Absorption and fluorescence spectroscopy - Advanced lab course physical chemistry

Introduction

UV/Vis spectroscopy is an important method for obtaining information on the electronic and vibrational structure of molecules. This lab course allows to connect quantum mechanical concepts to quantities that can be experimentally obtained using UV/Vis spectroscopy.

Transitions in UV/Vis spectroscopy

Here, we limit our discussion to the UV/Vis spectroscopy, which utilizes electromagnetic radian in the wavelength region from $\pu{250 nm}$ to $\pu{800 nm}$ to induce molecular transitions in electronically and vibrationally excited states. We do not consider gases or solids but focus only on molecules in liquid solvents.

First, we discuss the most important processes that are relevant for UV/Vis spectroscopy, including their influence on absorption and emission spectra. In the second part of this theory, we describe the process of absorption quantitatively.

The Franck-Condon principle

Left: Schematic potential energy of a diatomic molecule in its ground state $V_0(r)$ and in an electronically excited state $V_1(r)$ as function of the distance between the nuclei $r$ with equilibrium distance of the ground state $r_{\text{eq},0}$ and of the first excited state $r_{\text{eq},1}$ . Right: Jablonski diagram with $\text S_{1,2}\leftarrow \text S_{0,0}$ transition.

Due to their small mass, electrons move considerably faster than atomic nuclei. The Franck-Condon principle states that the duration of an electronic transition is sufficiently short to consider the position of the atomic nuclei as stationary.

The electronically-excited state differs from the electronic ground state by a larger energy and by a potential energy function with a larger equilibrium distance (figure ), since electrons are excited into anti-binding molecular states.

As a consequence of the shift in the molecular equilibrium distance, the atomic nuclei of the molecule are not in the configuration of minimal energy after the excitation. The energy difference is available to the molecule as vibrational energy (vertical line in figure ). Thus, the electronic excitation is coupled with a vibrational excitation.

Absorption and emission processes in a Jablonski diagram

Notation

We use the following notation to express the direction of transitions, the multiplicity and the energy of states. States are described according to their multiplicity $\text M$ with the term symbol $\text S$ for singlet and $\text T$ for triplet states.

The electronic and vibrational state are expressed using the notation $\text M_{ev}$ . The index $e$ states the electronic state and the index $v$ the vibrational state. Excited states (with larger energy) are denoted with a prime mark. The electronic and vibrational singlet ground state index is $0$ .

According to Hund's rule, the energetic ground state of most molecules is the singlet state $\text S_{0,0}$ . Triplet states are occupied only after an initial electronic excitation.

According to the Boltzmann distribution, most molecules occupy their electronic and vibrational ground state at room temperature.

We differentiate between transitions with and without radiation. For transitions where electromagnetic radiation is involved, a straight arrow is drawn. The direction of the arrow states whether an absorption

\text M_{e'v'} \leftarrow \text M_{ev}

or an emission

\text M_{e'v'} \rightarrow \text M_{ev}

is described. Radiationless transitions are denoted using a wavy arrow.

\text M_{e'v'} \rightsquigarrow \text M_{ev}

A Jablonski diagram allows to qualitatively visualize the absorption and emission processes. The ordinate shows the electronic states as horizontal lines. (The abscissa does not map a physical quantity.) Transitions are shown as arrows.

Transitions with radiation, i.e. absorption, fluorescence and phosphorescence are processes where a photon is either emitted or absorbed. These processes allow for a direct spectroscopic measurement.

Radiationless transitions

Vibrational relaxation

During intermolecular vibrational relaxation processes (SR) the energy of a molecule is transferred to its surroundings in the form of rotational, vibrational or translational energy contributions. This relaxation usually occurs from a vibrationally excited state in a lower vibrational state within the same electronic state. (figure ). The duration of a vibrational relaxation process takes is in the order of $\pu{e-12 s}$ .

Jablonski diagram of various vibration relaxation processes within a electronic state.

Internal conversion

In contrast, intramolecular internal conversion processes (IC) are iso-energetic processes. A transition happens between energetically equivalent vibrational states of equal multiplicity but with different electronic and vibrational levels (figure ). The energy of the electronic state is thus transferred to vibrational energy. The duration of internal conversion processes is as well in the order of $\pu{e-12 s}$ .

Inter-system crossing

The intramolecular inter-system crossing process (ISC) is iso-energetic as well. In contrast to the internal-conversion process the multiplicity changes upon transition (figure ). This transition is forbidden and thus happens rarely.

Transitions with radiation

Absorption

Upon absorption (A), a molecule is excited by electromagnetic radiation from its ground state to a vibrationally excited state (figure ). The duration of an absorption process is within the order of $\pu{e-15 s}$ .

Fluorescence

The term fluorescence (F) most commonly describes a transition from the first electronically excited state in the electronic ground state. The multiplicity does not change. The excited state is usually in its vibrational ground state, since radiationless processes such as vibrational relaxation take place faster than the fluorescence (Kasha rule, figure ). The lifetime of a molecule in its excited singlet state is in the order of $\pu{e-8 s}$ .

Phosphorescence

The term phosphorescence (P) describes a state transition from an electronically-excited triplet state to a singlet state (figure ). This transition is forbidden and happens only rarely. The lifetime of a molecule in its excited triplet state is approximately $\pu{1e-3}$ to $\pu{1 s}$ .

Shift of the 0-0-transition in absorption and fluorescence spectra

Upon absorption, a molecule can transition from its vibrational ground state to an arbitrary vibrational state. The transition into the first (index 0) vibrational level of the first electronically excited state — the 0-0 transition — is the lowest-energy transition in a absorption spectrum (figure ). Since vibrational relaxation and internal-conversion processes happen on a faster time-scale than fluorescence, fluorescence occurs mostly from the electronically excited vibrational ground state $\text S_{1,0}$ into some vibrational state of the electronic ground state $\text S_{0,v}$ . Consequently, the 0-0 transition within a fluorescence spectrum is the one with the largest energy difference (figure ).

Without additional considerations, 0-0 transitions within absorption and fluorescence spectra appear to be equal. However, if the molecule is surrounded by solvent molecules in a liquid, the solvent molecules adapt their position based on the electronic configuration of the molecule in its ground state (eq). Upon absorption the electronic structure of the molecule changes. Since the solvent molecules cannot react instantaneously to this change, the excited molecule is still surrounded by solvent molecules adapted to the ground state. This is called a Franck-Condon state. Due to solvent relaxation (LR) the solvent adapts towards the new structure of the molecule. This new solvent configuration is energetically more favorable.

Immediately after fluorescence, the solvent molecule has not adapted to the new ground state of the molecule. The Franck-Condon state after fluorescence is therefore energetically larger compared to the ground state from which absorption processes initiate ( figure ).

Therefore, 0-0 transitions in an absorption spectrum are shifted to larger energies when compared to an emission spectrum. The strength of the effect depends on the solvent.

Symmetry of absorption and emission spectra

Usually, the geometric structure of a molecule does not change significantly upon an electronic state transition. Therefore, the vibrational modes of the electronically excited state are similar to those in the ground state and the energy difference between the individual vibrational states are comparable. This explains the mirror symmetry usually observed when comparing a fluorescence emission spectrum and an adsorption spectrum in the range of the $\text S_1\leftarrow \text S_0$ transition.

Quantitative derivation of the absorption process

A quantum mechanical treatment of the absorption process is rather involved, since various branches of classical physics and quantum mechanics are needed to develop a simple model of the interaction between electromagnetic radiation and matter.

First, we consider the interaction between electromagnetic radiation and a charge distribution — such as a molecule — using classical electrodynamics. Here, we find an expression for the potential energy of a molecule in an electric field (section ).

Next, we employ quantum mechanics to describe the molecule as a system with discrete energy states. Using time-dependent perturbation theory, the potential energy of a molecule in an electric field can be considered as a perturbation. As a result, we will obtain that the energy state of the molecule can change upon the interaction with electromagnetic radiation (section ).

Considering the irradiated energy of the electromagnetic radiation in relation to the absorbed energy of the molecules in solution we are able to link the absorption process with intensity measurements that are experimentally available from UV/Vis spectra (section ).

Using the Franck-Condon approximation we consider electronic states and vibrational states separately. This allows to quantitatively describe electronic and vibrational transitions separately (section ). During this derivation, we introduce the quantities current density and density with respect to the energy of the electromagnetic radiation.

Classical electrodynamics

Electromagnetic radiation

Electromagnetic radiation can be described as an electric and a magnetic field, both of which oscillate as a function of time and propagate through space with the speed of light $c$ in a direction that is perpendicular to both fields. Here, we consider a setup where the electronic wave propagates in $x$ direction, and is polarized such that the electric field oscillates in the $y$ direction.

We do not consider interactions with the magnetic field.

The $y$ component of the electric field $g_y$ of an electromagnetic wave is a function of time $t$ and position $x$ . Since a molecule is typically orders of magnitudes smaller than the wavelength of the electromagnetic radiation used in most spectroscopy methods, the electromagnetic field strength of a single molecule is approximately only a function of time.

Spectral decomposition of electromagnetic radiation

Electromagnetic radiation can be considered as a superposition of individual harmonic electromagnetic waves. Each of the harmonic oscillations with angular frequency $\omega$ and a amplitude of $g_{0,y}$ can be expressed as

g_y(t, \omega) = g_{0,y}(\omega)\cos{(\omega t)}.

Here, we first consider individual harmonic oscillations and will later consider the superposition/integration over all angular frequencies. If a quantity relates to a single harmonic oscillation, we call it spectral, denote it with a lower case symbol and explicitly write its dependence on the angular frequency $\omega$ . The corresponding integrated (non-spectral) quantity is denoted by the corresponding upper case symbol.

Electromagnetic radiation carry energy. Since this energy transport is closely linked with the quantities that are available experimentally, we define here first the important quantities density and current density with respect to the energy of electromagnetic radiation.

Radiation energy density and radiation energy current density

The radiation energy density $U$ is the energy $E$ of electromagnetic radiation per volume $V$ .

The radiation energy current density $J$ is the energy $E$ of electromagnetic radiation with passes through an area $A$ within a time duration $t$ .

U = \frac{\mathrm dE}{\mathrm dV}\quad\quad\quad J = \frac{\mathrm dE}{A\mathrm dt}

Here we consider the important case where the radiation is homogeneous and propagates in parallel in a fixed direction. Then, then volume $V$ is the product of the area that is perpendicular to the direction of propagation $A$ within the distance traveled in the duration $t$ . This distance is the product of time $t$ and propagation speed $c$ . It follows for equation that

U = \frac{\mathrm dE}{\mathrm dV} = \frac{\mathrm dE}{A \mathrm dx} = \frac{1}{c}\frac{\mathrm dE}{A \mathrm dt} = \frac{1}{c}J

The spectral radiation energy density $u(\omega)$ is defined such that the expression $u(\omega)\mathrm d\omega$ corresponds to the radiation energy density in the angular frequency range $\omega \dots \omega+\mathrm d\omega$ .

The spectral radiation energy current density $j(\omega)$ is often called intensity. The expression $j(\omega) \mathrm d \omega$ states the radiation energy current density in the angular frequency range $\omega \dots \omega+\mathrm d\omega$ .

\begin{alignat*}{5} U &= \int_0^\infty u(\omega)\mathrm d\omega &\quad &\Leftrightarrow &\quad \mathrm d U &= u(\omega)\mathrm d\omega\\ J &= \int_0^\infty j(\omega)\mathrm d\omega & &\Leftrightarrow & \mathrm d J &= j(\omega)\mathrm d \omega. \end{alignat*}

Taking the derivative of both sides of equation with respect to $\omega$ yields equations

u(\omega)=\frac{1}{c}j(\omega).

From equation it follows with $\mathrm dx=c\mathrm d t$ that the change of the spectral energy density with time corresponds to the change of the spectral energy current density with position.

\frac{\mathrm du(\omega)}{\mathrm d t}=\frac{\mathrm d j(\omega)}{\mathrm d x}

Spectral energy density of electromagnetic radiation

The spectral radiation energy density $u(\omega)$ of an electromagnetic wave is given within classic electrodynamics with the permittivity $\varepsilon_0$ in vacuum as

u(\omega)=\frac{1}{2}\varepsilon_0 \left(g_{0,y}(\omega)\right)^2.

Rearranging yields an expression (needed later) for the square of the spectral amplitude $g_0(\omega)$ as function of the spectral radial energy density $u(\omega)$ .

\left(g_{0,y}(\omega)\right)^2=\frac{2}{\varepsilon_0}u(\omega)

Potential energy of molecules in an electrical field

A molecule exhibits a charge distribution. The electric field of this charge distribution can be calculated using the Maxwells equation within the framework of classical electrodynamics. This computation is generally complex and can be approximated with a series expansion (multipole expansion). The individual contributions to the series expansion correspond to electrical monopoles, dipoles and quadrupoles (as well as higher-order contributions).

In good approximation the terms which are of higher order than the dipole term can be ignored. Additionally, if the molecule does not carry a net charge, the monopole term is zero as well. Thus, in the dipole approximation we only consider the dipole moment of the molecule.

The potential energy $V_\text{dipole}$ of an individual dipole with dipole moment $\vec{\mu}$ in an electric field of strength $\vec{g}$ is given within classical electrodynamics as the dot product of dipole moment and electric field strength

V_\text{dipole} = \vec{\mu}\cdot\vec{g} = \mu_x g_x + \mu_y g_y + \mu_z g_z

The squared absolute value of the dipole moment is

\mu^2 := |\vec{\mu}|^2 = \mu_x^2+\mu_y^2+\mu_z^2.

The spatial orientation of molecules in solution is isotropic, which means that in solution a measurement of the dipole moment would reveal that $\mu_x=\mu_y=\mu_z$ . Using equation we can thus write

\frac{1}{3}\mu^2 = \mu_x^2= \mu_y^2= \mu_z^2.

Quantum mechanical basics

In the following we reproduce the formalism of time-dependent perturbation theory with a focus on spectroscopy.

Notation and mathematical foundations

All position coordinates, which are necessary to describe a system of $F$ particles are summarized in a $3F$ dimensional vector $\vec{x} = (x_1, x_2, \dots, x_{3F})$ . An integration over all position coordinates is expressed using the variable $\tau$ such that $\text d \tau = \text d x_1 \text d x_2 \dots \text d x_{3F}$ .

Partial derivatives of a quantity with respect to time $t$ are denoted by a dot above the quantity.

The Kronecker delta $\delta_{nm}$ is defined for discrete quantities $n$ and $m$ such that

\sum_{n=0}^{\infty} f_n\delta_{nm}=f_m

Which means that:

\begin{aligned} \delta_{nm} = \begin{cases} 0 & \text{for}\quad n\neq m \\ 1 & \text{for}\quad n=m \end{cases} \end{aligned}

Similarly, the $\delta$ distribution is defined for continuous variables $x$ and $x_0$ as

\int_{-\infty}^{\infty} f(x)\delta(x-x_0)\mathrm d x=f(x_0)\quad\Rightarrow\quad \int_{-\infty}^{\infty} \delta(x-x_0)\mathrm d x=1

This corresponds to a function with an arbitrarily sharp maximum at position $x_0$ . The behavior of physical quantities with a single sharp maximum can be approximated using the $\delta$ distribution.

Time-dependent perturbation theory

A system of atomic nuclei and electrons can be described using $N$ orthonormal states. In this unperturbed case each state is described by a wave function $\Psi_n(\vec{x},t)$ , which solves the time-dependent Schrödinger equation with the Hamilton operator of the unperturbed system $\hat{H}^{(0)}$ .

\hat{H}^{(0)}\Psi_n(\vec{x},t) = \mathrm i\hbar\dot{\Psi}_n(\vec{x},t)\quad\text{for}\quad n=1, \dots, N

If the Hamilton operator of the unperturbed system $\hat{H}^{(0)}$ does not depend on time, the wave function can be written as a product of a position-dependent wave function $\Phi_n(\vec{x})$ and of a time-dependent wave function $\Theta_n(t)$ .

\Psi_n(\vec{x},t)=\Phi_n(\vec{x})\Theta_n(t)\quad\text{for}\quad n=1, \dots, N

The substitution of this separation into the time-dependent Schrödinger equation (equation ) yields for the position-dependent wave function the following time-independent Schrödinger equation

\hat{H}^{(0)}\Phi_n(\vec{x}) = E_n \Phi_n(\vec{x})\quad\text{for}\quad n=1,\dots, N

and for the time-dependent wave function the expression

\Theta_n(t) = \exp{\left(-\frac{\mathrm i}{\hbar}E_n t\right)}\quad\text{for}\quad n=1,\dots, N.

The constant $E_n$ is the energy of the system in the state $\Psi_n$ .

A perturbed state $\Psi$ of the system can be described as a linear combination over all unperturbed states $\Psi_n$ . If the perturbation does depend on time, the coefficients $c_n$ of the linear combination can depend on time as well.

\Psi(\vec{x},t) = \sum_{n=1}^{N} c_n(t)\Psi_n(\vec{x},t) = \sum_{n=1}^{N} c_n(t)\Phi_n(\vec{x})\exp{\left(-\frac{\mathrm i}{\hbar}E_n t\right)}

The wave function $\Psi$ can be written using the time-dependent Schrödinger equation and the Hamilton operator $\hat{H}$ , which is composed of the unperturbed Hamilton operator $\hat{H}^{(0)}$ and the Hamilton operator of the perturbation $\hat{H}^{(1)}$ . If the perturbation is switched on at time $t=0$ , the Hamilton operator of the perturbation $\hat{H}^{(1)}$ is zero for all times $t<0$ .

\hat{H}(t)\Psi(\vec{x},t) = \mathrm i\hbar\dot{\Psi}(\vec{x},t)\quad\text{with}\quad\hat{H}(t)=\hat{H}^{(0)}+\hat{H}^{(1)}(t)

Substitution of the linear combination in equation yields

\begin{aligned} \sum_{n=1}^{N} &c_n(t)\underbrace{\hat{H}^{(0)}\Psi_n(\vec{x},t)}_{(\ast)} + \sum_{n=1}^{N}c_n(t)\hat{H}^{(1)}(t)\Psi_n(\vec{x},t)\\= \sum_{n=1}^{N}&c_n(t)\underbrace{\mathrm i\hbar\dot{\Psi}_n(\vec{x},t)}_{(\ast\ast)}+ \sum_{n=1}^{N}\dot{c}_n(t)\mathrm i\hbar\Psi_n(\vec{x},t). \end{aligned}

The parts marked with ( $\ast$ ) and ( $\ast\ast$ ) are equal according to equation . This allows to obtain the following expression from equation

\begin{aligned} \sum_{n=1}^{N} c_n(t)\hat{H}^{(1)}(t)\Psi_n(\vec{x},t)&= \sum_{n=1}^{N}\dot{c}_n(t)\mathrm i\hbar\Psi_n(\vec{x},t). \end{aligned}

Using the separation from equation and results in

\begin{aligned} &\sum_{n=1}^{N} c_n(t)\hat{H}^{(1)}(t)\Phi_n(\vec{x})\exp{\left(-\frac{\mathrm i}{\hbar}E_n t\right)}\\= &\sum_{n=1}^{N}\dot{c}_n(t)\mathrm i\hbar\Phi_n(\vec{x})\exp{\left(-\frac{\mathrm i}{\hbar}E_n t\right)} \end{aligned}

By multiplying both sides from the left with the complex-conjugate of an arbitrary wave function $\Phi_k^*(\vec{x})$ of the system and subsequent integration over all position coordinates yields

\begin{aligned} &\sum_{n=1}^{N} c_n(t)\underbrace{\int_{-\infty}^{\infty}\Phi_k^*(\vec{x})\hat{H}^{(1)}(t)\Phi_n(\vec{x})\text d\tau}_{=: H_{nk}^{(1)}(t)}\exp{\left(-\frac{\mathrm i}{\hbar}E_n t\right)}\\= &\sum_{n=1}^{N}\dot{c}_n(t)\mathrm i\hbar\underbrace{\int_{-\infty}^{\infty}\Phi_k^*(\vec{x})\Phi_n(\vec{x})\text d\tau}_{= \delta_{nk}}\exp{\left(-\frac{\mathrm i}{\hbar}E_n t\right)}. \end{aligned}

The term with the parenthesis on the left-hand side of the equation is denoted as $H_{nk}^{(1)}(t)$ . (The separation of the time-dependent Hamilton operator of the perturbation from the time-dependent solution of the wave function is not generally possible. Later in this script, the perturbation operator will be decomposed as the product of a position-dependent operator and a time-dependent function (see section , equation ), which means that the separation is possible in this case.) The term with the parenthesis on the right-hand side of the equation is either one or zero, since the wave functions are orthogonal and normalized. The right-hand side can be simplified such that

\sum_{n=1}^{N} c_n(t)H_{nk}^{(1)}(t)\exp{\left(-\frac{\mathrm i}{\hbar}E_n t\right)}=\dot{c}_k(t)\mathrm i\hbar\exp{\left(-\frac{\mathrm i}{\hbar}E_k t\right)}.

This equation allows to extract the time-derivative of the coefficients of state $k$ , denoted as $\dot{c}_k$ . The importance of the definition $\omega_{nk := \frac{E_k - E_n}{\hbar}}$ as resonance angular frequency will be obvious later.

\dot{c}_k(t)=-\frac{\mathrm i}{\hbar} \sum_{n=1}^{N} c_n(t)H_{nk}^{(1)}(t)\exp{\left(\mathrm i \omega_{nk} t\right)}

Integration over time from $0$ to $t$ yields an integral (equation ), which can only be solved analytically in a few cases (for example, for certain systems with only two states). An approximate solution can be obtained by recursively substituting the left-hand side (red) into the right-hand side (blue) During the substitution all indices are mapped according to $n\mapsto m$ and $k \mapsto n$ .

{\color{red}{c_k(t)}}=c_k(0)-\frac{\mathrm i}{\hbar} \int_0^t\text d t'\sum_{n=1}^{N} {\color{blue}{c_n(t')}}H_{nk}^{(1)}(t')\exp{\left(\mathrm i\omega_{nk}t'\right)}

Performing the substitution once yields the terms shown in equation .

\begin{aligned} c_k(t)= &+c_k(0)&\text{\small{Term 0}}&\qquad\\ &-\frac{\mathrm i}{\hbar}\sum_{n=1}^{N}\int_0^t\mathrm dt' c_n(0)H_{nk}^{(1)}(t')\exp{\left(\mathrm i \omega_{nk} t'\right)}&\text{\small{Term 1}}&\\ &+\left(\frac{\mathrm i}{\hbar}\right)^2 \sum_{n=1}^{N}\sum_{m=1}^{N}\int_0^t \mathrm dt'\int_0^{t'} \mathrm dt'' c_m(t'')H_{mn}^{(1)}(t'')H_{nk}^{(1)}(t')\exp{\left(\mathrm i \omega_{mn} t''+\mathrm i \omega_{nk} t'\right)}&\text{\small{Term 2}}&\\ &-\dots &\text{\small{further terms for multiple substitutions}}& \end{aligned}

The terms can be sorted according to the number of applications of the perturbation operator $\hat{H}^{(1)}$ .

Term 0 corresponds to a zero-order perturbation. The coefficient of the state $\Psi_k$ does not depend on time, which means the system is left unchanged and the perturbation is not accounted for.

Term 1 corresponds to a first-order perturbation, the system can transition from a state $\Psi_n$ to a different state $\Psi_k$ , the perturbation operator is applied once.

Term 2 describes a second-order perturbation. This allows for a transition from a state $\Psi_n$ via a state $\Psi_m$ to a state $\Psi_k$ , since the perturbation operator is applied twice.

If the influence of the perturbation operator $\hat{H}^{(1)}$ is small compared to the unperturbed Hamilton operator $\hat{H}^{(0)}$ , terms of higher order can be neglected. In der following, we only consider first-order perturbations ignoring terms including and following term 2 in equation .

Energy diagram of a system with three states. The state transitions from the ground state $\Psi_n$ which are covered by the corresponding perturbation model are shown with an arrow.

If we consider the system at time $t=0$ in state $\Psi_a$ , the coefficient $c_a(0)$ equals one and all other coefficients are zero. With $c_n(0) = \delta_{na}$ we can compute the sum in term 1 of equation and obtain

c_b(t)=\frac{1}{\mathrm i\hbar} \int_0^t H_{ab}^{(1)}(t')\exp{\left(\mathrm i \omega_{ab} t'\right)}\text d t'.

Oscillating perturbations

If the perturbation energy oscillates harmonically as function of time with the angular frequency $\omega$ and an amplitude, that can be written as the product of a time-independent perturbation operator $\hat{H}^{(1)'}$ and a spectral amplitude $s_0(\omega)$ , the time-dependent perturbation operator $\hat{H}^{(1)}(t)$ for a perturbation starting at time $t=0$ can be written as

\hat{H}^{(1)}(t) = \begin{cases}\hat{H}^{(1)'}s_0(\omega)\cos(\omega t)&\text{for}\quad t\ge0\\ 0&\text{for}\quad t<0 \end{cases}

For times $t\ge0$ we can write

\hat{H}^{(1)}(t)=\frac{\hat{H}^{(1)'}s_0(\omega)}{2}\left(\exp{\left(\mathrm i\omega t\right)}+\exp{\left(-\mathrm i\omega t\right)}\right).

Substitution into equation yields

\begin{aligned} c_b(t)&=\frac{s_0(\omega)H^{(1)'}_{ab}}{2\mathrm i\hbar} \int_0^t \left(\exp{\left(\mathrm i\left(\omega_{ab}-\omega\right) t'\right)}+\exp{\left(\mathrm i\left(\omega_{ab}+\omega\right) t'\right)}\right)\text d t'\\ &=\frac{s_0(\omega)H^{(1)'}_{ab}}{\mathrm 2i\hbar}\left(\frac{\exp{\left(\mathrm i\left(\omega_{ab}-\omega\right) t\right)}-1}{\mathrm i\left(\omega_{ab}-\omega\right)}+\frac{\exp{\left(\mathrm i\left(\omega_{ab}+\omega\right) t\right)}-1}{\mathrm i\left(\omega_{ab}+\omega\right)}\right). \end{aligned}

Two cases can be differentiated, namely $\omega\rightarrow\omega_\text{ab}$ and $\omega\rightarrow -\omega_\text{ab}$ . As an approximation we obtain

c_b(t)=\frac{-s_0(\omega)H^{(1)'}_{ab}}{2\hbar}\frac{\exp{\left(\mathrm i\left(\omega_{ab}-\omega\right) t\right)}-1}{\omega_{ab}-\omega}.

Transition probability

The squared absolute value of the coefficient $c_b$ can be interpreted as the probability for a transition of the system from state $\Psi_a$ to state $\Psi_b$ , the so-called spectral transition probability $p_{ab}(\omega)$ . Equation yields

\begin{aligned} p_{ab}(\omega) :&=\left|c_b(t)\right|^2\\&=\frac{\left(s_0(\omega)\right)^2\left|H_{ab}^{(1)'}\right|^2}{4\hbar^2}t^2\underbrace{\left(\frac{\sin{\left(\frac{t}{2}\left(\omega_{ab}-\omega\right)\right)}}{\frac{t}{2}\left(\omega_{ab}-\omega\right)}\right)^2}_{\mathclap{\text{sharp maximum at } \omega \rightarrow \omega_{ab}}}. \end{aligned}

The term in parenthesis can be approximated for large times $t$ as $\delta$ distribution (figure ), that only allows transitions between two systems with the same energy difference as the energy of the electromagnetic radiation, which is consistent with the preservation of energy.

Plot of the term in parenthesis in equation at variable time

t

. The function approximates the

\delta

distribution. The value range for the angular frequency matches the wavelength range

\pu{250-800nm}

In order to match the prerequisite for the $\delta$ distribution of equation , the integral over the term in the parenthesis over all angular frequencies $\omega_{ab}$ must be one. Using the substitution $u := \frac{t}{2}\left(\omega_{ab}-\omega\right)$ yields

\begin{aligned} \int_{-\infty}^{\infty}\left(\frac{\sin{\left(\frac{t}{2}\left(\omega_{ab}-\omega\right)\right)}}{\frac{t}{2}\left(\omega_{ab}-\omega\right)}\right)^2\mathrm d\omega_{ab} = \frac{2}{t} \int_{-\infty}^{\infty}\left(\frac{\sin{u}}{u}\right)^2\mathrm du\\ = \frac{2}{t}\pi\cdot 1= \frac{2\pi}{t}\underbrace{\int_{-\infty}^{\infty}\delta\left(\omega_{ab}-\omega\right)\mathrm d\omega_{ab}}_{=1}. \end{aligned}

By comparison we obtain

\left(\frac{\sin{\left(\frac{t}{2}\left(\omega_{ab}-\omega\right)\right)}}{\frac{t}{2}\left(\omega_{ab}-\omega\right)}\right)^2= \frac{2\pi}{t}\delta\left(\omega_{ab}-\omega\right)

and the spectral transition probability is

\begin{aligned} p_{ab}(\omega) &= \frac{\pi \left(s_0(\omega)\right)^2}{2\hbar^2}\left|H_{ab}^{(1)'}\right|^2 t\delta\left(\omega_{ab}-\omega\right). \end{aligned}

Fermi's golden rule

The change of the spectral transition probability $p$ with time is called spectral transition rate $w$ .

w_{ab} := \frac{\mathrm dp_{ab}}{\mathrm d t}

With equation we obtain Fermi's golden rule. (In the literature, one often finds that $s_0=1$ ).

w_{ab}=\frac{\pi \left(s_0(\omega)\right)^2}{2\hbar^2}\left|H_{ab}^{(1)'}\right|^2\delta(\omega_{ab}-\omega)

The transition probability is thus proportional to the squared absolute value of the integral

H_{ab}^{(1)'} = \int_{-\infty}^{\infty} \Phi_b^*(\vec{x}) \hat{H}^{(1)'}\Phi_a(\vec{x}) \mathrm d \tau.

Within the context of the first-order perturbation theroy, this equation represents a selection rule, since the transition from $\Psi_a$ to $\Psi_b$ is only possible if the integral $H_{ab}^{(1)'}$ of both states does not vanish.

Transitions due to interactions with electromagnetic radiation

According to the dipole approximation of section , the interaction energy of an electric field with the charge distribution of a molecule is given by the product of dipole moment and electric field strength. This energy can be considered using the perturbation operator $\hat{H}^{(1)}$ as a perturbation of the system. If the electric field oscillates in the $y$ direction (by irradiating electromagnetic radiation into the $x$ direction), we can write analogously to equation :

\hat{H}^{(1)} = \begin{cases}-\hat{\mu}_y(\vec{x})g_y(t, \omega)&\text{for}\quad t\geq0\\ 0&\text{for}\quad t<0 \end{cases}

Comparing equation with the ansatz for the oscillating perturbation (equation ) yields that $s_0 = -g_{0,y}$ and $\hat{H}^{(1)'} = \hat{\mu}_y$ . For an isotropic distribution of molecules according to equation and equation the transition probability is

w_{ab} = \frac{\pi}{6\hbar^2}\left|\mu_{ab}\right|^2\left(g_{0,y}(\omega)\right)^2\delta(\omega_{ab}-\omega).

The expectation value $\mu_{ab}$ is called transition dipole moment and is defined as

\mu_{ab}=\int_{-\infty}^\infty \Phi_b^*(\vec x) \hat\mu_y \Phi_a(\vec x)\mathrm d\tau.

It follows from equation by substituting $\left(g_{0,y}(\omega)\right)^2$ that

w_{ab} = \frac{\pi}{3\varepsilon_0\hbar^2}\left|\mu_{ab}\right|^2u(\omega) \delta(\omega_{ab}-\omega).

Integration the spectral transition rate $w_{ab}$ over all angular frequencies of the incoming radiation allows to obtain the transition rate $W_{ab}$ .

W_{ab} = \int_{0}^{\infty} w_{ab}(\omega)\mathrm d\omega = \int_{0}^{\infty}\mathrm d\omega \frac{\pi}{3\varepsilon_0\hbar^2}\left|\mu_{ab}\right|^2u(\omega) \delta(\omega_{ab}-\omega)

The $\delta$ distribution allows to perform the integration with the resonance angular frequency $\omega_{ab}$ . Therefore, the transition rate is a function of the spectral radiation density at a resonance angular frequency of $\omega_{ab}$ .

W_{ab} = \frac{\pi}{3\varepsilon_0\hbar^2}\left|\mu_{ab}\right|^2 u(\omega_{ab})

Absorption measurements

The expression in equation for the transition probability is not a directly measurable, experimentally available quantity. In the following, we derive a connection between absorption spectra and the transition probability.

Spectral molecular density

The resonance angular frequency $\omega_{ab}$ of a molecule can change due to different surroundings by solvent molecules. To capture this effect, we introduce the spectral molecular density $\rho_\text N(\omega_{ab})$ such that the expression $\rho_\text N(\omega_{ab}) \mathrm d \omega_{ab}$ yields the number of molecules per volume with a resonance angular frequency in the range of $\omega_{ab} \dots \omega_{ab}+\mathrm d\omega_{ab}$ . Integration over an absorption band yields the number of excitable molecules that might contribute to the absorption band for the transition $b\leftarrow a$ . The number of molecules per volume is available from Avogadro's constant $N_\text A$ and the molecule concentration $c_\text B$ .

\int_{\substack{\text{Absorption}\\b\leftarrow a}} \rho_\text N(\omega_{ab}) \mathrm d \omega_{ab} = N_\text A \frac{n_\text B}{V} = N_\text A c_\text B

The Lambert-Beer law

We consider here a sample substance dissolved in a non-absorbing solvent. This solution with known concentration $c_\text B$ is placed in a cuvette with thickness $d$ . The cuvette is irradiated with monochromatic electromagnetic radiation with angular frequency $\omega$ and intensity $j(\omega, x=0)$ . The intensity of the radiation exiting from the cuvette, $j(\omega, x=d)$ is detected.

According to the Lambert-Beer law the decrease of the intensity $\mathrm dj$ per thickness increment $\mathrm dx$ is proportional to the intensity $j$ and the molecule concentration $c_\text B$ with the proportionality constant $\kappa'$ .

\frac{\mathrm d j(\omega,x)}{\mathrm d x} = -\kappa'(\omega)c_\text B j(\omega)

By separation of the variables and subsequent integration we obtain an equation that allows to determine the molar decadic absorption coefficient $\kappa$ from the decadic absorption $A$ determined by, e.g., a spectrometer.

A := -\log\left(\frac{j(\omega,x=d)}{j(\omega,x=0)}\right) = \kappa(\omega)c_\text B d\quad\text{with}\quad \kappa := \frac{\kappa'}{\ln(10)}

Typically, spectrometer do not measure the intensity $j$ but the number of photons in a narrow angular frequency range per area and time, i.e., the spectral photon current density. Due to the proportionality the decadic absorption can be determined analogously.

Determination of the transition dipole moment by absorption measurements

During absorption, energy from the electric field is transferred to the molecules in solution. The time-derivative of the spectral radian energy density $u(\omega)$ can be determined as a product of transition rate $W_{ab}$ , of the energy of the absorbed electromagnetic radiation $\hbar\omega$ and of the spectral molecular density $\rho_\text N(\omega)$ . The sign is negative, since the spectral radiation energy density increases upon absorption. Using the transition rate from equation we obtain

\begin{align*} \frac{\mathrm du(\omega)}{\mathrm d t} &= -\omega \hbar W_{ab}\rho_\text N(\omega)\\ &=-\omega \frac{\pi}{3\varepsilon_0\hbar}\left|\mu_{ab}\right|^2 u(\omega)\rho_\text N(\omega). \end{align*}

From equation we obtain with equation and equation the following expression for the differential decrease of the intensity.

\begin{align*} \frac{\mathrm d j(\omega, x)}{\mathrm d x}=-\frac{\pi}{3\varepsilon_0c\hbar}\left|\mu_{ab}\right|^2\omega \rho_\text N(\omega)j(\omega) \end{align*}

Comparison of equation with the Lambert-Beer law in equation yields

\frac{\kappa(\omega)}{\omega} = \frac{\pi}{3\varepsilon_0c\hbar c_\text B\ln{10}}\left|\mu_{ab}\right|^2\rho_\text N(\omega).

Integration over one absorption band yields with equation that

\int_{\substack{\text{Absorption}\\b\leftarrow a}}\frac{\kappa(\omega)}{\omega}\mathrm d \omega=\frac{\pi N_\mathrm A}{3\varepsilon_0c\hbar\ln{10}}\left|\mu_{ab}\right|^2.

The integral on the left-hand side is called integral absorption and is proportional to the squared magnitude of the transition dipole moment.

The Franck-Condon approximation

Preparation

We assign to each position-dependent wave function $\Phi$ an electronic and a vibrational state. Instead of denoting the $n$ th energy level by writing $\Phi_n$ we introduce a new notation that assigns the electronic and vibrational state to the system. The notation $\Phi_{ev}$ species, that the system is in the $e$ th electronic state and in the $v$ th vibrational state. Additionally, we separate all position coordinates $\vec{x}$ in position coordinates of the nuclei $\vec{x}_\text K$ and of the electrons $\vec{x}_\text E$ . The integration over $\tau$ is similarly split over the nucleus coordinates $\tau_\text K$ and the electron coordinates $\tau_\text E$ .

\begin{aligned} \Phi_n(\vec{x}) &\mapsto \Phi_{ev}(\vec{x}_\text K,\vec{x}_\text E)\\ \mathrm d \tau &\mapsto \mathrm d \tau_\text K \mathrm d \tau_\text E \end{aligned}

For general vibronic transitions, both the electronic and the vibrational state can change:

\begin{aligned} \text{Transition}\ a\ \text{to}\ b &\mapsto \text{Transition}\ e\ \text{to}\ e'\ \text{and}\ v\ \text{to}\ v' \end{aligned}

In the Franck-Condon approximation, each position-dependent wave function $\Phi_{ev}(\vec{x}_\text K,\vec{x}_\text E)$ is written as the product of a vibrational wave function $\xi_{ev}$ and an electronic wave function $\phi_e$ .

According to Condon , a separation similar to the Born-Oppenheimer approximation is performed, such that the vibrational wave function depends only on the nucleus coordinates and the electronic wave function for a set of given, fixed nuclei coordinates only depends on the electron coordinates. The latter parametric dependency of the electronic wave function on the coordinates of the nuclei is denoted with the semicolon:

\Phi_{ev}(\vec{x}_\text K,\vec{x}_\text E) = \xi_{ev}(\vec x_\text K) \phi_e(\vec x_\text E; \vec x_\text K)

For the further calculation we need to separate the dipole operator as well as a sum where the first summand depends on the electron coordinates $\hat{\mu}_\text E(x_\text E)$ and the second on the coordinates of the nuclei $\hat{\mu}_\text K(x_\text K)$ .

\hat{\mu}(\vec{x}) = \hat{\mu}_\text E(x_\text E)+\hat{\mu}_\text K(x_\text K)

The transition dipole moment in the Franck-Condon approximation

From the above separation (equation and ) we obtain for the transition dipole moment the expression shown in equation .

> \begin{aligned} \mu_{ab}&=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \xi_{e'v'}^*(\vec x_\text K) \phi_{e'}^*(\vec x_\text E; \vec x_\text K) \left(\hat{\mu}_\text K(\vec x_\text K) + \hat{\mu}_\text E(\vec x_\text E)\right)\xi_{ev}(\vec x_\text K) \phi_{e}(\vec x_\text E; \vec x_\text K)\mathrm d \tau_\text E \mathrm d \tau_\text K\\ &=\int_{-\infty}^{\infty} \xi_{e'v'}^*(\vec x_\text K)\underbrace{\left(\int_{-\infty}^{\infty} \phi_{e'}^*(\vec x_\text E; \vec x_\text K) \hat{\mu}_\text E(\vec x_\text E) \phi_{e}(\vec x_\text E; \vec x_\text K) \mathrm d \tau_\text E\right)}_{=:\mu_{ee'}^\text{E}} \xi_{ev}(\vec x_\text K)\mathrm d \tau_\text K +\int_{-\infty}^{\infty} \xi_{e'v'}^*(\vec x_\text K) \hat{\mu}_\text K(\vec x_\text K)\underbrace{\left(\int_{-\infty}^{\infty}\phi_{e'}^*(\vec x_\text E; \vec x_\text K)\phi_{e}(\vec x_\text E; \vec x_\text K)\mathrm d \tau_\text E\right)}_{=\delta_{ee'}} \xi_{e'v'}^*(\vec x_\text K)\mathrm d \tau_\text K \end{aligned}

The first term with parenthesis is called electronic transition dipole moment $\mu_{ee'}^\text{E}$ and the second term in parenthesis vanishes for a vibronic transition, since the electronic wave functions are orthogonal.

In good approximation the electronic transition dipole moment $\mu_{ee'}^\text E$ does not depend on the coordinates of the nuclei and can be moved outside of the integral. The resulting expression contains the integral overlap of the vibrational wave functions $S_{(ev)(e'v')}$ .

\begin{aligned} \mu_{ab}&=\mu_{ee'}^\text{E}\int_{-\infty}^{\infty} \xi_{e'v'}^*(\vec x_\text K) \xi_{ev}(\vec x_\text K) \mathrm d \tau_\text K\\ &=\mu_{ee'}^\text{E}S_{(ev)(e'v')} \end{aligned}

From equation it follows for the integration over a vibronic transition in a spectrum that

\begin{aligned} \int_{\substack{\text{vibronic}\\\text{absorption}\\e'\leftarrow e\\v'\leftarrow v}}\frac{\kappa(\omega)}{\omega}\mathrm d \omega&=\frac{\pi N_\mathrm A}{3\varepsilon_0c\hbar\ln{10}}\left|\mu_{ee'}^\text E\right|^2\left|S_{(ev)(e'v')}\right|^2. \end{aligned}

The absolute square value of the overlap integral $\left|S_{(ev)(e'v')}\right|^2$ is called Franck-Condon factor.

Under the assumption that all molecules are in their electronic and vibrational ground state $e=0,\ v=0$ , it is possible to determine the electronic transition dipole moment by integration over all vibrational absorption bands of an electronically excited state $e'$ , since the sum over all squared of the overlap integral is one, due to their normalization. If $k$ vibrational transitions into the electronic state $e'$ are observed, the integration over all vibrational transitions yields

\sum_{v'=0}^{k-1}\left|S_{(0,0)(e'v')}\right|^2 = 1

and it thus follows that

\begin{aligned} \int_{\substack{\text{electronic}\\\text{absorption}\\e'\leftarrow 0}}\frac{\kappa(\omega)}{\omega}\mathrm d \omega&=\frac{\pi N_\mathrm A}{3\varepsilon_0c\hbar\ln{10}}\left|\mu_{0,e'}^\text E\right|^2\sum_{v'=0}^{k-1}\left|S_{(0,0)(e'v')}\right|^2\\ &=\frac{\pi N_\mathrm A}{3\varepsilon_0c\hbar\ln{10}}\left|\mu_{0,e'}^\text E\right|^2. \end{aligned}

As a result, absorption measurement allow to determine the electronic transition dipole moment for a transition from the ground state to en electronically excited state.

Experimental setup

The setup of an absorption spectrometer is shown in figure . The switching of the beam path between the cuvette containing the sample and the cuvette containing a reference substance is achieved using a beam splitter. Using a photomultiplier, the intensity of the radiation exiting the cuvette $j(\omega, x=d)$ or the reference intensity $j(\omega, x=0)$ at the angular frequency determined by the excitation monochromator can be measured.

The setup of a fluorometer is shown in figure . The intensity of the emitted electromagnetic radiation $j_\text{F}(\omega)$ is determined at an angle of 90° relative to the incoming radiation. The reference intensity $j(\omega, x=0)$ is determined by a photomultiplier. The beam path can be switched again with a beam splitter. Using the excitation monochromator, electromagnetic radiation within a certain wavelength region is sent to the sample. The intensity of the emitted light is determined after it has passed through an emission monochromator as a function of the wavelength.

Fluorescence emission spectrum

For recording a fluorescence emission spectrum electromagnetic radiation with a fixed wavelength is irradiated on the sample and the intensity of the resulting light is measured as function of wavelength.

The wavelength of the excitation light (excitation wave length) has to be selected such that the radiation is absorbed by the molecules. Of course the energy of the incoming radiation is always larger than the possible spectral energy of the emitted light, which needs to be considered as well. Thus, the excitation wavelength has to be smaller than the lowest emission wavelength that we wish to detect. Additionally, The excitation wavelength should be reduced by the spectral band width of the excitation monochromator to reduce Rayleigh scattering.

Fluorescence excitation spectrum

For recording a fluorescence excitation spectrum light with a variable wavelength is irradiated on the sample and the intensity of the emitted light at a fixed wavelength is detected.

The wavelength at which the intensity is detected should be larger than the maximum wave length of the spectrum and within a spectral region with a strong emission of the sample.

Spectral bandwidth of the monochromator

By adjusting the emission slit width of the excitation monochromator the spectral bandwidth — the width of the energy range of the almost-monochromatic electromagnetic radiation — is determined. The smaller the exit slit width the smaller the spectral bandwidth of the light, and the smaller the intensity of the emitted light.

Similarly, the exit slit width of the emission monochromator allows to determine the spectral bandwidth of the emitted electromagnetic light.

Measured quantities when recording a fluorescence spectrum

Electromagnetic radiation with intensity $j(\omega, x=0)$ passes though the cuvette with a dissolved sample and exits the cuvette with intensity $j(\omega, x=d)$ . The fluorescence intensity $j_\text{F}(\omega)$ is proportional to the number of excited molecules, i.e.

j_\text{F}(\omega)\propto j(\omega, x=0)-j(\omega, x=d).

The quantity $j(\omega, x=d)$ can be expressed using the Lambert-Beer law in equation . The exponential function can be approximated for small concentrations $c_\text B$ and for small decadic absorption coefficients $\kappa$ in a Taylor series up to the linear term.

\begin{aligned} j(\omega, x=d)&=j(\omega, x=0)\exp{\left(-\kappa(\omega)c_\text B d\ln{10}\right)}\\ &= j(\omega, x=0)\left(1-\left(\kappa(\omega)c_\text B d\ln{10}\right)\right) \end{aligned}

Substitution of equation in equation yields

j_\text F(\omega)\propto j(\omega, x=0) \kappa(\omega)

Thus, the fluorescence intensity $j_\text F(\omega)$ depends on the intensity of the incoming excitation light. Using the reference photomultiplier inside of the fluorometer, this reference quantity is determined. Rearranging yields a direct proportionality between the intensity ratio given by the fluorometer and the decadic absorption coefficient $\kappa(\omega)$ .

\frac{j_\text F(\omega)}{j(\omega, x=0)}\propto \kappa(\omega)

The proportionality constant can be adjusted such that the absorption spectrum and the fluorescence spectra can be plotted in a common graphical plot for a simplified comparison.

Instructions

Go to virtual lab course

Preparation

Prepare a fluorescence cuvette (thickness $\pu{10 mm}$ , 4 clear sides) with pure cyclohexane and with 2-chloronaphtalene solution (concentration $c_{\text B}= \pu{0.23 mol/m3}$ ).

Measurement of the absorption spectrum

Record an absorption spectrum in order to compute the Franck-Condon factors and the electronic transition dipole moments.

Turn on the absorption spectrometer with the upper switch and start the computer. Login with user Praktikum and password Praktikum.
Start the software Methode 2-Chlornaphthalin. Confirm that you login as administrator. After the program has loaded, select Folder List on the left to open various parameter views.
First, select the view Sample Info. Rename your sample using Sample ID with your group number and assign the substance name using Description.
Next, in the Folder List left select Data Collection. Note down the listed parameters (wave length region, spectral band width of the excitation monochromator, step width). The default values are suitable to collect an overview spectrum of the UV/Vis region.
Start the measurement by clicking Data Collection $\rightarrow$ Start.
Confirm the popup window with OK, such that the absorption spectrometer automatically records a baseline (Autozero).
After the measurement of the baseline, a new window appears. Place the sample in the first and the reference in the second cuvette holder and confirm with OK.
Save the obtained spectrum by selecting File $\rightarrow$ Export. Change the folder to D:\Praktikum\Semester\Group. Confirm by pressing Export and check, whether the files are actually saved in the corresponding directory.
Record an additional absorption spectrum. Set the step widht to $\pu{0.1 nm}$ and the scan range from $\pu{350 nm}$ to $\pu{240 nm}$ .
Transfer your files to your USB drive. Switch off the spectrometer and the computer.

Measurement of a fluorescence emission spectrum

Settings of the fluorometer for measuring the fluorescence emission spectrum.
	Parameter	Value
Mode	Scan type	Emission
Start wave length	Start	$\pu{250 nm}$
Final wave length	End	first $\pu{800 nm}$ , then $\pu{400 nm}$
Scan speed	Speed	$\pu{100 nm min-1}$
Excitation wavelength	Ex. WL	$\pu{240 nm}$
Spectral bandwidth of the excitation monochromator	Ex. Slit	$\pu{10 nm}$
Spectral bandwidth of the emission monochromator	Em. Slit	$\pu{2.5 nm}$ ; $\pu{5 nm}$ ; $\pu{7.5 nm}$ ; $\pu{10 nm}$
	Gain	Medium
	Auto Lamp	On

Switch on the fluorometer with the left switch and start the computer. Login with username Praktikum and password Praktikum.

Start the software BLAcquisition. Select File $\rightarrow$ Load Method and open the method Methode Fluoreszenzspektrum from the desktop.

Please check, if the values from table are used. For the final wave length, first use $\pu{800 nm}$ to record an overview spectrum. Later, reduce the end wave length for all further measurements to $\pu{400 nm}$ .

Repeat the following steps for acquiring fluorescence spectra at various spectral band widths of the emission monochromator.

Set the spectral band width of the emission monochromator.
Press the green arrow and wait until a window opens.
Insert the reference into the fluorometer and assign a name to the sample (including your group number, the word reference and the spectral band width of the emission monochromator. Start the measurement by clicking Start.
After the measurement is finished, the window opens again. Insert the sample and assign a name to the sample (this time including the word sample). Again press Start.
After the measurement finished, press Stop.
Save both spectra using File $\rightarrow$ Save Data. Choose the file type Ascii Data (*.txt) and place the files in the directory on the Desktop named Semester\Group.
Press File $\rightarrow$ Clear Experiment and confirm with OK.

Measurement of the fluorescence excitation spectrum

Settings of the fluorometer for measuring the fluorescence absorption spectrum.
	Parameter	Value
Mode	Scan type	Excitation
Start excitation wavelength	Start	$\pu{240 nm}$
Final excitation wavelength	End	$\pu{335 nm}$
Scan speed	Speed	$\pu{100 nm min-1}$
Emission wavelength	Em. WL	$\pu{338 nm}$
	Gain	Medium
	Auto Lamp	On

Repeat the following steps to measure fluorescence spectra at various spectral bandwidths of the emission and excitation monochromator. Change the settings according to table . Conduct three measurements with the following combinations of spectral bandwidths:

Ex. Slit: $\pu{5 nm}$ ; Em. Slit: $\pu{5 nm}$
Ex. Slit: $\pu{2.5 nm}$ ; Em. Slit: $\pu{20 nm}$
Ex. Slit: $\pu{2.5 nm}$ ; Em. Slit: $\pu{5 nm}$

Repeat the following steps:

Set the spectral bandwidth of the excitation monochromator and the spectral bandwidth of the emission monochromator.
Click the green arrow and wait for a window to appear.
Insert the sample into the fluorometer and assign a sample name in the popup window with your group number and the spectral bandwidths of the emission and excitation monochromator. Start the measurement by clicking Start.
If the measurement is finished, the window re-appears. Press Stop.
Save the recorded spectrum using File $\rightarrow$ Save Data. Select the file type Ascii Data (*.txt) and save the file into the directory Semester\Group on the desktop.
Press File $\rightarrow$ Clear Experiment and confirm with OK.

Transfer your files to your USB drive. Switch off the fluorometer and the computer.

Data analysis

Go to analysis of the virtual data

In the theoretical part of your lab report, use only the equations you actually use in your data analysis. Assume, that the uncertainty of the data reported by the spectrometer is negligible.

Absorption spectrum

From the decadic absorption $A$ given by the absorption spectrometer, calculate the molar decadic absorption coefficient $\kappa$ and plot the quantity against the wavelength $\lambda$ . The absorption spectrometer has automatically corrected the spectrum using the baseline and the reference. Use the Lambert-Beer law

A(\lambda) = \kappa(\lambda)c_\text B d.

Interpret the absorption spectrum regarding the electronic and vibrational states. Please consider, that various electronic transitions can differ significantly in their electronic transition dipole moment. According to equation a significant change (by roughly one order of magnitude) in the plot of the molar decadic absorption coefficient against the wavelength allows to conclude that an electronic transition took place. Smaller changes of the molar decadic absorption coefficient are an indication of various vibrational levels within an electronic state.

Enlarge the low-energy part of the spectrum by plotting it in an extra plot to better see the vibrational structure of the first electronic state.

Summarize the wavelengths of the local maxima and the molar decadic absorption coefficient in a table. Compare the values with the literature values from and (see table and table ).

Calculate the electronic transition dipole moment $\mu^\text{E}_{0,e'}$ in units of $\pu{Cm}$ for transitions from the electronic ground state to the other electronic states. Since the spectrometer outputs the decadic absorption $A$ as function of the wavelength $\lambda$ , the substition $\omega = \frac{2\pi c}{\lambda}$ in equation yields

\begin{aligned} \int_{\substack{\text{electronic}\\\text{absorption}\\e'\leftarrow 0}}-\frac{\kappa(\lambda)}{\lambda}\mathrm d \lambda&=\frac{\pi N_\mathrm A}{3\varepsilon_0c\hbar\ln{10}}\left|\mu_{\text E,0,e'}\right|^2. \end{aligned}

Please state precisely how (and why) you have selected the limits for the integration and determine an error/uncertainty for the obtained integral. Calculate, according to Gaussian error propagation the error of the obtained electronic transition dipole moment.

Determine the Franck-Condon factor for the 0-0 vibrational transition of the electronic transition with the lowest energy. Use equation and your value for the transition dipole moment. Determine the uncertainty of the Franck-Condon factor.

Assignment of the vibrational bands of the first excited electronic state within the absorption spectrum of 2-chloronaphtalene in light petroleum at a temperature of approximately $\pu{77 K}$ , after .
Transition	$\lambda\,/\,\pu{nm}$
$\text S_{1,0}\leftarrow \text S_{0,0}$	321
$\text S_{1,1}\leftarrow \text S_{0,0}$	316
$\text S_{1,2}\leftarrow \text S_{0,0}$	314
$\text S_{1,3}\leftarrow \text S_{0,0}$	311
$\text S_{1,4}\leftarrow \text S_{0,0}$	307
$\text S_{1,5}\leftarrow \text S_{0,0}$	305
$\text S_{1,6}\leftarrow \text S_{0,0}$	304
$\text S_{1,7}\leftarrow \text S_{0,0}$	300
$\text S_{1,8}\leftarrow \text S_{0,0}$	297

Assignment of the vibrational bands of the second excited electronic state within the absorption spectrum of 2-chloronaphtalene, after.
Transition	$\lambda\,/\,\pu{nm}$
$\text S_{2,0}\leftarrow \text S_{0,0}$	289
$\text S_{2,1}\leftarrow \text S_{0,0}$	278
$\text S_{2,2}\leftarrow \text S_{0,0}$	267
$\text S_{2,3}\leftarrow \text S_{0,0}$	260

Fluorescence emission spectra

Correct the measured fluorescence emission spectra by subtracting for each of the four measurements the spectrum of the reference from the spectrum of the sample.

Argue whether the used parameters (excitation wave length, wave length region) have been appropriately chosen.

Plot the corrected fluorescence emission spectra as a function of the wavelength. State the intensity ratio given by the fluorometer as a dimensionless quantity. Determine the optimal settings for the spectral bandwidth of the emission monochromator.

Compare the fluorescence emission spectra recorded with the optimal settings with the absorption spectrum in one graph. Scale the fluorescence emission spectra by a suitable factor such that both spectra are comparable.

What would a mirror symmetry between the absorption and the fluorescence emission spectrum tell you about the energy level of the vibration states in the ground state and in the first excited state?

Assign all peaks within the fluorescence emission spectrum to a state transition and summarize your assignment in a table.

Discuss the shift of the 0-0-transition between the absorption and the fluorescence emission spectrum.

Fluorescence excitation spectrum

Plot all measured fluorescence excitation spectra in a single graph against the wavelength. State the intensity ratio given by the fluorometer as a dimensionless number. Determine the optimal parameters for the spectral bandwidth of the emission and the excitation monochromator and justify your choice.

Describe whether the used parameters (excitation wavelength, wave length region) have been selected reasonably.

Compare the fluorescence excitation spectrum (obtained at the most suitable parameters) and the absorption spectrum in a single graph and discuss your data.

Summary

Summarize your analysis by stating which of the theoretical concepts described in the theoretical section could be observed experimentally.

Preparation for the entrance exam

The following topics may be part of the entrance exam.

Setup of the absorption spectrometer and of the fluorometer
Lambert-Beer law
Visualization and approximate duration of all state transitions using a Jablonski diagram
Kasha rule
Franck-Condon factor
Franck-Condon principle and potential curve of diatomic molecules
Transition dipole moment
Integral absorption
Spectral bandwidth of the monochromators

Introduction

Transitions in UV/Vis spectroscopy

The Franck-Condon principle

Absorption and emission processes in a Jablonski diagram

Notation

Radiationless transitions

Vibrational relaxation

Internal conversion

Inter-system crossing

Transitions with radiation

Absorption

Fluorescence

Phosphorescence

Shift of the 0-0-transition in absorption and fluorescence spectra

Symmetry of absorption and emission spectra

Quantitative derivation of the absorption process

Classical electrodynamics

Electromagnetic radiation

Spectral decomposition of electromagnetic radiation

Radiation energy density and radiation energy current density

Spectral energy density of electromagnetic radiation

Potential energy of molecules in an electrical field

Quantum mechanical basics

Notation and mathematical foundations

Time-dependent perturbation theory

Oscillating perturbations

Transition probability

Fermi's golden rule

Transitions due to interactions with electromagnetic radiation

Absorption measurements

Spectral molecular density

The Lambert-Beer law

Determination of the transition dipole moment by absorption measurements

The Franck-Condon approximation

Preparation

The transition dipole moment in the Franck-Condon approximation

Experimental setup

Fluorescence emission spectrum

Fluorescence excitation spectrum

Spectral bandwidth of the monochromator

Measured quantities when recording a fluorescence spectrum

Instructions

Preparation

Measurement of the absorption spectrum

Measurement of a fluorescence emission spectrum

Measurement of the fluorescence excitation spectrum

Data analysis

Absorption spectrum

Fluorescence emission spectra

Fluorescence excitation spectrum

Summary

Preparation for the entrance exam

Bibliography