Appendix
Atomic Orbital Overlap and Resonance Between Standard and
“Long-Bond” Lewis Structures
As well as relative energy considerations, atomic orbital overlap via the off-diagonal matrix elements ( ij
H ) of the secular equations (Section 1-3) helps promote
the importance of “long-bond” structures for N-centre electron-rich bonding
units
1
. We shall demonstrate this by consideration of 4-electron 3-centre bonding,
for which the standard and “long-bond” Lewis structures are structures (1), (2) and
(3) of Section 23-1. With Heitler-London type wave-functions for the electronpair bonds, the wave-functions for these structures are the 1
, 2
and 3
of
Eqn. 23-2.
The extent to which j
will linearly combine with i
, and the magnitude of
the concomitant resonance stabilization energy, depend on the magnitude of the
Hamiltonian matrix element ij
H for i ≠ j, as well as on the energy separation
–
jj
ii
H
H . The off-diagonal ij
H is atomic orbital overlap dependent, directly
through the overlap integrals ya
S , ab
S and yb
S , and indirectly through the core
Hamiltonian and electron repulsion integrals of the general types
o
H , (μν|λλ) and
(μν|λσ). (The μ, ν, λ and σ are any of the atomic orbitals y, a and b, with μ ≠ ν; see
for example Ref. 2 for integral definitions). With the i
of Eqn. 23-2, it may be
deduced (see for example Ref. 3 for procedure) that the dominant terms for 13
H
and 23
H are functions of ya
S and ab
S , respectively. However because the atomic
orbitals y and b are located on non-adjacent centres, the magnitude of the overlap
integral yb
S is small, and therefore the dominant terms for 12
H are functions of
the product ya ab
S S . Consequently, because all overlap integrals are less than unity
in magnitude, 13
H and 23
H will have appreciably larger magnitudes than has
12
H .
Ó Springer International Publishing Switzerland 2016
R.D. Harcourt, Bonding in Electron-Rich Molecules,
Lecture Notes in Chemistry 90, DOI 10.1007/978-3-319-16676-6
325
Atomic Orbital Overlap and Resonance Between Standard and
“Long-Bond” Lewis Structures
As well as relative energy considerations, atomic orbital overlap via the off-diagonal matrix elements ( ij
H ) of the secular equations (Section 1-3) helps promote
the importance of “long-bond” structures for N-centre electron-rich bonding
units
1
. We shall demonstrate this by consideration of 4-electron 3-centre bonding,
for which the standard and “long-bond” Lewis structures are structures (1), (2) and
(3) of Section 23-1. With Heitler-London type wave-functions for the electronpair bonds, the wave-functions for these structures are the 1
, 2
and 3
of
Eqn. 23-2.
The extent to which j
will linearly combine with i
, and the magnitude of
the concomitant resonance stabilization energy, depend on the magnitude of the
Hamiltonian matrix element ij
H for i ≠ j, as well as on the energy separation
–
jj
ii
H
H . The off-diagonal ij
H is atomic orbital overlap dependent, directly
through the overlap integrals ya
S , ab
S and yb
S , and indirectly through the core
Hamiltonian and electron repulsion integrals of the general types
o
H , (μν|λλ) and
(μν|λσ). (The μ, ν, λ and σ are any of the atomic orbitals y, a and b, with μ ≠ ν; see
for example Ref. 2 for integral definitions). With the i
of Eqn. 23-2, it may be
deduced (see for example Ref. 3 for procedure) that the dominant terms for 13
H
and 23
H are functions of ya
S and ab
S , respectively. However because the atomic
orbitals y and b are located on non-adjacent centres, the magnitude of the overlap
integral yb
S is small, and therefore the dominant terms for 12
H are functions of
the product ya ab
S S . Consequently, because all overlap integrals are less than unity
in magnitude, 13
H and 23
H will have appreciably larger magnitudes than has
12
H .
Ó Springer International Publishing Switzerland 2016
R.D. Harcourt, Bonding in Electron-Rich Molecules,
Lecture Notes in Chemistry 90, DOI 10.1007/978-3-319-16676-6
325