Electronic Spectroscopy

1. Introduction: UV-Vis Spectroscopy and Electronic Transitions

Electronic spectroscopy probes transitions between electronic states of atoms and molecules using ultraviolet and visible (UV-Vis) radiation, typically spanning wavelengths from approximately 200 to 800 nm (energies of 1.5–6.2 eV). When a photon of the appropriate energy is absorbed, an electron is promoted from an occupied molecular orbital to an unoccupied (or partially occupied) one, changing the electronic configuration and energy of the system.

Unlike rotational and vibrational spectroscopy, which probe the motion of nuclei on a single potential energy surface, electronic spectroscopy involves transitions between different potential energy surfaces. Each electronic state has its own set of vibrational and rotational levels, leading to broad bands with rich vibrational fine structure in the gas phase. In solution, collisional broadening typically produces smooth, featureless absorption bands.

The key quantity in electronic spectroscopy is the transition dipole moment:

A transition is allowed (and typically strong) if $\boldsymbol{\mu}_{fi} \neq 0$. If the integral vanishes by symmetry, the transition is said to be forbidden. However, as we shall see, several mechanisms can relax these strict selection rules and make nominally forbidden transitions weakly observable.

The types of electronic transitions commonly encountered include:

2. Derivation: The Beer-Lambert Law

The Beer-Lambert law is the fundamental quantitative relationship in absorption spectroscopy. We derive it from the physical principle that the fraction of light absorbed by an infinitesimally thin layer of an absorbing medium is proportional to the number of absorbers in that layer.

2.1 Differential Attenuation

Consider a monochromatic beam of intensity $I$ passing through a homogeneous absorbing medium of concentration $c$ (in mol/L). As the beam traverses a thin slab of thickness $dx$, the decrease in intensity $dI$ is proportional to the current intensity, the concentration of absorbers, and the thickness of the slab:

where $\alpha$ is a proportionality constant that characterizes the intrinsic absorption strength of the material at the given wavelength. The negative sign indicates that intensity decreases as the light propagates through the medium.

2.2 Integration and Transmittance

Separating variables and integrating across the full path length $l$:

Exponentiating both sides yields the transmittance $T$:

2.3 Absorbance and the Molar Absorption Coefficient

It is conventional to convert from natural logarithms to base-10 logarithms and define the absorbance (optical density) $A$:

Defining the molar absorption coefficient (molar absorptivity) as $\varepsilon = \alpha / \ln 10$, we arrive at the celebrated Beer-Lambert law:

Here $\varepsilon$ has units of L mol$^{-1}$ cm$^{-1}$ when $c$ is in mol/L and $l$ in cm. Typical values range from $\varepsilon \sim 10$ for weak ($d$–$d$) transitions to $\varepsilon > 10^5$ for strongly allowed charge-transfer transitions.

2.4 Additivity for Mixtures

A crucial property of absorbance is that it is additive. For a mixture of $N$ species, each with its own concentration $c_j$ and molar absorptivity $\varepsilon_j(\lambda)$, the total absorbance at wavelength $\lambda$ is:

This follows directly from the fact that the attenuation coefficients add: the differential equation for the mixture is $dI = -\left(\sum_j \alpha_j c_j\right) I\, dx$, and the same integration procedure yields the sum of individual absorbances. This additivity is the foundation of quantitative multi-component analysis and is valid as long as the absorbing species do not interact (no ground-state complexes, no scattering, etc.).

3. Derivation: The Franck-Condon Principle

The Franck-Condon principle governs the intensity distribution among vibrational bands in an electronic transition. Its quantum-mechanical formulation arises naturally from the Born-Oppenheimer approximation.

3.1 Born-Oppenheimer Factorization

Within the Born-Oppenheimer approximation, the total molecular wavefunction factorizes as:

where $\psi_{\text{el}}$ is the electronic wavefunction (depending parametrically on nuclear coordinates $\mathbf{R}$), $\chi_{\text{vib}}$ is the vibrational wavefunction, and $\phi_{\text{rot}}$ is the rotational wavefunction. The transition dipole moment between initial state $|i\rangle = |\psi_{\text{el}}^{\prime\prime}\rangle|\chi_{v^{\prime\prime}}\rangle$ and final state $|f\rangle = |\psi_{\text{el}}^{\prime}\rangle|\chi_{v^{\prime}}\rangle$ is:

3.2 Separating Electronic and Nuclear Contributions

The nuclear part of the dipole operator, $\hat{\boldsymbol{\mu}}_{\text{nuc}} = \sum_A Z_A e \mathbf{R}_A$, depends only on nuclear coordinates. Because the electronic wavefunctions for different electronic states are orthogonal at each nuclear configuration, the nuclear contribution vanishes:

For the electronic part, we separate the integration over electronic and nuclear coordinates:

The bracketed integral is the electronic transition dipole moment, which varies slowly with $\mathbf{R}$and can be evaluated at the equilibrium geometry $\mathbf{R}_0$ (the Condon approximation):

3.3 The Franck-Condon Factor

The transition probability (proportional to the square of the transition dipole moment) therefore factorizes into an electronic part and a vibrational overlap integral:

The quantity $\left|\langle \chi_{v^{\prime}} | \chi_{v^{\prime\prime}} \rangle\right|^2$ is the Franck-Condon factor (FCF). It represents the probability that the nuclei, which were in vibrational state $v^{\prime\prime}$ on the initial electronic surface, will be found in vibrational state $v^{\prime}$ on the final electronic surface immediately after the electronic transition.

The Franck-Condon factors satisfy the completeness relation:

3.4 Vibrational Progressions

If the equilibrium bond length changes significantly upon electronic excitation (i.e., the upper and lower potential energy curves are displaced by $\Delta R_e$), then the vibrational wavefunctions are centered at different positions. The overlap integral $\langle \chi_{v^{\prime}} | \chi_{v^{\prime\prime}} \rangle$is largest when $\chi_{v^{\prime}}$ has significant amplitude in the region where$\chi_{v^{\prime\prime}}$ is concentrated. This produces a progression of bands with an intensity envelope governed by the FCFs.

For displaced harmonic oscillators with dimensionless displacement parameter $\Delta$(proportional to $\Delta R_e \sqrt{\mu\omega/\hbar}$), the FCFs for transitions from $v^{\prime\prime}=0$ follow a Poisson distribution:

where $S$ is the Huang-Rhys factor. The most probable transition goes to $v^{\prime} \approx S$, and the progression is broader for larger displacements.

3.5 Classical Turning Point Argument

The classical version of the Franck-Condon principle (due to Franck himself) is based on the argument that electronic transitions occur much faster ($\sim 10^{-16}$ s) than nuclear motion ($\sim 10^{-13}$ s). Therefore, during the transition, the nuclear positions and momenta are essentially unchanged — the transition is vertical on the potential energy diagram.

In the classical picture, the nuclei spend most of their time near the turning points of their vibrational motion, where the kinetic energy is zero and the velocity vanishes. For the ground vibrational level ($v^{\prime\prime}=0$), the nuclei are most likely found near the equilibrium position$R_e^{\prime\prime}$. A vertical line drawn at $R = R_e^{\prime\prime}$intersects the upper potential curve at a point whose energy determines the vibrational level of the excited state that is most strongly populated. This is the classical turning point argument.

More rigorously, at the classical turning point $R_{\text{tp}}$ of vibrational level $v^{\prime}$ on the upper surface, the kinetic energy vanishes and the total energy equals the potential energy:

The vibrational wavefunction has maximum amplitude near the turning points, so the overlap with the ground-state wavefunction (peaked at $R_e^{\prime\prime}$) is maximized when a turning point of $\chi_{v^{\prime}}$ coincides with $R_e^{\prime\prime}$. This gives the same result as the full quantum calculation in the limit of large quantum numbers.

4. Derivation: Selection Rules for Electronic Transitions

The selection rules for electronic transitions are derived by analyzing which transitions yield a non-zero transition dipole moment $\boldsymbol{\mu}_{fi}$. These rules constrain the changes in spin, orbital, and symmetry quantum numbers.

4.1 Spin Selection Rule: $\Delta S = 0$

The electric dipole operator $\hat{\boldsymbol{\mu}} = -e\sum_i \mathbf{r}_i$ does not act on spin coordinates. Therefore the transition dipole moment factorizes:

The spin eigenfunctions for different total spin quantum numbers are orthogonal:

Therefore $\boldsymbol{\mu}_{fi} = 0$ unless $S^{\prime} = S^{\prime\prime}$ and$M_S^{\prime} = M_S^{\prime\prime}$, giving the spin selection rule:

This rule strictly forbids singlet-triplet transitions. In practice, spin-orbit coupling (the $\hat{H}_{\text{SO}} = \xi(r)\,\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}$ interaction) mixes states of different $S$, relaxing this selection rule. The effect scales as $Z^4$ (where $Z$ is the nuclear charge), making spin-forbidden transitions more prominent in heavy-atom compounds.

4.2 Orbital Selection Rules from Symmetry

For atoms, the orbital angular momentum selection rules follow from the commutation properties of the electric dipole operator with the angular momentum operators. The position vector $\mathbf{r}$transforms as a rank-1 spherical tensor ($l=1$). By the Wigner-Eckart theorem, the matrix element $\langle n^{\prime}, l^{\prime}, m_l^{\prime} | r_q^{(1)} | n, l, m_l \rangle$ is non-zero only if:

For multi-electron atoms, the corresponding rules for the total orbital angular momentum quantum number $L$ are:

For diatomic molecules, the projection of orbital angular momentum on the internuclear axis is the relevant quantum number $\Lambda$. The selection rule becomes:

Thus $\Sigma \leftrightarrow \Sigma$, $\Sigma \leftrightarrow \Pi$, and $\Pi \leftrightarrow \Delta$ transitions are allowed, while $\Sigma \leftrightarrow \Delta$is forbidden. For $\Sigma$ states, an additional rule applies: $\Sigma^+ \leftrightarrow \Sigma^+$and $\Sigma^- \leftrightarrow \Sigma^-$, but $\Sigma^+ \nleftrightarrow \Sigma^-$.

4.3 The Laporte Rule ($g \leftrightarrow u$)

In systems with a center of inversion (centrosymmetric atoms, homonuclear diatomics, octahedral complexes), the parity of the wavefunction is a good quantum number. The electric dipole operator has odd parity ($u$), so the matrix element is non-zero only if the initial and final states have opposite parity:

This is the Laporte rule. It explains why $d$–$d$ transitions in octahedral transition metal complexes are very weak ($\varepsilon \sim 1$–$100$), since all$d$ orbitals are gerade ($g$).

4.4 Relaxation Mechanisms

Two principal mechanisms make nominally forbidden transitions observable:

5. Derivation: Term Symbols and Molecular States

5.1 Atomic Term Symbols

For a multi-electron atom in the Russell-Saunders (LS) coupling scheme, the electronic state is specified by a term symbol of the form:

where $S$ is the total spin quantum number ($\mathbf{S} = \sum_i \mathbf{s}_i$),$L$ is the total orbital angular momentum quantum number ($\mathbf{L} = \sum_i \mathbf{l}_i$), and $J$ is the total angular momentum quantum number ($\mathbf{J} = \mathbf{L} + \mathbf{S}$). The quantity $2S+1$ is the spin multiplicity. The letter code for $L$ is:

The possible values of $J$ range from $|L - S|$ to $L + S$ in integer steps. Each value of $J$ gives a distinct energy level (fine structure), with the ordering determined by Hund's third rule: for a less-than-half-filled subshell, the lowest $J$ lies lowest in energy (normal ordering); for more-than-half-filled, the highest $J$ lies lowest (inverted ordering).

Example: Carbon ($2p^2$) — Two electrons in the $2p$ subshell. The possible terms are found by coupling $l_1=1$ and $l_2=1$:

Applying the Pauli exclusion principle (antisymmetric total wavefunction for equivalent electrons) yields the allowed terms: $ {}^3P$, $ {}^1D$, $ {}^1S$. By Hund's rules, the ground state is $ {}^3P_0$ (highest $S$, then highest $L$, then lowest $J$ for less-than-half-filled).

5.2 Diatomic Molecular Term Symbols

For diatomic molecules, the cylindrical symmetry of the internuclear axis replaces the spherical symmetry of atoms. The projection of the total electronic orbital angular momentum onto the internuclear axis is quantized:

The molecular term symbol has the general form:

where $\Lambda$ is denoted by Greek letters ($\Sigma, \Pi, \Delta, \Phi, \ldots$for $\Lambda = 0, 1, 2, 3, \ldots$), $\Omega = |\Lambda + \Sigma_z|$ is the projection of total angular momentum (with $\Sigma_z = M_S$), the superscript $\pm$applies only to $\Sigma$ states and indicates the symmetry of the wavefunction under reflection in a plane containing the bond axis, and $g/u$ applies only to homonuclear diatomics (symmetry under inversion).

5.3 Applications to Specific Molecules

6. Derivation: Fluorescence and Phosphorescence

After absorbing a UV-Vis photon and being promoted to an excited electronic state, a molecule can return to the ground state via several competing pathways. The Jablonski diagram organizes these processes systematically.

6.1 Kasha's Rule

Kasha's rule (1950) states that luminescence (whether fluorescence or phosphorescence) occurs from the lowest excited state of a given multiplicity. This is because internal conversion (IC) between higher excited states of the same multiplicity is extremely fast ($\sim 10^{-12}$–$10^{-13}$ s) due to the small energy gaps and large vibrational overlap between neighboring electronic states.

Formally, if a molecule is excited to $S_n$ ($n > 1$), it rapidly relaxes non-radiatively through the cascade:

Fluorescence then occurs from $S_1$, regardless of the initial excitation wavelength. This explains why the fluorescence spectrum is generally independent of the excitation wavelength (though the fluorescence intensity depends on the absorption at the excitation wavelength).

6.2 Quantum Yield and Lifetime

From $S_1$, the molecule can undergo fluorescence (radiative decay with rate constant $k_f$) or non-radiative decay (rate constant $k_{\text{nr}}$, which includes internal conversion to $S_0$ and intersystem crossing to $T_1$). The fluorescence quantum yield is defined as the fraction of excited molecules that emit a photon:

The fluorescence lifetime is the average time a molecule spends in $S_1$ before decaying:

To derive these, consider the population $[S_1](t)$ of the excited state. After a short excitation pulse, the decay is governed by first-order kinetics:

The fluorescence intensity is proportional to the radiative decay rate:$I_f(t) = k_f [S_1](t) = k_f [S_1]_0 e^{-t/\tau}$. Integrating over all time:

The natural (radiative) lifetime is $\tau_0 = 1/k_f = \tau / \phi_f$. It represents the lifetime the molecule would have if fluorescence were the only decay pathway.

6.3 The Stern-Volmer Equation

In the presence of a quencher Q at concentration $[Q]$, collisional (dynamic) quenching introduces an additional decay pathway with rate $k_q[Q]$. The modified quantum yield is:

Taking the ratio of the unquenched quantum yield $\phi_0 = k_f/(k_f + k_{\text{nr}})$to the quenched quantum yield:

Recognizing that $\tau_0 = 1/(k_f + k_{\text{nr}})$ is the unquenched lifetime, we define the Stern-Volmer constant $K_{\text{SV}} = k_q \tau_0$ and arrive at:

A plot of $\phi_0/\phi$ versus $[Q]$ yields a straight line with slope $K_{\text{SV}}$and intercept 1. If $\tau_0$ is known independently (from time-resolved measurements), the bimolecular quenching rate constant $k_q$ can be extracted. For diffusion-controlled quenching,$k_q \approx 10^{10}$ M$^{-1}$s$^{-1}$.

6.4 Intersystem Crossing and Phosphorescence

Intersystem crossing (ISC) is the non-radiative transition from $S_1$ to $T_1$(the lowest triplet state). This process requires a spin flip and is formally forbidden, but it is facilitated by spin-orbit coupling. The ISC rate constant $k_{\text{ISC}}$ is significant when:

Once in $T_1$, the molecule can decay radiatively to $S_0$ by emitting a phosphorescence photon. Since $T_1 \to S_0$ is spin-forbidden, the radiative rate constant $k_p$ is very small, and the phosphorescence lifetime is correspondingly long ($10^{-6}$ to $10^{2}$ s). Because $T_1$ lies below $S_1$(by Hund's rule), phosphorescence is always red-shifted relative to fluorescence.

The phosphorescence quantum yield depends on both the ISC efficiency and the competition between radiative and non-radiative decay from $T_1$:

7. Applications of Electronic Spectroscopy

8. Historical Context

9. Python Simulation: Franck-Condon Factors

This simulation computes Franck-Condon factors by numerically evaluating the overlap integrals between harmonic oscillator wavefunctions for two displaced potential energy surfaces. The code uses pure numpy (no scipy) and plots the potential energy curves, FC factors, and the resulting simulated absorption spectrum with Gaussian broadening. The key parameter is the dimensionless displacement$\Delta$, which controls the shift between the equilibrium positions of the ground and excited states.

10. Fortran Simulation: Term Symbols and Transition Properties

This Fortran program computes atomic and molecular term symbols, transition energies, and oscillator strengths for hydrogen atom transitions. It also tabulates the selection rules for both atomic and molecular electronic transitions, and demonstrates the Stern-Volmer equation for fluorescence quenching with a numerical example.