52
Chapter 3 Wave-Functions and Valence-Bond Structures for 1-Electron Bonds, …
Table 3-3: Atomic orbital charges and bond orders for A · B , A—B , A · B
and A B
.
aa
P
bb
P
ab
P
1
ab
( )
2
1/ (1
)
k
2
2
/ (1
)
k
k
2
/ (1
)
k
k
2
ab
( )
2
2 / (1
)
k
2
2
2 / (1
)
k
k
2
2 / (1
)
k
k
ab
2
* 1
ab
( ) ( )
2
1 1/ (1
)
k
2
2
1
/ (1
)
k
k
2
/ (1
)
k
k
ab
2
* 2
ab
( ) ( )
2
2
0
For the Pauling “3-electron bond”
A · B
A B
A B
, we may write
1
2
ab
2
* 1
1
1
1
2
1
1
2
2
ab
ab
( ) ( ) (a) ( ) (b) {(a) (b)
(a) (b) } / (1
)
k
k , from which it may
be deduced that the weights for the component structures
A B and
A B are
2
1/ (1
)
k
and
2
2
/ (1
)
k
k
. These weights correspond
20 to the odd-electron
charges for the b and a atomic orbitals of
A · B , which arise from single occupancy of the antibonding
ab
*
orbital in
ab
2
* 1
ab
( ) ( )
. This result will be required
in Section 14-3.
More elaborate definitions of atomic orbital charges, bond-orders and valencebond structural weights are needed when atomic orbital overlap integrals are
included in normalization constants and orthogonality relationships. These are not
required for the considerations of this book.
References
1 J.W. Linnett, J. Amer. Chem. Soc., 83, 2643 (1961).
2 K. Ruedenberg, (a) Revs. Modern Phys., 34, 326 (1962); (b) Localization and Delocalization in Quantum Chemistry (o. Chalvet et al., eds.) Reidel, Dordrecht-Holland. Vol. I,
p. 223 (1975).
3 M.J. Feinberg, K. Ruedenberg and E.L. Mehler in “Advances in Quantum Chemistry”,
(P.O. Lowdin, ed.), Academic Press, New York, 5, 27 (1970). also M.J. Feinberg and
K. Ruedenberg, J. Chem. Phys., 54, 1495 (1971).
4 M. Kutzelnigg, Angewandte. Chem. (int. Ed.) 12, 546 (1973). F. Dreissler and W. Kutzelnigg, Theor. Chim. Acta, 43, 1 (1976)
5 M.W. Schmidt, J. Ivanic and K. Ruedenberg, J. Chem. Phys. 140, 204104 (2014) and
Refs. 9 and 11 therein. See R.D. Harcourt, H. Solomon, J. Beckworth and L. Chislett,
Amer. J. Phys. 50, 557 (1982) for a Bohr orbit demonstration of the Ruedenberg theory.
6 J.A. Pople, Quart. Revs. 11, 273 (1957), and references therein.
7 J.W. Linnett. J. Chem. Soc., 275 (1956).
8 L. Pauling, (a) J. Amer. Chem. Soc., 53, 1367, 3225 (1931); (b) The Nature of the
Chemical Bond (Cornell, 1960), Chapter 10.
Chapter 3 Wave-Functions and Valence-Bond Structures for 1-Electron Bonds, …
Table 3-3: Atomic orbital charges and bond orders for A · B , A—B , A · B
and A B
.
aa
P
bb
P
ab
P
1
ab
( )
2
1/ (1
)
k
2
2
/ (1
)
k
k
2
/ (1
)
k
k
2
ab
( )
2
2 / (1
)
k
2
2
2 / (1
)
k
k
2
2 / (1
)
k
k
ab
2
* 1
ab
( ) ( )
2
1 1/ (1
)
k
2
2
1
/ (1
)
k
k
2
/ (1
)
k
k
ab
2
* 2
ab
( ) ( )
2
2
0
For the Pauling “3-electron bond”
A · B
A B
A B
, we may write
1
2
ab
2
* 1
1
1
1
2
1
1
2
2
ab
ab
( ) ( ) (a) ( ) (b) {(a) (b)
(a) (b) } / (1
)
k
k , from which it may
be deduced that the weights for the component structures
A B and
A B are
2
1/ (1
)
k
and
2
2
/ (1
)
k
k
. These weights correspond
20 to the odd-electron
charges for the b and a atomic orbitals of
A · B , which arise from single occupancy of the antibonding
ab
*
orbital in
ab
2
* 1
ab
( ) ( )
. This result will be required
in Section 14-3.
More elaborate definitions of atomic orbital charges, bond-orders and valencebond structural weights are needed when atomic orbital overlap integrals are
included in normalization constants and orthogonality relationships. These are not
required for the considerations of this book.
References
1 J.W. Linnett, J. Amer. Chem. Soc., 83, 2643 (1961).
2 K. Ruedenberg, (a) Revs. Modern Phys., 34, 326 (1962); (b) Localization and Delocalization in Quantum Chemistry (o. Chalvet et al., eds.) Reidel, Dordrecht-Holland. Vol. I,
p. 223 (1975).
3 M.J. Feinberg, K. Ruedenberg and E.L. Mehler in “Advances in Quantum Chemistry”,
(P.O. Lowdin, ed.), Academic Press, New York, 5, 27 (1970). also M.J. Feinberg and
K. Ruedenberg, J. Chem. Phys., 54, 1495 (1971).
4 M. Kutzelnigg, Angewandte. Chem. (int. Ed.) 12, 546 (1973). F. Dreissler and W. Kutzelnigg, Theor. Chim. Acta, 43, 1 (1976)
5 M.W. Schmidt, J. Ivanic and K. Ruedenberg, J. Chem. Phys. 140, 204104 (2014) and
Refs. 9 and 11 therein. See R.D. Harcourt, H. Solomon, J. Beckworth and L. Chislett,
Amer. J. Phys. 50, 557 (1982) for a Bohr orbit demonstration of the Ruedenberg theory.
6 J.A. Pople, Quart. Revs. 11, 273 (1957), and references therein.
7 J.W. Linnett. J. Chem. Soc., 275 (1956).
8 L. Pauling, (a) J. Amer. Chem. Soc., 53, 1367, 3225 (1931); (b) The Nature of the
Chemical Bond (Cornell, 1960), Chapter 10.