Bachelor of Arts
Physics and Astronomy
Dimensional scaling, Hydrogen, Dihydrogen cation, Hydrogen molecule ion
The hydrogen molecule ion is the simplest molecule, consisting of only two protons and an electron. As such, understanding this problem is essential in order to extend quantum mechanical techniques to more complex molecules such as the next simplest hydrogen molecule. The non-ionized hydrogen molecule represents the simplest system with only axial symmetry exhibiting Pauli exclusion principle effects due to the two identical electrons (fermions) in the neutral molecule. Both molecules have been treated in great detail both experimentally and theoretically and the nature of their solutions and energies are well understood.
Dimensional scaling of the problem can provide insight into the nature of the exact solutions to a system. For example, the problem may be solvable in certain dimensions other than three due to the simplicity of the problem or some symmetry that is present in other dimensionalities. In the present work, the former results in the hydrogen molecule ion being exactly solvable in closed form in one dimension. Solutions for the energies for a scaling of the hydrogen molecule ion Hamiltonian done by Herschbach et. al. and by Lopez et. al. [M. Lopez-Cabrera, A. L. Tan, and J. C. Loeser, J. Phys. Chem., 1993, 97, 2467-2478. and D. D. Frantz and D. R. Herschbach, J. Chem. Phys., 1990, 92, 6668-6686.] results in the energy for the three-dimensional problem being bounded by the D→1 and D→ ∞ limits, both of which can be solved in closed form. [T. C. Scott, M. Aubert-Frecon, and J. Grotendorst, Chemical Physics, 2006, 324, 323-338.]
In the present work, a model of the one-dimensional hydrogen molecule ion is developed in which the charge distributions and electric fields are both mathematically fully described in one dimension. The wavefunctions governing the spacial coordinate for this model were found to be combinations of Airy functions of the first type and the wavefunctions for a free particle (sine and cosine functions) and the energies were found to be similarly governed by the Airy function and trigonometric functions.
Various physical interpretations of this model are introduced with example numerical calculations. In one interpretation, the model describes a single electron bound between two plates of positive charge. The results of this problem assume that the plates are fixed in space and have a relatively simple function governing the energies. Another interpretation assumes that the particles in one dimension are uniform in charge and area, making itappropriate for application to the hydrogen molecule and for comparison to the hydrogen atom. Numerical analysis of these results show that the molecule will have lower energy than un-bonded hydrogen atoms, suggesting that this molecule will bond.
The scaling of units with the dimensional scaling performed is briefly discussed in the process. There are some difficulties associated with the dimensional scaling of units of charge, energy, and mass in a physically reasonable way that solves the problem. Some elegant mathematical relationships that help provide insight into possible solutions for this problem are presented, but the problem is left unresolved, resulting in a barrier for generalization of the model to dimensionalities greater than three
Suggestions for other potentially illuminating extensions on the work are made. One is some possibilities for extension of the physical interpretation of the problem to the hydrogen molecule based on a change of variables suggested by Goldman [S.P. Goldman, Phys. Rev. A, 1998, 57, R677-R680.]. Others include techniques for three dimensions and beyond for the hydrogen molecule and hydrogen molecule ion respectively.
Galamba, Joseph, "Model of the One-Dimensional Molecular Hydrogen Cation" (2012). Honors Papers. 353.