3 TABLE I: Population probabilities P n of the $^{1}$S bound states of helium after the β decay of a T − anion. Also given are the corresponding energies E n (in atomic units) obtained in the present work. n E n P n (%) P n (%) P n (%) (this work) a (this work) (Ref. 17) (Ref. 18) 1 $_{−}$2 . 903 4572 22 . 98998 22 . 993764 19 . 147 2 $_{−}$2 . 1459 527 46 . 86960 46 . 867404 21 . 149 3 $_{−}$2 . 0612 659 0 . 01320 0 . 135 0 . 27 4 $_{−}$2 . 03358 41 0 . 18363 0 . 21 0 . 143 5 $_{−}$2 . 02117 49 0 . 09220 − 0 . 07 6 $_{−}$2 . 01456 04 0 . 05262 − 0 . 039 7 $_{−}$2 . 01062 35 0 . 03275 − 0 . 024 8 $_{−}$2 . 00809 09 0 . 02175 − 0 . 016 9 $_{−}$2 . 00636 79 0 . 01522 − 0 . 011 10 $_{−}$2 . 00514 07 0 . 01111 − 0 . 008 11 $_{−}$2 . 00423 55 0 . 00848 − − 12 $_{−}$2 . 00355 07 0 . 00666 − − 13 $_{−}$2 . 00302 05 0 . 00513 − − 14 $_{−}$2 . 002 5951 0 . 00469 − − 15 $_{−}$2 . 0022 570 0 . 00273 − − ∑ P n 70 . 30975 70 . 206168 40 . 877 a The bold digits agree with the results in Ref. 25. cussed above in the limit of exact eigenstates the density P ( E, θ ) becomes independent of the complex-scaling pa- rameter θ . A variation of θ for approximate eigenstates provides the possibility to determine an optimal θ$_{opt}$ with highest stability. The best approximation of P ( E, θ ) is then obtained according to Furthermore, the θ dependence of the spectra gives an indication for the convergence of the results. ∂P ( E, θ ) ∂θ ∣ θ$_{opt}$ = min. → P ( E ) := P ( E, θ$_{opt}$ ) . (10) III. RESULTS In Table I the calculated transition probabilities for 15 $^{1}$S bound states of He are listed. The results reveal that almost every second T − decay will end in the first excited state of He . The next probable final state is the He ground state with nearly 23%. With a summed probability of 0 . 45% the higher excited He states are rarely populated after β decay of T $^{−}$. The sum over all calculated bound states yields 70 . 3%. The summa- tion over all calculated states (discrete and discretized continuum states) yields the expected value of 100 . 00%, since the same basis is used for initial and all final states, but indicates the proper numerical implementation. The excellent agreement of the energy eigenvalues at the or- der of µ hartree with the very accurate data in [25] as- sures on the other hand the high quality of the basis set adopted in the present work and its ability to describe many states simultaneously with high precision. A closer view on the energies shows that the degree of accuracy of the present results follows the expected trends. First, the accuracy increases with n , since the importance of cor- relation decreases, if the state becomes more asymmet- ric and the two electrons have smaller spatial overlap. For even higher values of n the states become increas- ingly diffuse and thus it is very difficult to describe them properly without running into numerically caused linear dependencies. angle θ can in principle be chosen arbitrarily within 0 ◦ ≤ θ ≤ 45 $^{◦}$. In the limit of an infinite basis all observ- ables calculated with the aid of complex scaling should become independent of θ . Since only finite basis sets can be applied in practice, only approximate eigenstates can be obtained that may depend on θ . The angle θ can thus be understood as a variational parameter that modifies the adopted basis as can be seen from the inverse rela- tion between basis-set exponents and the scaling angle discussed, e. g., in [29]. A diagonalization of the Hamil- tonian (8) in the basis described by the Eqs. (4) and (5) yields the complex-scaled energies E$_{j}$ ( θ ) and wavefunc- tions Ψ$_{j}$( θ ) where the latter are still defined by Eq. (6), but with complex coefficients c$_{jk}$ ( θ ). With the aid of the complex-scaled energies and wave- functions the transition-probability density into the elec- tronic continuum can be extracted from the complex- scaled resolvent according to [9] A comparison to the final-state probabilities reported by Frolov [17] and Harston and Pyper [18] is also given in Table I. Especially for the highly populated ground and first excited states the results of this work confirm the expectedly very accurate results of Frolov [17] that were obtained with explicitly correlated basis functions. The agreement for the third excited state ( n = 4) is, however, less good, and for the second excited state ( n = 3) there is even an order of magnitude difference. All attempts to improve the basis set for this state failed to yield a better agreement. This could be an indication for a typographical error (a missing zero after the decimal point) in [17]. The 〈 Ψ( θ $^{∗}$) | is the biorthonormal eigenstate to | Ψ( θ ) 〉 . It is obtained from the latter by a transposition and com- plex conjugation of the angular part, while the radial part is only transposed but not complex conjugated. The sum over k includes all complex-scaled eigenstates and eigen- values calculated by solving the generalized complex sym- metric, but non-hermitian eigenvalue problem. As dis- The comparison with the results in [18] that were obtained with a relativistic MCDF (multi-configuration Dirac-Fock) method shows on the other hand pronounced P ( E, θ ) = 1 π Im { ∑ k 〈 Ψ T − i ( θ $^{∗}$) | Ψ He k ( θ ) 〉〈 Ψ He k ( θ $^{∗}$) | Ψ T − i ( θ ) 〉 E He k ( θ ) − E } . (9)