My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 2.
MECHANICS
THE SIMPLIFICATION OF MANY-ELECTRON QUANTUM
as whole entities. Secondly, it is assumed that electrons and nuclei do not move at
relativistic speeds. 2 At this point, the Hamiltonian operator may be written as
Ĥ({r i }, {d α }) = − 1 ∑
∇ 2 r
2
i
+ 1 ∑ 1
2 |r
i
i − r j | − ∑ Z α
|d
i≠j
i,α α − r i |
− 1 ∑ 1
∇ 2 d
2 M α
+ 1 ∑
(2.2)
Z α Z β
α 2 |d α − d β | .
α
Here, {r i } and {d α } represent electronic and nuclear positions respectively, while
{Z α } are the nuclear charges.
The next step, the Born-Oppenheimer or adiabatic approximation, relies on the
fact that the nuclear masses M α are very large. The electrons are considered to react
immediately to any change in the nuclear coordinates, and the relaxed electronic
configuration then provides a potential for the heavy, slow-moving nuclei. The wave
function is written as a product:
α≠β
Ψ({x i }, {d α }, t) = ψ({x i }; {d α })χ({d α })e −iEt . (2.3)
The dependence of ψ on {d α } is assumed to be parametric: ψ must vary smoothly
as a function of the nuclear coordinates. The x i represent electron spins as well as
positions.
This approximation reduces equation (2.1) to two separate equations involving
the nuclear and electronic coordinates respectively:
(
− 1 ∑
∇ 2 r
2
i
+ 1 ∑ 1
2 |r
i
i − r j | − ∑ )
Z α
ψ n = E n ({d α })ψ n (2.4)
|d
i≠j
i,α α − r i |
(
)
− 1 ∑ 1
∇ 2 d
2 M α
+ 1 ∑ Z α Z β
α α 2 |d α − d β | + E n({d α }) χ nλ = E nλ χ nλ (2.5)
α≠β
In equation (2.4), the nuclear coordinates {d α } are fixed. The electronic coordinates
{x i } do not enter into equation (2.5); the link is provided by the electronic energy
eigenvalue E n ({d α }), which is equivalent to a potential for the nuclei.
2 In fact, it is not strictly necessary [5, 38, 29] to exclude relativistic effects and magnetic
interactions, which become important in certain situations. However, in many other cases, they
do not play a significant rôle and may be neglected. They are not relevant to the topics discussed
in this thesis.
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