Transition dipole moment

Three wave functions as special solutions of the time-dependent Schrödinger equation for an electron in a potential of a harmonic oscillator . Left: real part (blue) and imaginary part (red) of the wave function. Right: The probability of finding the electron in a certain position. The upper row shows an eigenstate with low energy, the middle row shows an energy state with higher energy and the lower row shows the quantum mechanical superposition of the two upper states. The two upper right figures, in contrast to the lower right figure, form stationary states from. The lower right figure shows that the electron moves back and forth in the superpositioned state. This oscillating movement of the electron is exactly the cause of an oscillating, electrical dipole moment , which leads to the emission of electromagnetic waves. It is therefore directly proportional to the transition probability between the two eigenstates.

The transition dipole moment (also transition matrix element ) is a measure of the ability of an atom , molecule or solid to absorb electromagnetic radiation or, in the case of fluorescent substances, to emit it . ${\displaystyle {\vec {M}}_{ik}}$

With absorption, for example, an atom changes from its energetic ground state (or generally from a lower state) to an excited state , with the atom oscillating back and forth between the two states over a finite time. During this time the atom is in a quantum mechanical superposition of both states; Depending on the duration, it contains parts of both the ground and the excited state, with the latter increasing over time. Since the two states differ in the local distribution of the particle density , a local oscillation with a defined frequency takes place over the period, which corresponds exactly to a classic dipole . If electromagnetic radiation falls on the atom in the form of a photon with exactly the same frequency, the photon can be absorbed by the atom.

The transition dipole moment is a complex , vector quantity . The square of its amount is proportional to the probability of the transition ; the direction of the transition dipole moment indicates how the incident light must be polarized so that absorption can take place.

Physical background

For a neutral atom or molecule that is in a homogeneous, electric field E , the forces on the individual, differently charged parts (positive nucleus and negatively charged electrons ) cancel each other out; nevertheless, the forces act on the individual parts in different places, so that u. a. a torque can result. If the electrostatic potential contains e.g. B. the energy operator of a hydrogen atom a perturbation term ${\displaystyle \phi }$ ${\displaystyle {\mathcal {H}}={\mathcal {H}}^{0}}$

${\displaystyle {\begin{aligned}{\mathcal {H}}^{1}&=e\,\phi \,({\vec {r}}_{\text{Kern}})-e\,\phi \,({\vec {r}}_{\text{Elektron}}),\end{aligned}}}$

where is the elementary charge . If the distance between nucleus and electron is much smaller than the length scale over which it changes, (e.g. small compared to the wavelength of the radiation used), then this disturbance can be described to a good approximation by the linear term, the through ${\displaystyle e}$ ${\displaystyle r=|{\vec {r}}_{\text{Kern}}-{\vec {r}}_{\text{Elektron}}|}$ ${\displaystyle \phi }$ ${\displaystyle \lambda }$ ${\displaystyle r}$

${\displaystyle {\begin{aligned}H_{\text{dipol}}^{1}&=e\,({\vec {r}}_{\text{Kern}}-{\vec {r}}_{\text{Elektron}})\cdot {\vec {\nabla }}\phi (r_{\text{Kern}})\\&=-{\vec {\mu }}_{e}\cdot {\vec {E}}(r_{\text{Kern}}).\end{aligned}}}$

given is. This is the "dipole approximation" (or also "long-wave approximation") of the coupling to the electric field and is the operator of the electric dipole moment of the hydrogen atom. It represents the first link in a Taylor expansion of in um . ${\displaystyle {\vec {\mu }}_{e}=-e\,({\vec {r}}_{\text{Elektron}}-{\vec {r}}_{\text{Kern}})}$ ${\displaystyle H^{1}}$ ${\displaystyle r/\lambda }$ ${\displaystyle r/\lambda =0}$

This means that there is an interaction between the dipole moment and the E field. In terms of quantum mechanics , a transition between two states and when ${\displaystyle |\Psi _{i}\rangle }$ ${\displaystyle |\Psi _{k}\rangle }$

${\displaystyle {\vec {M}}_{ik}=\langle \Psi _{i}|{\vec {\mu }}_{e}|\Psi _{k}\rangle \neq 0}$.

This secondary diagonal element (or transition element ) of the dipole moment operator is called the transition dipole moment . If the transition dipole moment is zero, the transition is called "dipole-forbidden" and higher multipole moments must be considered in order to describe the transition.

The transition probability between the two states is then proportional to its absolute square:

${\displaystyle w(i\to k)\equiv A_{ik}\sim |{\vec {M}}_{ik}\cdot {\vec {E}}|^{2}}$

or for emission in any spatial direction:

${\displaystyle A_{ik}\sim |{\vec {M}}_{ik}|^{2}}$.

Although the classical absorption spectra had already been researched so precisely that a number of selection rules between permitted and forbidden transitions were known, they were only explained through the quantum mechanical consideration. Two comments should be made here:

• The transition probability cannot be expressed solely with classical quantities, such as the diplomas of the two states. Rather, the states and oscillate with phases or , for which there is no classic analog. ${\displaystyle |\Psi _{i}\rangle }$ ${\displaystyle |\Psi _{k}\rangle }$ ${\displaystyle e^{i\omega _{i}t}}$ ${\displaystyle e^{i\omega _{k}t}}$
• In particular, the transition dipole moment is not the difference between the dipole moments of the two states, even if the name could be misunderstood. Rather, it is a secondary diagonal element of the dipole moment operator.

Semiclassical view

The exact consideration of the interaction between electromagnetic radiation and an atom or molecule requires the formalism of quantum field theory . In the following, only the atomic part is treated quantum mechanically for the sake of simplicity; electromagnetic fields are considered classically . This semiclassical approximation delivers good results, but relativistic and quantum field theoretical corrections have to be used for a higher accuracy .

The electric dipole moment of a charge distribution is classically defined as . ${\displaystyle \rho ({\vec {r}})}$ ${\displaystyle {\vec {p}}=\int \rho ({\vec {r}})\,{\vec {r}}\,d^{3}r}$

In quantum mechanics this corresponds . ${\displaystyle \langle {\vec {p}}\rangle =e\langle {\vec {r}}\rangle =e\Psi ^{*}{\vec {r}}\Psi }$

For a mixed state , the phases in and straight out. In contrast, the transition element oscillates , where is given by with Planck's reduced quantum of action . The mixed state thus resonates . Since and in general have different local functional courses and thus particle densities, the particle density of the mixed state also oscillates locally back and forth. The mixed state therefore represents a Hertzian dipole that also radiates. ${\displaystyle |\Psi \rangle =a_{i}|\Psi _{i}\rangle +a_{k}|\Psi _{k}\rangle }$ ${\displaystyle \Psi _{i}^{*}{\vec {r}}\Psi _{i}}$ ${\displaystyle \Psi _{k}^{*}{\vec {r}}\Psi _{k}}$ ${\displaystyle \Psi _{i}^{*}{\vec {r}}\Psi _{k}\sim \sin((\omega _{ik})t)}$ ${\displaystyle \omega _{ik}}$ ${\displaystyle \hbar \,\omega _{ik}=|E_{i}-E_{k}|=\Delta E_{ik}}$ ${\displaystyle \hbar }$ ${\displaystyle \omega _{ik}}$ ${\displaystyle |\Psi _{i}\rangle }$ ${\displaystyle |\Psi _{k}\rangle }$ ${\displaystyle |\Psi |^{2}}$ ${\displaystyle \omega _{ik}}$

The average emitted radiation power of a Hertzian dipole is:

${\displaystyle {\bar {P}}={\frac {2}{3}}\,{\frac {p^{2}\,\omega ^{4}}{4\,\pi \,\varepsilon _{0}\,c^{3}}}}$

in which

• ${\displaystyle \varepsilon _{0}}$the electric field constant ,
• ${\displaystyle c}$the speed of light and
• ${\displaystyle p=q\,L}$is the amplitude of the dipole moment.

Time averaged is to be set. The radiation power emitted during the transition is obtained ${\displaystyle p^{2}=2|M_{ik}|^{2}}$ ${\displaystyle |i\rangle \rightarrow |k\rangle }$

${\displaystyle {\bar {P}}_{ik}={\frac {4}{3}}\,{\frac {\omega _{ik}^{4}}{4\,\pi \,\varepsilon _{0}\,c^{3}}}|M_{ik}|^{2}}$.

${\displaystyle N_{i}}$Atoms in the state emit the radiation power on average with the transition . ${\displaystyle |i\rangle }$ ${\displaystyle |i\rangle \rightarrow |k\rangle }$ ${\displaystyle \omega _{ik}}$ ${\displaystyle {\bar {P}}=N_{i}\,{\bar {P_{ik}}}}$

The probability that an atom in the state of transition with emission of a photon takes place in a time interval of one second is given by the Einstein coefficient . With this the radiation power becomes: ${\displaystyle |i\rangle }$ ${\displaystyle |i\rangle \rightarrow |k\rangle }$ ${\displaystyle A_{ik}}$

${\displaystyle {\bar {P}}=N_{i}\,A_{ik}\,\hbar \,\omega _{ik}}$.

Comparing this equation with the expression for , it follows: ${\displaystyle {\bar {P}}_{ik}}$

${\displaystyle A_{ik}={\frac {4}{3}}\,{\frac {\omega _{ik}^{3}}{4\,\pi \,\varepsilon _{0}\,\hbar \,c^{3}}}|M_{ik}|^{2}}$.

The last equation therefore gives a relationship between the transition dipole moment and the probability of the corresponding transition. ${\displaystyle M_{ik}}$ ${\displaystyle A_{ik}}$

Relation to selection rules

The selection rules , whether a transition is allowed or forbidden, are generally derived from, where these are the atomic numbers , or for electrons is −1. A transition is forbidden if the integral vanishes, otherwise it is allowed. The exact value of the transition dipole moment is of no interest for the selection rules. For idealized models such as the harmonic oscillator , the rigid rotator and the hydrogen atom (but also other atoms and dipole molecules) numerous, vanishing matrix elements can be found by simple symmetry considerations. ${\displaystyle M_{ik}=e\langle \Psi _{i}|\sum _{j}Z_{j}{\vec {r_{j}}}|\Psi _{k}\rangle }$ ${\displaystyle Z_{j}}$

As an example: reverses its sign for reflections, i.e. has negative parity . The transition element therefore disappears if and have the same parity. This explains why no dipole transitions for the hydrogen , , , , are allowed, ..., but rather , , , ... ${\displaystyle {\vec {r}}}$ ${\displaystyle |i\rangle }$ ${\displaystyle |k\rangle }$ ${\displaystyle |s\rangle \rightarrow |s\rangle }$ ${\displaystyle |p\rangle \rightarrow |p\rangle }$ ${\displaystyle |d\rangle \rightarrow |d\rangle }$ ${\displaystyle |d\rangle \rightarrow |s\rangle }$ ${\displaystyle |f\rangle \rightarrow |p\rangle }$ ${\displaystyle |p\rangle \rightarrow |s\rangle }$ ${\displaystyle |d\rangle \rightarrow |p\rangle }$ ${\displaystyle |f\rangle \rightarrow |d\rangle }$

If a transition is prohibited according to this rule, electrical quadrupole or magnetic dipole transitions etc. are still possible in a higher order of perturbation theory. Thus, for the transition of the hydrogen atom, the electric quadrupole moment also vanish (but not for reasons of parity, since there is even parity) and all higher electric multipole moments. The magnetic dipole moment only disappears in the non-relativistic limit case. ${\displaystyle |2s\rangle \rightarrow |1s\rangle }$ ${\displaystyle x^{2}}$

Web links

Wiktionary: transition dipole moment  - explanations of meanings, word origins, synonyms, translations

• Michael Komma: Quantum transition of the mixed state of atomic orbitals. Retrieved on December 1, 2018 (quantum transition of the mixed state of atomic orbitals - the quantum leap). 
• Giles Henderson, John C. Wright, Jon L. Holmes: How a Photon Is Created or Absorbed. Retrieved December 2, 2018 ( How a Photon is Created or Absorbed is an electronic version of a paper by the same title published in this Journal in 1979 (J. Chem. Educ. 1979 56 631-635)). 

literature

• Wolfgang Demtröder : Atoms, Molecules and Photons. An Introduction to Atomic, Molecular and Quantum Physics. Springer, Berlin et al. 2006, ISBN 3-540-20631-0 .
• J. Michael Hollas: Modern Spectroscopy. 4th edition. John Wiley and Sons, Chichester 2004, ISBN 0-470-84416-7 .
• R. Stephen Berry , Stuart A. Rice , John Ross: Physical Chemistry. 2nd edition. Oxford University Press, New York NY et al. 2000, ISBN 0-19-510589-3 .
• Martin Klessinger, Josef Michl: Excited States and Photochemistry of Organic Molecules. VCH, New York NY et al. 1995, ISBN 1-56081-588-4 .
• JJ Sakurai : Advanced Quantum Mechanics. Addison-Wesley, Reading MA et al. 1967, ISBN 0-201-06710-2 (Chapter: Emission and Absorption of Photons by Atoms ).

Individual evidence

1. ↑ For a derivation of the dipole coupling starting with the Hamilton operator of the minimal coupling to the (quantized) electromagnetic field see z. B. Rodney Loudon: The Quantum Theory of Radiation . 3. Edition. Oxford University Press, 2000, §4.8 (English).