CASINO manual - Theory of Condensed Matter

Suppose the DMC simulation is equilibrated, so that f(R, t) = Ψ(R)φ 0 (R). Then
∫ ∫
W (t + τ) = G(R ← R ′ , τ)f(R ′ , τ) dR ′ dR
∫ ∫
= Ψ(R)〈R| exp[−τ(Ĥ − E T )]|R ′ 〉Ψ −1 (R ′ )Ψ(R ′ )φ 0 (R ′ ) dR dR ′
= 〈Ψ| exp[−τ(Ĥ − E T )]|φ 0 〉 = W (t) exp[−τ(E 0 − E T )].
So
E 0 = − 1 ( )
W (t + τ)
τ log exp(−τET )
W (t)
≈ − 1 τ log ( exp[−ET (m + 1)τ]M tot (m + 1)
M tot (m)
)
. (54)
This is the single-iteration growth estimator.
By taking the expectation value of the argument of the logarithm and using our estimate of the
effective time step, we obtain a much less noisy estimate of the ground state 22 :
⎛∑ ( )
m
1
E growth (m) = −
τ EFF (m) log m
⎝
′ =1 Π(m′ , T p )M tot (m ′ ) exp[−ET
⎞
(m ′ +1)τ EFF(m ′ )]M tot(m ′ +1)
M tot(m ′ )
∑ m
⎠
m ′ =1 Π(m′ , T p )M tot (m ′ )
(∑ m
1
= −
τ EFF (m) log m ′ =1 Π(m′ , T p ) exp[−E T (m ′ + 1)τ EFF (m ′ )]M tot (m ′ )
+ 1)
∑ m . (55)
m ′ =1 Π(m′ , T p )M tot (m ′ )
Equation (55) is used to evaluate the growth estimator of the energy in casino if the
growth estimator flag is set to T in the input file. The error bar on the growth estimator is always
much larger than the error on the mixed estimator in practice, so we do not normally use the growth
estimator.
13.9 Automatic block-resetting
Numerous schemes for preventing population control catastrophes due to the occurrence of ‘persistent
electrons’ have been investigated. Of these, the one that seems to perform best in practice involves
returning to an earlier point in the simulation and changing the random number sequence.
If the dmc trip weight input variable is set to a nonzero value, then a config.backup file will be
created. This contains a copy of the config.out file from the beginning of the previous block. If the
total weight at any given iteration exceeds dmc trip weight, then the data from config.backup
will be read in, the last block of lines will be erased from the dmc.hist file and the random number
generator will be called a few times so that the configurations go off on new random walks, hopefully
avoiding the catastrophe that led to the population explosion in the first place. (If dmc trip weight
is exceeded in the first or second blocks, then the initial config.in file will be read in instead of
config.backup.)
Note that if the accumulation of expectation values other than the energy is flagged in input, and
the calculation is in a DMC statistics accumulation phase, then the resulting expval.data file will be
subject to the same treatment (through a saved ‘expal.backup’ file).
If a block has to be reset more than max rec attempts times then the program will abort with an
error.
Great care should be taken when choosing a value for dmc trip weight. It should be sufficiently large
that it cannot interfere with normal population fluctuations: this would lead to population control
biasing. (Note that the population often grows rapidly at the start of equilibration: again, it must
be ensured that automatic block resetting does not interfere with this natural process.) On the other
hand, dmc trip weight should be sufficiently small that persistence is dealt with quickly and that
there is insufficient time for a population of configurations containing a persistent electron to stabilize.
Choosing larger block lengths (by decreasing the value of dmc equil nblock and dmc stats nblock)
allows the program to return to an earlier point in the simulation, increasing the likelihood that the
catastrophe will be avoided.
22 Note that by bringing the average inside the logarithm we introduce a small bias into E growth .
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