Franck–Condon principle

The Franck -Condon principle is a quantum mechanical laws that help can be made between different vibrational states of a molecule statements about the probabilities of transitions. The principle is based on the case where change in addition to the vibrational state and the electronic excitation of the molecule, and is used for example, in Molecular Physics, spectroscopy, and as the active medium of molecular lasers ( such as dye and molecular gas laser). It is named after the physicists James Franck and Edward Condon.

Physical background

The internal state of a molecule can assume only certain discrete values ​​, according to quantum mechanics. A state is described by a wave function and an associated energy value that the molecule adopts in the state. In a molecule, a change in state, that is, a change takes place of the inner excitation in three ways:

• Electronic excitation ( through various excited states of the electrons in the molecule),
• Vibration excitation ( due to the vibration of the atomic nuclei of the molecule ),
• Rotational excitation ( by the rotation of the molecule, this plays only a minor role for the Franck -Condon principle).

Figure 1 shows two different electronic states of a molecule are schematically shown ( here the example of the simplest case: that of a diatomic molecule ). Shown below is the ground state and an excited electronic state above. Both electronic states are divided into different vibrational states (respectively) of the molecule, which are named with numerical values ​​of 0 and larger.

Between these states can occur transitions by absorption, fluorescence, or by collisions of the molecule with electrons, atoms or other molecules. A transition, as shown in the figure between the two different vibrational states of electronic states takes place, it is called a vibronic transition.

Statement

The Franck -Condon principle is based on the fact that the exchange of electrons between different states takes place so fast ( about 10-15 seconds) that the internuclear distance does not change during the excitation ( A core oscillation period takes about 10-13 s ). This high speed of the electronic transition towards nuclear movement is made possible by the low mass of the electron (analogous to the Born- Oppenheimer approximation ).

When a molecule now moves from an electronic state to another, so this transition is more likely, the more the vibration wave functions of the two conditions are compatible with each other (that is as similar as possible to the original core coordinate ). Some vibronic transitions are thus more likely than others. With the given by the Franck -Condon principle formalism, the intensities of these transitions can be calculated, as they are used as for spectroscopy.

The example of the states shown in Figure 1, this means: From vibration ground state () in the ground state is the most probable transition into the electronically excited state of the one of the ends in the vibration state. Transitions to other vibrational states can also take place, however, is the probability lower.

An example of such an intensity distribution is shown in Figure 2 vibronic transitions from the electronic ground state to the excited state ( absorption) are blue, the reverse transitions ( fluorescence), displays in green. The narrow lines are observed when the molecules are present in a gaseous form. However, the dashed lines show the case that the molecules are in liquid or solid phase: This is a so-called line widening instead. A special case is the transition ( ), because here alone is equal to the energy difference between the upper and lower state for fluorescence such as absorption, both transitions are thus observed at the same energy or frequency.

Quantum mechanical formulation

The initial state of the transition is composed of an electronic share ( ε ) and a vibration ratio (v ) and is denoted in the Bra- Ket notation with. For an exact treatment in addition to the spin should still be taken into account, however, is neglected here for reasons of clarity. The final state is denoted with analog.

A transition between two states is described by the dipole, composed of the elementary charge e and the locations of the electrons, as well as the charges EZJ and locations of the atomic nuclei:

The transition probability is given by the scalar product to

While the intensity I of the transition, the square of this transition probability is:

In order to calculate it is exploited that the wave function can be expressed approximately by the product of electronic and vibrational wave function:

Wherein the electronic wave function solely depends on the coordinates of the electron wave function and the vibration of which the cores alone.

This separation of the wave functions is to be understood in analogy to the Born- Oppenheimer approximation. It is the fundamental assumption in the Franck -Condon principle. In summary, an equation for the calculation of the intensities:

This simplification was made, namely

,

Which is only allowed as long as the over the electron coordinates continuous scalar product is truly independent of the position of the nuclei. This is in reality not be the case, but may be sufficient for the present case.

The second term in the above- shown equation disappears because the electron wave functions of various states are orthogonal to each other.

What remains is a product of two terms: the square of the first term ( the overlap integral ) is of the so-called Franck-Condon factor, while the second term represents the probability amplitude, which determines the selection rule of transition.

The Franck -Condon principle makes statements about allowed vibronic transitions between two different electronic states, with a further quantum mechanical selection rules or modify this may even completely prohibit the probabilities of these transitions. Selection rules with respect to the rotation of the molecule have been neglected here. They play a role in gases, while they are negligible in liquids and solids.

It is noteworthy that the quantum mechanical formulation of the Franck- Condon principle is the result of a series of approximations, especially the dipole approximation and the Born- Oppenheimer approximation. The predictable with their aid intensities may differ in reality, for example, if additional magnetic dipole transitions or electric quadrupole transitions need to be considered or the factorization described in an electronic as well as vibrational and spin- share is not sufficiently allowed.