234 Finally, by replacing a 2by 1 - b 2 ,one obtains
+ (CSn)'[qr + 2(crm)2]-r + 4ab(crmcs"/3)(c,m)2qsf - (crm)2(csn)2?2 I P , - EA, + I P , - EA, - 2(crm)2(csn)21'
AEm,' = 2b'{IPn - EA,
E
(AI) The interaction between all other combinations of orbitals can be found easily by a similar treatment.
x 8 Y ( 1
1 - t)[qr Rr and with the following transformations
T
E,* = I P ,
AEmnf
E
AEmn =
M(m,m)
+ M(n,n) +
+
d [ ~ m , m) ~(n,n)12 4~(m,n)2
M(n,m)
M(n,n)
=
I P , - EA,
=
=
I P , - EA,
0
- (crm)2(csn)2~ [qs - 2 b 2 ~ s ( ~ s n (A2) )2]
The final perturbation energy becomes (after correction for double account of electron-electron interaction) AE,,
where the matrix elements M are M(m,m)
F x F (1 - k)[qr + 2b2xr(crm)2]
+ AsoIv,~
where AE,,' has the value derived' in Appendix I and Asolv,, that of eq 5. The variational treatment applied to this expression leads to the following equation for the perturbational energy
+
E,* - E," b2[(Em* - En*)ba - (Em* - En*)bs=~]
=
d ( E , * - En*)2
crmcSn/3
+ (c,")2(q, + 2(crm)9-re -
(crm)lqsT r - ( c r m ) 2 ( c s n ) 2 1' ( ~1) - b 2 [ I P , - EA,
+
+ 2bzxr(crm)2]
+ qs(Crrn)'- r -
Appendix I1 Minimization of the electron transfer energy with respect to the variational parameters a and b is included here. AEmn =
+
- ;)[qs - 2 b 2 ~ s ( ~ s n ) 2 ]
+
+
(A3) When the perturbation is small, i.e., 4/32
