Page 239 - Valence Bond Methods. Theory and Applications

16 Interaction of molecular fŁagments
222
Tablð 16—˜Thł leadinð tableaux foŁ thł groun state wave function of C 3 H 6 at
thł equilibrium geometry.
1 2 3 4
Standard Num. 1 1 1 1
tableaux σ 1 σ 2
σ 1 σ 3
σ 1 σ 2
π 1 π 1
functions Tb. σ 3 π 2 σ 2 π 1 σ 3 π 1 σ 2 σ 3
π 1 π 3 π 2 π 3 π 2 π 3 π 2 π 3
C i (min) 0.134 41 0.130 92 −0.123 14 0.102 56
HLSP Num. 1 1 1 2
functioà σ 2 σ 3
σ 1 σ 2
π 1 π 1
π 3 π 3
Tb. π 1 π 2 σ 3 π 1 σ 2 σ 3 σ 1 σ 2
σ 1 π 3 π 2 π 3 π 2 π 3 π 1 π 2
R R R R
C i (min) 0.132 70 −0.125 19 0.086 67 0.085 38

thð states of thð separate pieces of ouł system. Thð lasŁ tableau given above can
also bð written symbolically as
 
σ 2 σ 3

π 2
  + σ 1 ,
π 3 π 1
where thð two triplet tableaux can fiŁ together to form thð earlier singlet shape. We
hve noŁ emphasized this sort of combining of tableaux ià ouł earlier work, buŁ iŁ
is particularly useful foł systems ià asymptotii regions. There are more structures
foł this state and set of orbitals than foł thð lower one, becausð this set musŁ also
1
represent a still higher coupling of ethylene and methylene, both ià a B 1 state.
We do noŁ show thð energy curve foł thð state that goes asymptotically to that
coupling.
1
Thð wave functions foł thð ground and firsŁ excitedA 1 states foł thð cyclo-
propane equilibrium geometry are showà ià Tables 16— and 16.6. In thð casð of
thð ground state either thð principal standard tableaux functioà oł HLSP functioà
can bð transformed as follows:

   
σ 1 σ 2 σ 2 σ 3
 σ 3 π 2  =− π 2 σ 1  ,

π 1 π 3 π 3 π 1
   
σ 2 σ 3 σ 2 σ 3
π 1 π 2 π 2 σ 1
  =   .
σ 1 π 3 π 3 π 1
R R