A theoretical approach to molecular single-electron transistors

Abstract

We present theoretical methods and computational strategies of the effects of nanoparticles on linear optical properties of molecules. We present quantum mechanical-molecular mechanics response methods for calculating electromagnetic properties of molecules interacting with nanoparticles and we report strategies for calculating electronic and redox states of molecules sandwiched between gold nanoparticles.

Appendix

The Ehrenfest method used in the response theory section can be derived in the following manner where the time development of the expectation value of the following operator is represented by

$$ \hat{Q} (t) = {{\rm e}}^{-\hat{\kappa} (t)} \hat{Q} {{\rm e}}^{\hat{\kappa}(t)} $$

(106)

Considering the expectation value of the commutator between $\hat{Q} (t)$ and the operators in the Schrödinger equation yields

$$ \langle t \vert [\hat{Q}(t), \hat{H}(t) ] \vert t \rangle = \langle t \vert [\hat{Q}(t), i \frac{\partial}{\partial t}] \vert t \rangle $$

(107)

with $\vert t \rangle$ being the time-dependent wave function. As mentioned in the section on response theory, $\vert t \rangle = {{\rm e}}^{\hat{\kappa} (t)} \vert 0 \rangle$ and Eq. 107 can be expanded as

$$ \left\langle 0 \left\vert {{\rm e}}^{\hat{\kappa} (t)} \left( \begin{array}{c} {{\rm e}}^{- \hat{\kappa} (t)} \hat{Q} {{\rm e}}^{\hat{\kappa}(t)} (\hat{H}_{(0)} + \hat{V}(t)) - \\ (\hat{H}_{(0)} + \hat{V}(t)){{\rm e}}^{- \hat{\kappa} (t)} \hat{Q} {{\rm e}}^{\hat{\kappa}(t)} \end{array} \right) {{\rm e}}^{-\hat{\kappa}(t) } \right\vert 0 \right\rangle = $$

(108)

$$ \left\langle 0 \left\vert {{\rm e}}^{\hat{\kappa} (t)} \left( \begin{array}{c} {{\rm e}}^{- \hat{\kappa} (t)} \hat{Q} {{\rm e}}^{\hat{\kappa}(t)} i \frac{\partial}{\partial t} - \\ i \frac{\partial}{\partial t} {{\rm e}}^{- \hat{\kappa} (t)} \hat{Q} {{\rm e}}^{\hat{\kappa}(t)} \end{array} \right) {{\rm e}}^{-\hat{\kappa} (t)} \right\vert 0 \right\rangle $$

(109)

and by rearranging, we obtain

$$ \langle 0 \vert \hat{Q} {{\rm e}}^{\hat{\kappa} (t)} (\hat{H}_{(0)} + \hat{V} (t)) {{\rm e}}^{- \hat{\kappa} (t)} - {{\rm e}}^{\hat{\kappa} (t)} (\hat{H}_{(0)} + \hat{V}(t)) {{\rm e}}^{- \hat{\kappa} (t)} \hat{Q} \vert 0 \rangle = $$

(110)

$$ \langle 0 \vert \hat{Q} {{\rm e}}^{\hat{\kappa} (t)} i \frac{\partial}{\partial t} {{\rm e}}^{- \hat{\kappa} (t)} - {{\rm e}}^{\hat{\kappa} (t)} i \frac{\partial}{\partial t} {{\rm e}}^{- \hat{\kappa} (t)}\hat{Q} \vert 0 \rangle $$

(111)

and we rewrite as

$$ \langle 0 \vert \left[ \hat{Q}, {{\rm e}}^{\hat{\kappa} (t)} (\hat{H}_{(0)} + \hat{V}(t)){{\rm e}}^{- \hat{\kappa} (t)} \right] \vert 0 \rangle = \langle 0 \vert \left[ \hat{Q} , {{\rm e}}^{\hat{\kappa} (t)} i \frac{\partial}{\partial t} {{\rm e}}^{- \hat{\kappa} (t)} \right] \vert 0 \rangle $$

(112)

Collecting the terms on the left-hand side

$$ \left\langle 0 \left\vert \left[ \hat{Q}, {{\rm e}}^{\hat{\kappa} (t)} \left( \hat{H}_{(0)} + \hat{V}(t) - i \frac{\partial}{\partial t} \right) {{\rm e}}^{- \hat{\kappa} (t)} \right] \right\vert 0 \right\rangle =0 $$

(113)

and remembering that the exponential function can be expanded as ${{\rm e}}^{\hat{\kappa} (t)} = 1 + \hat{\kappa} (t) + \cdots$ we obtain

$$ \left\langle 0 \left\vert \left[ \hat{Q}, \left( \hat{H}_{(0)} + \hat{V}(t) - i \frac{\partial}{\partial t} \right) \right] \right\vert 0 \right\rangle + \left\langle 0 \left\vert \left[ \hat{Q}, \left[ \hat{\kappa} (t) , \left( \hat{H}_{(0)} + \hat{V}(t) - i \frac{\partial}{\partial t} \right) \right] \right] \right\vert 0 \right\rangle + \cdots =0 $$

(114)

In the following, it will be useful to know the following commutator

$$ \begin{aligned} \left[ \hat{\kappa} (t) , -i \frac{\partial}{\partial t} \right] \left\vert t \right\rangle &= \hat{\kappa} (t) \cdot \left(-i \frac{\partial}{\partial t} \right) \left\vert t \right\rangle + i \frac{\partial}{\partial t} \cdot \hat{\kappa} (t) \left\vert t \right\rangle \\ &= \hat{\kappa} (t) \cdot \left(-i \frac{\partial}{\partial t} \right) \left\vert t \right\rangle + i \kappa^\bullet (t) \vert t\rangle \quad + i \hat{\kappa} (t) \cdot \left( \frac{\partial}{\partial t} \right) \left\vert t \right\rangle \\ &= i \kappa^\bullet (t) \vert t \rangle \end{aligned} $$

(115)

where $\kappa^{\bullet} (t) = \frac{\partial \kappa}{\partial t }.$

By expanding Eq. 114 in orders of first-order perturbation and neglecting higher-order terms, we obtain:

$$ \left\langle 0 \left\vert \left[ \hat{Q}, \left( \hat{H}_{(0)}^{(1)} + \hat{V}(t) \right) \right] \right\vert 0 \right\rangle + \left\langle 0 \left\vert \left[ \hat{Q}, \left[ \hat{\kappa}^{(1)} (t) , \hat{H}_{(0)}^{(0)} - i \frac{\partial}{\partial t} \right] \right] \right\vert 0 \right\rangle =0 $$

(116)

where it should be remembered that $\hat{V}(t)$ is of first order and we rewrite this as:

$$ \left\langle 0 \left\vert \left[ \hat{Q}, \left( \hat{H}_{(0)}^{(1)} + \hat{V}(t) \right) \right] \right\vert 0 \right\rangle + \left\langle 0 \left\vert \left[ \hat{Q} , \left[ \hat{\kappa}^{(1)} (t), \hat{H}_{(0)}^{(0)} \right] + \left[\hat{\kappa}^{(1)} (t) , - i \frac{\partial}{\partial t} \right] \right] \right\vert 0 \right\rangle = 0 $$

(117)

Using Eq. 115, it can be written as

$$ \left\langle 0 \left\vert \left[ \hat{Q}, \left( \hat{H}_{(0)}^{(1)} + \hat{V}(t) \right) \right] \right\vert 0 \right\rangle + \left\langle 0 \left\vert \left[ \hat{Q} , \left[ \hat{\kappa}^{(1)} (t) , \hat{H}_{(0)}^{(0)} \right] \right\vert 0 \right\rangle + i \left\langle 0 \left\vert \left[ \hat{Q} , \kappa^\bullet (t) \right] \right] \right\vert 0 \right\rangle = 0 $$

(118)

Since the frequency domain is of interest, we perform a Fourier transform but first we start by rearranging the equation:

$$ \begin{aligned} \left\langle 0 \left\vert \left[ \hat{Q}, - \left[ \hat{H}_{(0)}^{(0)} , \hat{\kappa}^{(1)} \right] + \hat{H}_{(0)}^{(1)} \right] \right\vert 0 \right\rangle + i \left\langle 0 \left\vert \left[ \hat{Q} , \kappa^{\bullet} (t) \right] \right\vert 0 \right\rangle &= - \left\langle 0 \left\vert \left[ \hat{Q}, \hat{V} (t) \right] \right\vert 0 \right\rangle \\ \left\langle 0 \left\vert \left[ \hat{Q}, \left[ \hat{H}_{(0)}^{(0)} , \hat{\kappa}^{(1)} \right] - \hat{H}_{(0)}^{(1)} \right] \right\vert 0 \right\rangle - i \left\langle 0 \left\vert \left[ \hat{Q} , \kappa^{\bullet} (t) \right] \right\vert 0 \right\rangle &= \left\langle 0 \left\vert \left[ \hat{Q}, \hat{V} (t) \right] \right\vert 0 \right\rangle \end{aligned} $$

(119)

The Fourier transform of the equation brings it into the frequency domain we start by multiplying by ${{\rm e}}^{i\omega t}$ and thereby taking the integral one the time-dependent terms.

$$ \left\langle 0 \left\vert \left[ \hat{Q}, \left[ \hat{H}_{(0)}^{(0)} , \hat{\kappa}^{\omega} \right] - \hat{H}^{\omega} \right] \right\vert 0 \right\rangle - i \left\langle 0 \left\vert \left[ \hat{Q} , \int {\rm d}t\,{{\rm e}}^{i \omega t} \kappa^{\bullet} (t) \right] \right\vert 0 \right\rangle = \left\langle 0 \left\vert \left[ \hat{Q}, \hat{V}^{\omega} \right] \right\vert 0 \right\rangle $$

(120)

The second term can be solved considering the partial integration and remembering that the first term can be neglected

$$ \begin{aligned} \int {\rm d}t\,{{\rm e}}^{i \omega t} \kappa^{\bullet} (t) &= - \int \left[ \frac{\partial}{\partial t} {{\rm e}}^{i \omega t} \right] \kappa (t) dt \\ &= - i \omega \int {{\rm e}}^{i \omega t} \kappa (t) {\rm d}t \\ &= - i \omega \kappa^{\omega} \end{aligned} $$

(121)

Eq. 120 can now be written as:

$$ \left\langle 0 \left\vert \left[ \hat{Q}, \left[ \hat{H}_{(0)}^{(0)} , \hat{\kappa}^{\omega} \right] - \hat{H}^{\omega} \right] \right\vert 0 \right\rangle - \omega \left\langle 0 \left\vert \left[ \hat{Q} , \kappa^{\omega} \right] \right\vert 0 \right\rangle = \left\langle 0 \left\vert \left[ \hat{Q}, \hat{V}^{\omega} \right] \right\vert 0 \right\rangle. $$

(122)