The Displaced Harmonic Oscillator

Abstract

In this chapter, we discuss a more specific model for the transition between the vibrational manifolds using parallel displaced harmonic normal modes, for which the time-correlation function can be evaluated explicitly. We consider the limit of high frequency modes (or low temperature) where vibrational progressions appear and the limit of low frequencies (or high temperature) where the lineshape becomes Gaussian where position and width only depend on the total reorganization energy.

In this chapter, we discuss a more specific model for the transition between the vibrational manifolds using parallel displaced harmonic normal modes, for which the time-correlation function can be evaluated explicitly. We consider the limit of high frequency modes (or low temperature) where vibrational progressions appear and the limit of low frequencies (or high temperature) where the lineshape becomes Gaussian where position and width only depend on the total reorganization energy.

1 The Time-Correlation Function in the Displaced Harmonic Oscillator Approximation

We apply the harmonic approximation (17.11) for the nuclear motion to the zero-order Hamiltonian (18.31)

$$\begin{aligned} H_{0}=|\psi _{e}>\bigg (\!E_{e}^{0}+\sum _{r}\hbar \omega _{r}^{e}b_{r}^{e\dagger }b_{r}^{e}\!\bigg )<\psi _{e}|+|\psi _{g}>\bigg (\! E_{g}^{0}+\sum _{r}\hbar \omega _{r}^{g}b_{r}^{g\dagger }b_{r}^{g}\!\bigg )<\psi _{g}|. \end{aligned}$$

(19.1)

Fig. 19.1

Displaced oscillator model. The displaced oscillator model assumes that the normal mode eigenvectors are the same in both electronic states involved. Then the different modes are still independent. The figure shows the potential energy along one such normal mode Q. The minima at $Q_{g}^{0}$ and $Q_{e}^{0}$ are shifted relative to each other by a distance $d=Q_{e}^{0}-Q_{g}^{0}$. The elongation of the normal mode is denoted as $q_{g(e)}=Q-Q_{g(e)}^{0}$. The curvature of the two parabolas is the same. Thus neglecting frequency changes in the excited state, the vibrationless transition energy $\hbar \omega _{00}$ equals the pure electronic transition energy $E_{e}^{min}-E_{g}^{min}$. The reorganization energy $E_{R}=\frac{1}{2}\omega ^{2}d^{2}$ is the amount of energy which can be released in the excited state after a vertical transition from the vibronic groundstate

In a simplified but popular model, we neglect mixing of the normal modes (parallel mode approximation, the eigenvectors ($u_{j}^{r}$ in 17.13) are the same) and frequency changes ( $\omega _{r}^{g}=\omega _{r}^{e}=\omega _{r}$) in the excited state but allow for a shift of the equilibrium position $(q_{r}^{e}=q_{r}^{g}+d_{r}$).¹ The potential energy for the two states then is approximated by (Fig. 19.1)

$$\begin{aligned} E_{g}=E_{g}^{min}+\frac{1}{2}\sum _{r}\omega _{r}^{2}q_{r}^{2} \end{aligned}$$

(19.2)

$$\begin{aligned} E_{e}=E_{e}^{min}+\frac{1}{2}\sum _{r}\omega _{r}^{2}(q_{r}^{e})^{2}=E_{e}^{min}+\frac{1}{2}\sum _{r}\omega _{r}^{2}(q_{r}+d_{r})^{2}. \end{aligned}$$

(19.3)

The vertical excitation energy is²

$$\begin{aligned} E_{e}(q_{r}=0)-E_{g}(q_{r}=0)=E_{e}^{min}+\frac{1}{2}\sum _{r}\omega _{r}^{2}d_{r}{}^{2}-E_{g}^{min}=\hbar \omega _{00}+E_{R} \end{aligned}$$

(19.4)

with the reorganization energy

$$\begin{aligned} E_{R}=\frac{1}{2}\sum _{r}\omega _{r}^{2}d_{r}^{2}. \end{aligned}$$

(19.5)

We introduce the ladder operators by substituting

$$\begin{aligned} q_{r}=\sqrt{\frac{\hbar }{2\omega _{r}}}\bigg (b_{r}^{g}+b_{r}^{g\dagger }\bigg ) \end{aligned}$$

(19.6)

$$\begin{aligned} q_{r}^{e}=\sqrt{\frac{\hbar }{2\omega _{r}}}\bigg (b_{r}^{e}+b_{r}^{e\dagger }\bigg )=\sqrt{\frac{\hbar }{2\omega _{r}}}\bigg (b_{r}^{g}+b_{r}^{g\dagger }\bigg )+d_{r}. \end{aligned}$$

(19.7)

Since $d_{r}$ is real valued we find

$$ b_{r}^{e}=b_{r}^{g}+\frac{1}{2}\sqrt{\frac{2\omega _{r}}{\hbar }}d_{r}=b_{r}^{g}+\sqrt{\frac{\omega _{r}}{2\hbar }}d_{r}=b_{r}^{g}+g_{r} $$

with the vibronic coupling parameter

$$ g_{r}=\sqrt{\frac{\omega _{r}}{2\hbar }}d_{r}. $$

From

$$\begin{aligned} \hbar \omega _{r}\bigg (b_{r}^{e\dagger }b_{r}^{e}+\frac{1}{2}\bigg )&=\hbar \omega _{r}\left( (b_{r}^{g\dagger }+g_{r})(b_{r}^{g}+g_{r})+\frac{1}{2}\right) \\&=\hbar \omega _{r}\bigg (b_{r}^{g\dagger }b_{r}^{g}+\frac{1}{2}\bigg )+\hbar \omega _{r}g_{r}\bigg (b_{r}^{g}+b_{r}^{g\dagger }\bigg )+\hbar \omega _{r}g_{r}^{2} \end{aligned}$$

we obtain the “displaced harmonic oscillator” model (DHO)

$$\begin{aligned} H_{g}=\sum _{r}\hbar \omega _{r}b_{r}^{+}b_{r} \end{aligned}$$

(19.8)

$$\begin{aligned} H_{e}&=\sum _{r}\hbar \omega _{r}b_{r}^{e\dagger }b_{r}^{e}\nonumber \\&=H_{g}+\sum _{r}g_{r}\hbar \omega _{r}(b_{r}^{\dagger }+b_{r})+\sum _{r}g_{r}^{2}\hbar \omega _{r} \end{aligned}$$

(19.9)

where the superscript g is omitted from now and the last term is the reorganization energy

$$\begin{aligned} E_{R}=\sum _{r}g_{r}^{2}\hbar \omega _{r}. \end{aligned}$$

(19.10)

The correlation function (18.50)

$$\begin{aligned} F(t)=\left\langle \mathrm{e}^{-\frac{\mathrm{i}t}{\hbar }H_{g}}\mathrm{e}^{\frac{\mathrm{i}t}{\hbar }H_{e}}\right\rangle _{g}=Q^{-1}\text{ tr }\left( \mathrm{e}^{-H_{g}/k_{B}T}\mathrm{e}^{-\frac{\mathrm{i}t}{\hbar }H_{g}}\mathrm{e}^{\frac{\mathrm{i}t}{\hbar }H_{e}}\right) \end{aligned}$$

(19.11)

with

$$\begin{aligned} Q=\text{ tr }(\mathrm{e}^{-H_{g}/k_{B}T}) \end{aligned}$$

(19.12)

factorizes in the parallel mode approximation

$$\begin{aligned} F(t)=\prod _{r}F_{r}(t) \end{aligned}$$

(19.13)

$$\begin{aligned} F_{r}(t)= & {} Q_{r}^{-1}\text{ tr }\left( \mathrm{e}^{-\hbar \omega _{r}b_{r}^{\dagger }b_{r}/k_{B}T}\mathrm{e}^{-\mathrm{i}t\omega _{r}b_{r}^{\dagger }b_{r}}\mathrm{e}^{\mathrm{i}t\omega _{r}(b_{r}^{\dagger }+g_{r})(b_{r}+g_{r})}\right) \nonumber \\= & {} \left\langle \mathrm{e}^{-\mathrm{i}t\omega _{r}b_{r}^{\dagger }b_{r}}\mathrm{e}^{\mathrm{i}t\omega _{r}(b_{r}^{\dagger }+g_{r})(b_{r}+g_{r})}\right\rangle . \end{aligned}$$

(19.14)

As shown in the appendix this can be evaluated as

$$\begin{aligned}&F_{r}(t)=\exp \bigg (g_{r}^{2}\left[ (\mathrm{e}^{\mathrm{i}\omega _{r}t}-1)(\overline{n_{r}}+1)+(\mathrm{e}^{-\mathrm{i}\omega _{r}t}-1)\overline{n_{r}}\right] \bigg ) \\ \nonumber&=\exp \bigg (g_{r}^{2}(2\overline{n}_{r}+1)(\cos \omega _{r}t-1)+\mathrm{i}g_{r}^{2}\sin \omega _{r}t\bigg ) \end{aligned}$$

(19.15)

with the average phonon numbers

$$\begin{aligned} \overline{n_{r}}=\frac{1}{\mathrm{e}^{\hbar \omega _{r}/k_{B}T}-1}. \end{aligned}$$

(19.16)

Expression (19.14) contains phonon absorption (positive frequencies) and emission processes (negative frequencies). We discuss two important limiting cases.

2 High Frequency Modes

In the limit $\hbar \omega _{r}\gg k_{B} T$ the average phonon number

$$\begin{aligned} \overline{n_{r}}=\frac{1}{\mathrm{e}^{\hbar \omega /k_{B}T}-1} \end{aligned}$$

(19.17)

is small and the correlation function becomes

$$\begin{aligned} F_{r}(t)\rightarrow \exp \left( g_{r}^{2}(\mathrm{e}^{\mathrm{i}\omega t}-1)\right) . \end{aligned}$$

(19.18)

Expansion of $F_{r}(t)$ as a power series of $g_{r}^{2}$ gives

$$\begin{aligned} F_{r}(t)=\sum _{j}\frac{g_{r}^{2j}}{j!}\mathrm{e}^{-g_{r}^{2}}\mathrm{e}^{\mathrm{i}(j\omega _{r})t} \end{aligned}$$

(19.19)

which corresponds to a progression of transitions $0\rightarrow j\,\omega _{r}$ with Franck–Condon factors (Fig. 19.2)

$$\begin{aligned} FC(0,j)=\frac{g_{r}^{2j}}{j!}\mathrm{e}^{-g_{r}^{2}}. \end{aligned}$$

(19.20)

Fig. 19.2

Progression of a low frequency mode. The Fourier transform of (19.15) is shown for typical values of $\hbar \omega =50\,\mathrm{cm}^{-1}$, 1, $E_{r}=50\,\mathrm{cm}^{-1}$ and (a) $kT=10\, \mathrm{cm}^{-1}$ (b) $kT=200\,\mathrm{cm}\,^{-1}$. A small damping was introduced to obtain finite linewidths

Fig. 19.3

Time-correlation function. $|F_{r}(t)|=\exp \left\{ g_{r}^{2}(2\overline{n}_{r}+1)(\cos (t)-1)\right\}$ is shown for $g_{r}^{2}(2\overline{n}_{r}+1)=1$(broken curve) and $g_{r}^{2}(2\overline{n}_{r}+1)=10$(full curve)

Fig. 19.4

Gaussian envelope

3 Low Frequency Modes

In the high temperature limit ($\hbar \omega _{r}\ll k_{B}T$) the time-correlation function of one oscillator (19.15) has peaks at $t=0,\pm \frac{2\pi }{\omega _{r}},\dots$ which become very sharp for large $\overline{n}_{r}\approx k_{B}T/\hbar \omega _{r}$ ³ (Fig. 19.3). The product correlation function of many oscillators is non vanishing only around $t=0$, i.e. the correlation function decays rapidly and can be approximated by the Taylor series (in this context also known as short time approximation)

$$\begin{aligned} F(t)\approx \exp \left\{ -t^{2}\sum _{r}(\overline{n}_{r}+\frac{1}{2})g_{r}^{2}\omega _{r}^{2}+it\sum _{r}g_{r}^{2}\omega _{r}\right\} \approx \exp \left\{ -t^{2}\frac{E_{R}k_{B}T}{\hbar ^{2}}+\frac{it}{\hbar }E_{R}\right\} . \end{aligned}$$

(19.21)

The lineshape is approximately given by a Gaussian (Fig. 19.4)

$$\begin{aligned} FCD(\hbar \omega )&=\frac{1}{2\pi \hbar }\int _{-\infty }^{\infty }dt\, e^{-i\omega _{0}t}\exp \left\{ -t^{2}\frac{E_{R}k_{B}T}{\hbar ^{2}}+\frac{it}{\hbar }E_{R}\right\} \\ \nonumber&=\frac{1}{2\pi \hbar }\sqrt{\frac{\pi \hbar ^{2}}{E_{R}k_{B}T}}\exp \left\{ -\frac{(\hbar \omega -E_{R})^{2}}{4E_{R}k_{B}T}\right\} \\ \nonumber&=\sqrt{\frac{1}{4\pi E_{R}k_{B}T}}\exp \left\{ -\frac{(\hbar \omega -E_{R})^{2}}{4E_{R}k_{B}T}\right\} \end{aligned}$$

(19.22)

with the reorganization energy

$$\begin{aligned} E_{r}=\sum _{r}g_{r}^{2}\hbar \omega _{r} \end{aligned}$$

(19.23)

and the variance

$$\begin{aligned} \varDelta ^{2}=2E_{R}k_{B}T. \end{aligned}$$

(19.24)