1 Summary
•
Applying the Variatonal principle in a coordinate basis to a system in state
|Ψ⟩
, which within the Hartree
approximation can be written as a product of states
|Ψ⟩
=
|ϕ1⟩ |ϕ2⟩
, we arrive at solving a system of two
equations:
h1(x1)ϕ1(x1) + Zdx2w(x1, x2)|ϕ2(x2)|2ϕ1(x1) = ϵ1ϕ1(x1)
h2(x2)ϕ2(x2) + Zdx1w(x1, x2)|ϕ1(x1)|2ϕ2(x2) = ϵ2ϕ2(x2)
The interaction terms (yellow) can be interpreted as a mean field interaction. That is, the potential that one
system in a fixed state feels due to the weighted average interaction with the other system over all its states.
•
In order to reduce the dimensonality of the problem we are solving, we make use of the anti-symmetric nature
of Ψ(
r1, ..., rN
)with respect to exchange of electrons to reduce the number of integrals to compute. This
introduces the electronic density ρ(r)related to the probability of finding one electron at position r
•
The Hohenberg-Kohn theorem states that the energy
E
of a system is a functional of the electronic density
ρ
:
E[ρ] = *ΨX
i
hi+X
iX
j
1
|ˆri−ˆrj|
Ψ+
which forms the basis in finding Ewithin a problem of reduced dimensionality
2 Question
•
Can we arrive at a formulation of using the electronic density to compute the energy using a functionals similar
to that derived in class when also considering quadrupolar interactions in addition to pairwise interactions?
1
