Variational Calculation of the Energy for Helium‐like Ground State from Z = 2 to 5

Abstract

The non-relativistic en er gies of the ground states in the he lium-like sys tems from Z = 2 to 5 are cal cu lated by us ing three dif fer ent ba sis func tions, the Slat er-type, the or dered co or di nates and the mod i fied an gu lar wave func tions in which the an gu lar variational pa ram e ters are in cluded. In cal cu la tions with the lat ter two types of ba sis func tions, we only used 168 terms in eight l com po nents. The best value of en ergy for neu tral he lium atom in this work is ob tained by us ing the mod i fied an gu lar wave func tions. It is de ter mined to be -2.9037161 a.u., which is about 0.00017 a.u. lower than that ob tained with the 221-term Slat er-type ex pansions. In cases of the he lium-like ions from Z = 3 to 5, the com par i son of our cal cu lated re sults us ing these three types of ba sis func tions in di cates that the re sults ob tained with the mod i fied an gu lar wave func tions are better than those with the other two types of ba sis func tions. The con ver gence pat tern in en e rgy with the increas ing num ber of terms in the an gu lar par tial wave func tions is also an a lyzed. IN TRO DUC TIONHe lium is one of the sim plest many-body sys tems in atomic phys ics. The the o ret i cal de scrip tion of its ground state has been a sub ject in ten sively in ves ti gated since the birth of quan tum me chan ics. Al though the eigenfunction and the eigenvalue of the Hamiltonian for a two-electron atom can not be ex pressed in terms of ex act an a lytic forms, many ap prox imate an a lytic wave func tions have been pro posed to give a reason able de scrip tion of the cor re la tion ef fect be tween two electrons in this sys tem. In this as pect, some im por tant works should be men tioned. Hylleraas ini ti ated a pi o neer ing work in which the dis tance be tween two elec trons, r12, was ex plic itly in cluded to de scribe the in ter nal mo tion in the sys tem.1 Since then sev eral im proved Hylleraas-type wave func tions have been ex ten sively stud ied in the lit er a ture. For ex am ple, Pekeris sug gested a Sturmian-type ba sis wave func tion. 2 It has the advan tage that the nor mal iza tion ma trix is a sparse ma trix, so that a gen er al ized eigenequation has been sim pli fied in the diagonalizing pro ce dure for any num ber of ba sis func tions. 2,3By the use of 246-term Hylleraas-type ex pan sion con tain ing the log a rith mic terms such as ln(r1 + r2), Frankowski et al., 4 Freund et al. (1984), 5 and Baker 6 ob tained the ac cu rate non-relativistic en ergy of the ground state for two-electron sys tems. Their results show that the con ver gence of en ergy is very fast. Schwartz also de duced that a more rapid con ver gence of energy could be achieved if some ba sis func tions with the fractional pow ers of (r 1 + r 2 ) and r 12 are in cor po rated in the trial wave func tions of a variational cal cu la tion. 7 Thakkar and Koga used a more gen eral ex pan sion, in which the pow ers of (r1 + r2) and r12 were al lowed to take real val ues and they did not need to be ...