Essentials of Computational Chemistry

572 APPENDIX C
by removing the undesirable higher spin states through a process known as projection or
annihilation. Consider the case where the UHF wave function for the desired state s is
contaminated by the next higher possible spin state (s+1) , i.e.,
UHF = cs s + c(s+1) (s+1) (C.30)
where each pure spin wave function is normalized and the sum of the squares of the coefficients
c is 1 for normalization of UHF . When the annihilation operator of Eq. (14.19) is
applied to the spin-contaminated wave function we have
As+1 UHF S
= cs
2 −{(s + 1)[(s + 1) + 1]} s
[s(s + 1)] −{(s + 1)[(s + 1) + 1]}
S
+ c(s+1)
2 −{(s + 1)[(s + 1) + 1]} (s+1)
[s(s + 1)] −{(s + 1)[(s + 1) + 1]}
[s(s + 1)] −{(s + 1)[(s + 1) + 1]} s
= cs
[s(s + 1)] −{(s + 1)[(s + 1) + 1]}
{(s + 1)[(s + 1) + 1]}−{(s + 1)[(s + 1) + 1]} (s+1)
+ c(s+1)
[s(s + 1)] −{(s + 1)[(s + 1) + 1]}
= cs · 1 · s + c(s+1) · 0 · (s+1)
= cs s (C.31)
Thus, the annihilation operator completely removes the next higher spin state and delivers
a wave function that is a pure s spin state. Note, however, that it is not a normalized
wave function, since cs < 1 (otherwise the original wave function would not have been spin
contaminated). Normalization is simple in this case, since we have
〈 UHF |As+1| UHF 〉=〈cs s + c(s+1) (s+1) |cs s 〉
=〈cs s |cs s 〉+〈c(s+1) (s+1) |cs s 〉
= c 2 s 〈s | s 〉+csc(s+1)〈 (s+1) | s 〉
= c 2 s
(C.32)
With the annihilated wave function in hand, any property may be computed in the usual
fashion as an expectation value of the appropriate operator. The Hamiltonian operator is a
particularly simple operator to work with because we can make good use of the original
UHF wave function in evaluating the expectation value. Thus
EPUHF = 〈UHF |HAs+1| UHF 〉
〈 UHF |As+1| UHF 〉
= 〈cs s + c(s+1) (s+1) |H |cs s 〉
〈 UHF |As+1| UHF 〉