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