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On the validity range of the Born-Oppenheimer approximation: A semiclassical study for all-particle quantization of three-body Coulomb systems
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Image of FIG. 1.
FIG. 1.

Semiclassical spectra for three-body vibrational motion which is performed by two protons and one negative charge of (a) AFC-II spectrum obtained from 100 trajectories. Peaks shown in the positive energy region must be all spurious, since is taken for the three-body-breakup limit. (b) Renormalized AFC spectrum obtained from the same 100 trajectories as in panel (a). About 6000 copies of these individual trajectories are generated by rescaling in the range , with step size being 0.0005.

Image of FIG. 2.
FIG. 2.

Comparison between the BO and non-BO energy levels with variable , which is shown at the upper left corner of each panel. The vertical broken lines indicate the positions of the low-lying energy levels estimated with the BO approximation. The spectral peaks are for the vibronic states given by the non-BO three-body motion. (In panel of , the peak height for the lowest non-BO state happens to be so low that it is invisible in this scale of the figure.)

Image of FIG. 3.
FIG. 3.

The ground and first excited state adiabatic potential curves for . The energy of the third vibrational state on the ground electronic state overlaps with that of the first vibrational state on the excited state. But, the first vibrational state on the ground potential curve is well separated from the others.

Image of FIG. 4.
FIG. 4.

Log-log relation between the mass ratios and the error in the BO energy for the ground energy levels, where . Plots are found to lie on a straight line, whose gradient is 1.5. More than one plot are shown for each value of because the choice of the scaling parameter range affects the resolution of the Fourier transform of the renormalized AFC.

Image of FIG. 5.
FIG. 5.

The solid and dashed curves represent and , respectively.


Generic image for table
Table I.

Energy of the vibronic ground state given by the BO and non-BO for various . The energies are measured from that of the three-body-breakup limit.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: On the validity range of the Born-Oppenheimer approximation: A semiclassical study for all-particle quantization of three-body Coulomb systems