XVIth International Workshop on
Quantum Systems in
Chemistry and Physics
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Solving the Schrödinger and Dirac-Coulomb equations with and without magnetic fields
Hiroyuki Nakashima and Hiroshi Nakatsuji
Quantum Chemistry Research Institute (QCRI) and JST-CREST, Kyoto, Japan
The free complement (FC) method is a new established method with a different concept from ordinary molecular orbital theory to solve the Schrödinger equation of atoms and molecules very accurately [1]. We have extensively studied and developed the FC method toward getting extreme high accurate wave functions for general atoms and molecules [2]. On the other hand, due to its generality of the Hamiltonian generation, the FC method is applicable to solving the relativistic Dirac-Coulomb equations [3] and the system in extremely strong magnetic fields [4] etc. It is rather important, for instance, astronomical physics and interstellar chemistry for the direct comparison with astronomical observations.
For correctly solving the Dirac-Coulomb equation, we have considered several theoretical requirements against the variational collapse problem and such as a treatment of resonance state. In the FC method, the Hamiltonian can provide a correct relationship of the multi-dimensional Dirac-Coulomb wave function (FC balance). The inverse Hamiltonian method and H-square quantity also with the complex scaling method [5] make numerically stable calculation possible even for the problematic Dirac-Coulomb equation.
We applied our methods to the systems in extremely strong magnetic fields with both nonrelativistic and relativistic levels. The Universe’s strongest magnetic field was observed on Magnetar object and the quantum mechanical calculations in magnetic fields become realistically important. In such very strong magnetic fields, we have to rely on observations in space and highly reliable theoretical studies because it is impossible to perform any experiment on the earth. We could obtain very accurate wave functions even with a strong competition between spin magnetic interaction and ordinary Coulomb force. We hope our way becomes an accurate theoretical methodology for studying unknown interesting phenomena under strong magnetic fields.

References
[1] H. Nakatsuji, J. Chem. Phys. 113, 2949 (2000). H. Nakatsuji and E. R. Davidson, J. Chem. Phys. 115, 2000 (2001). H. Nakatsuji, Phys. Rev. A 65, 052122 (2002). H. Nakatsuji, Phys. Rev. Lett. 93, 030403 (2004). H. Nakatsuji, Phys. Rev. A 72, 062110 (2005).
[2] H. Nakatsuji, H. Nakashima, Y. Kurokawa, and A. Ishikawa, Phys. Rev. Lett. 99, 240402 (2007).
[3] H. Nakatsuji and H. Nakashima, Phys. Rev. Lett. 95, 050407 (2005).
[4] H. Nakashima and H. Nakatsuji, Astrophys. J. 725, 528 (2010).
[5] G. Pestka, M. Bylicki, and J. Karwowski, J. Phys. B 39, 2979 (2006).


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