Solving the Schro ̈dinger Equation of Atoms and Molecules without Analytical Integration Based on the Free Iterative-Complement-Interaction Wave Function

 

Hiroyuki Nakashima and Hiroshi Nakatsuji^a)

 

• Quantum Chemistry Research Institute, Kyodai Katsura Venture Plaza 106, Goryo Oohara 1-36, Nishikyo-ku, Kyoto 615-8245, Japan and Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan

• ^a)Author to whom correspondence should be addressed. Electronic mail: h.nakatsuji@qcri.or.jp

 

 

ABSTRACT

 

The Schrödinger equation was solved very accurately for helium atom and its isoelectronic ions $(Z=1–10)$ with the free iterative complement interaction (ICI) method followed by the variational principle. We obtained highly accurate wave functions and energies of helium atom and its isoelectronic ions. For helium, the calculated energy was $-\mathbf{2.903}\mathbf{724}\mathbf{377}\mathbf{034}\mathbf{119}\mathbf{598}\mathbf{311}\mathbf{159}\mathbf{245}\mathbf{194}\mathbf{404}\mathbf{446}\mathbf{696}\mathbf{9}0537\mathrm{a.u.}$, correct over 40 digit accuracy, and for ${\mathrm{H}}^{-}$, it was $\mathbf{-}\mathbf{0.527}\mathbf{751}\mathbf{016}\mathbf{544}\mathbf{377}\mathbf{196}\mathbf{590}\mathbf{814}\mathbf{566}\mathbf{747}\mathbf{511}38304502\mathrm{a.u.}$ These results prove numerically that with the free ICI method, we can calculate the solutions of the Schrödinger equation as accurately as one desires. We examined several types of scaling function $g$ and initial function ${\psi }_0$ of the free ICI method. The performance was good when logarithm functions were used in the initial function because the logarithm function is physically essential for three-particle collision area. The best performance was obtained when we introduce a new logarithm function containing not only $r_1$ and $r_2$ but also $r_{12}$ in the same logarithm function.

http://www.qcri.or.jp/pdfs/PRL.99.240402.PDF 

 

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/84646/1/1209.pdf