Modern Physics: 1-D Diatomic Molecule

In class, we have looked at the bound states of the finite square well. If we place two finite square wells near to each other, we get a crude, one-dimensional model of a diatomic molecule. In our experiment, we will use a DOS-based computer program, BOUND1D, to numerically simulate a homonuclear diatomic molecule. The eigenvalues (energy levels) that the program finds for you correspond to that of an electron in the molecular potential of this molecule.

Your problem is to determine, by detailed analysis of the energy eigenvalues for varying separation distances of square wells (with constant width), whether the action of an electron in this molecular potential stabilizes the molecule. After all, the Coulomb interaction between a pair of atomic nuclei is repulsive since they are both of positive charge. For a diatomic molecule to exist, something must create a net attractive potential for the nuclear cores. Do the electrons produce this attractive potential, keeping the nuclei from flying apart? That is, does the binding energy of the molecular ground state get deeper as the separation distance between the wells decreases, or does it get shallower?

The BOUND1D program may be downloaded with this link by pressing "SHIFT" while clicking on the link. The executable you retrieve is a self-extracting zip archive. Place it in its own subdirectory and run it to extract the program file.

BOUND1D must be run within a DOS window. The program has many capabilities, accessible via the "Parts" menu, but we will only need "Part 1: Finding Eigenvalues." The double square well potential can be chosen under the "Potential" menu. The well parameters, especially the well width and separation, can be accessed in the "Parameter" menu, under "Vary well parameters."

The program's algorithm can be directed to find the energy eigenvalues under the "Spectrum" menu by selecting "Find eigenvalues." This will cause the energy levels to be graphically displayed on the screen. To get the actual numeric binding energies, you can use "See wave functions" under the "Spectrum" menu, obtaining the spatial wavefunction and its energy eigenvalue to record.

Vary the well separation at constant well width, and consider the variation in bound state energies that results. Plot the ground state energy as a function of well separation, for the square well paramters that you choose. Diagram your potential, and label the parameters that you are reporting. Does the presence of an "electron" stabilize the diatomic molecule?