CASINO manual - Theory of Condensed Matter

these weights can be obtained from the asymptotic number of descendants of a walker r j . We then
give a description of the FW algorithm that is now implemented in casino.
35.1 Derivation of the FW method
We show that the weight ω j in Eq. (435) can be interpreted as the asymptotic number of descendents
from the walker r j . We write the importance-sampled Schrödinger equation as
∫
f(r, τ) = G(r ← r ′ , τ)f(r ′ , 0)dr ′ , (436)
where G is the importance-sampled Green’s function. When the initial walker r j is represented by a
δ-function, f(r ′ , 0) = δ(r ′ − r j ), Eq. (436) reduces to
f(r, τ) = G(r ← r j , τ). (437)
This can be interpreted as the transition probability of the walker to move from r j to r in time τ. We
write the importance-sampled Green’s function in its spectral expansion [10],
f(r, τ) = G(r ← r j , τ) = Ψ T(r)
Ψ T (r j )
∞∑
exp[−τ(E n − E Ref )]Φ n (r)Φ n (r j ), (438)
n=0
where Φ n and E n are eigenfunctions and eigenvalues of Ĥ, respectively, and E Ref is a constant. When
considering the limit of τ → ∞ and integrating over the final position r, we obtain
∫
lim f(r, τ) dr = 〈Ψ T|Φ 0 〉 exp[−τ(E 0 − E Ref )] Φ 0(r j )
(439)
τ→∞ Ψ T (r j )
= 〈Ψ T |Φ〉 Φ 0(r j )
Ψ T (r j ) . (440)
When E Ref = E 0 , all contributions from the excited-states decay away in the penultimate equation
in the limit τ → ∞. The left-hand side of the penultimate equation can be interpreted as the number
of descendents from walker r j for asymptotic τ,
∫
N(τ → ∞) = lim f(r, τ) dr. (441)
τ→∞
Combining the last two equations, we find
N(τ → ∞) = 〈Ψ T |Φ〉 Φ(r j)
Ψ T (r j ) , (442)
where 〈Ψ T |Φ〉 is a constant. This is the important relationship between the ratio Φ/Ψ T and the
asymptotic number of walkers descended from the initial walker r j . When this relationship is inserted
in the pure estimator of Eq. (434), the constants cancel in the numerator and denominator. Hence,
the weights ω j can be calculated in a DMC simulation from the asymptotic number of walkers. Since
this technique involves taking information from a later time in the simulation to evaluate quantities
at an earlier time, this method is called future-walking or forward-walking method. We introduce the
future-walking time τ FW (or N FW when given in time steps) that is necessary to project out the ratio
of wave functions in Eq. (442). See Sec. 35.3 for a discussion of τ FW .
35.2 The FW algorithm
Different implementations exist to calculate the asymptotic number of descendents in FW. The
tagging algorithm introduced by Barnett et al. [112], for example, assigns a label to each walker which
uniquely identifies all its ancestors. The asymptotic number of descendents of a walker r j at time t is
then determined by searching through all labels of the walkers at time t+τ fw and counting the walkers
that descend from r j . A more elegant algorithm was proposed by Casulleras and Boronat (CB) [113]
which evaluates the product A j ω j and the sum ∑ j ω j in Eq. (434) instead of calculating the weight
ω j and quantity A j for each walker individually. We use this idea for the FW implementation in
casino but chose a slightly improved version to the one originally proposed by CB.
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