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Kato [97] showed that these series have a nonzero radius of convergence. This radius is determined by the distance from the origin to the nearest singularity in the complex plane là . The study of the radius of convergence, là , and whether or not this is the same as the critical value of lc , has a long history with controversial results [98–101]. Recently, Morgan and co-workers [102] have performed a 401-order perturbation calculation to resolve this controversy over the radius of convergence of the l ¼ 1=Z expansion for the ground-state energy.

15. The crossing points between two different sizes N and N þ 1 give a series for flðNÞ g. By systematically increasing the order N, one can reach a critical point lc ¼ 1:2286 Æ 0:0005. Here lc and the error are estimated using the final minimum and maximum values and their difference over 48 < N < 60. This value is in agreement with the calculation using the first-order method [11]. For the critical exponent a, starting from the series faðNÞ ðlÞg and following the direct approach of finite-size scaling Eq.

The ratio between the ground-state energy and the second lowest eigenvalue of the two-electron atom raised to a power N as a function of l for N ¼ 6; 7; . . ; 13. As expected, the second derivative will develop a delta-function-like behavior as N is getting larger, as shown in Fig. 8c. The behavior of the ground-state energy and its first and second derivatives resembles the behavior of the free energy at a first-order phase transition. For the two-electron atoms, when l < lc the nuclear charge is large enough to bind two electrons; this situation remains until the system reaches a critical point lc , which is the maximum value of l for which the Hamiltonian has a bound state or the minimum charge necessary to bind two electrons.