# Quantum theory of rotational isomerism and Hill equation

## Abstract

The process of rotational isomerism of linear triatomic molecules is described by the potential with two different-depth minima and one barrier between them. The corresponding quantum-mechanical equation is represented in the form that is a special case of the Hill equation. It is shown that the Hill-Schroedinger equation has a Klein's quadratic group symmetry which, in its turn, contains three invariant subgroups. The presence of these subgroups makes it possible to create a picture of energy spectrum which depends on a parameter and has many merging and branch points. The parameter-dependent energy spectrum of the Hill-Schroedinger equation, like Mathieu-characteristics, contains branch points from the left and from the right of the demarcation line. However, compared to the Mathieu-characteristics, in the Hill-Schroedinger equation spectrum the 'right' points are moved away even further for some distance that is the bigger, the bigger is the less deep well. The asymptotic wave functions of the Hill-Schroedinger equation for the energy values near the potential minimum contain two isolated sharp peaks indicating a possibility of the presence of two stable isomers. At high energy values near the potential maximum, the height of two peaks decreases, and between them there appear chaotic oscillations. This form ofmore »

- Authors:

- I. Javakhishvili Tbilisi State University, 3, I. Chavchavadze Avenue, 0179 Tbilisi (Georgia)
- Institut fuer Physik, Martin-Luther Universitat Halle-Wittenberg, Heinrich-Damerow-Str. 4, 06120 Halle (Germany)

- Publication Date:

- OSTI Identifier:
- 22093609

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 53; Journal Issue: 6; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; CHAOS THEORY; ENERGY SPECTRA; HILL EQUATION; ISOMERIZATION; ISOMERS; MOLECULES; OSCILLATIONS; QUANTUM MECHANICS; ROTATION; SCHROEDINGER EQUATION; SYMMETRY; THREE-BODY PROBLEM; WAVE FUNCTIONS

### Citation Formats

```
Ugulava, A., Toklikishvili, Z., Chkhaidze, S., Abramishvili, R., and Chotorlishvili, L.
```*Quantum theory of rotational isomerism and Hill equation*. United States: N. p., 2012.
Web. doi:10.1063/1.4729247.

```
Ugulava, A., Toklikishvili, Z., Chkhaidze, S., Abramishvili, R., & Chotorlishvili, L.
```*Quantum theory of rotational isomerism and Hill equation*. United States. doi:10.1063/1.4729247.

```
Ugulava, A., Toklikishvili, Z., Chkhaidze, S., Abramishvili, R., and Chotorlishvili, L. Fri .
"Quantum theory of rotational isomerism and Hill equation". United States. doi:10.1063/1.4729247.
```

```
@article{osti_22093609,
```

title = {Quantum theory of rotational isomerism and Hill equation},

author = {Ugulava, A. and Toklikishvili, Z. and Chkhaidze, S. and Abramishvili, R. and Chotorlishvili, L.},

abstractNote = {The process of rotational isomerism of linear triatomic molecules is described by the potential with two different-depth minima and one barrier between them. The corresponding quantum-mechanical equation is represented in the form that is a special case of the Hill equation. It is shown that the Hill-Schroedinger equation has a Klein's quadratic group symmetry which, in its turn, contains three invariant subgroups. The presence of these subgroups makes it possible to create a picture of energy spectrum which depends on a parameter and has many merging and branch points. The parameter-dependent energy spectrum of the Hill-Schroedinger equation, like Mathieu-characteristics, contains branch points from the left and from the right of the demarcation line. However, compared to the Mathieu-characteristics, in the Hill-Schroedinger equation spectrum the 'right' points are moved away even further for some distance that is the bigger, the bigger is the less deep well. The asymptotic wave functions of the Hill-Schroedinger equation for the energy values near the potential minimum contain two isolated sharp peaks indicating a possibility of the presence of two stable isomers. At high energy values near the potential maximum, the height of two peaks decreases, and between them there appear chaotic oscillations. This form of the wave functions corresponds to the process of isomerization.},

doi = {10.1063/1.4729247},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 6,

volume = 53,

place = {United States},

year = {2012},

month = {6}

}