Introduction

The energies included when molecules build transitions via changing their electronic distributions are of the order of numerous electron volts and so the photons emitted or absorbed lie in the visible and ultraviolet regions of the electromagnetic spectrum. Electronic spectra give information about the electronic structures of molecules, and by analyzing them we are able to deduce the energies of electrons in different molecular orbitals and therefore to test the theories of molecular structure.

Definition of Electronic Spectroscopy

If the electron arrangement in a molecule changes then the energy of the molecule is generally changed. The spectra due to transitions between molecular states that differ in electron arrangement are said electronic spectra. The ro-vibrational picture using the Harmonic Oscillator (by corrections for anharmonicity) and Rigid Rotor (through corrections for rotation-vibration coupling and centrifugal distortion) is an excellent description for the ground state of a diatomic molecule (NIST, 2009).

Fig: polyatomic molecules

In common, polyatomic molecules will have 3N-6 vibrational modes, and be asymmetric rotors by 3 different inertial axes IA, IB, and IC. For each of the 3N-6 normal coordinate vibrations, a potential well will exist by a rotational energy level ladder that resembles the picture at left from McQuarrie & Simon. Each electronic state has its own ro-vibrational potential energy curve (diatomic)/surface (polyatomic).The anhamonic potential describing the ro-vibrational dynamics of a molecule is in common dissimilar for each electronic state. Excitation of a bound electron from the Highest Occupied Molecular Orbital (HOMO) to the smallest Unoccupied Molecular Orbital (LUMO) enhances the spatial extent of the electron distribution, making the total electron density larger and more diffuse, and often more polarizable. Therefore, in a diatomic one discovers that the bond strength between the 2 atoms is weaker.

Fig: slightly weaker bond

A slightly weaker bond denotes that the force steady for vibrations will be lower, and the relation between the force constant and the 2^nd

derivative of the potential V, k=-d²V/dx² specifies that the weaker force constant in the excited state implies not only a lower vibrational frequency, but simultaneously a broader spatial extent for the potential energy curve. This is demonstrated in the figure at right:

As we know that the highest lying state is well above the dissociation threshold. If this particular electronic state for a diatomic molecule were populated

(By absorption of an energetic photon, or energetic collision with another high energy particle) then the molecule would dissociate.

Electronic excited states imply vibrational excitation

Another quite important feature to note is that the centers of each potential well are not aligned vertically. As explained below, this is why we have electronic transitions to different vibrational quantum numbers. By analogy to the situation where a vibrational excited state of a molecule must as well enclose rotational excitation, electronic excited states of molecules must as well be vibrationally excited.

Electronic Excitations Occur Vertically on the Potential Energy Curve Surface: The Franck-Condon Principle

The Born-Oppenheimer approximation is based on the fact that the proton or neutron mass is roughly in the year 1,870 times that of an electron and that though nuclei move rapidly on a human time scale, the electrons move about 1,850 times faster. Therefore, electronic motions whenever viewed from the perspective of the nuclear coordinates take place as if the nuclei were fixed in place. One can say that the electrons adiabatically, (or smoothly and continuously) follow the nuclei.

Applying the Born-Oppenheimer approximation to transitions between electronic energy levels led James Franck and R. Condon to formulate the Franck-Condon Principle, which states that: An obvious 1^st question might be:

Whenever a stable ground state molecule absorbs ultraviolet/visible light, it might undergo excitation from its HOMO to an excited state such as the LUMO. The main effect is that the electronic configuration of the molecule will have changed on excitation, so that the bonding properties of the molecule will have just converted. As a result, the nuclei must shift to reorganize to the new electronic configuration, which instantaneously places up a molecular vibration. This is why electronic transitions can't occur in the absence of vibrational dynamics.

Elementary chemical reactions are a much more compound version of electronic evolutions, and the nuclei of all products (and surrounding solvent, if a solution reaction) must reorganize around the new electronic configuration. This is why vibrational dynamics are an excellent way to learn fundamental reaction dynamics procedures, and why a Nobel Prize was awarded in the year 1999 to Prof. A. Zewail for the learn of chemical reaction dynamics; he and his many co-workers illustrated that the complex interplay between electronic conversions and molecular vibrations can give a real-time movie of chemical reactions as they take place.

Electronic Transitions and Frank-Condon Factors

If an evolution between molecular quantum conditions is to be stimulated via absorption or emission of light, the transition dipole moment must be non-zero. To describe the Franck-Condon Principle more mathematically, we can compute the conversion dipole matrix elements. To simplify a extremely complicated problem as much as possible, we will discuss only a diatomic molecule for the time being. For electronic energy levels, the total wave function is a divisible product of 4 types of motion:

Nuclear (with electrons following the nuclear displacements...)

Translational

Rotational

Vibrational

Electronic (with nuclei immobile on the relevant time scales for motion, via the B.-O. approximation)

The wave functions must be written as a product of each of the 4 component wave functions. The translational wave functions for the molecule are given via particle-in-a-box eigenfunctions. Rotational dynamics are explained via the Rigid Rotor wave functions through corrections for bond length transforms caused through vibrational excitations (for instance, rotation-vibration coupling) and through corrections for centrifugal distortion if the molecule is extremely excited rotationally. Vibrational dynamics are described via Harmonic Oscillator wave functions by corrections for anharmonicity. The electronic wave functions specified via the relevant molecular orbitals describing the ground or excited electronic states. Therefore, the overall wave function for a diatomic molecule must be written as:

 Ψ_total - Ψ_trans * Ψ_rot  * Ψ_vib * Ψ_elect

To explain the Franck-Condon factors, we will simplify the problem via considering only the largest energy parts of the problem, which are the last two terms: the vibrational and electronic energy. Thus, though the true statement of the Franck-Condon Principle is given by, (Ψ_{total final}|μˆ|Ψ_{total initial} we will instead consider the simpler case of

(Ψ_{elect vib final}|μˆ| Ψ_{elect vib initial}) where Ψ_{elect vib i} - Ψ_{elect i}* Ψ_{vib i}

In this case, we must consider the total molecular dipole for the operator μˆ, now including the charges and positions of both electrons and nuclei: 

μˆ = -e Σ_i r_i  + e Σ_i z_i R_i

Where ri are the electronic coordinates, Zi are the nuclear charges, and Ri are the nuclear coordinates. Now the problem of stating the Franck-Condon Principle mathematically is to evaluate.

(Ψ_{elect final} * Ψ_{vib final}|(-eΣ_i r_i + eΣ_i zi Ri)| Ψ_{elect initial} * Ψ_{vib initial})

= (Ψ_{elect final} |-eΣ_i r_i | Ψ_{elect initial}) * (Ψ _{vib final}|Ψ_{vib initial}) + eΣ_i Z_i (Ψ_{elect final}| Ψ_{elect initial} ) * (Ψ_{vib final}|R_i| Ψ_{vib initial})

The Born-Oppenheimer approximation tells us that the nuclear coordinate wave function for the vibrations, Ψ_{vib I} , factors out from the first term (via the B.-O. approximation) because the nuclei (that are vibrating) are moving so much more slowly than the electrons. The second term on the right-hand side of the equation must be identically zero, because the overlap between different electronic states is zero because of orthogonality. Thus, the Franck-Condon Principle can now be written as

(Ψ_{elect vib final}|μˆ| Ψ_{elect vib initial}) - (Ψ_{elect final}| -v Σ_i r_i | Ψ_{elect initial}) x(Ψ_{elect final}| Ψ_{vib initial})

= -e Σ_i i (Ψ_{elect final}|r_i| Ψ_{elect initial}) * (Ψ_{vib final} | Ψ_{vib initial})

         -p_{elect initial} * S_{[vib, final, initial]}

Where S(vib, final, initial) - (Ψ_{vib final}| Ψ_{vib initial})

Is defined as the Franck-condon factor.

Physical Significance of the Franck-Condon Factor

The Franck-Condon factor determines the probability for electronic transitions, and therefore determines the intensity of spectral lines. To have a huge absorption cross section or high probability that the molecule will absorb/emit UV-visible light, then there must be great overlap between the vibrational states in the initial and last electronic states. The case when good overlap is attained is illustrated in the figure at right.

The ground, or initial state will most often be in  the v = 0 vibrational level, since the  vibrational energy will normally be 1,000-3000 cm^-1, many times the thermal energy kT, which at room temperature is about 200 cm^-1.

The most probable position for which the transition can occur is from the equilibrium bond distance for a diatomic molecule, Re, since with only zero-point vibrations occurring, the vibrational wave function is a symmetric even function with no nodes.

Fig: Franck-Condon principle

Since the Franck-Condon principle tells us that the electronic transitions occur with no immediate motion of the nuclei, the excited state at the moment of creation isn't vibrating. Thus, the probability for creation of an electronic excited state will be maximized if the excited state vibrational wave function is located vertically above the ground state potential energy curve so as to make creation of the excited state occur at a classical turning point where the molecule is as compressed as can be compatible with the excited state vibrational wave function. This is illustrated in the figure to the left. In the example shown at right (both from Atkins) it appears that the electronic transition occurs from v=0 in the ground state to v' ~ 9 in the excited state.

Vibronic Progressions Demonstrate How Electronic Spectra involve many Vibrational Levels

Electronic transitions are probable for a broad range of vibrational levels inside the initial and final electronic states. We refer to amalgamation of vibrational and electronic quantum states as the molecular vibronic manifold. An instance of how the ground electronic state, through v=0 can have non-zero overlap by the excited electronic state vibrational levels is revealed in the figure at left from McQuarrie and Simon.

In assured ultra-high resolution spectroscopies, these as molecular beam free-jet expansions, the molecules become cooled to temperatures near zero Kelvin. Under such conditions, the individual rotational levels inside the vibronic manifold turn into resolvable, and one examines the ro-vibronic transitions. An example of the vibronic progression for the electronic and vibrational absorption spectrum of molecular iodine (I₂) is shown below.

A medium sized molecule often has absorption fluorescence resembling those below, by absorption the solid line on the left, and fluorescence emission the dashed line on the right. In the energy diagram at the top of the figure, the energies for transitions from the ground electronic state occur from the v = 0 vibrational level. The ground electronic state for most neutral organic species will contain a single configuration, and is signified S₀ to show that it is a ground state. When we determine the vibronic  manifold  as  in  the  figures  above  and  below,  we  add  the vibrational quantum number to the state description, now termed  S0,v=0. The excitations revealed are for the transitions S1,v=4 <= S0, v=0 down to S1, v=0  <= S0,  v=0. The latter transition is frequently termed the '0-0' transition, because it happens between the v=0 vibrational levels, and is the lowest energy excited state

Fig: Kasha's Rule proposed

Kasha's Rule proposed by Prof.  Michael Kasha in the year 1953 states that emission will always take place from the S1, v=0 state. In the next section, we will discuss the rapid energy flow that happens to funnel excitations from any higher-lying excited states to the S1, v=0 state, principally vibrational relaxation and internal conversion.  Kasha's Rule explains the mirror image symmetry between the vibronic structure of the stimulated absorption and spontaneous emission transitions (NIST, 2009).

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