Morse potential

The Morse potential (blue) compared to the square potential of the harmonic oscillator (green).
The energy levels are also drawn in, which are equidistant in the harmonic oscillator ( ), but in the Morse code, with increasing energy, they have less and less distance to the binding energy , which is greater than the energy actually required to escape from the potential trough , since the zero point energy is greater than zero . ${\displaystyle \hbar \omega }$ ${\displaystyle D_{e}}$ ${\displaystyle D_{0}}$ ${\displaystyle \left(\nu =0\right)}$

The Morse potential is a term from molecular physics . The 1929 by American physicist Philip McCord Morse proposed context describes the course of the electronic potential of a diatomic molecule as a function of core bond distance by an exponential approximation: ${\displaystyle V}$ ${\displaystyle R}$

${\displaystyle V(R)=D_{\text{e}}\cdot \left(1-e^{-a\cdot (R-R_{\text{e}})}\right)^{2}}$

With

• ${\displaystyle D_{\text{e}}}$the (spectroscopic) dissociation energy
• ${\displaystyle R_{\text{e}}}$the core distance with the lowest potential energy and
• ${\displaystyle a}$ a constant (sometimes referred to as "stiffness of potential")

These quantities are characteristic of the molecule under consideration.

Since one usually defines the potential at infinity as zero:

${\displaystyle V(\infty )=0}$

the Morse potential is often given in the alternative form:

${\displaystyle V(R)-D_{\text{e}}=D_{\text{e}}\cdot \left(e^{-2a\cdot (R-R_{\text{e}})}-2e^{-a\cdot (R-R_{\text{e}})}\right)}$

This shifts the zero point potential . This shift enables the definition of a cutoff radius from which the potential is no longer taken into account. ${\displaystyle -D_{\text{e}}}$

The Schrödinger equation can be solved analytically with the Morse code. This is how the vibration energies can be calculated: ${\displaystyle E_{\nu }}$

${\displaystyle E_{\nu }=h\omega _{0}\cdot \left(\nu +{\frac {1}{2}}\right)-{\frac {h^{2}\omega _{0}^{2}}{4D_{\text{e}}}}\cdot \left(\nu +{\frac {1}{2}}\right)^{2}}$

With

• the Planck's quantum of action ${\displaystyle \ h}$
• the vibration quantum number ${\displaystyle \nu }$
• the frequency that is linked to the constant of the Morse potential via the particle mass ${\displaystyle \omega _{0}}$ ${\displaystyle m}$ ${\displaystyle a}$

${\displaystyle \omega _{0}={\frac {a}{2\pi }}{\sqrt {\frac {2D_{\text{e}}}{m}}}}$

Nowadays, the RKR potential (RKR stands for Ragnar Rydberg , Oskar Klein and Lloyd Rees ) or the Lennard-Jones potential are used to calculate vibration energies .

literature

• Wolfgang Demtröder: Molecular Physics: Theoretical Foundations and Experimental Methods . Oldenbourg Wissenschaftsverlag, 2003, ISBN 978-3-486-24974-3 , p. 93-94 .
• Ludwig Bergmann, Clemens Schaefer, Wilhelm Raith, with contributions from H. Kleinpoppen , M. Fink, N. Risch: Components of matter: Atoms, molecules, atomic nuclei, elementary particles . Walter de Gruyter, 2003, ISBN 978-3-11-016800-6 , p. 460-462 .
• Gerd Otter, Raimund Honecker: Atoms - Molecules - Cores: Molecular and Nuclear Physics . Vieweg + Teubner, 1996, ISBN 978-3-519-03220-5 , pp. 152-154 .

Individual evidence

1. ^ Philip M. Morse: Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels . In: Physical Review . tape 34 , no. 1 , June 1, 1929, p. 57 , doi : 10.1103 / PhysRev.34.57 .
2. ^ Ingolf V. Hertel, C.-P. Schulz: Atoms, Molecules and Optical Physics 2: Molecules and Photon Spectroscopy and Scattering Physics . Springer, 2011, ISBN 978-3-642-11972-9 , pp. 13 .