Application of the Hylleraas- B -spline basis set: Nonrelativistic Bethe logarithm of helium

Abstract

In this work, we report an application of Hylleraas-B-spline basis set to the nonrelativistic Bethe logarithm calculation of helium. The Bethe logarithm for n 1 S, n up to 10, states of helium are calculated with a precision of 7-9 significant digits in two gauges, which greatly improves the accuracy of the traditional B-spline basis set. In addition, to deal with the numerical linear correlation problem in Bethe logarithm calculation, we developed a multiple-precision generalized symmetric eigenvalue problem solver (MGSEPS). This program may be very useful to precision calculations.

Application of the correlated B-spline basis functions to the leading relativistic and QED corrections of helium

“…is for both the 2 3 S and 3 3 S states. Yang et al [25] have implemented the correlated B-splines to calculate the helium atomic energy level and their non-relativistic ground state energy is −2.903 724 377 1(2) a.u.. A knot distribution optimization was performed for any individual states and present values of energies for the 1 1 S, 2 1 S, 2 3 S and 3 3 S states are listed in Table II. The optimized result of −2.903 724 377 034 0(2) a.u.…”

“…Compared with the relativistic correction, the more difficult to calculate in the leading QED correction are Bethe logarithm and Araki-sucher correction. The Bethe logarithms for the 1 1 S, 2 1 S, 2 3 S and 3 3 S state of the helium atom are summarized in Table I calculated by Zhang et al [22] using traditional Bspline function and Yang et al [26] using the C-BSBF, respectively, which based on the Drake-Goldman's method. The Korobov's results listed in the last column of Table I based on the integral representation method of Schwartz are the benchmarks.…”

“…Then Zhang et al extended it to calculate the Bethe logarithms for the S state of the helium atom [22], in which the Bethe logarithms for the triplet state with weak electron correlation can be reached with five to eight accurate figures, but the precision is limited for the single state as the electron correlation effect is not included in the basis set. Therefore, Yang et al have developed the explicitly correlated B-spline basis method and successfully applied it to the calculation of energy levels, static dipole polarizabilities [25], and Bethe logarithms [26] for the singlet states of the helium atom. The nonrelativistic ground state energy has reached −2.903 724 377 1(2) [25], which is six orders of magnitude better than the result of Lin et al [23].…”

“…Therefore, Yang et al have developed the explicitly correlated B-spline basis method and successfully applied it to the calculation of energy levels, static dipole polarizabilities [25], and Bethe logarithms [26] for the singlet states of the helium atom. The nonrelativistic ground state energy has reached −2.903 724 377 1(2) [25], which is six orders of magnitude better than the result of Lin et al [23]. Moreover, they have been able to obtain static dipole polarizabilities with a relative error of 10 −9 and Bethe logarithms with a relative error of 10 −7 , respectively, which shows that the correlated B-spline basis functions(C-BSBF) can describe well the electronic correlation of the singlet states and effectively improve the numerical convergence rates.…”

Theory and computational methods for high precision spectra of few-electron atomic systems

“…在文献 [108]中, Kor-obov还给出了一个新的计算氦原子里德堡态Bethe logarithm值的近似公式. 近几年, 国内的一些研究者, 如Tang等人 [109] 、Yang等人 [110] 、Zhang等人 [111] [50,86] ; 严格的计算直到 2003年才由Yan和Drake [112] 完成, 他们用Hylleraas坐标 下的赝态方法, 计算了锂原子2S和3S态的Bethe logarithm. 在同一年, Pachucki和Komasa [113] 用ECG计算了 锂原子2S态的Bethe logarithm, 其精度与Yan和Drake的…”