1. Computational methods for Coulomb four-body systems
Zong-Chao Yan
University of New Brunswick
Canada
Collaborators:
G. W. F. Drake
Liming Wang
Chun Li
NSERC, SHARCnet, ACEnet, CAS/SAFEA
August 24-27, 2015, Trento

2. Atomic physics method: Proposed by Drake in 90s for isotope shifts
Shiner et al 3,4He
Riis,..,Drake, 6,7Li+
Extended to radioactive isotopes: in the past 10 years
6He, 8He, 11Li, 11Be (Argonne, GSI, Drake, Pachucki et al.)
Currently: 8B, one-proton halo (Argonne, GSI)

3. b b a a A A’
No experiment can separate MS and FS so that we have to reply on theory
to determine MS accurately.

4. Isotope
shift MS FS 1 Z

5. Why isotope shifts?
Finally, nuclear polarizability: Several nuclides have a halo in the excited state not in the ground state (Pachucki et al)

6. Absolute measurement

7. Theoretical background
For low-Z systems, we use perturbation theory:
Variational principle:
then

8. Rayleigh-Ritz Method:
Choose a basis set
Then
Now
Letting
we have a generalized eigenvalue equation

9. Hylleraas basis set:
The basis is generated according to
The nonlinear parameters are optimized by

10. Perkins expansion:
If all
are odd, then the integral becomes an infinite series:
In terms of W integrals:

11. Ground state of lithium

13. Li 2s-2p oscillator strengthΩ
length
velocity
accelera. 10 11 12 13 14 15

14. Relativistic and QED corrections
The Bethe logarithm is very difficult to calculate.

15. Drake-Goldman Method: Can. J. Phys. 77, 835 (1999)
Works for atoms: H, He, Li
Molecule: H2+ (converged to digits)

16. (N1, N2)
Ratio
(4172, 875)
(4172, 1452)  
(4172, 2445)   
8.7
(4172, 4109)   
5.6
(4172, 6809)
2.5
Extrapolation
(4)
(3)
P-K-P (2013)
(4)
(3)
Puchalski, Kedziera, Pachucki PRA 87, (2013)

17. Slow convergence when:
Li, Wang, Yan, Int. J. Quantum Chem. 113,1307(2013)

19. Singular integral: type I
Our approach:
Expand etc. into infinite series
Perform multiple summation with convergence accelerators
Absolutely numerically stable
Recursion relations with quadrature
Pachucki’s approach:

20. Singular integral: type II

21. Other methods
Explicitly correlated Gaussian
Extensively used by Adamowicz et al and Pachucki et al
(sometimes mixed use with Hylleraas) up to Be
b) Hylleraas-CI
Sims and Hagstrom, He, Li, Be, but for nonrelativistic case