ChargeTransport (version 1.0.2)

Marcus: Marcus Charge Transfer Rates

Description

Computes charge transfer (CT) rates using the following semi-classical Marcus expression: $$k = \frac{2\pi}{\hbar}J^2\sqrt{\frac{1}{4\pi\lambda k_{B}T}}exp \left( -\frac{ \left( \lambda + \Delta E^{0} + \Delta E_{Field} \right) ^{2}}{4\lambda k_{B}T} \right)$$ where $J$ and $\DeltaG0 = Ef - Ei$ are respectively the electronic coupling and the site energy difference between the initial and final electronic states involved in the charge transfer reaction, and $\lambda$ is the total reorganization energy. $dEField$ is an additional contribution to the site energy difference due to an external electric field and $T$ is the temperature.

Usage

Marcus(J, lambda, dE0 = 0, dEField = 0, temp = 300)

Arguments

J

a scalar, a vector, a matrix or an array containing the electronic couplings (in eV) used to calculate the CT rates.

lambda

a scalar, a vector, a matrix or an array containing the total reorganization energies (in eV) used to calculate the CT rates.

dE0

a scalar, a vector, a matrix or an array containing the site energy differences (in eV) used to calculate the CT rates. By default self-exchange reactions are considered (dE0=0).

dEField

a scalar, a vector, a matrix or an array containing an additional contribution to the site energy differences due to an external electric field (in eV). By default no electric field is applied (dEField=0).

temp

a scalar giving the temperature (in Kelvin) at which to evalute the CT rates. By default CT rates are evaluated at room temperature (temp=300).

Value

Depending on the dimension of the objects passed to the function a scalar, a vector, a matrix or an array containing the Marcus CT rates (in s-1) is returned.

Details

The arguments of these function can be scalars, vectors, matrices or arrays. Mixing scalar values with vectors, matrices or arrays is allowed but in all other cases the arguments must have the same dimensions and lengths. Using matrices or arrays is useful to compute simultaneously several charge transfer rates for different pairs of molecules, structures ...

References

R.A. Marcus, Journal of Chemical Physics, 24:966, 1956

See Also

energyConversion, dEField, MarcusLevichJortner, LandauZener, KMC

Examples

## Produce a map of the decimal logarithm of the Marcus,
## Marcus-Levich-Jortner and Landau-Zener rate expressions for:
nuN <- 1445 # effective vibrational mode wavenumber in cm-1
lambdaI <- 0.14  # internal reorganization energy in eV
lambdaS <- 36E-3 # external reorganization energy in eV

N  <- 301
J  <- seq( 0  , 65,length.out=N)*1E-3 # eV
dE <- seq(-0.5,0.5,length.out=N)      # eV
G  <- expand.grid(J, dE)
J  <- G[,1]
dE <- G[,2]

kMLJ    <- MarcusLevichJortner(
           J = J, lambdaI = lambdaI, lambdaS = lambdaS,
           hBarW = centimeterMinusOne2electronVolt(nuN), dE0 = dE)
kMarcus <- Marcus(
           J = J, lambda = lambdaI+lambdaS, dE0 = dE)
kLZ     <- LandauZener(
           J = J, lambda = lambdaI+lambdaS,
           nuN = centimeterMinusOne2Hertz(nuN), dE0 = dE)

kMLJ    <- matrix(kMLJ   , nrow = N, ncol = N)
kMarcus <- matrix(kMarcus, nrow = N, ncol = N)
kLZ     <- matrix(kLZ    , nrow = N, ncol = N)

addAxis <- function(bottom = TRUE, left = FALSE, above = FALSE, right = FALSE){
  useless <- lapply(1:4,axis, labels=FALSE)
  if(bottom) axis(1, labels = TRUE)
  if(left  ) axis(2, labels = TRUE)
  if(above ) axis(3, labels = TRUE)
  if(right ) axis(4, labels = TRUE)
  if(bottom) mtext(side=1,line=1.2, text=expression( abs(J)/eV), cex=par("cex"))
  if(left  ) mtext(side=2,line=1.2, text=expression(Delta*E/eV), cex=par("cex"))
  if(right ) mtext(side=4,line=1.2, text=expression(Delta*E/eV), cex=par("cex"))
  box()
}

layout(matrix(1:3, ncol=3))
par(cex=2, lwd=1.5, pty="s", mgp=c(1.1,0.1,0), tck=0.02, mar=rep(0.7,4), oma=rep(2,4))
contour(unique(J), unique(dE), log10(kMLJ   ),
        zlim = c(1,15), levels = -15:15, xaxt="n", yaxt="n", labcex=3)
addAxis(TRUE, TRUE, FALSE, FALSE)
title("Marcus-Levich-Jortner", line=1)
contour(unique(J), unique(dE), log10(kMarcus),
        zlim = c(1,15), levels = -15:15, xaxt="n", yaxt="n", labcex=3)
addAxis(TRUE, FALSE, FALSE, FALSE)
title("Marcus", line=1)
contour(unique(J), unique(dE), log10(kLZ    ),
        zlim = c(1,15), levels = -15:15, xaxt="n", yaxt="n", labcex=3)
addAxis(TRUE, FALSE, FALSE, TRUE)
title("Landau-Zener", line=1)