The electron-electron guiding function
The integral of interest is
I   d 3r1
  dx
12
 
dy12 dz12 f  r1 ,r2 
1
2

1
0
0

 
  d r1  d  d  r12 2 f  r1 ,r12  dr
Eqn 1
3
Let
g r    r .5 
Eqn 2
Figure 1
And then make a change of variables to y defined by
r
 g r'  dr'
y
0

 g r'  dr'

0

r .5 1  1  y
.5r .5r 2
 4 r  r 2 
.25.5x .25

Eqn 3
So that
dy 
g r  dr

 g r'  dr'
 F  r  dr
Eqn 4
0
Making
1
 1
 2   1
 d
I   d r1  d 

 2  0 2 0
0

f r1 , r  y 
1
3




2
g r  y  /  4r  y   g x  dx 


0


dy
Eqn 5
Which gives the weight as
g r 
w

4r
2
 g r'  dr'
0

 1  2r 
r 2
Eqn 6