Second-order Fermi’s golden rule rate for a two-acceptor electron transfer model

A vibronic Hamiltonian¹ of a 3-state electron-transfer system with a single donor and two acceptors is $\hat{H}_0+\hat{V}$.
\begin{align} \hat{H}_{0}&=\hat{h}_{a}\left|a\right\rangle \left\langle a\right|+\hat{h}_{b}\left|b\right\rangle \left\langle b\right|+\hat{h}_{c}\left|c\right\rangle \left\langle c\right|\\ V&=g_{ab}\left|a\right\rangle \left\langle b\right|+g_{ac}\left|a\right\rangle \left\langle c\right|+g_{bc}\left|b\right\rangle \left\langle c\right|+h.c. \end{align}
In this Hamiltonian, $\ket{a}$ is the donor state and $\ket{b}$ and $\ket{c}$ are the two acceptor states. If the electronic couplings among $a$, $b$, and $c$ are small, the electron transfer rate can be calculated by using Fermi’s golden rule. To do this, we turn to the interaction picture and calculate the propagator up to the second order.

The Hamiltonian in the interaction picture $V_{I}(t)=e^{i\hat{H}_0t}\hat{V}e^{-i\hat{H}_0t}$ is
\begin{align} V_{I}(t) & = g_{ab}e^{i\hat{h}_{a}t}e^{-i\hat{h}_{b}t}\left|a\right\rangle \left\langle b\right|+g_{ab}^{*}e^{i\hat{h}_{b}t}e^{-i\hat{h}_{a}t}\left|b\right\rangle \left\langle a\right|\\ +& g_{ac}e^{i\hat{h}_{a}t}e^{-i\hat{h}_{c}t}\left|a\right\rangle \left\langle c\right|+g_{ac}^{*}e^{i\hat{h}_{c}t}e^{-i\hat{h}_{a}t}\left|c\right\rangle \left\langle a\right|\\ +& g_{bc}e^{i\hat{h}_{b}t}e^{-i\hat{h}_{c}t}\left|b\right\rangle \left\langle c\right|+g_{bc}^{*}e^{i\hat{h}_{c}t}e^{-i\hat{h}_{b}t}\left|c\right\rangle \left\langle b\right|. \end{align}

The corresponding propagator is
\[ \hat{U}_{I}(t)=1-i\int_{0}^{t}\hat{V}_{I}(t)dt'-\int_{0}^{t}dt'\int_{0}^{t'}dt''\hat{V}_{I}(t')\hat{V}_{I}(t'')+\cdots \]
The population at the state $\left|a\right\rangle $ is (assuming the initial nuclear equilibrium at the potential energy surface of $\left|a\right\rangle $)
\[ P_{a}(t)=\left\langle \phi_{0}\right|\left\langle a\right|\hat{U}_{I}^{\dagger}(t)\left|a\right\rangle \left\langle a\right|\hat{U}_{I}(t)\left|a\right\rangle \left|\phi_{0}\right\rangle \]
The quantity in the middle is
\begin{align*} \left\langle a\right|\hat{U}_{I}(t)\left|a\right\rangle = &1-i\int_{0}^{t}\left\langle a\right|\hat{V}_{I}(t)\left|a\right\rangle dt'\\ & -\int_{0}^{t}dt'\int_{0}^{t'}dt''\left\langle a\right|\hat{V}_{I}(t')\hat{V}_{I}(t'')\left|a\right\rangle \\ & +\cdots \end{align*}
The expectation value in the first integral is
\[ \left\langle a\right|\hat{V}_{I}(t)\left|a\right\rangle =0. \]
To calculate the expectation value in the second integral, the operator product $\hat{V}_{I}(t')\hat{V}_{I}(t'')$ is worked out.
\begin{align*} &\hat{V}_{I}(t')\hat{V}_{I}(t'') \\= & g_{ab}^{2}e^{i\hat{h}_{a}t'}e^{-i\hat{h}_{b}(t'-t'')}e^{-i\hat{h}_{a}t}\left|a\right\rangle \left\langle a\right| +\\ & g_{ac}^{2}e^{i\hat{h}_{a}t}e^{-i\hat{h}_{c}(t'-t'')}e^{-i\hat{h}_{a}t}\left|a\right\rangle \left\langle a\right| + \\ & \mathrm{something}\left|b\right\rangle \left\langle b\right| +\mathrm{something}\left|c\right\rangle \left\langle c\right|\\ \end{align*}
Thus, the expectation integral is
\[ \int_{0}^{t}dt'\int_{0}^{t'}dt''\left(g_{ab}^{2}e^{i\hat{h}_{a}t'}e^{-i\hat{h}_{b}(t'-t'')}e^{-i\hat{h}_{a}t}+g_{ac}^{2}e^{i\hat{h}_{a}t}e^{-i\hat{h}_{c}(t'-t'')}e^{-i\hat{h}_{a}t}\right) \]
Each term in the integral is identical to a 2-state expression (i.e., when the third state doesn’t exist). This is the rate constant of the electron transfer reaction.

Infinite-order correction in terms of $g_{bc}$

We can further assume $\hat{h}_{b}=\hat{h}_{c}$ and calculate higher order corrections in terms of $g_{bc}$ to the (second-order) rate constant above. The first correction is order-3 for $\hat{V}_{I}(t)$ and we only select the terms that are order-2 for $g_{ab}$ and order-1 for $g_{ac}$:

\begin{align} & (i)^{3}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}\left\langle a\right|V_{I}(t_{1})V_{I}(t_{2})V_{I}(t_{3})\left|a\right\rangle \\ \sim & (i)^{3}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}\underbrace{g_{ab}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}t_{1}}g_{bc}e^{i\hat{h}_{b}t_{2}}e^{-i\hat{h}_{c}t_{2}}g_{ac}^{*}e^{i\hat{h}_{c}t_{3}}e^{-i\hat{h}_{a}t_{3}}}_{ab|bc|c^{*}a^{*}}\\ & +(i)^{3}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}\underbrace{g_{ac}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{c}t_{1}}g_{bc}^{*}e^{i\hat{h}_{c}t_{2}}e^{-i\hat{h}_{b}t_{2}}g_{ab}^{*}e^{i\hat{h}_{b}t_{3}}e^{-i\hat{h}_{a}t_{3}}}_{ac|c^{*}b^{*}|b^{*}a^{*}} \end{align}
The first integral is (after using $\hat{h}_{b}=\hat{h}_{c}$)
\begin{align} & (i)^{3}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}g_{ab}g_{bc}g_{ac}^{*}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}t_{1}}e^{i\hat{h}_{b}t_{3}}e^{-i\hat{h}_{a}t_{3}}\\ = & (i)^{3}g_{ab}g_{bc}g_{ac}^{*}\int_{0}^{t}dt_{1}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}t_{1}}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}e^{i\hat{h}_{b}t_{3}}e^{-i\hat{h}_{a}t_{3}}\\ = & (i)^{3}g_{ab}g_{bc}g_{ac}^{*}\int_{0}^{t}dt_{1}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}t_{1}}\int_{0}^{t_{1}}dt_{3}\int_{t_{3}}^{t_{1}}dt_{2}e^{i\hat{h}_{b}t_{3}}e^{-i\hat{h}_{a}t_{3}}\\ = & (i)^{3}g_{ab}g_{bc}g_{ac}^{*}\int_{0}^{t}dt_{1}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}t_{1}}\int_{0}^{t_{1}}dt_{3}(t_{1}-t_{3})e^{i\hat{h}_{b}t_{3}}e^{-i\hat{h}_{a}t_{3}}\\ = & (i)^{3}g_{ab}g_{bc}g_{ac}^{*}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{3}(t_{1}-t_{3})e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}(t_{1}-t_{3})}e^{-i\hat{h}_{a}t_{3}}\\ = & (i)^{3}g_{ab}g_{bc}g_{ac}^{*}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}(t_{1}-t_{2})e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}(t_{1}-t_{2})}e^{-i\hat{h}_{a}t_{2}} \end{align}
which is almost the same as the integral in Eq. (1) except for an additional time difference term $(t_{1}-t_{3})$. From the second line to the third line the integration order is changed. In the last line $t_{3}$ is replaced by $t_{2}$.

Continuing this process to calculate higher-order corrections in terms of $g_{bc}$ up until the infinite order will take effect of the coupling between $\left|b\right\rangle $ and $\left|c\right\rangle $ fully into account. The expression of the correction is
\begin{align} \mathrm{correction}= & g_{ab}g_{ac}^{*}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{3}e^{ig_{bc}(t_{1}-t_{2})}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}(t_{1}-t_{2})}e^{-i\hat{h}_{a}t_{2}}\\ +& g_{ab}^{*}g_{ac}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{3}e^{ig_{bc}(t_{1}-t_{2})}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}(t_{1}-t_{2})}e^{-i\hat{h}_{a}t_{2}}. \end{align}

Adding this correction to the rate equation in Eq. (1), we have the final expression of Fermi’s golden rule electron transfer rate constant for a two-acceptor system
\begin{align} & \int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left(g_{ab}^{2}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}(t_{1}-t_{2})}e^{-i\hat{h}_{a}t_{2}}+g_{ac}^{2}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}(t_{1}-t_{2})}e^{-i\hat{h}_{a}t_{2}}\right)\\ +& \int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}g_{ab}^{*}g_{ac}e^{ig_{bc}(t_{1}-t_{2})}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}(t_{1}-t_{2})}e^{-i\hat{h}_{a}t_{2}}\\ +& \int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}g_{ab}^{*}g_{ac}e^{ig_{bc}(t_{1}-t_{2})}e^{i\hat{h}_{a}t_{1}}e^{-i\hat{h}_{b}(t_{1}-t_{2})}e^{-i\hat{h}_{a}t_{2}}. \end{align}
Further assuming $g_{ab}=g_{ac}$, the rate expression above is reduced to
\[ 2\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left(g_{ab}^{2}e^{i\hat{h}_{a}t_{1}}e^{-i(\hat{h}_{b}+g_{ab}^{2})(t_{1}-t_{2})}e^{-i\hat{h}_{a}t_{2}}\right). \]

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