My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
where λ is a Lagrange multiplier, and thus
(
ˆT + ˆV ext + ∑ ∫
n(r ′ )[v E (r i , r ′ ) − f(r i , r ′ )] dr ′
i
+ ∑ )
f(r i , r j ) − λ Ψ(r 1 . . . r N ) = 0. (6.7)
i>j
This is an eigenvalue equation for Ψ:
( ) ˆT + ˆVext + ĤMPC e−e Ψ = λΨ, (6.8)
where the required term in the Hamiltonian is
Ĥe−e
MPC = ∑ ∫
n(r ′ )[v E (r i , r ′ ) − f(r i , r ′ )] dr ′ + ∑
i
i>j
f(r i , r j ). (6.9)
However, the eigenvalue λ does not correspond to the energy:
λ = 〈Ψ| ˆT + ˆV ext + ĤMPC e−e |Ψ〉 (6.10)
≠
The relationship between ĤMPC e−e
∫∫
E MPC
e−e
= 〈 〉
Ĥe−e
MPC 1 −
2
cell
E[Ψ].
and E MPC
e−e
is the following:
n(r)n(r ′ )[v E (r − r ′ ) − f(r − r ′ )] dr dr ′ . (6.11)
In a DMC simulation, the modified Hamiltonian term ĤMPC e−e
should be used to
calculate the drift vector and the branching probability; this will ensure that the
distribution converges to the correct wave function. However, when the goal is to
estimate the ground-state energy, equation (6.11) should be used.
To evaluate ĤMPC e−e
or E MPC
e−e
during a simulation requires a knowledge of n(r),
the electron density. In general, this is not known exactly before the simulation
begins. However, a good approximation may be obtained from the independentparticle
calculation, which is already required for generating the orbitals in the trial
wave function.
When using this approximation, it is possible for the resultant QMC density
to differ from the approximate density used to calculate the electron-electron interaction
energy during the simulation; the calculation is then not self-consistent.
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