TY - JOUR
T1 - Exact solution of coupled equations and the hyperspherical formalism
T2 - Calculation of expectation values and wavefunctions of three Coulomb-bound particles
AU - Haftel, M. I.
AU - Mandelzweig, V. B.
PY - 1983/10/1
Y1 - 1983/10/1
N2 - Exact solutions of one-dimensional coupled differential equations are developed by substituting in power series. The properties of these solutions and the possibility of their application to the few-body problem in the framework of the hyperspherical method are studied. The necessity of logarithmic terms in the nonrelativistic many-body wavefunctions, as well as their absence in the relativistic case, is stressed. Explicit form of the solution of the one-dimensional hyperspherical matrix equation corresponding to the three-body Coulomb problem is found and used to obtain Schroedinger and Faddeev bound state wavefunctions, correlation integrals and probabilities of different hyperspherical states. The results of calculations with inclusion of up to 25 hyperspherical harmonics (Km = 16) for the ground and excited state of the helium atom, the ground state of the positronium ion and the negative hydrogen ion are given and compared with those obtained by the multiconfigurational Hartree-Fock and variational methods as well as with other hyperspherical calculations. We find that generally the correlation integrals converge as the energies, that is, as 1 Km4. While the method is essentially exact, computer round-off error limits the precision for Km > 12 in the positronium calculations.
AB - Exact solutions of one-dimensional coupled differential equations are developed by substituting in power series. The properties of these solutions and the possibility of their application to the few-body problem in the framework of the hyperspherical method are studied. The necessity of logarithmic terms in the nonrelativistic many-body wavefunctions, as well as their absence in the relativistic case, is stressed. Explicit form of the solution of the one-dimensional hyperspherical matrix equation corresponding to the three-body Coulomb problem is found and used to obtain Schroedinger and Faddeev bound state wavefunctions, correlation integrals and probabilities of different hyperspherical states. The results of calculations with inclusion of up to 25 hyperspherical harmonics (Km = 16) for the ground and excited state of the helium atom, the ground state of the positronium ion and the negative hydrogen ion are given and compared with those obtained by the multiconfigurational Hartree-Fock and variational methods as well as with other hyperspherical calculations. We find that generally the correlation integrals converge as the energies, that is, as 1 Km4. While the method is essentially exact, computer round-off error limits the precision for Km > 12 in the positronium calculations.
UR - http://www.scopus.com/inward/record.url?scp=0003057776&partnerID=8YFLogxK
U2 - 10.1016/0003-4916(83)90004-0
DO - 10.1016/0003-4916(83)90004-0
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AN - SCOPUS:0003057776
SN - 0003-4916
VL - 150
SP - 48
EP - 91
JO - Annals of Physics
JF - Annals of Physics
IS - 1
ER -