**Density Functional Theory to modeling electron-transfer reactions**

**The electron-transfer reaction**

**A theory to describe the system**

General description

Two states

Donor (D)

Acceptor (A)

**What is an electron-transfer reaction?**

“The transfer of an electron from one molecular entity to another, or between two localized sites in the same molecular entity.” (PAC, 1994, 66, 1077. Glossary of terms used in physical organic chemistry (IUPAC Recommendations 1994))

Examples of ET reactions:

Anaerobic respiration

Photosynthesis

Vision

The electron is transfer from donor to acceptor

The charge is externally introduced/extracted:

The charge is separated:

The electronic coupling between D and A

The quantum-mechanical approach

**The Marcus Theory**

Based on the ideas:

Fast movement of the electrons

The Frank-Condom principle applied to electron-transfer reactions

Conservation of the total energy of the system

Dielectric continuum theory to solvent molecules

Defined by:

Depending on its value

Adiabatic electron-transfer reactions

It causes a split in the intersection point

Diabatic electron-transfer reactions

Relatively strong electron coupling

Reaction occurs as a "jump"

Small probability per passage through the intersection region

Very weak electronic coupling

**Modern approaches to the problem**

**What is an electron reaction?**

Long-range ET reactions

Tunneling is the main mechanism

Well described by diabatic states

The classical expression does not account for tunneling effects

New expression is needed

High temperature limit

Equal to the classical equation

It account with tunnel effects

Conclusion:

Reorganization energy

Driving Force

Electronic coupling

Constrained DFT

Frozen Density Embedded

Density Functional Theory

1964

Hohenberg-Kohn theorems

The density is uniquely determined by an external potential

The ground state energy is a functional of the density

1965

The Kohn-Sham formalism

Density of a non-interacting system

The Kohn-Sham energy functional

LDA

GGA

LDA+U

Meta-GGA

Hybrids

Approximations to

Exc

TD-DFT

ET reaction rates calculated with TD-DFT

Main issues:

Diabatic states obtained from adiabatic states (ground and CT states need to be computed)

Impossible to generate charge-localized states (SIE)

Too low charge-transfer excitation energies

Wrong asymptotic behavior at long distances

Results:

Poor description of ET reactions

We need a way to generate charge-localized states

Impose a constraint into the system

Generate charge-localized states

Obtain the information needed to calculate the ET reaction rate

Defining a constraint

weight function defining the property

constrained observable

The system is optimized under the constraint

Looking for stationary points yield

Initial guess ρ

Calculate

Calculate orbitals with the KS equation

Recalculate the orbitals with the above equation

Converged?

NO

Yes

outer loop

inner loop

How to define the fragments

Atomic populations schemes are used to define the wight element

(Mulliken, Löwdin, Becke or Hirshfeld)

Charge localized states obtained from two constrained DFT calculations

First charge-localized state

N = -1

N = 1

Subsystem density formulation

Generate charge-localized states

Obtain diabatic states

Subsystem density formulation

Total density of the system

A functional for the energy

Nonadditive kinetic energy term

In general, a non linear function satisfies :

The same is applicable for a nonlinear functional

The nonadditive term represents

In addition to

we need to approximate

Similar to KS one-electron equations

The effective potential is defined as

It is solved self-consistently

obtained from a KS calculation for the isolated subsystem

Converged?

No

Yes

freeze-and-than cycle

How to define define the fragments

Subsystem B

Subsystem A

Subsystem B defined with an extra charge

It enter in the optimization process with the charge already constrained within this fragment

We obtain the density of one charge-localized state

The same procedure to generate the charge-localized state of products

CDFT

FDE

Constraint

Subsystems

DFT calculations of the whole system

DFT calculations on smaller subsystems

Two ground-state DFT calculations

Four ground-state DFT calculations (minimum)

Approximate

Approximate

Large systems are computationally demanding

Large system are calculated adding more subsystems

Charge-Transfer excited states avoided

and

or

Activated complex

The system remains in the GS

Introduced through a Lagrangian into the inner loop

Defined as the difference in charge

Constraint property

from KSCED eqn.

Second charge-localized state

The total energy is obtained:

Freezing subsystem B

Freezing subsystem A

Which terms can we extract?

Two charge-localized states:

They are difference because states A and D are not orthogonal

The electronic coupling is obtained after diagonalizing the Hamiltonian matrix

Which terms do we obtain?

The two charge-localized states are obtained

The CT excitation energy

Electronic coupling