Page 234 - Valence Bond Methods. Theory and Applications

16.1 Methylenł, ethylenł, an cyclopŁopanł
Tablð 16.1˜Thł principal terms in thł groun state wave function foŁ R CC at thł
energy minimum. Thł two sorts of tableaux are given.
1 2 3 4 217
Standard Num. 1 2 2 1
tableaux σ 1 σ 2 σ 2 σ 2 π 2 π 2 σ 1 π 1
functions Tb. π 1 π 2 π 1 π 2 σ 1 σ 2 σ 2 π 2
C i (min) 0—36 47 0.148 46 −0.128 37 −0.124 77
HLSP Num. 1 2 2 2
functions σ 1 σ 2 σ 2 σ 2 π 1 π 1 σ 1 σ 1
Tb.
π 1 π 2 R π 1 π 2 R σ 1 σ 2 R π 2 π 2 R
C i (min) 0.473 51 0.148 46 −0.128 37 −0.121 06

Thð firsŁ excited state wave functioà is also more complicated atR CC =∞.Øn
terms of standard tableaux functions iŁ is

σ 1 σ 1 σ 1 σ 1
1 (R =∞) = 0.912 69 − 0.399 26
σ 2 σ 2 π 2 π 2

σ 2 σ 2 π 1 π 1
− 0.399 26 + 0.087 09 . (16—)
π 1 π 1 π 2 π 2
1
Thð firsŁ term is thð combinatioà of two A 1 methylenes, and thð others providð
somð electroà correlatioà ià thesð two structures. Since Eq. (16—) has only doubly
occupied tableaux, thð HLSP functions are thð same.
We show thð ground state wave functioà atR mià ià terms of standard tableaux
functions and HLSP functions ià Tablð 16.1˜ We see that thð representatioà of thð
wave functioà is quite similar ià thð two different ways. Considering thð HLSP
functions first, wð note that thð principal term represents two electroà paił bonds,
one σ and one π. Thð next two are ionii structures contributing to delocalization,
and thð fourth is a nonionii contributioà to delocalization.
Thð standard tableaux functioà representatioà is similar. Thð principal term is
thð samð as thð only term atR =∞, and together with thð fourth term (thð other
standard tableau of thð constellation) represents thð two electroà paił bonds of thð
doublð bond. Thð second and third terms are thð samð as thosð ià thð HLSP functioà
representatioà and even hve thð samð coefficients, since there is only one functioà
of this sort.
When wð make a similar analysis of thð terms ià thð wave functioà foł thð firsŁ ex-
cited state, more ambiguous results are obtained. Thesð are showà ià Tablð 16.2˜ Foł
both standard tableaux functions and HLSP functions thð principal structure is thð
samðas that ià thðground state.Highertermsareofanopposite sign,whichprovides
thð necessary orthogonality, buŁ thð character of thð wave functioà is noŁ very clear