Franck-Condon principle

The Franck-Condon principle is a quantum mechanical law that can be used to make statements about the probabilities of transitions between different vibrational states of a molecule . The principle relates to the case that in addition to the oscillation state, the electronic excitation of the molecule also changes, and is used, for example, in molecular physics , spectroscopy and as an active medium for molecular lasers (such as dye and molecular gas lasers). It is named after physicists James Franck and Edward Condon .

Physical background

According to quantum mechanics, the internal state of a molecule can only take on certain, discrete values. A state is described by a wave function and an associated energy value that the molecule assumes in the state. In a molecule, a change in state, i.e. a change in internal excitation, can take place in three ways:

• electronic excitation (through different excitation states of the electrons in the molecule),
• Vibration excitation (through the vibration of the atomic nuclei of the molecule),
• Rotational excitation (through the rotation of the molecule; this only plays a subordinate role for the Franck-Condon principle).

Figure 1: Schematic representation of two electronic states using the example of a diatomic molecule. The two arrows represent two vibronic transitions between these two states.

Figure 1 shows two different electronic states of a molecule (here using the example of the simplest case: that of a two-atom molecule). The basic electronic state is shown below and an electronically excited state above. Each electronic state is subdivided into different vibration states ( or ) of the molecule, which are named with numerical values ​​of 0 and larger. ${\displaystyle v'}$ ${\displaystyle v''}$

Transitions between the electronic states can take place through absorption , fluorescence, or through collisions of the molecule with electrons, atoms or other molecules. If there is a transition, as shown in the figure, between the vibration states of two different electronic states, this is called a vibronic transition.

statement

The Franck-Condon principle is based on the fact that the change of electrons between different states takes place so quickly (in approx. 10 ⁻¹⁵  seconds ) that the nuclear distance does not change during the excitation (one nuclear oscillation period lasts approx. 10 ⁻¹³  s.). This high speed of the electronic transition compared to the nuclear motion is made possible by the low mass of the electrons (analogous to the Born-Oppenheimer approximation ).

If a molecule changes from one electronic state to another, the more the vibration wave functions of the two states are compatible with each other (i.e. are as similar as possible at the original nuclear coordinate), the more likely this transition is . Some vibronic transitions are therefore more likely than others, namely those in which the core distance does not change. A vertical arrow is drawn for these transitions (see Figure 1), which is why the term “vertical transitions” is also used. With the formalism given by the Franck-Condon principle, the intensities of these transitions can be calculated, such as those used for spectroscopy .

Using the example of the states in Figure 1, this means: From the vibration ground state ( ) in the electronic ground state, the most likely transition to the electronically excited state is the one that ends in the vibration state . Transitions into other vibration states can also take place, but the probability of this is lower. ${\displaystyle v''=0}$ ${\displaystyle v'=2}$

Figure 2: Schematic representation of the intensity distribution of vibronic transitions according to the Franck-Condon principle. The numbers indicate the initial and final state of the vibration. ${\displaystyle v''\rightarrow v'}$ ${\displaystyle v''}$ ${\displaystyle v'}$

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 shown in blue, while the reverse transitions (fluorescence) are shown in green. The narrow lines are observed when the molecules are in gas form. The dashed lines, on the other hand, show the case that the molecules are in a liquid or solid phase : what is known as line broadening takes place here. The transition ( ) is a special case , because here alone the energy difference between the upper and lower state for fluorescence and absorption is the same, so both transitions can be observed at the same energy or frequency. ${\displaystyle v''=0\leftrightarrow v'=0}$

Quantum mechanical formulation

The initial state of the transition consists of an electronic component ( ε ) and a vibration component ( v ) and is denoted by in the Bra-Ket notation . For an exact treatment, the spin would also have to be taken into account, but this is neglected here for reasons of clarity. The final state is denoted by analogously . A quantum mechanical description outside of the Bra-Ket notation can also be found in the literature. ${\displaystyle \left|\psi \right\rangle }$ ${\displaystyle \left|\psi '\right\rangle }$

A transition between the two states is described by the dipole operator , which is composed of the elementary charge −e and the locations of the electrons, as well as the charges eZ _j and locations of the atomic nuclei: ${\displaystyle {\boldsymbol {\mu }}}$ ${\displaystyle {\boldsymbol {r}}_{\mathrm {i} }}$ ${\displaystyle {\boldsymbol {R}}_{\mathrm {j} }}$

${\displaystyle {\boldsymbol {\mu }}={\boldsymbol {\mu }}_{\epsilon }+{\boldsymbol {\mu }}_{K}=-e\sum \limits _{i}{{\boldsymbol {r}}_{i}}+e\sum \limits _{j}{Z_{j}{\boldsymbol {R}}_{j}}}$

The transition probability from to is given by the scalar product ${\displaystyle \left|\psi \right\rangle }$ ${\displaystyle \left|\psi '\right\rangle }$

${\displaystyle P=\left\langle \psi '\right|{\boldsymbol {\mu }}\left|\psi \right\rangle }$,

while the intensity I of the transition is the square of this transition probability:

${\displaystyle I=\left|P\right|^{2}}$.

To calculate this, it is used that the wave function can be expressed approximately by a product of the electronic and vibration wave function:

${\displaystyle \psi \ =\psi _{\epsilon }({\boldsymbol {r}}_{i})\psi _{v}({\boldsymbol {R}}_{j})}$,

where the electronic wave function depends only on the coordinates of the electrons and the vibration wave function only on those of the nuclei. ${\displaystyle \psi _{\epsilon }}$ ${\displaystyle \psi _{v}}$

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

${\displaystyle {\begin{aligned}P&=\left\langle \psi _{\epsilon }'\psi _{v}'\right|{\boldsymbol {\mu }}\left|\psi _{\epsilon }\psi _{v}\right\rangle \\&=\left\langle \psi _{\epsilon }'\psi _{v}'\right|{\boldsymbol {\mu }}_{\epsilon }+{\boldsymbol {\mu }}_{K}\left|\psi _{\epsilon }\psi _{v}\right\rangle \\&=\left\langle \psi _{\epsilon }'\psi _{v}'\right|{\boldsymbol {\mu }}_{\epsilon }\left|\psi _{\epsilon }\psi _{v}\right\rangle +\left\langle \psi _{\epsilon }'\psi _{v}'\right|{\boldsymbol {\mu }}_{K}\left|\psi _{\epsilon }\psi _{v}\right\rangle \\&=\underbrace {\left\langle \psi _{v}'|\psi _{v}\right\rangle } _{\mathrm {{\ddot {U}}berlappungsintegral} \ }\cdot \underbrace {\left\langle \psi _{\epsilon }'|{\boldsymbol {\mu }}_{\epsilon }|\psi _{\epsilon }\right\rangle } _{\mathrm {{\ddot {U}}bergangsdipolmoment} }+\underbrace {\left\langle \psi _{\epsilon }'|\psi _{\epsilon }\right\rangle } _{=\,0}\left\langle \psi _{v}'|{\boldsymbol {\mu }}_{K}|\psi _{v}\right\rangle \\\end{aligned}}}$

A simplification has been made here, namely

${\displaystyle \left\langle \psi _{\epsilon }'\psi _{v}'\right|{\boldsymbol {\mu }}_{\epsilon }\left|\psi _{\epsilon }\psi _{v}\right\rangle \approx \left\langle \psi _{v}'|\psi _{v}\right\rangle \left\langle \psi _{\epsilon }'|{\boldsymbol {\mu }}_{\epsilon }|\psi _{\epsilon }\right\rangle }$,

which is only permissible as long as the scalar product going over the electron coordinates is really independent of the position of the nuclei. In reality, this is not exactly the case, but it is often a sufficiently good approximation. The second summand in the equation shown above disappears because the electron wave functions of different electronic states are mutually orthogonal . ${\displaystyle \left\langle \psi _{\epsilon }'|{\boldsymbol {\mu }}_{\epsilon }|\psi _{\epsilon }\right\rangle }$

What remains is a product of two terms: The square of the first term ( overlap integral ) is the Franck-Condon factor , while the second term indicates the transition dipole moment, which is determined by the orbital angular momentum and spin selection rule for electrical dipole radiation.

The overlap integral depends exclusively on the core coordinates. It shows the overlap between the vibratory wave functions of the electronic states involved in the transition (ground and excited state). It follows from this that vibronic transitions are more intense, the better the vibration wave functions overlap. This can be seen in the size of the Franck-Condon factors, which ranges from (maximum overlap) to (no overlap). ${\displaystyle S=1}$ ${\displaystyle S=0}$

The Franck-Condon principle makes statements about permitted vibronic transitions between two different electronic states, whereby further quantum mechanical selection rules modify the probabilities of these transitions or even prohibit them entirely. The vibration selection rule no longer applies here, since it is vibration wave functions of different electronic states. Rather, the vibration selection rule is more or less replaced by the Franck-Condon factors. Selection rules regarding the rotation of the molecule have been neglected here. They play a role in gases, while they are negligible in liquids and solids. ${\displaystyle \Delta v=\pm 1}$

It should be emphasized that the quantum mechanical formulation of the Franck-Condon principle is the result of a series of approximations, above all the dipole approximation and the Born-Oppenheimer approximation . The intensities that can be calculated with their help can deviate in reality, for example if magnetic dipole transitions or electrical quadrupole transitions have to be taken into account or the described factorization into an electronic, vibration and spin component is not sufficiently permissible.

literature

Original publications by Franck and Condon in professional journals:

• J. Franck: Elementary processes of photochemical reactions . In: Trans. Faraday Soc . No. 21 , 1926, pp. 536-542 , doi : 10.1039 / TF9262100536 .
• EU Condon: A Theory of Intensity Distribution in Band Systems . In: Phys. Rev . No. 28 , 1926, pp. 1182-1201 , doi : 10.1103 / PhysRev.28.1182 .
• EU Condon: Nuclear Motions Associated with Electron Transitions in Diatomic Molecules . In: Phys. Rev . No. 32 , 1928, pp. 858-872 , doi : 10.1103 / PhysRev.32.858 .

Textbooks on the topic:

• Peter W. Atkins, Ronald S. Friedman: Molecular Quantum Mechanics . 4th edition. Oxford University Press, 2004, ISBN 0-19-927498-3 .
• Gerhard Herzberg: Molecular Spectra and Molecular Structure . 2nd Edition. Krieger Publishing Company, 1992, ISBN 0-89464-789-X .

Individual evidence

1. ↑ J. Michael Hollas: Modern methods in spectroscopy. Vieweg, Braunschweig / Wiesbaden 1995, ISBN 3-528-06600-8 , pp. 226-230
2. ^ Karl Hensen: Molecular Structure and Spectra. Steinkopff, Darmstadt 1973, ISBN 978-3-7985-0607-7 , pp. 127-130
3. ^ David J. Willock: Molecular Symmetry. Wiley, Chichester 2009, ISBN 978-0-470-85348-1 , pp. 339-342
4. ^ Daniel C. Harris, Michael D. Bertolucci: Symmetry and Spectroscopy. Dover, New York 1978, ISBN 0-486-66144-X , pp. 330-332
5. ^ Robert L. Brooks: The Fundamentals of Atomic and Molecular Physics. Springer, New York / Heidelberg 2013, ISBN 978-1-4614-6677-2 , p. 151
6. ↑ P. Atkins, J. de Paula: Physical Chemistry. Freeman, New York 2006, ISBN 0-7167-8759-8 , p. 486
7. ↑ Gerd Wedler: Textbook of Physical Chemistry. Wiley-VCH, Weinheim 2004, ISBN 978-3-527-31066-1 , p. 621