Link to the article : https://doi.org/10.1080/00268976.2019.1632950 | DOI for code :
Set of data on the static and dynamic invariants of polarizability (mean polarizability and anisotropy) with programs for obtaining the derivatives of these invariants (for H2, HD and D2) for specific inter-nuclear distance defined by rovibrational wavefunction for some state.
This repository contains :
- Internuclear distance dependent polarizability from which the two non-zero invariants are accessible for the analysis of the derivatives of and Taylor series expansion. These invariants are
| Property | Definition |
|---|---|
| Mean polarizability |  |
| Polarizability anisotropy |  |
The above properties are available as parameter for H2 HD and D2.
- Rovibrational wavefunctions for H2, HD and D2 for v=0--2 and J=0--15. (# This repository deals with computation of the derivatives of the invariants and their Taylor series expansions. For seamlessly performing this set of computation with minimal introduction of error, the inter-nuclear distance is truncated to 0.5--3.0 a.u. for v=0-2 and 0.7--3.8 for v=3-4 a.u. )
- A python module is included which performs the above computation for the wavelength range : 182.26 to 1320.6 nm.
The programs for computation of matrix element (which includes cubic spline interpolation and numerical integration) are written in the Python program pol_derivative.py.
Clone this repository or download as a zip file. According to the program of choice, refer to the README.md for the repository and the README.mdin the Python-module folder.
- Parameter (mean polarizability or anisotropy) is interpolated to the asked wavelength. This is followed by fitting it over inter-nuclear distance using a polynomial function. Fit coefficients are reported in the output.
- The expectation value (re) of the inter-nuclear distance is computed for specific rovibrational state of H2, HD and D2 , in the electronic ground state.
- Using the coefficients of polynomial function the derivatives of the parameter are computed at re. Set of derivatives up to 7th order are reported in the output.
- Taylor series expansions are generated using the derivatives centered at re. Subsequently, the matrix elements of the Taylor series expansions of the parameter are computed along with that of the original parameter. The set of these matrix elements are reported in the output.
- The matrix elements are computed as the following integral over the inter-nuclear distance.
where, 
= 0.5 a.u. and 
= 3.0 a.u.
Cubic spline interpolation procedure used in FORTRAN and python codes has been adapted from Numerical Recipes in FORTRAN, William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Michael Metcalf, Cambridge University Press; 2nd edition.
For evaluation of the definite integral the Adaptive Gausssian Quadrature implemented in SciPy has been used.
References on the evaluation of the definite integral and implementation:
- T. N. L. Patterson, Math. Comput. 22, 847 (1968)
- T. N. L. Patterson, Math. Comput. 23, 892 (1969)
- R. Piessens, E. de Doncker-Kapenga, C. Uberhuber, and D. Kahaner, Quadpack - A Sub-routine Package for Automatic Integration (Springer-Verlag Berlin Heidelberg, 1983)
Python code by Ankit Raj (NCTU, Taiwan).
This work has been published in the following article:
Ankit Raj, Henryk A. Witek & Hiro-o Hamaguchi
Vibration–rotation interactions in H2, HD and D2 : centrifugal distortion factors and the derivatives of polarisability invariants
Molecular Physics
DOI: https://doi.org/10.1080/00268976.2019.1632950
Citing this code :