4.9 Application: Hückel Theory
89
The first term in the brackets is the usual kinetic energy term, while the second term
produces the orbital Zeeman effect. The third term describes the second-order interactions corresponding to the atomic contribution to the susceptibility. The second
term can easily be converted into the more familiar form of the Zeeman operator as
follows:
e
m
(A − A j ) ·
i
∇
=
e
2m
B ·
(r − R j ) ∧ p
=
e
2m
l · B
=−m · B
(4.127)
Here, p is the momentum operator of the electron in atom j , l is the corresponding
angular momentum operator, and m is the magnetic dipole operator. These operators
are related by
m =−
e
2m
l =−
μ B
l
(4.128)
Here μ B is the Bohr magneton. Angular momentum is thus expressed in units of ,
and the magnetic moment in units of the Bohr magneton. The basis atomic orbitals
will be eigenfunctions of the first two operators. So to first order the London basis
orbitals are eigenfunctions of the total Hamiltonian. Moreover, for a p z -orbital, the
Zeeman effect for a magnetic field along the z-axis vanishes. As a result, the on-site
parameter α is independent of the London gauge:
χ j |H|χ j =αχ j |χ j =α
(4.129)
However, the inter-site integrals, which depend on the potential energy, V ,a r ei n -
fluenced by the gauge factors:
χ i |V |χ j =
φ i
V exp
i
e
(A i − A j ) · r
φ j
(4.130)
At this point London introduced an important approximation by replacing the variable position vector in this equation by the position vector (relative to the arbitrary
origin) of the center of the bond between the two atoms:
r = (R i + R j )/2
(4.131)
In this approximation the phase factor is turned into a constant, which can be removed from the brackets. One has:
1
2
(A i − A j ) · (R i + R j ) =
1
4
(B ∧ R i ) · R j − (B ∧ R j ) · R i
=
1
2
B · (R i ∧ R j )
= B · S ij
(4.132)
89
The first term in the brackets is the usual kinetic energy term, while the second term
produces the orbital Zeeman effect. The third term describes the second-order interactions corresponding to the atomic contribution to the susceptibility. The second
term can easily be converted into the more familiar form of the Zeeman operator as
follows:
e
m
(A − A j ) ·
i
∇
=
e
2m
B ·
(r − R j ) ∧ p
=
e
2m
l · B
=−m · B
(4.127)
Here, p is the momentum operator of the electron in atom j , l is the corresponding
angular momentum operator, and m is the magnetic dipole operator. These operators
are related by
m =−
e
2m
l =−
μ B
l
(4.128)
Here μ B is the Bohr magneton. Angular momentum is thus expressed in units of ,
and the magnetic moment in units of the Bohr magneton. The basis atomic orbitals
will be eigenfunctions of the first two operators. So to first order the London basis
orbitals are eigenfunctions of the total Hamiltonian. Moreover, for a p z -orbital, the
Zeeman effect for a magnetic field along the z-axis vanishes. As a result, the on-site
parameter α is independent of the London gauge:
χ j |H|χ j =αχ j |χ j =α
(4.129)
However, the inter-site integrals, which depend on the potential energy, V ,a r ei n -
fluenced by the gauge factors:
χ i |V |χ j =
φ i
V exp
i
e
(A i − A j ) · r
φ j
(4.130)
At this point London introduced an important approximation by replacing the variable position vector in this equation by the position vector (relative to the arbitrary
origin) of the center of the bond between the two atoms:
r = (R i + R j )/2
(4.131)
In this approximation the phase factor is turned into a constant, which can be removed from the brackets. One has:
1
2
(A i − A j ) · (R i + R j ) =
1
4
(B ∧ R i ) · R j − (B ∧ R j ) · R i
=
1
2
B · (R i ∧ R j )
= B · S ij
(4.132)