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

DFTB can provide input parameters, such as charge transfer integrals,
that are needed in approximate methods that model charge transport
properties in the hopping regime. Note that DFTB is an approximate
method. You might use ADF to calculate more accurate charge transfer
integrals, or use specifically optimized DFTB parameters.

See also:

  * ADF tutorial on charge transfer integrals

  * Example: Charge transfer integrals Alq3 dimer

In theoretical models of charge transport in organic materials, see
Refs. [1] [2] [3], the whole system is divided into fragments, in
which an electron or hole is localized on a fragment, and can hop from
one fragment to another. In the tight-binding approximation that is
used in these models the electron or hole is approximated with 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 (also referred to as charge transfer
integrals or hopping matrix elements) and site energies (energy of a
charge when it is localized at a particular molecule) as a function of
the geometric conformation of adjacent molecules. 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
"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. DFTB can also calculate transfer
integrals based on the direct method by using a fragment approach. The
charge transfer integrals obtained in this way may 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. Also, the direct method is applicable in cases
where an orbital on one molecule couples with two or more orbitals on
another molecule.


Charge transfer integrals direct method
=======================================

In this method the matrix elements of the molecular DFTB Hamiltonian
H^{DFTB} in the basis of fragment orbitals is 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, the molecular system typically should be built from
two fragments. See also the section on Fragment orbital analysis. In
the "Engine DFTB" block, specify

   Properties
      Fragments
         File frag1.results/dftb.rkf
         File frag2.results/dftb.rkf
         TransferIntegrals
      End
   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
of those fragment orbitals are close to the HOMO (LUMO) of that
fragment.

If two fragments are used, the electronic coupling V (also known as
effective (generalized) transfer integrals 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^{DFTB} | \phi_1 \rangle \\ e_2 & =
   \langle \phi_2 | H^{DFTB} | \phi_2 \rangle \\ J   & = \langle
   \phi_1 | H^{DFTB} | \phi_2 \rangle \\ S   & = \langle \phi_1 |
   \phi_2 \rangle \\ V   & = \frac{J-S(e1+e2)/2}{1-S^2}

In case of electron mobility calculations \phi_1 is the LUMO of
fragment 1 and \phi_2 is the LUMO of fragment 2. In case of hole
mobility calculations \phi_1 is the HOMO of fragment 1 and \phi_2 is
the HOMO of fragment 2. The electronic coupling between the HOMO of
the donor fragment and the LUMO of the acceptor fragment and vice-
versa is 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}

   Properties
      Fragments
         Analysis Yes/No
         EMax float
         Emin float
         File string
         TIDegeneracyThreshold float
         TransferIntegrals Yes/No
      End
   End

"Properties"
   "Fragments"
      Type:
         Block

      Description:
         Fragment files

      "Analysis"
         Type:
            Bool

         Default value:
            Yes

         GUI name:
            Fragment analysis

         Description:
            Mulliken population analysis in terms of fragment
            orbitals.

      "EMax"
         Type:
            Float

         Default value:
            0.25

         Unit:
            Hartree

         Description:
            Upper end of the energy interval for which the orbitals
            are analyzed.

      "Emin"
         Type:
            Float

         Default value:
            -0.75

         Unit:
            Hartree

         Description:
            Lower end of the energy interval for which the orbitals
            are analyzed.

      "File"
         Type:
            String

         Recurring:
            True

         Description:
            Path (either absolute or relative) of fragment file

      "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.

      "TransferIntegrals"
         Type:
            Bool

         Default value:
            No

         GUI name:
            Charge transfer integrals

         Description:
            Calculate the charge transfer integrals, spatial overlap
            integrals and site energies.
            Charge transfer integrals can be used in models that
            calculate transport properties.

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

[2] 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).

[3] 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)
