CASINO manual - Theory of Condensed Matter

where the numerator is the average of the measured values of its arguments over the configurations
indexed by 1 ≤ t ′ ≤ N − t, and ˆσ H 2 is the variance of these measures. One possibility for setting the
cutoff is to check against the self-consistent inequality t max < 3τ(t max ) while computing the sum, and
truncate it as soon as it stops being true. This allows an estimate of the error in above expression to
be calculated:
√
2(2tmax + 1)
ɛ τ (t max ) = τ
. (190)
N
The reblocking method and the correlation time are in principle equally valid methods for estimating
the error in the energy. Each of them has its own disadvantages, though: the plot of reblocked standard
errors can often become noisy before the plateau is reached, preventing accurate determination of the
optimal reblocking length, whereas the error in the correlation time decays very slowly with the
number of energies in the sample. It is recommended that both measures of serial correlation be taken
into account for optimal results.
24.3 Estimating equilibration times and correlation periods
The root-mean-square distance diffused by a particle in a period T of imaginary time is √ 2N D DAT ,
where A is the move acceptance ratio (which is usually close to 1 in DMC and 1/2 in VMC), N D is the
dimensionality of the system (which is usually 3, unless a strict 2D or 1D system is being studied) and
D = 1/2m is the diffusion constant, where m is the particle mass (note that D = 1/2 for electrons).
We expect that correlation effects will disappear when the particles have diffused through distances
in excess of the largest physically relevant length-scale λ. Let T = N move × τ, where N move is the
number of moves and τ is the time step. Then the number of moves over which we expect correlation
effects to be present is
λ 2
N move =
2N D DτA . (191)
The number of equilibration moves should be substantially larger than the above estimate of the
correlation period in order to ensure that all of the transient effects due to the initial distribution die
away. The required equilibration period is often greater than one might expect by simply examining
the variation of the total energy with time.
In practical QMC calculations, with sensible choices of time step, we often find the VMC correlation
period to be about 5 configuration moves and the DMC correlation period to be about 1000 moves.
25 Wave-function optimization
Optimization of the trial wave function is a crucial part of a VMC or DMC calculation. casino
allows optimization of the parameters in the Jastrow factor, the coefficients of the determinants in
a multideterminant wave function, the parameters in the backflow functions, pairing parameters in
electron–hole gases and parameters in the orbitals for certain electron and electron–hole phases as
well as modification functions for atomic orbitals. All optimizable parameters are contained in the
file correlation.data. Furthermore, each optimizable parameter is followed by a flag indicating
whether the parameter is fixed (‘0’) or free to be optimized (‘1’). See Sec. 7.4 for information on
correlation.data.
There are two methods available within casino for wave function optimization: variance minimization
and energy minimization. Both methods can be used to optimize any or all of the parameters
mentioned above. In addition to the ‘standard’ variance-minimization method, there also exists a
much faster version, which can be used when only parameters which appear linearly in the Jastrow
factor are being optimized.
25.1 Variance minimization: the standard method
Consider a real trial wave function Ψ(R), where R is a point in the electron configuration space. In
VMC the energy is written as
∫
|Ψ(R)| 2 E L (R) dR
E = ∫
|Ψ(R)|2 dR , (192)
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