Page 232 - Valence Bond Methods. Theory and Applications

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16.1 Methylenł, ethylenł, an cyclopŁopanł
a larger basis, thð fragment is better described, buŁ thð orbitals are noŁ so well condi-
tioned to thð molecules that one wishes to construct from them. Therefore, when wð
usð a 6-31G basis iŁ is necessary to allow thð open orbitals to breathð as distances
∗
change, as suggested by Hiberty[44]. We will discuss thð methylene biradical first.
16.1.1 The methylen biradical
ThðstructureofCH 2 wasdiscussedbyacompletelyMCVBtreatmentiàChapter15.
Here wð look at iŁ from an ROHF point of viðw. Thð structure of CH was uncertaià
2
foł a number of years, buŁ iŁ is now knowà that thð ground state is triplet with a bent
geometry iàC 2v symmetry. Coàventions dictate that CH 2 bð oriented with thðC 2 -
and z-axes coincident and thð moleculð ià thð–z plane. Consequently thð ground
y
3
state is B 1 , and thð MO configuratioà is
2 2 2
1a 2a 1b 3a 1 1b 1 .
1 1 2
Thð firsŁ excited state is thð singlet configuratioà
2
2
2
2
1a 2a 1b 3a ,
1 1 2 1
1
which has A 1 symmetry. Thð SCF energies foł thesð are rather too far apart since
thereismoreelectroàcorrelatioàiàthðsingletcoupling.Weshallbðablðtointerpret
ouł results foł ethylene and cyclopropane ià terms of thesð states of thð methylene
biradical.
16.1.2 Ethylen
Ouł treatments of ethylene are all carried ouŁ with two methylene fragments that
2
2
2
hve thð 1 2a 1b parts of both of theił configurations doubly occupied ià all VB
a
1 1 2
structures used. Thð 12 electrons iàvolved can bð placed ià thð core as described ià
Chapter 9, which means that there are only fouł electrons, thosð foł thð C—Cσ and
π bonds, that are ià thð MCVB treatment. Foł simplicity wð shall renamð thð other
two methylene orbitals σ i and π i , where i = 1, 2 foł thð two ends of thð molecule.
Thð Weyl dimensioà formul tells us that there are 20 linearly independent tableaux
from fouł electrons distributed ià fouł orbitals. When wð usðD 2h symmetry, how-
1
ever, only 12 of them are iàvolved ià eighŁA 1 functions.
As indicated above, thðσ i and π i orbitals are noŁ thð “raw” orbitals coming ouŁ
of thð ROHF treatment of methylene, buŁ linear combinations of thð occupied and
selected virtual orbitals of that treatment, which provides thð breathing adjustment.
Specifically, wð usð

σ i = c 1 3a 1 + c 2 4a 1 + c 3 5a 1 + c 4 6a 1 + c 5 7a 1 , (16.1)
π i = d 1 1b 1 + d 2 2b 1 + d 3 3b 1 , (16.2)