The ro-vibrational picture using the Harmonic Oscillator (with corrections for anharmonicity) and Rigid Rotor (with corrections for rotation-vibration coupling and centrifugal distortion) is an excellent description for the ground state of a diatomic molecule.
In general, polyatomic molecules will have 3N-6 vibrational modes, and be asymmetric rotors with three different inertial axes IA, IB, and IC. For each of the 3N-6 normal coordinate vibrations, a potential well will exist with a rotational energy level ladder that resembles the picture at left from McQuarrie & Simon.
The anhamonic potential describing the ro-vibrational dynamics of
a molecule is in general different for each electronic state.
Excitation of a bound electron from the Highest Occupied Molecular
Orbital (HOMO) to the Lowest Unoccupied Molecular Orbital (LUMO)
increases the spatial extent of the electron distribution, making the
total electron density larger and more diffuse, and often more
polarizable. Thus, in a diatomic, one finds that the bond strength
between the two atoms is weaker.
A slightly weaker bond means that the force constant for
vibrations will be lower, and the relation between the force constant
and the second derivative of the potential V,
indicates
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 illustrated in the
figure at right:
Note 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.
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 also contain rotational excitation, electronic excited states of molecules must also be vibrationally excited.
The Born-Oppenheimer approximation is based on the fact that the proton or neutron mass is roughly 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. Thus, electronic motions when viewed from the perspective of the nuclear coordinates occur 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
Electronic transitions occur much faster than nuclei can respond.
An obvious first question might be: Why must the nuclei move in response to an electronic transition?
When a stable ground state molecule absorbs ultraviolet/visible light, it may 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 changed. Therefore, the nuclei must move to reorganize to the new electronic configuration, which instantaneously sets up a molecular vibration. This is why electronic transitions cannot occur in the absence of vibrational dynamics.
Elementary chemical reactions are a much more complex version of electronic transitions, 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 study fundamental reaction dynamics processes, and why a Nobel Prize was awarded in 1999 to Prof. A. Zewail for the study of chemical reaction dynamics; he and his many co-workers showed that the complex interplay between electronic transitions and molecular vibrations can provide a real-time movie of chemical reactions as they occur.
If a transition between molecular
quantum states is to be stimulated by absorption or emission of
light, the transition dipole moment must be non-zero. To
explain the Franck-Condon Principle more mathematically, we
can calculate the transition dipole matrix elements
.
To simplify a very complicated problem as much as possible, we will
discuss only a diatomic molecule for the time being. For electronic
energy levels, the total wavefunction is a separable product of four
kinds 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 wavefunctions must be written as a product of each of the four component wavefunctions. The translational wavefunctions for the molecule are given by particle-in-a-box eigenfunctions. Rotational dynamics are described by the Rigid Rotor wavefunctions with corrections for bond length changes caused by vibrational excitations (i.e., rotation-vibration coupling) and with corrections for centrifugal distortion if the molecule is highly excited rotationally. Vibrational dynamics are described by Harmonic Oscillator wavefunctions with corrections for anharmonicity. The electronic wavefunctions given by the relevant molecular orbitals describing the ground or excited electronic states. Thus, the overall wavefunction for a diatomic molecule must be written as:
To describe the
Franck-Condon factors, we will simplify the problem by considering
only the largest energy parts of the problem, which are the last two
terms: the vibrational and electronic energy. (Recall that
relative energies for translations are << 1 cm-1,
rotations on the order of 1 cm-1, vibrations about 1,000
cm-1, and electronic energies in the 10,000 to 100,000
range). Thus, though the true statement of the Franck-Condon
Principle is given by
,
we will instead consider the simpler case of
In this case, we
must consider the total molecular dipole for the operator
,
now including the charges and positions of both electrons and nuclei:
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
.
The
Born-Oppenheimer approximation tells us that the nuclear coordinate
wavefunction for the vibrations,
,
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
The Franck-Condon factor determines the probability for electronic transitions, and thus determines the intensity of spectral lines. To have a large absorption cross section, or high probability that the molecule will absorb/emit UV-visible light, then there must be large overlap between the vibrational states in the initial and final electronic states. The case when good overlap is obtained 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 wavefunction is a symmetric even function with no nodes.
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 is not vibrating. Thus, the probability for creation of an electronic excited state will be maximized if the excited state vibrational wavefunction 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 wavefunction. 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.
Electronic transitions are possible for a wide range of vibrational levels within the initial and final electronic states. We refer to combination of vibrational and electronic quantum states as the molecular vibronic manifold. An example of how the ground electronic state, with v=0 can have non-zero overlap with the excited electronic state vibrational levels is shown in the figure at left from McQuarrie and Simon.
In certain ultra-high resolution spectroscopies, such as molecular beam free-jet expansions, the molecules become cooled to temperatures near zero Kelvin. Under these conditions, the individual rotational levels within the vibronic manifold become resolvable, and one observes 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, with 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 have a singlet configuration, and is denoted S0 to indicate that it is a ground state. When we resolve the vibronic manifold as in the figures above and below, we add the vibrational quantum number to the state description, now called S0,v=0. The excitations shown are for the transitions S1,v=4 <= S0, v=0 down to S1, v=0 <= S0, v=0. The latter transition is often called the '0-0' transition, since it occurs between the v=0 vibrational levels, and is the lowest energy excited state.
Kasha's Rule proposed by Prof. Michael Kasha in 1953 states that emission will always occur from the S1, v=0 state. In the next section, we will discuss the rapid energy flow that occurs 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.