It is possible to include the effect of the environment on CDFT
  diabats by coupling it with an FDE calculation [4].

* The performance of ElectronTransfer in the evaluation of the hole
  transfer coupling was benchmarked against wave function methods,
  with an error below 7%. PBE, PW91, B3LYP and PBE0 functionals, with
  PW91k for the non-additive component of the kinetic energy and a TZP
  basis set, are recommended to obtain the FDE-derived localized
  states [5].

* The code was parallelized in ADF2018.

**Limitations**

* Hybrid functionals are not yet supported

* adf.rkf (TAPE21) files with charge or spin-localized states with
  specific names are needed

These two limitations do not apply to the method with the
TRANSFERINTEGRALS key, but the TRANSFERINTEGRALS method has many other
limitations. Like the method with the TRANSFERINTEGRALS key, fragments
are needed.

**ELECTRONTRANSFER input**

The minimum input for the ELECTRONTRANSFER key is:

   FRAGMENTS
    frag1 FragFile1
    ...
    fragN FragFileN
   END
   ELECTRONTRANSFER
    NumFrag N
   END

where frag1 ... fragN are the labels of the fragments in the
calculation, and FragFile1 ... FragFileN are the adf.rkf (TAPE21)
files of spin-restricted calculations of the isolated fragments. N is
the total number of fragments employed in the calculation.

**Files and file names**

The fragment files to be used in the ELECTRONTRANSFER calculation are
generally different from the adf.rkf (TAPE21) files used in the
FRAGMENTS key block. Two types of fragment adf.rkf (TAPE21) files are
needed by the calculation:

* The isolated closed-shell TAPE files for the FRAGMENTS key block

* The adf.rkf (TAPE21) files of the charge- or spin-localized states
  (which can be obtained with an FDE calculation as done in the
  example below)

There are two charge localized states. They are labeled with A and B.
The respective adf.rkf (TAPE21) files must be named as follows:

   fragA1.t21, fragA2.t21, ... , fragAN.t21 (for state A)
   fragB1.t21, fragB2.t21, ... , fragBN.t21 (for state B)

The above files should be copied to the working folder of the ADF
calculation prior to executing ADF.

**Options**

   ELECTRONTRANSFER
    NumFrag N
    {Joint|Disjoint}
    {Debug}
    {Print EIGS|SAB}
    {FDE}
    {INVTHR threshold}
    {CDFT}
    {KNADD}
   END

"Joint|Disjoint"
   The default is Joint. Joint is always recommended. The Disjoint
   formalism is described in Ref. [2] and is much faster than the
   Joint formalism when more than 2 fragments are considered. Joint
   and Disjoint are equivalent for systems composed of only 2
   fragments. Disjoint should only be used if the fragment files are
   obtained in an FDE calculation (see FDE below).

"Debug"
   The code performs additional checks (determinants,
   diagonalizations, inversions, traces, etc.). A substantial increase
   in the output should be expected.

"Print"
   If EIGS, it will print the (unformatted) matrix of the MO
   coefficients in the AO representation. If SAB, it will print the
   (unformatted) diagonal and full transition overlap matrix in the MO
   representation.

"FDE"
   An FDE calculation including more than two fragments must include
   the following key block:

      ELECTRONTRANSFER
        FDE
      END

   and the numerical integration precision in the *last* FDE
   calculation for every subsystem should be set to no less than:

      BeckeGrid
         Quality Good
      End

   if in the subsequent ELECTRONTRANSFER calculation the Disjoint
   subkey is used.

"Invthr threshold"
   The default is 1.0e-3. It is a threshold for the Penrose inversion
   of the transition overlap matrix. If warnings about density fitting
   are printed, invthr may be increased up to 1.0e-2. Larger invthr
   might affect the quality of the calculated couplings and excitation
   energies.

"CDFT"
   If Disjoint is selected, this keyword must be selected if the
   evaluation of the electronic coupling is sought between diabats
   located on the same CDFT fragment. In this case, the CDFT fragment
   always has to be first under the ATOMS keyword.

"KNADD"
   If Disjoint is selected, this allows the kinetic energy of each
   fragment to be obtained locally without using the full grid. This
   keyword is recommended when there are many tens of subsystems, such
   as systems with several solvent molecules.

**Output**

The output of the example in
$AMSHOME/examples/adf/ElectronTransfer_FDE_H2O is discussed here. This
example involves the calculation of electronic coupling, site energies
and charge-transfer excitation energy for the hole transfer in a water
dimer.

   ============  Electron Transfer RESULTS ===================

   Electronic Coupling =         0.000000 eV
   Electronic Coupling =        -0.003569 cm-1
   H11-H22             =        -1.396546 eV
   Excitation Energy   =         1.396546 eV
   Overlap             =         0.000000
   H11 H22 H12 =  -152.443000816341  -152.391678701092  -151.743979368040 Eh
   S11 S22 S12 =     0.981795415192     0.981006454450    -0.000000023700


   =========== END Electron Transfer RESULTS ================

Due to symmetry, the overlap is almost diagonal (Overlap = 0.00), thus
the transition density is evaluated with one less electron as
explained in Ref. [2].

The electronic coupling between the state with a positive charge
localized on one water molecule and another with the charge localized
on the other water molecule is given by "Electronic Coupling" and is
reported in eV and cm^-1.

"H11-H22" is the difference of the site energies in eV. Values of the
site energies are given by the first two values of "H11 H22 H12" in
atomic units.

"Excitation Energy" reports the value of the transfer excitation
energy as calculated by diagonalization of the 2x2 generalized
eigenvalue problem in the basis of the charge-localized states, see
Refs. [2] [9].

"S11 S22 S12" are the values of the non-normalized overlaps.


Charge transfer integrals with FOCDFT
=====================================

FOCDFT can be used to calculate charge transfer integrals. FOCDFT is a
fragment-orbital-based method for constraining (integer) charges and
spins on fragments. It uses the (localized) Löwdin orbitals
constructed by ADF to constrain charges and spins on fragments. This
method can be used to specify two different electron configurations of
the same complex, such that a charge on the first fragment in the
first calculation is transferred to the second fragment in the second
calculation.

The two electron configurations calculated with FOCDFT will lead to
two one-determinantal wave functions with occupied orbitals, which are
not necessarily mutually orthonormal. Löwdin's pairing theorem (see
for example Ref. [10]) is used to make a set of occupied orbitals that
are orthonormalized and 'paired', and which do not change the
resulting two one-determinantal wave functions. Because FOCDFT uses
the (localized) Löwdin orbitals constructed by ADF, and charge is
transferred from one fragment to the other, the two one-determinantal
wave functions are orthonormal and there is at least one pair of
'corresponding orbitals' that have zero overlap. These two
'corresponding orbitals' a_i (of the first calculation) and b_i (of
the second calculation) are used to calculate an approximation to the
electronic coupling by calculating

   <a_i| 0.5 F^{A} + 0.5 F^B| b_i>

where F^{A} is the Fock matrix in the first calculation and F^B is the
off-diagonal Fock matrix in the second calculation. The off-diagonal
Fock matrix is the off-diagonal part of the Fock matrix between Löwdin
orbitals on the first and second fragments, before it is set to zero
in FOCDFT. The overlap of the other paired orbitals is taken into
account in the final electronic coupling.

In FOCDFT electron transfer, the effects of orbital relaxation due to
localized charges are included, and the effects of polarization due to
molecules in the environment can be included. FOCDFT electron-transfer
calculations can treat both hole and electron transfer. Hybrid
functionals are supported. FOCDFT electron transfer is not benchmarked
against wave function methods.

For the calculation of electronic couplings between locally excited
states and charge separated states see the section on exciton transfer
integrals with FOCDFT.

With the proper input, ADF will perform the two FOCDFT calculations on
the two diabatic states on the same complex consecutively and print
the site energies and electronic coupling in the output. For hole
charge transfer in the ADF part of the input, one should use input
such as:

   fragments
      Frag1 Frag1.results/adf.rkf
      Frag2 Frag2.results/adf.rkf
   end
   spinpolarization 1
   unrestricted
   FOCDFT
     charge 1 0
     spinpolarization 1 0
     electrontransfer
       charge 0 1
       spinpolarization 0 1
     end
   end

The FOCDFT and FOCDFT%ElectronTransfer keys in more detail:

   FOCDFT
      Charge float_list
      SpinPolarization float_list
      ElectronTransfer
         Charge float_list
         OffDiagonalHF Yes/No
         OffDiagonalHFValue float
         SpinPolarization float_list
      End
   End

"FOCDFT"
   Type:
      Block

   Description:
      Localize orbitals on fragments and constrain charge and spin
      polarization on fragments.

   "Charge"
      Type:
         Float List

      Description:
         Constrain charge on fragments. Number of charges should be
         the same as number of fragments, and sum should add up to
         total charge.

   "SpinPolarization"
      Type:
         Float List

      Description:
         Constrain spin polarization on fragments. Number of spin
         polarization values should be the same as number of
         fragments, and sum should add up to total spin polarization.
         Calculation should be spin unrestricted.

   "ElectronTransfer"
      Type:
         Block

      Description:
      "Charge"
         Type:
            Float List

         Description:
            Constrain charge on fragments for the second calculation.
            Number of charges should be the same as number of
            fragments, and sum should add up to total charge.

      "OffDiagonalHF"
         Type:
            Bool

         Default value:
            No

         GUI name:
            Off diagonal HF

         Description:
            Off diagonal HF matrix elements.

      "OffDiagonalHFValue"
         Type:
            Float

         Default value:
            1.0

         GUI name:
            Off diagonal HF value

         Description:
            Amount of HF in case off diagonal HF matrix elements.

      "SpinPolarization"
         Type:
            Float List

         Description:
            Constrain spin polarization on fragments for the second
            calculation. Number of spin polarization values should be
            the same as number of fragments, and sum should add up to
            total spin polarization. Calculation should be spin
            unrestricted.

[1] P. Ramos, M. Mankarious, M. Pavanello, *A critical look at methods
    for calculating charge transfer couplings fast and accurately*, in
    Practical Aspects of Computational Chemistry IV, Jerzy Leszczynski
    and Manoj Shukla (eds.), 2016, Springer

[2] M. Pavanello, T. Van Voorhis, L. Visscher, and J. Neugebauer, *An
    accurate and linear-scaling method for calculating charge-transfer
    excitation energies and diabatic couplings*, Journal of Chemical
    Physics 138, 054101 (2013)

[3] A. Solovyeva, M. Pavanello, J. Neugebauer, *Describing Long-Range
    Charge-Separation Processes with Subsystem Density-Functional
    Theory*, J. Chem. Phys. 140, 164103 (2014)

[4] P. Ramos, M. Pavanello, *Constrained Subsystem Density Functional
    Theory*, Physical Chemistry Chemical Physics 18, 21172 (2016).

[5] P. Ramos, M. Papadakis, M. Pavanello, *Performance of Frozen
    Density Embedding for Modeling Hole Transfer Reactions*, Journal
    of Physical Chemistry B 119, 7541 (2015)

[6] M.D. Newton, *Quantum chemical probes of electron-transfer
    kinetics: the nature of donor-acceptor interactions*, Chemical
    Reviews 91, 767 (1991).

[7] K. Senthilkumar, F.C. Grozema, F.M. Bickelhaupt, and L.D.A.
    Siebbeles, *Charge transport in columnar stacked triphenylenes:
    Effects of conformational fluctuations on charge transfer
    integrals and site energies*, Journal of Chemical Physics 119,
    9809 (2003).

[8] K. Senthilkumar, F.C. Grozema, C. Fonseca Guerra, F.M.
    Bickelhaupt, F.D. Lewis, Y.A. Berlin, M.A. Ratner, and L.D.A.
    Siebbeles, *Absolute Rates of Hole Transfer in DNA*, Journal of
    the American Chemical Society 127, 14894 (2005)

[9] M. Pavanello and J. Neugebauer, *Modelling charge transfer
    reactions with the frozen density embedding formalism*, Journal of
    Chemical Physics 135, 234103 (2011)

[10] I. Mayer, *Löwdin's pairing theorem and some of its
     applications*, Molecular Physics, 108, 3273 (2010)
