88
4 Representations
origin, the relevant relations are as follows:
B = ∇ ∧ A
A =
1
2
B ∧ r
(4.122)
This implies that the divergence of the vector potential is zero, and hence A and ∇
commute: [∇, A]=0. The introduction of the magnetic field will add an extra term
in the kinetic energy operator, which becomes
T =
1
2m
i
∇ + eA
2
=−
h 2
8π 2 m
∇ + i
e
A
2
=−
h 2
8π 2 m
+ i
e
A · ∇ + i
e
∇ · A −
e 2
2 A
2
=−
h 2
8π 2 m
+ 2i
e
A · ∇ −
e 2
2 A
2
(4.123)
where we have taken into account that the “del” (or nabla) operator and the vector
potential commute. The electron charge is −e. London proposed that the atomic
basis functions should be multiplied by a phase factor, which explicitly depends on
the vector potential [10]. In this London gauge the atomic orbitals are rewritten as
|χ j =exp
−i
e
A j · r
|φ j
(4.124)
where A j is the vector potential at the position of the j th atom. The effect of this
phase factor is to move the origin of the vector potential from an arbitrary origin to
the local position of atom j . The action of the del operator and Laplacian on this
gauge is given by
∇ exp
−i
e
A j · r
= exp
−i
e
A j · r
−i
e
A j + ∇
exp
−i
e
A j · r
= exp
−i
e
A j · r
−
e 2
2 A
2
j − 2i
e
A j · ∇ +
(4.125)
Combining this result with Eqs. (4.123) and (4.124) yields
T |χ j =−
h 2
8π 2 m
exp
−i
e
A j · r
×
+ 2i
e
(A − A j ) · ∇ −
e 2
2 (A − A j ) · (A − A j )
|φ j (4.126)