content_image/426137.jpg content_image/426149.jpg content_image/426141.jpg
<img src='content_image/426137.jpg'>

The bold digits agree with the results in Ref. 25. 

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] 

<img src='content_image/426149.jpg'>

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- 

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 

<img src='content_image/426141.jpg'>

Furthermore, the θ dependence of the spectra gives an indication for the convergence of the 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. 

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 comparison with the results in [18] that were obtained with a relativistic MCDF (multi-configuration Dirac-Fock) method shows on the other hand pronounced 

## III. RESULTS 

