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Electron Transfer

The NWChem electron transfer (ET) module calculates the electronic coupling energy (also called the electron transfer matrix element) between ET reactant and product states. The electronic coupling (VRP), and nuclear reorganization energy (λ) are all components of the electron transfer rate defined by Marcus’ theory, which also depends on the temperature (see Reference 1 below):

\[{k_{ET}}= \frac{2\pi}{\hbar} V_{RP}^{2} \frac{1}{\sqrt{4\pi \lambda k_{B}T}} \exp \left( \frac{- \Delta G^{*}}{k_{B} T} \right)\]

The ET module utilizes the method of Corresponding Orbital Transformation to calculate VRP. The only input required are the names of the files containing the open-shell (UHF) MO vectors for the ET reactant and product states (R and P).

The basis set used in the calculation of VRP must be the same as the basis set used to calculate the MO vectors of R and P. The magnitude of VRP depends on the amount of overlap between R and P, which is important to consider when choosing the basis set. Diffuse functions may be necessary to fill in the overlap, particularly when the ET distance is long.

The MO’s of R and P must correspond to localized states. for instance, in the reaction A- + B → A + B- the transferring electron is localized on A in the reactant state and is localized on B in the product state. To verify the localization of the electron in the calculation of the vectors, carefully examine the Mulliken population analysis. In order to determine which orbitals are involved in the electron transfer, use the print keyword “mulliken ao” which prints the Mulliken population of each basis function.

An effective core potential (ECP) basis can be used to replace core electrons. However, there is one caveat: the orbitals involved in electron transfer must not be replaced with ECP’s. Since the ET orbitals are valence orbitals, this is not usually a problem, but the user should use ECP’s with care.

Suggested references are listed below. The first two references gives a good description of Marcus’ two-state ET model, and the appendix of the third reference details the method used in the ET module.

  1. R.A. Marcus, N. Sutin, Biochimica Biophysica Acta 35, 437, (1985).
  2. J.R. Bolton, N. Mataga, and G. McLendon in “Electron Transfer in Inorganic, Organic and Biological Systems” (American Chemical Society, Washington, D.C., 1991)
  3. A. Farazdel, M. Dupuis, E. Clementi, and A. Aviram, J. Am. Chem. Soc., 112, 4206 (1990).

VECTORS: input of MO vectors for ET reactant and product states

 VECTORS [reactants] <string reactants_filename>  
 VECTORS [products ] <string products_filename>

In the VECTORS directive the user specifies the source of the molecular orbital vectors for the ET reactant and product states. This is required input, as no default filename will be set by the program. In fact, this is the only required input in the ET module, although there are other optional keywords described below.

FOCK/NOFOCK: method for calculating the two-electron contribution to VRP

  <string (FOCK||NOFOCK) default FOCK>

This directive enables/disables the use of the NWChem’s Fock matrix routine in the calculation of the two-electron portion of the ET Hamiltonian. Since the Fock matrix routine has been optimized for speed, accuracy and parallel performance, it is the most efficient choice.

Alternatively, the user can calculate the two-electron contribution to the ET Hamiltonian with another subroutine which may be more accurate for systems with a small number of basis functions, although it is slower.

TOL2E: integral screening threshold

 TOL2E <real tol2e default max(10e-12,min(10e-7, S(RP)*10e-7 )>

The variable tol2e is used in determining the integral screening threshold for the evaluation of the two-electron contribution to the Hamiltonian between the electron transfer reactant and product states. As a default, tol2e is set depending on the magnitude of the overlap between the ET reactant and product states (SRP), and is not less than 1.0d-12 or greater than 1.0d-7.

The input to specify the threshold explicitly within the ET directive is, for example:

 tol2e 1e-9


The following example is for a simple electron transfer reaction, He → He+. The ET calculation is easy to execute, but it is crucial that ET reactant and product wavefunctions reflect localized states. This can be accomplished using either a fragment guess, or a charged atomic density guess. For self-exchange ET reactions such as this one, you can use the REORDER keyword to move the electron from the first helium to the second.

Example input :

basis "ao basis" 
 * library aug-cc-pvtz

 He 0 0 0

charge 1

 tol2e 1d-9
 vectors output HeP.movecs
task scf

charge 0

 vectors output He.movecs
task scf

geometry noautosym noautoz
  He 0.0   0.0   0.0
  He 5.0   0.0   0.0

charge 1
#ET reactants:
  doublet; uhf; vectors input fragment HeP.movecs He.movecs output HeA.movecs
task scf

#ET products:
  doublet; uhf; vectors input HeA.movecs reorder 2 1 output HeB.movecs
task scf

 vectors reactants HeA.movecs
 vectors products HeB.movecs
task scf et

Here is what the output looks like for this example:

                           Electron Transfer Calculation

 MO vectors for reactants: HeA.movecs
 MO vectors for products : HeB.movecs

 Electronic energy of reactants     H(RR)      -5.2836825646
 Electronic energy of products      H(PP)      -5.2836825646

 Reactants/Products overlap S(RP) : -4.20D-04

 Reactants/Products interaction energy:    
 One-electron contribution         H1(RP)       0.0027017960

 Beginning calculation of 2e contribution
 Two-electron integral screening (tol2e) : 4.20D-11

 Two-electron contribution         H2(RP)      -0.0004625156
 Total interaction energy           H(RP)       0.0022392804

 Electron Transfer Coupling Energy |V(RP)|      0.0000220152
                                                       4.832 cm-1
                                                    0.000599 eV
                                                       0.014 kcal/mol

The overlap between the ET reactant and product states (SRP) is small, so the magnitude of the coupling between the states is also small. If the fragment guess or charged atomic density guess were not used, the Mulliken spin population would be 0.5 on both He atoms, the overlap between the ET reactant and product states would be 100% and an infinite VRP would result.