Computational methods for Coulomb
four-body systems
Zong-Chao Yan
University of New Brunswick
Canada
Collaborators:
G. W. F. Drake
Liming Wang
Chun Li
August 24-27, 2015, Trento
NSERC, SHARCnet, ACEnet, CAS/SAFEA
The rms nuclar charge radius :
1
rc    c ( r ) r 2d 3r
Z
where  c is the nuclear charge density, which requires
2
the nuclear wa ve function.
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)
Etheory  Enr   2 Erel   3 EQED    Enuc
2 Z rc
Enuc 
3
2
  r 
i
i
a b
a b
a b
a b
Etheory
 Enra b   2Erel
  3EQED
   Enuc
for a transiti on a  b

A,A'
IS
 

A,A'
MS
M A  M A'

, mass shift
M A M A'
A,A'
MS
 
A,A'
FS
, for two isotopes A and A'
A,A'
 FS
 Z  (0)  ( rc2 )( A, A' ), field shift
2
b
b
a
a
A
No experiment can separate MS and FS so that we have to reply on theory
to determine MS accurately.
A’
Isotope
shift
MS
1
FS
Z
Why isotope shifts?
Mass independen t terms
1,  2 ,  3 ,  4
are canceled exactly for IS
2 rc
Since Enuc 
3
4

M
2
 r 
i
Mass dependent terms
( M ),  M  ,  2 ( M ),  2 ( M )2 ,  3 ( M )

 2
are known to high precision
 1 MHz
i
 10 kHz  1 MHz
ignored as a theoreti cal uncertaint y
In particular  4 ~ 1 MHz, but cancelled for IS
Finally, nuclear polarizability: Several nuclides have a halo in the excited state not
in the ground state (Pachucki et al)
Absolute measurement
Theoretical background
For low-Z systems, we use perturbation theory:
H  H    H rel   H QED  


H    E 
Etot  E      H rel        H QED    
Variational principle:
 tr H 0  tr
Etr 
 tr  tr
H    E 
then
Etr  E0
Rayleigh-Ritz Method: Choose a basis set
{ p , p  , , N }
Then
Now
Letting
 tr 
N
c
p 

p
p
Etr  Etr (c,c ,  ,cN )
Etr
0
c p
we have a generalized eigenvalue equation
Hc  Oc
H ij   i H 0  j
Oij   i  j
 1 2
Z 
 
  2 i  r 

i 1 
i 
3
H0
 1




i   j 



M
i  j  rij

3
Hylleraas basis set:
j1
j2
j3
j12
j23
j31  r1   r2  r3
r1 r2 r3 r12 r23 r31 e
          
The basis is generated according to
j  j  j  j  j  j  
The nonlinear parameters are optimized by
E


   | H |
 E  |



LM
l1l2 l12 ,l3
Y
rˆ1, rˆ2 , rˆ3 
I   dr1dr2 dr3 r1 r2 r3 r12 r23 r31 e
j1
j2
j3
j1 2
j3 1  r1   r2  r3
j2 3
Perkins expansion:
M 
L 
q 
k 
r   Pq cos   C jqk rq  k rj q   k
j


j , L 
j  q for even j



 j   for odd j
  , L 

M  
M 
If all
I 

j , j , j
are odd, then the integral becomes an infinite series:
 T q 
q 
In terms of W integrals:
W l , m , n; ,  ,  
l!

    



l
dxx e
x


x
m
dyy e
 y


y
dzz n e z
l  m  n  p    ! 


p   l    p  !l  m    p  

l  m  n  

 
  F 

,
l

m

n

p


;
l

m

p


;

   






  




p
Ground state of lithium
Li 2s-2p oscillator strength
Ω
10
11
12
13
14
15
length
0.7469568293363
0.7469568131213
0.7469568104873
0.7469568101163
0.7469568100337
0.74695681002
velocity
0.7469573407
0.7469568605
0.7469568221
0.7469568120
0.7469568102
0.7469568101
accelera.
0.7472526
0.7469435
0.7469664
0.7469573
0.7469581
0.7469572
Relativistic and QED corrections
H rel




  i   j   rij  (rij   i ) j   
   
 i
rij
i j 

 rij

  ri  ln     nLS  


 EQED    Z
Q




i
 lim r a 

i j
 nLS  
a 

n
ij
pn


  
 
  ln  

  
 
  rij  
i j
    ln a  rij 
En  E  ln En  E
pn

 E n  E 
n
The Bethe logarithm  nLS  is very difficult to calculate.

Q

Drake-Goldman Method: Can. J. Phys. 77, 835 (1999)
 nLS  

n
0pn

2
En  E0  ln En  E0
0pn
2
 E n  E0 
n
j1
j2
j3
j1 2
j2 3
rˆ1 , rˆ2 , rˆ3 
YlLM
1l2 l1 2 ,l3
j3 1 [  Z r1  ( Z 1) g K r2 ( Z  2 ) r3 / n ]
r1 r2 r3 r12 r23 r31 e
j1  j2  j3  j12  j23  j31    K  2
K  0,, 
g  10
Works for atoms: H, He, Li
Molecule: H2+ (converged to 9-10 digits)
M  
 MP
( 0)



 MP  ln( Z 2  / M )
M
 (  M( 0 )   ( 0 ) ) /(  / M )  1
(N1, N2)
 (0)
Rati
o
 MP
(4172, 875)
2.980036890
0.113984800
(4172, 1452)
2.980822654
0.113843054
(4172, 2445)
2.980912900
8.7
0.113802133
(4172, 4109)
2.980930379
5.6
0.113797473
(4172, 6809)
2.980937153
2.5
0.113802599
Extrapolation
2.980941(4)
0.11383(3)
P-K-P (2013)
2.980944(4)
0.11381(3)
Puchalski, Kedziera, Pachucki PRA 87, 032503 (2013)

l!
l  m  n  p  2 !  
W ( l , m , n;  ,  ,  ) 

l  mn 3 
     
p 0 l  1  p  ! l  m  2  p      



2 F1 1, l  m  n  p  3; l  m  p  3;
   

Slow convergence when:




~1
   
Li, Wang, Yan, Int. J. Quantum Chem. 113,1307(2013)



p
Singular integral: type I
I   dr1dr2dr3r1 r2 r3 r12 r23 r31 e
j1
j2
j3
2
j23
j31 r1  r2 r3
Our approach:
2
• Expand r etc. into infinite series
• Perform multiple summation with convergence
accelerators
• Absolutely numerically stable
rj
Pachucki’s approach:
• Recursion relations with quadrature
Singular integral: type II
I   dr1dr2dr3r1 r2 r3 r12 r
j1
j2
j3
2
2
j3 1 r1  r2 r3
23
31
r
e
This integral appears in  using global operator
4
method. We currently calculate this using the
asymptotic - expansion technique twice.
King et al : direct expansion method
Pachucki et al : resursive methods
Other methods
a)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