CASINO manual - Theory of Condensed Matter

13.7 Evaluating expectation values of observables
The reference energy E T is varied to maintain a reasonably steady population. However, this procedure
can result in a bias in the estimate of expectation values, especially for small populations. To remove
this one can evaluate expectation values using the method of UNR [19].
Using the label m for time step, Eq. (17) becomes
∫
f(R, m) = G DMC (R ← R ′ , τ)f(R ′ , m − 1) dR ′ . (49)
Clearly, in the absence of the accept/reject step, the effect of including the (time-step-dependent)
reference energy E T (m) in G DMC can be ‘undone’ by multiplying the right-hand-side of Eq. (49) by
exp[−τE T (m)]. In a similar fashion the effect of including the reference energy from the previous
time step can be eliminated by multiplying by exp[−τE T (m − 1)]. When the accept/reject step is
present, we can approximately undo the effect of the reference energy by using our best estimate of
the effective time step τ EFF (See Sec. 13.5) in the ‘undoing’ factors.
Continuing this process, we may eliminate the effect of changing the reference energy from f(m) by
multiplying it by
Π(m) = ∏
exp [−τ EFF E T (m − m ′ )] , (50)
m ′ =0
where in principle the product runs over all previous time steps. In practice it is sufficient to include
T p (=tpdmc) terms in the product, provided that T p is greater than the number of iterations over
which the DMC data are correlated by fluctuations in the reference energy: T p = NINT(10/τ) is
generally sufficient (the estimation of correlation periods is discussed in Sec. 24.3). Let Π(m, T p ) =
∏ Tp
m ′ =0 exp[−τ EFF(m)E T (m − m ′ )]. Then the mixed estimator of the expectation value of a (local)
operator  may be written as:
〈Ψ|Â|Φ〉
〈Ψ|Φ〉
=
≈
∫
Ψ(R) Â(R)Φ(R) dR
∫
Ψ(R)Φ(R) dR
∑ m
m ′ =1 Π(m′ , T p ) ∑ N config (m ′ )
α=1 w α (m ′)Â(α, m′ )
∑ m
m ′ =1 Π(m′ , T p ) ∑ , (51)
N config (m ′ )
α=1 w α (m ′ )
where w α (m ′ ) is the weight of configuration α at the end of time step m ′ . (For unweighted DMC, w α
is simply the branching factor.) Note that if we choose Â(α, m) = Ψ−1 (R α,m )ĤΨ(R α,m) then Eq.
(51) gives us our mixed estimator of the ground state energy at time step m. This is used as our ‘best
estimate’ of the ground state energy, E best (see Sec. 13.4).
The terms in the Π weights are exponential functions of the reference energy; hence the Π weights
are potentially very large (or small). However, it can be seen that any constant contribution to the
reference energy will cancel in Eq. (51). Therefore, in practice, we evaluate the Π-weights as:
Π(m, T p ) =
T
∏ p−1
m ′ =0
exp [τ EFF (1)E V − τ EFF (m)E T (m − m ′ )] , (52)
where E V is the variational energy. This is necessary in order to avoid floating point errors.
Note that in practice population-control bias is usually negligible if more than a few hundred configurations
are used, and that the Π weights are an additional source of statistical noise. For this
reason we do not usually use the UNR Π-weighting scheme [i.e., Π(m, T p ) = 1 for all m in all formulae
involving the Π weights]. The scheme is not used when tpdmc is set to 0, which is the default.
13.8 Growth estimator of the energy
The total weight of a DMC simulation at time t = mτ is given by:
∫
W (t) ≡
f(R, t) dR ≈
N config (m)
∑
α=1
w α (m) ≡ M tot (m). (53)
129