# A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

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## Abstract

Validity ranges of Lie canonical perturbation theory (LCPT) are investigated in terms of non-blow-up regions. We investigate how the validity ranges depend on the perturbation order in two systems, one of which is a simple Hamiltonian system with one degree of freedom and the other is a HCN molecule. Our analysis of the former system indicates that non-blow-up regions become reduced in size as the perturbation order increases. In case of LCPT by Dragt and Finn and that by Deprit, the non-blow-up regions enclose the region inside the separatrix of the Hamiltonian, but it may not be the case for LCPT by Hori. We also analyze how well the actions constructed by these LCPTs approximate the true action of the Hamiltonian in the non-blow-up regions and find that the conventional truncated LCPT does not work over the whole region inside the separatrix, whereas LCPT by Dragt and Finn without truncation does. Our analysis of the latter system indicates that non-blow-up regions do not necessarily cover the whole regions inside the HCN well. We propose a new perturbation method to improve non-blow-up regions and validity ranges inside them. Our method is free from blowing up and retains the same normal form as the conventional LCPT. We demonstrate our method in the two systems and show that the actions constructed by our method have larger validity ranges than those by the conventional and our previous methods proposed in Teramoto and Komatsuzaki (J Chem Phys 129:094302, 2008; Phys Rev E 78:017202, 2008).

## Keywords

Lie canonical perturbation theory Non-blow-up region HCN isomerization## Notes

### Acknowledgments

We would like to dedicate this article to continuous stimulating and pioneering works by Professor Greg. Ezra in the research field of classical and semiclassical chemical dynamics. HT would like thank Professor Kazuyuki Yagasaki and Zin Arai for their valuable comments on the definition of the action in the Hamiltonian with one degree of freedom and Professor Turgay Uzer for his comment on the Padé approximation as an alternative method for improving the validity range. This work has been supported by JSPS, the Cooperative Research Program of “Network Joint Research Center for Materials and Devices”, Research Center for Computational Science, Okazaki, Japan, and Grant-in-Aid for challenging Exploratory Research (to TK), and Grant-in-Aid for Scientific Research (B) (to TK) from the Ministry of Education, Culture, Sports, Science and Technology, and Nara Women’s University Intramural Grant for Project Research (to MT), Grant-in-Aid for challenging Exploratory Research (to MT), and Grant-in-Aid for Scientific Research (C) (to MT) from the Ministry of Education, Culture, Sports, Science and Technology.

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