Molecular physics
The molecular physics is a branch of physics that (eg, energy levels ) and the behavior is concerned with the study of the chemical structure (eg, bond lengths and angles ), the properties ( eg, reaction processes) of molecules. Therefore, the molecular physics can be understood as an interface between physics and chemistry. Research objects and methods are similar to those of physical chemistry.
Basics are the discoveries of atomic physics and quantum mechanics. An important model to calculate the molecular properties is the Born- Oppenheimer approximation. The important in atomic physics orbital model is extended in molecular physics to molecular orbitals.
One of the most important methods of measurement of molecular physics is the vibrational spectroscopy and molecular spectroscopy, in which not only the electronic energy states, but also vibrational and rotational states occur.
Molecular spectra
In the molecules ( known from the atoms ) are the electronic energy levels of vibration and rotational states of these further sub-divided. The energy gaps between electronic states are the largest and are at a few electron volts, the associated radiation is approximately in the visible range. The radiation of the vibration transitions lies in the middle infrared (approximately 3-10 microns ), the rotation of transitions in the far infrared ( approximately 30-150 microns ). The spectrum of a molecule is usually from more than lines of an atom. To a change in the electronic state belongs to a so-called belt system, each single band corresponding to a simultaneous vibrational transition in the electronic transition. Each band again consists of individual spectral lines, to each of which a parallel with the electronic transition and vibrational transition taking place rotational transition belongs.
Here in particular diatomic molecules can be considered, in which the conditions can be represented easily.
Rotation of a diatomic molecule
Approximately ( low rotational quantum numbers, that is, if the molecule is not as fast rotated so that the core distance significantly increases) can be seen the molecule to be rigid, that is, the distance between the nuclei is constant. It comes to the quantization of angular momentum ( called, although the associated quantum number is called ):
With the rotational quantum numbers and derived from Planck 's constant size. The rotational energy is
With the inertia and the so-called rotational constant
Of the molecule. Spectroscopic measurements can determine the rotational constant and so close to the moment of inertia and the core distance.
The spacing between the energy levels and is, therefore increases with increasing number of quanta. The selection rule for transitions having absorption or emission of a photon, in addition, the molecule must possess a permanent dipole moment, which is not the case for molecules with two identical atoms; therefore, these molecules have no pure rotational spectrum.
The energy differences between the rotation levels within the range of the typical thermal energy of particles at room temperature. In thermal equilibrium, the energy states are occupied according to the Boltzmann statistics. One must note, however, that it is the state with the quantum number J is actually to degenerate states ( with the magnetic quantum numbers). The population density is therefore proportional. The rotational spectrum is also still dependent on the transition probabilities between states. If they are about the same, then reflect the intensities of the spectral densities of the cast again. Typically, the population density of off with increasing J initially increases by a factor of up to a maximum, then decreases by the exponential factor again; so then see often from rotational spectra. The distances between the spectral lines are all the same, because the distances between the energy levels with an increase of J by 1 always go around and through the selection rule, only transitions to the nearest level are possible.
Vibrations of a diatomic molecule
The atoms of a diatomic dumbbell- shaped molecule can vibrate against each other. The simplest approximation is a harmonic oscillation here; the potential energy of an atom to another atom must increase quadratically with the distance of this one equilibrium distance r0. The energy levels of a quantum harmonic oscillator ( quantum number ) are equidistant:
However, real molecules deviate strongly from this behavior that the potential is not harmonic ( anharmonic oscillator ) and increases when approaching the other atom to much more than distance - here it asymptotically approaches the dissociation energy of the molecule. 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 best-fitting harmonic potential.
This function is the real potential of significantly better. The Schrödinger equation for the Morse potential can be solved analytically and the energy levels found to be:
In contrast to the harmonic oscillator which allowed adjacent vibrational states now are not equally spaced, but reduce their distance approximately with. It is also noted that only finite, there are many bonded states, is given by.
The selection rules for transitions between vibrational levels in the dipole approximation are for the harmonic oscillator, anharmonic oscillator are also for the decreasing probability allowed. When stretching vibrations in addition a rotational transition must take place, it is paid so. A distinction is the so-called P- and R - branch, where P and R respectively. In flexion vibrations also a transition without changing the rotational state is possible is called the Q- branch.
Rotational-vibrational interaction
Because the moment of inertia of the molecule varies by the vibrations, you must still add the rotation-vibration interaction energy to the energy of the molecule at a closer look. After the following approach for the total energy
You can adjust the so-called Dunham coefficients to the experimental results.
The effective potential for the molecular vibration in the diatomic molecule is increased by the rotation ( the potential without rotation):
This results in higher rotational quantum numbers for the formation of a so-called barrier to rotation: With increasing distance of the nuclei, the effective potential increases from a minimum ( equilibrium position ) to a maximum ( the rotation of the barrier), which already lies above the dissociation to then again drop to the dissociation. Thus, the molecule can be in a state of vibration "behind" the rotational barrier are whose energy is higher than the dissociation energy. This can result in dissociation by the tunnel effect. At very high rotational quantum numbers, the potential minimum is raised above the dissociation energy, at higher rotational quantum numbers, there are finally no minimum and thus no stable states more.
Electronic states in the diatomic molecule
Similar to atoms again, the state of an electron is n by a principal quantum number, and specifies a orbital angular momentum quantum number I, wherein the different values of L are as assigned to the atom letter (s, p, d, f, ...). However, the electric field is no longer spherically symmetric, so the angular momentum must be set relative to the internuclear axis. The projection of the orbital angular momentum on the internuclear axis is called with the corresponding quantum number λ. For different values of λ to write Greek letters corresponding to the Roman letters in the values of l ( σ, π, δ, φ, ...). It then writes, for example, the ground state of the hydrogen molecule ( 1sσ ) 2: two electrons are in the ground state n = 1, l = 0, λ = 0
The coupling of the individual angular momenta of a molecule angular momentum is expediently carried out depending on the strengths of the interactions in a different order, it is called Hund's coupling cases ( a) - (e ) ( by Friedrich Hund ).
The sum of the projections onto the internuclear axis orbital angular momenta is called the quantum number Λ; the sum of the electron intrinsic angular momentum ( spin ) is called as the atom with the quantum number S; the projection of that total spin on the internuclear axis is called the quantum number Σ; the sum of and is called the quantum number Ω. The mechanical angular momentum of the molecule interacts with the electronic angular momenta even in interaction.
For the electronic states often other terms are used: X stands for the ground state, A, B, C, ... stand for higher and higher excited states (small letters a, b, c, ... denote a rule triplet states ).
Hamiltonian for molecules
It is customary to write the Hamiltonian not in SI units, but in so-called atomic units, since this entails the following advantages:
- Since constants of nature no longer explicitly appear, the results are easier to write down in atomic units and regardless of the accuracy of the involved constants of nature. The calculated in atomic units sizes but can be easily calculated back in SI units.
- Numerical solution method of the Schrödinger equation behave more comfortable because the numbers to be processed much closer to the number 1, as is the case in SI units.
The Hamiltonian is given by
With
- , The kinetic energy of the electron
- , The kinetic energy of the nuclei
- , Of the potential energy of the interaction between the electrons
- , Of the potential energy of the interaction between the cores
- , Of the potential energy of the interaction between the electrons and atomic nuclei.
Here, and the indexes of the electron, and the indexes of the atomic nuclei of the distance between the i-th and the j-th electron of the distance between the - th and -th atomic nucleus, and the distance between the -th electron and the -th nucleus, the atomic number of the -th nucleus.
The time-independent Schrödinger equation is then given by, although in practice, the total Schrödinger equation using the Born-Oppenheimer approximation is split into an electronic Schrödinger equation ( with fixed nuclear coordinates ) and a nuclear Schrödinger equation. The solution of the nuclear Schrödinger equation relies requires the solution of the electronic Schrödinger equation for all (relevant) core geometries, since the electronic energy as a function of the nuclear geometry is received there. The electronic Schrödinger equation is formally by setting.