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For further needs we also introduce the functions G o .r;R/ and G a .r;R/
defined as
Z
dx e x p x C ˇ j U.r/ j :
2
p
G o .r;R/ D
ˇjU.R/j
Z
2
p
dx e x .r r /.x C x 0 /
G a .r;R/ D
ˇjU.R/j
r x C ˇ j U.r/ j .x C ˇ j U.r / j / r 2
r 2 :
Using the identities F.R/ D G a .R;R/ and G o .R;R/ D e ˇjU.R/jj
yields
n.R/
1 C e ˇjU.R/j F.R/ e ˇjU.R/j ŒG o .r;R/ C G a .r;R/:
n.r/ D
(3.167)
In the case of repulsion, the conditions L 2 .r/ > 0 and L 2 .a/ > 0 determine
the lower limits of integration in the expression for n.r/. Following the route of
derivation of Eq. 3.164 we obtain
2
2
3=2
Z
dx e x p x ˇU.r/
1
p
4
C.R/e ˇU.R/
n.r/ D
ˇU.r/
3
dx e x r x ˇU.r/ .x ˇU.a// a 2
r 2
Z
5
C
;
ˇU.a/
where, again, C.R/ is given by Eq. 3.165 . The definitions of F.R/, G o ,andG a
change. Instead of Eq. 3.166 we have
dx e x r x ˇU.R/ .x ˇU.a// a 2
Z
2
p
F rep .R/ D
R 2 :
ˇU.a/
Next,
Z
dx e x p x ˇU.r/ D e ˇU.r/ ;
2
p
G o .r;R/ D
ˇU.r/             Search WWH ::

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