J . Phys. Chem. 1985,89, 925-930
can be taken outside of the integral, leaving us to only consider the integral over 1"'. Clearly, the integrals of odd functions of X about X = 0 will vanish. The only nonzero terms in the sum will thus be for n - 1 even, i.e., n odd, leaving us with Im G(E') c- d(2"+1) dE(2"+1)
"-0
I
1:X2"
Im G(E')
C
dE'=
dE(2n+')
The number of terms in the sum that are necessary for it to converge is clearly a function of how small 6 is, Le., how closely one approaches the pole. The principal value integral is thus computed in three parts: leading up to the pole, about the pole, and beyond the pole
dX
E'=€
"Im G(E')
where we have taken the integral about X = 0 from -6 to 6. The evaluation of this integral is straightforward: 262n+l dX = 2n 1
1:X2"
925
dE'=
Im G(E')
+
E'-E-dIm G(E? dE' E'- E dE'+
-
+
d(Z"+l)Im G(E')l
,,=o
dE
