bound states spectra for the D states in the three-body .... 2 J. S. Sims, D. R. Parmer, and J. M. Reese, J. Phys. B 15 ... A. Alexander and H. J. Monkhorst, Phys.
Bound states in Coulomb threebody symmetric systems David M. Bishop and Alexei M. Frolov Citation: J. Chem. Phys. 96, 7186 (1992); doi: 10.1063/1.462533 View online: http://dx.doi.org/10.1063/1.462533 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v96/i9 Published by the American Institute of Physics.
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NOTES Bound states in Coulomb three-body symmetric systems David M. Bishop and Alexei M. Frolova ) Department o/Chemistry, University o/Ottawa, Ottawa, Canada K1N 6N5
(Received 13 December 1991; accepted 23 January 1992)
In a previous paper I we studied the general theory of bound states spectra for the D states in the three-body Coulomb systems with unit charges and arbitrary masses. In this short paper we give results for some specific systems (COHe, ddp, and ttp); for ddp and ttp there have, hitherto, been no highly accurate calculations, however, these do exist for the helium atom. 2-4 We follow the methodology we have used previouslyl,5 but with much larger basis sets (up to 1350 functions); in Ref. 1 only 700 functions were used for the model systems discussed there and in Ref. 5 only 350:-400 functions were used for the calculations on ddp, ttp, and the helium atom. Our results are given in Tables I-III. For the mesomolecular calculations we have used the numerical constants which are given in Ref. 6 (set II): m,.. = 206.768 262 me, md = 3670.483 014 me> and m t = 5496.921 58 me' Small changes in the muonium, deuterium, and tritium masses (m,... md, and mt) can be accommodated by the following approximate total energy formulas in (mesoatomic units
m,..= 1):
(2)3He, MeHe) (a) singlet !l
=
5495.8852 me,
= 3.739 14X 10- 4
a.u.,
(b) triplet !l=3.73967X1O-4 a.u. Note that we calculate the isotopic shifts directly and do not use perturbation theory to account for the mass polarization operator. In conclusion, we wish briefly to discuss the problem related to the presence of the 1D state in the Ps - ion. This state is obviously unbounded, 1 since it has the ratio A = m,/(me + me + me) ::::::0.3333, while the threshold ratio Ate D) ::::::0.0845. At is the value of the mass ratio when the binding energy is zero. As in Refs. 7 and 8, where the unbound 3P state for the Ps - system was discussed, we have made a number of calculations (with N:::::: 50-125) for he unbound 1D state. The preliminary results testify to the presence of a quasistationary I D state with energy 0.225 ±0.OO5 a.u. However, the width of this resonance (rr)' and, therefore its presence in experiments, can not be evaluated by our procedure. Probably, however, the shape resonance in the 1D series for the Ps - ion does exist.
and TABLE I. The total energies (in mesoatomic units m" = 1) and the binding energies (in electron volts) of the ground D state in the dd[.1 system. a.b
where the coefficients 1.128 and 1.452 have been determined from separate energy calculations. The helium atom results are given in Table III. They are comparable with the results of Refs. 2 and 3, but the results of Drake4 for the 3 ID and 3 3D states of ""He are significantly better. Since we have taken the nonlinear parameters for our method from Ref. 5, and these are not necessarily optimal for He, we expect that in future work, after reoptimization of these parameters, we will obtain better results. Note that in Ref. 5 we used an incorrect symmetrization for the 3 I D state of the helium atom and some of the basis functions were a mixture of 3 I D and 3 3D configurations. The isotopic shifts with respect to ooHe for the 3D states of the He atom are calculated to be 00
NC
Total energy
Binding energy
1000 1200 1350
-
-
00
d
0.488 708 245 5 0.488 708 295 4 0.488 708 311 1 0.488 708 345 ± (2.5 X 10 - 8)
am.a.u. =9.014 575 6X1O- 16 J and 1 eV = 1.602177 3X10- 19 J. bIn this system the threshold energy equals E, = - 0.4733357155 m.a.u. (m" = 1) or - 2663.201193 6 eV. eN is the number of basis functions. dThese are extrapolated values found by fitting E(N) to the formula E(N) =E(oo) +AINY. TABLE II. The total energies (in mesoatomic units m" = 1) and the binding energies (in electron volts) of the ground D state in the tt[.1 system.' N
Total energy
Binding energy
1000 1200 1350 b 00
-
- 172.699 382 -172.700 153 - 172.700 609 -172.7010±(2.5X1O- 4)
0.512 568 3500 0.5125684870 0.512 568 568 0 0.512568 70± (5X 10 - 8)
(a) singlet !l = 2.817 38X 10- 4 a.u., (b) triplet !l = 2.817 78X 10- 4 a.u., 7186
J. Chern. Phys. 96 (9). 1 May 1992
86.492 818 7 86.493 099 5 86.493 1878 86.493 3± (1 X 10- 4 )
'In this system the threshold energy equals E, = - 0.4818741667 m.a.u. (m" = 1) or - 2711.242389 eV. bSee footnote d of Table I. 0021-9606/92/097186·02$006.00
@ 1992 American Institute of Physics
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Letters to the Editor
TABLE III. The total energies (in atomic units) of the ground 3 ID and 3 3D states in the "He atom.' N
3 ID state C
3 3D state
1000
-
-
1200 1350 00
b
2.055 6207240 2.055 620 725 0 2.055 620 725 5 2.055620 735± (5X 10 -9)
2.055 2.055 2.055 2.055
636 275 1 636 287 9 636 293 3 636 31 ± (1 X 10 - 8)
"In the case of this system the threshold energy equals E, = - 2.0 a.u. or E, = - 54.422 792 2 eV. !>gee footnote d of Table I. cE. = 4.359 748 2X 10- 18 J.
If
r r
