Molecular physics

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The molecular physics is a branch of physics that deals with the study of the chemical structure (. Eg bond lengths and angles), the properties (eg. As energy levels) and behavior (eg., Reaction processes) of molecules employed. Therefore, molecular physics can also be viewed as a border area between physics and chemistry . Objects and methods of investigation largely correspond to those of physical chemistry .

The basis is the knowledge of atomic physics and quantum mechanics . The Born-Oppenheimer approximation is an important model for calculating the molecular properties . The orbital model , which is important in atomic physics, is expanded to include molecular orbitals in molecular physics .

One of the most important measurement methods in molecular physics is vibration spectroscopy or molecular spectroscopy , in which not only the electronic energy states , but also vibration and rotation states occur.

Molecular spectra

In molecules, the electronic energy levels (known from atoms ) are further subdivided into vibrational states and those from rotational states. The energy gaps between electronic states are greatest and are around a few electron volts , the associated radiation is roughly in the visible range. The radiation from vibrational transitions is in the mid- infrared (approximately between 3 and 10  µm ), that of rotational transitions in the far infrared (approximately between 30 and 150 µm). The spectrum of a molecule usually consists of many more lines than that of an atom. A so-called band system is part of a change in the electronic state, with each individual band corresponding to a simultaneous oscillation transition during the electronic transition. Each band in turn consists of individual spectral lines, each of which includes a rotation transition that takes place parallel to the electronic transition and the vibrational transition.

Especially diatomic molecules are considered here, where the states can be represented more easily.

Rotation of a diatomic molecule

As an approximation (for low rotational quantum numbers, that is, when the molecule does not rotate so fast that the distance between the nuclei increases noticeably), the molecule can be regarded as a rigid rotator , that is, the distance between the atomic nuclei is constant. The angular momentum is quantized ( called although the associated quantum number is mentioned):

with the rotational quantum numbers and the quantity derived from Planck's quantum of action . The rotational energy is

with the moment of inertia and the so-called rotation constant

of the molecule. Spectroscopic measurements can be used to determine the constant of rotation and thus to deduce the moment of inertia and the core distance.

The distance between the energy levels and is , therefore, increases with increasing quantum number. The selection rule for transitions with absorption or emission of a photon is , in addition, the molecule must have a permanent dipole moment, which is not the case with molecules with two similar atoms; therefore these molecules do not have a pure rotation spectrum.

The energy differences between rotation levels are in the range of the typical thermal energies of particles at room temperature. In thermal equilibrium, the energy states are occupied according to the Boltzmann statistics . However, one has to note that the state with the quantum number J is actually a degenerate state (with the directional quantum numbers ). The occupation density is therefore proportional to . The rotation spectrum is also dependent on the transition probabilities between the states. If these are roughly the same, then the intensities of the spectral lines reflect the population densities. Typically, the population density increases from weg with increasing J initially by the factor up to a maximum, and then decreases again by the exponential factor; this is what rotation spectra often look like. The distances between the spectral lines are all the same, because the distances between the energy levels always increase when J is increased by 1 and the selection rule only allows transitions to the next level.

Vibrations of a diatomic molecule

The atoms of a diatomic, dumbbell-shaped molecule can also vibrate against each other. The simplest approximation here is a harmonic oscillation; the potential energy of an atom must increase quadratically with the distance from an equilibrium distance r 0 to the other atom. The energy levels of a quantum mechanical harmonic oscillator (quantum number ) are equidistant:

However, real molecules deviate strongly from this behavior, the potential is not harmonic ( anharmonic oscillator ) and increases much more when approaching the other atom than when moving away - here it approaches the dissociation energy of the molecule asymptotically . A better approximation than the harmonic potential is the so-called Morse potential .

is the zero point energy of the potential, a parameter:

here is the spring constant of the most suitable harmonic potential.

Rotational vibration spectrum of gaseous hydrogen chloride at room temperature. The stretching oscillation is shown here, which is why there is no Q branch.

This function depicts the real potential much better. The Schrödinger equation can be solved analytically in a quadratic approximation ( Taylor expansion of the Morse potential). The energy levels can be calculated like this:

In contrast to the harmonic oscillator, the permitted neighboring oscillation states are no longer equidistant, but rather reduce their spacing approximately . It should also be noted that there are only a finite number of bound states, where is given by .

The selection rules for transitions between oscillation levels in the dipole approximation are allowed for the harmonic oscillator, for the anharmonic oscillator are also allowed with decreasing probability. In the case of stretching vibrations, a rotational transition must also take place, so it applies here . A distinction is made between the so-called P and R branches, where P and R denote. In the case of flexural oscillations, a transition is also possible without changing the state of rotation, which is called the Q branch.

Rotational-vibration interaction

Vibration-rotation transitions and the resulting spectrum

Because the moment of inertia of the molecule fluctuates due to the vibrations, one must add the rotational-vibration interaction energy to the energy of the molecule on closer inspection. According to the following approach for the total energy

one can adapt the so-called Dunham coefficients to the experimental results.

The effective potential for the molecular oscillation in the diatomic molecule is increased by the rotation ( is the potential without rotation):

This leads to the formation of a so-called rotation barrier at higher rotational quantum numbers: As the distance between the atomic nuclei increases, the effective potential grows from a minimum (equilibrium position) to a maximum (the rotation barrier), which is already above the dissociation energy and then falls back to the dissociation energy. As a result, the molecule can be in a state of oscillation “behind” the rotation barrier, the energy of which is higher than the dissociation energy. Dissociation can then occur due to the tunnel effect . With very high rotational quantum numbers, the potential minimum is also raised above the dissociation energy, with even higher rotational quantum numbers there is finally no minimum and thus no more stable states.

Electronic states in the diatomic molecule

Similar to atoms, the state of an electron is indicated by a principal quantum number n and an orbital angular momentum quantum number l , whereby letters are assigned to the various values ​​of l as for the atom (s, p, d, f, ...). However, the electric field is no longer spherically symmetrical, so the orbital angular momentum must be set relative to the core connection axis. The projection of the orbital angular momentum onto the core connection axis is called with the associated quantum number λ. Greek letters are written for the different values ​​of λ, which correspond to the Roman letters for the values ​​of l (σ, π, δ, φ, ...). One then writes e.g. B. the ground state of the hydrogen molecule (1sσ) 2 : 2 electrons are in the ground state n  = 1, l  = 0, λ = 0.

The coupling of the individual angular momentum to form a molecular angular momentum expediently takes place in different order depending on the strength of the interactions; one speaks of Hund's coupling cases (a) - (e) (after Friedrich Hund ).

The sum of the orbital angular momenta projected onto the core connection axis is called with the quantum number Λ; the sum of the intrinsic electron angular momentum (spins) is called the same as for the atom with the quantum number S; the projection of this total spin onto the nuclear connection axis is called with the quantum number Σ; the sum of and is called with the quantum number Ω. The mechanical angular momentum of the molecule also interacts with the electronic angular momentum.

Other designations are often used for the electronic states: X stands for the ground state, A, B, C, ... stand for the increasingly higher excited states (small letters a, b, c, ... usually indicate triplet states).

Hamilton operator for molecules

It is common not to write the Hamilton operator in SI units, but in so-called atomic units, as this has the following advantages:

  • Since natural constants no longer appear explicitly, the results are easier to write down in atomic units and are independent of the accuracy of the natural constants involved. The quantities calculated in atomic units can still be easily calculated back in SI units.
  • Numerical solution methods of the Schrödinger equation behave numerically more stable, since the numbers to be processed are much closer to the number 1 than is the case in SI units.

The Hamilton operator results in

With

  • , the kinetic energy of the electrons
  • , the kinetic energy of the atomic nucleus
  • , the potential energy of the interaction between the electrons
  • , the potential energy of the interaction between the nuclei
  • , the potential energy of the interaction between electrons and atomic nuclei.

Here and are the indices about the electrons or the indices about the atomic nuclei, the distance between the i-th and the j-th electron, the distance between the -th and the -th atomic nucleus and the distance between the -th electron and the -th atomic nucleus, the atomic number of the -th atomic nucleus.

The time-independent Schrödinger equation then results in , although in practice the overall Schrödinger equation is divided into an electronic Schrödinger equation (with fixed core coordinates) and a Kernschrödinger equation using the Born-Oppenheimer approximations . The solution of the Kernschrödinger equation requires the solution of the electronic Schrödinger equation for all (relevant) core geometries, since the electronic energy is included there as a function of the core geometry. The electronic Schrödinger equation is formally obtained by setting .

See also

Commons : Molecular Physics  - Collection of images, videos, and audio files

literature

  • Hermann Haken, Hans-Christoph Wolf: Molecular Physics and Quantum Chemistry: Introduction to the experimental and theoretical basics. Springer 2006, ISBN 978-3-540-30315-2 .