Convergence of an s‐Wave calculation of the He ground state

Abstract

ABSTRACT:The configuration interaction (CI) method using a large Laguerre basis restricted to ᐉ ϭ 0 orbitals is applied to the calculation of the He ground state. The maximum number of orbitals included was 60. The numerical evidence suggests that the energy converges aswhere N is the number of Laguerre basis functions. The electron-electron ␦-function expectation converges as ⌬␦ N Ϸ A/N 5/ 2 ϩ B/N 6/ 2 ϩ . . . , and the variational limit for the ᐉ ϭ 0 basis is estimated as 0.1557637174(2) a 0 3 . It was seen that extrapolation of the energy to the variational limit is dependent on the basis dimension at which the exponent in the Laguerre basis was optimized. In effect, it may be best to choose a nonoptimal exponent if one wishes to extrapolate to the variational limit. An investigation of the natural orbital asymptotics revealed the energy converged as These studies have investigated the convergence of the energy with respect to the number of partial waves included in the wave function, and also with respect to the dimension of the radial basis.It has been known since 1962 [9] that the energy converges slowly with respect to J, the maximum angular momentum of any orbital included in the CI expansion. In particular, the leading term to the energy increment is known to behave as (1) at high J. Later work [1][2][3][4]6] showed that the energy increments can be written more generally as

Application of Löwdin's canonical orthogonalization method to the Slater-type orbital configuration-interaction basis set

“…It is clearly seen that the energy is more converged than the one‐group STO‐CI calculations and the reduction of the basis is also smaller. At n max = 60, the relative error of our two‐group STO‐CI calculation is about which is quite close to the value of obtained by the largest LTO‐CI basis of Mitroy et al The previous STO‐CI calculation of Jitrik and Bunge achieves to an accuracy of at a full CI level where all the parameters of STOs are optimized. The three‐group STO‐CI calculation of Sims and Hagstrom in an extended real × 24 arithmetic gets .…”

“…The value of ε was chosen at in all of these calculations. The orthogonal LTO‐CI basis calculations performed by Mitroy et al with the same number of basis functions are included for comparison. Other theoretical calculations based on an extrapolation procedure to infinite number of basis functions are also included at the bottom of Table .…”

Rate of Convergence of the Configuration Interaction Model for the Helium Ground State

“…It is notable that this expansion is also slowly convergent and, due to the increasing number of terms, more slowly convergent as increases. While its rate of convergence has been discussed [17,3,5], there are no rigorous results, which, when combined with the results of this paper, would give a rate of convergence in terms of one-electron orbitals. We note however that the expansion of the radial part in r < := min{r 1 , r 2 } and r > := max{r 1 , r 2 } converges rapidly [4,21].…”

“…A related topic of interest is the rate of convergence with respect to the radial basis. The cases of a Laguerre basis and the natural orbital basis for a CI calculation of the = 0 part of the ground state of helium have been investigated numerically [17]. A similar rigorous analysis of these asymptotics could lead to insight into the important terms of the expansion, as well as providing rigorous extrapolation formulas.…”

Convergence of the partial wave expansion of the He ground state

“…It should be noted that the δ‐function operator also appears in the Breit–Pauli relativistic correction as the two‐body Darwin interaction 7, 29. The present work builds on an earlier investigation that studied the convergence of the radial basis in a simplified model of the helium atom, which only included l = 0 orbitals 30.…”