My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
However, as long as the density and wave function are close to their ground-state
forms, this lack of self-consistency is not important; the energy calculated in QMC is
not sensitive to small errors in the approximated density, as the following argument
shows.
The estimated density will be denoted ñ; the QMC energy now depends on both
ñ and Ψ:
E[Ψ; ñ] = 〈 Ψ ∣ ∣ ˆT + ˆVext + ĤMPC e−e [ñ] ∣ ∣Ψ 〉 − 1 2
∫∫
cell
ñ(r)ñ(r ′ )[v E (r, r ′ ) − f(r, r ′ )] dr dr ′ .
(6.12)
When the ground-state density is estimated correctly (ñ = n 0 ), the energy is minimised
by the true ground-state wave function:
( )
δ
δΨ E[Ψ; n 0] = 0. (6.13)
Ψ=Φ 0
This is just a restatement of the variational principle. A similar condition applies
to the estimated density:
δ
δñ(r)
∫cell
E[Ψ; ñ] = [n(r ′ ) − ñ(r ′ )][v E (r − r ′ ) − f(r − r ′ )] dr ′ , (6.14)
where n is the QMC density corresponding to Ψ. It follows that when the calculation
is self-consistent
( δ
δñ
)ñ=n
E[Ψ; ñ] = 0. (6.15)
The implication of equations (6.13) and (6.15) is that when Ψ = Φ 0 and ñ = n 0 ,
the energy is stationary with respect to both Ψ and ñ; therefore, when both these
functions are close to their ground-state forms, the error in the calculated QMC
energy is second-order in (Ψ − Φ 0 ) and (ñ − n 0 ).
Equation (6.9) illustrates the reason for the improvement in speed achieved by the
MPC interaction. Two-body interactions require O [N 2 ] operations, while one-body
interactions require only O [N]; the only two-body term in the MPC interaction is
f, which is a much simpler function to evaluate than the costly v E . The remaining
term in equation (6.9) is effectively a one-body potential.
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