Quantum Monte Carlo Methods¶
qmc
factory element:
Parent elements
simulation, loop
type selector
method
attributetype options:
vmc
Variational Monte Carlo
linear
Wavefunction optimization with linear method
dmc
Diffusion Monte Carlo
rmc
Reptation Monte Carlo
shared attributes:
Name
Datatype
Values
Default
Description
method
text
listed above
invalid
QMC driver
move
text
pbyp, alle
pbyp
Method used to move electrons
gpu
text
yes/no
dep.
Use the GPU
trace
text
no
???
checkpoint
integer
1, 0, n
1
Checkpoint frequency
record
integer
n
0
Save configuration ever n steps
target
text
???
completed
text
???
append
text
yes/no
no
???
Additional information:
move
: There are two ways to move electrons. The more used method is the particlebyparticle move. In this method, only one electron is moved for acceptance or rejection. The other method is the allelectron move; namely, all the electrons are moved once for testing acceptance or rejection.gpu
: When the executable is compiled with CUDA, the target computing device can be chosen by this switch. With a regular CPUonly compilation, this option is not effective.checkpoint
: This enables and disables checkpointing and specifying the frequency of output. Possible values are:[1] No checkpoint (default setting).
[0] Dump after the completion of a QMC section.
[n] Dump after every \(n\) blocks. Also dump at the end of the run.
The particle configurations are written to a .config.h5
file.
<qmc method="dmc" move="pbyp" checkpoint="0">
<parameter name="timestep"> 0.004 </parameter>
<parameter name="blocks"> 100 </parameter>
<parameter name="steps"> 400 </parameter>
</qmc>
The checkpoint flag instructs QMCPACK to output walker configurations. This also
works in VMC. This outputs an h5 file with the name projectid.runnumber.config.h5
.
Check that this file exists before attempting a restart.
To continue a run, specify the mcwalkerset
element before your VMC/DMC block:
<mcwalkerset fileroot="BH.s002" version="0 6" collected="yes"/>
<qmc method="dmc" move="pbyp" checkpoint="0">
<parameter name="timestep"> 0.004 </parameter>
<parameter name="blocks"> 100 </parameter>
<parameter name="steps"> 400 </parameter>
</qmc>
BH
is the project id, and s002
is the calculation number to read in the walkers from the previous run.
In the project id section, make sure that the series number is different from any existing ones to avoid overwriting them.
Variational Monte Carlo¶
vmc
method:
parameters:
Name
Datatype
Values
Default
Description
walkers
integer
\(> 0\)
dep.
Number of walkers per MPI task
blocks
integer
\(\geq 0\)
1
Number of blocks
steps
integer
\(\geq 0\)
1
Number of steps per block
warmupsteps
integer
\(\geq 0\)
0
Number of steps for warming up
substeps
integer
\(\geq 0\)
1
Number of substeps per step
usedrift
text
yes,no
yes
Use the algorithm with drift
timestep
real
\(> 0\)
0.1
Time step for each electron move
samples
integer
\(\geq 0\)
0
Number of walker samples for DMC/optimization
stepsbetweensamples
integer
\(> 0\)
1
Period of sample accumulation
samplesperthread
integer
\(\geq 0\)
0
Number of samples per thread
storeconfigs
integer
all values
0
Show configurations o
blocks_between_recompute
integer
\(\geq 0\)
dep.
Wavefunction recompute frequency
spinMoves
text
yes,no
no
Whether or not to sample the electron spins
spinMass
real
\(> 0\)
1.0
Effective mass for spin sampling
Additional information:
walkers
The number of walkers per MPI task. The initial default number of ixml{walkers} is one per OpenMP thread or per MPI task if threading is disabled. The number is rounded down to a multiple of the number of threads with a minimum of one per thread to ensure perfect load balancing. One walker per thread is created in the event fewerwalkers
than threads are requested.blocks
This parameter is universal for all the QMC methods. The MC processes are divided into a number ofblocks
, each containing a number of steps. At the end of each block, the statistics accumulated in the block are dumped into files, e.g.,scalar.dat
. Typically, each block should have a sufficient number of steps that the I/O at the end of each block is negligible compared with the computational cost. Each block should not take so long that monitoring its progress is difficult. There should be a sufficient number ofblocks
to perform statistical analysis.warmupsteps
warmupsteps
are used only for equilibration. Property measurements are not performed during warmup steps.steps
steps
are the number of energy and other property measurements to perform per block.substeps
For each substep, an attempt is made to move each of the electrons once only by either particlebyparticle or an allelectron move. Because the local energy is evaluated only at each full step and not each substep,substeps
are computationally cheaper and can be used to reduce the correlation between property measurements at a lower cost.usedrift
The VMC is implemented in two algorithms with or without drift. In the nodrift algorithm, the move of each electron is proposed with a Gaussian distribution. The standard deviation is chosen as the time step input. In the drift algorithm, electrons are moved by Langevin dynamics.timestep
The meaning of time step depends on whether or not the drift is used. In general, larger time steps reduce the time correlation but might also reduce the acceptance ratio, reducing overall statistical efficiency. For VMC, typically the acceptance ratio should be close to 50% for an efficient simulation.samples
Seperate from conventional energy and other property measurements, samples refers to storing whole electron configurations in memory (“walker samples”) as would be needed by subsequent wavefunction optimization or DMC steps. A standard VMC run to measure the energy does not need samples to be set.\[\texttt{samples}= \frac{\texttt{blocks}\cdot\texttt{steps}\cdot\texttt{walkers}}{\texttt{stepsbetweensamples}}\cdot\texttt{number of MPI tasks}\]samplesperthread
This is an alternative way to set the target amount of samples and can be useful when preparing a stored population for a subsequent DMC calculation.\[\texttt{samplesperthread}= \frac{\texttt{blocks}\cdot\texttt{steps}}{\texttt{stepsbetweensamples}}\]stepsbetweensamples
Because samples generated by consecutive steps are correlated, havingstepsbetweensamples
larger than 1 can be used to reduces that correlation. In practice, using larger substeps is cheaper than usingstepsbetweensamples
to decorrelate samples.storeconfigs
Ifstoreconfigs
is set to a nonzero value, then electron configurations during the VMC run are saved to files.blocks_between_recompute
Recompute the accuracy critical determinant part of the wavefunction from scratch: =1 by default when using mixed precision. =0 (no recompute) by default when not using mixed precision. Recomputing introduces a performance penalty dependent on system size.spinMoves
Determines whether or not the spin variables are sampled following
spinMass
If spin sampling is on usingspinMoves
== yes, the spin mass determines the rate
of spin sampling, resulting in an effective spin timestep \(\tau_s = \frac{\tau}{\mu_s}\).
An example VMC section for a simple VMC run:
<qmc method="vmc" move="pbyp">
<estimator name="LocalEnergy" hdf5="no"/>
<parameter name="walkers"> 256 </parameter>
<parameter name="warmupSteps"> 100 </parameter>
<parameter name="substeps"> 5 </parameter>
<parameter name="blocks"> 20 </parameter>
<parameter name="steps"> 100 </parameter>
<parameter name="timestep"> 1.0 </parameter>
<parameter name="usedrift"> yes </parameter>
</qmc>
Here we set 256 walkers
per MPI, have a brief initial equilibration of 100 steps
, and then have 20 blocks
of 100 steps
with 5 substeps
each.
The following is an example of VMC section storing configurations (walker samples) for optimization.
<qmc method="vmc" move="pbyp" gpu="yes">
<estimator name="LocalEnergy" hdf5="no"/>
<parameter name="walkers"> 256 </parameter>
<parameter name="samples"> 2867200 </parameter>
<parameter name="stepsbetweensamples"> 1 </parameter>
<parameter name="substeps"> 5 </parameter>
<parameter name="warmupSteps"> 5 </parameter>
<parameter name="blocks"> 70 </parameter>
<parameter name="timestep"> 1.0 </parameter>
<parameter name="usedrift"> no </parameter>
</qmc>
Wavefunction optimization¶
Optimizing wavefunction is critical in all kinds of realspace QMC calculations because it significantly improves both the accuracy and efficiency of computation. However, it is very difficult to directly adopt deterministic minimization approaches because of the stochastic nature of evaluating quantities with MC. Thanks to the algorithmic breakthrough during the first decade of this century and the tremendous computer power available, it is now feasible to optimize tens of thousands of parameters in a wavefunction for a solid or molecule. QMCPACK has multiple optimizers implemented based on the stateoftheart linear method. We are continually improving our optimizers for robustness and friendliness and are trying to provide a single solution. Because of the large variation of wavefunction types carrying distinct characteristics, using several optimizers might be needed in some cases. We strongly suggested reading recommendations from the experts who maintain these optimizers.
A typical optimization block looks like the following. It starts with method=”linear” and contains three blocks of parameters.
<loop max="10">
<qmc method="linear" move="pbyp" gpu="yes">
<! Specify the VMC options >
<parameter name="walkers"> 256 </parameter>
<parameter name="samples"> 2867200 </parameter>
<parameter name="stepsbetweensamples"> 1 </parameter>
<parameter name="substeps"> 5 </parameter>
<parameter name="warmupSteps"> 5 </parameter>
<parameter name="blocks"> 70 </parameter>
<parameter name="timestep"> 1.0 </parameter>
<parameter name="usedrift"> no </parameter>
<estimator name="LocalEnergy" hdf5="no"/>
...
<! Specify the correlated sampling options and define the cost function >
<parameter name="minwalkers"> 0.3 </parameter>
<cost name="energy"> 0.95 </cost>
<cost name="unreweightedvariance"> 0.00 </cost>
<cost name="reweightedvariance"> 0.05 </cost>
...
<! Specify the optimizer options >
<parameter name="MinMethod"> OneShiftOnly </parameter>
...
</qmc>
</loop>
 Loop is helpful to repeatedly execute identical optimization blocks.
 The first part is highly identical to a regular VMC block.
 The second part is to specify the correlated sampling options and
define the cost function.
 The last part is used to specify the options of different optimizers,
which can be very distinct from one to another.
VMC run for the optimization¶
The VMC calculation for the wavefunction optimization has a strict requirement
that samples
or samplesperthread
must be specified because of the optimizer needs for the stored samples
.
The input parameters of this part are identical to the VMC method.
Recommendations:
Run the inclusive VMC calculation correctly and efficiently because this takes a significant amount of time during optimization. For example, make sure the derived
steps
per block is 1 and use largersubsteps
to control the correlation betweensamples
.A reasonable starting wavefunction is necessary. A lot of optimization fails because of a bad wavefunction starting point. The sign of a bad initial wavefunction includes but is not limited to a very long equilibration time, low acceptance ratio, and huge variance. The first thing to do after a failed optimization is to check the information provided by the VMC calculation via
*.scalar.dat files
.
Optimizers¶
QMCPACK implements a number of different optimizers each with different priorities for accuracy, convergence, memory usage, and stability. The optimizers can be switched among “OneShiftOnly” (default), “adaptive,” “descent,” “hybrid,” and “quartic” (old) using the following line in the optimization block:
<parameter name="MinMethod"> THE METHOD YOU LIKE </parameter>
OneShiftOnly Optimizer¶
The OneShiftOnly optimizer targets a fast optimization by moving parameters more aggressively. It works with OpenMP and GPU and can be considered for large systems.
This method relies on the effective weight of correlated sampling rather than the cost function value to justify a new set of parameters.
If the effective weight is larger than minwalkers
, the new set is taken whether or not the cost function value decreases.
If a proposed set is rejected, the standard output prints the measured ratio of effective samples to the total number of samples
and adjustment on minwalkers
can be made if needed.
linear
method:
parameters:
Name
Datatype
Values
Default
Description
shift_i
real
\(> 0\)
0.01
Direct stabilizer added to the Hamiltonian matrix
shift_s
real
\(> 0\)
1.00
Initial stabilizer based on the overlap matrix
Additional information:
shift_i
This is the direct term added to the diagonal of the Hamiltonian matrix. It provides more stable but slower optimization with a large value.shift_s
This is the initial value of the stabilizer based on the overlap matrix added to the Hamiltonian matrix. It provides more stable but slower optimization with a large value. The used value is autoadjusted by the optimizer.
Recommendations:
Default
shift_i
,shift_s
should be fine.For hard cases, increasing
shift_i
(by a factor of 5 or 10) can significantly stabilize the optimization by reducing the pace towards the optimal parameter set.If the VMC energy of the last optimization iterations grows significantly, increase
minwalkers
closer to 1 and make the optimization stable.If the first iterations of optimization are rejected on a reasonable initial wavefunction, lower the
minwalkers
value based on the measured value printed in the standard output to accept the move.
We recommended using this optimizer in two sections with a very small minwalkers
in the first and a large value in the second, such as the following.
In the very beginning, parameters are far away from optimal values and large changes are proposed by the optimizer.
Having a small minwalkers
makes it much easier to accept these changes.
When the energy gradually converges, we can have a large minwalkers
to avoid risky parameter sets.
<loop max="6">
<qmc method="linear" move="pbyp" gpu="yes">
<! Specify the VMC options >
<parameter name="walkers"> 1 </parameter>
<parameter name="samples"> 10000 </parameter>
<parameter name="stepsbetweensamples"> 1 </parameter>
<parameter name="substeps"> 5 </parameter>
<parameter name="warmupSteps"> 5 </parameter>
<parameter name="blocks"> 25 </parameter>
<parameter name="timestep"> 1.0 </parameter>
<parameter name="usedrift"> no </parameter>
<estimator name="LocalEnergy" hdf5="no"/>
<! Specify the optimizer options >
<parameter name="MinMethod"> OneShiftOnly </parameter>
<parameter name="minwalkers"> 1e4 </parameter>
</qmc>
</loop>
<loop max="12">
<qmc method="linear" move="pbyp" gpu="yes">
<! Specify the VMC options >
<parameter name="walkers"> 1 </parameter>
<parameter name="samples"> 20000 </parameter>
<parameter name="stepsbetweensamples"> 1 </parameter>
<parameter name="substeps"> 5 </parameter>
<parameter name="warmupSteps"> 2 </parameter>
<parameter name="blocks"> 50 </parameter>
<parameter name="timestep"> 1.0 </parameter>
<parameter name="usedrift"> no </parameter>
<estimator name="LocalEnergy" hdf5="no"/>
<! Specify the optimizer options >
<parameter name="MinMethod"> OneShiftOnly </parameter>
<parameter name="minwalkers"> 0.5 </parameter>
</qmc>
</loop>
For each optimization step, you will see
The new set of parameters is valid. Updating the trial wave function!
or
The new set of parameters is not valid. Revert to the old set!
Occasional rejection is fine. Frequent rejection indicates potential
problems, and users should inspect the VMC calculation or change
optimization strategy. To track the progress of optimization, use the
command qmca q ev *.scalar.dat
to look at the VMC energy and
variance for each optimization step.
Adaptive Organizer¶
The default setting of the adaptive optimizer is to construct the linear method Hamiltonian and overlap matrices explicitly and add different shifts to the Hamiltonian matrix as “stabilizers.” The generalized eigenvalue problem is solved for each shift to obtain updates to the wavefunction parameters. Then a correlated sampling is performed for each shift’s updated wavefunction and the initial trial wavefunction using the middle shift’s updated wavefunction as the guiding function. The cost function for these wavefunctions is compared, and the update corresponding to the best cost function is selected. In the next iteration, the median magnitude of the stabilizers is set to the magnitude that generated the best update in the current iteration, thus adapting the magnitude of the stabilizers automatically.
When the trial wavefunction contains more than 10,000 parameters, constructing and storing the linear method matrices could become a memory bottleneck. To avoid explicit construction of these matrices, the adaptive optimizer implements the block linear method (BLM) approach. [ZN17] The BLM tries to find an approximate solution \(\vec{c}_{opt}\) to the standard LM generalized eigenvalue problem by dividing the variable space into a number of blocks and making intelligent estimates for which directions within those blocks will be most important for constructing \(\vec{c}_{opt}\), which is then obtained by solving a smaller, more memoryefficient eigenproblem in the basis of these supposedly important blockwise directions.
linear
method:
parameters:
Name
Datatype
Values
Default
Description
max_relative_change
real
\(> 0\)
10.0
Allowed change in cost function
max_param_change
real
\(> 0\)
0.3
Allowed change in wavefunction parameter
shift_i
real
\(> 0\)
0.01
Initial diagonal stabilizer added to the Hamiltonian matrix
shift_s
real
\(> 0\)
1.00
Initial overlapbased stabilizer added to the Hamiltonian matrix
target_shift_i
real
any
1.0
Diagonal stabilizer value aimed for during adaptive method (disabled if \(\leq 0\))
cost_increase_tol
real
\(\geq 0\)
0.0
Tolerance for cost function increases
chase_lowest
text
yes, no
yes
Chase the lowest eigenvector in iterative solver
chase_closest
text
yes, no
no
Chase the eigenvector closest to initial guess
block_lm
text
yes, no
no
Use BLM
blocks
integer
\(> 0\)
Number of blocks in BLM
nolds
integer
\(> 0\)
Number of old update vectors used in BLM
nkept
integer
\(> 0\)
Number of eigenvectors to keep per block in BLM
Additional information:
shift_i
This is the initial coefficient used to scale the diagonal stabilizer. More stable but slower optimization is expected with a large value. The adaptive method will automatically adjust this value after each linear method iteration.shift_s
This is the initial coefficient used to scale the overlapbased stabilizer. More stable but slower optimization is expected with a large value. The adaptive method will automatically adjust this value after each linear method iteration.target_shift_i
If set greater than zero, the adaptive method will choose the update whose shift_i value is closest to this target value so long as the associated cost is within cost_increase_tol of the lowest cost. Disable this behavior by setting target_shift_i to a negative number.cost_increase_tol
Tolerance for cost function increases when selecting the best shift.nblocks
This is the number of blocks used in BLM. The amount of memory required to store LM matrices decreases as the number of blocks increases. But the error introduced by BLM would increase as the number of blocks increases.nolds
In BLM, the interblock correlation is accounted for by including a small number of wavefunction update vectors outside the block. Larger would include more interblock correlation and more accurate results but also higher memory requirements.nkept
This is the number of update directions retained from each block in the BLM. If all directions are retained in each block, then the BLM becomes equivalent to the standard LM. Retaining five or fewer directions per block is often sufficient.
Recommendations:
Default
shift_i
,shift_s
should be fine.When there are fewer than about 5,000 variables being optimized, the traditional LM is preferred because it has a lower overhead than the BLM when the number of variables is small.
Initial experience with the BLM suggests that a few hundred blocks and a handful of and often provide a good balance between memory use and accuracy. In general, using fewer blocks should be more accurate but would require more memory.
<loop max="15">
<qmc method="linear" move="pbyp">
<! Specify the VMC options >
<parameter name="walkers"> 1 </parameter>
<parameter name="samples"> 20000 </parameter>
<parameter name="stepsbetweensamples"> 1 </parameter>
<parameter name="substeps"> 5 </parameter>
<parameter name="warmupSteps"> 5 </parameter>
<parameter name="blocks"> 50 </parameter>
<parameter name="timestep"> 1.0 </parameter>
<parameter name="usedrift"> no </parameter>
<estimator name="LocalEnergy" hdf5="no"/>
<! Specify the correlated sampling options and define the cost function >
<cost name="energy"> 1.00 </cost>
<cost name="unreweightedvariance"> 0.00 </cost>
<cost name="reweightedvariance"> 0.00 </cost>
<! Specify the optimizer options >
<parameter name="MinMethod">adaptive</parameter>
<parameter name="max_relative_cost_change">10.0</parameter>
<parameter name="shift_i"> 1.00 </parameter>
<parameter name="shift_s"> 1.00 </parameter>
<parameter name="max_param_change"> 0.3 </parameter>
<parameter name="chase_lowest"> yes </parameter>
<parameter name="chase_closest"> yes </parameter>
<parameter name="block_lm"> no </parameter>
<! Specify the BLM specific options if needed
<parameter name="nblocks"> 100 </parameter>
<parameter name="nolds"> 5 </parameter>
<parameter name="nkept"> 3 </parameter>
>
</qmc>
</loop>
The adaptive optimizer is also able to optimize individual excited states directly. [ZN16] In this case, it tries to minimize the following function:
The global minimum of this function corresponds to the state whose energy lies immediately above the shift parameter \(\omega\) in the energy spectrum. For example, if \(\omega\) were placed in between the ground state energy and the first excited state energy and the wavefunction ansatz was capable of a good description for the first excited state, then the wavefunction would be optimized for the first excited state. Note that if the ansatz is not capable of a good description of the excited state in question, the optimization could converge to a different state, as is known to occur in some circumstances for traditional ground state optimizations. Note also that the ground state can be targeted by this method by choosing \(\omega\) to be below the ground state energy, although we should stress that this is not the same thing as a traditional ground state optimization and will in general give a slightly different wavefunction. Excited state targeting requires two additional parameters, as shown in the following table.
Excited state targeting:
parameters:
Name
Datatype
Values
Default
Description
targetExcited
text
yes, no
no
Whether to use the excited state targeting optimization
omega
real
real numbers
none
Energy shift used to target different excited states
Excited state recommendations:
Because of the finite variance in any approximate wavefunction, we recommended setting \(\omega=\omega_0\sigma\), where \(\omega_0\) is placed just below the energy of the targeted state and \(\sigma^2\) is the energy variance.
To obtain an unbiased excitation energy, the ground state should be optimized with the excited state variational principle as well by setting
omega
below the ground state energy. Note that using the ground state variational principle for the ground state and the excited state variational principle for the excited state creates a bias in favor of the ground state.
Descent Optimizer¶
Gradient descent algorithms are an alternative set of optimization methods to the OneShiftOnly and adaptive optimizers based on the linear method. These methods use only first derivatives to optimize trial wave functions and convergence can be accelerated by retaining a memory of previous derivative values. Multiple flavors of accelerated descent methods are available. They differ in details such as the schemes for adaptive adjustment of step sizes. [ON19] Descent algorithms avoid the construction of matrices that occurs in the linear method and consequently can be applied to larger sets of optimizable parameters. Parameters for descent are shown in the table below.
descent
method:
parameters:
Name
Datatype
Values
Default
Description
flavor
text
RMSprop, Random, ADAM, AMSGrad
RMSprop
Particular type of descent method
Ramp_eta
text
yes, no
no
Whether to gradually ramp up step sizes
Ramp_num
integer
\(> 0\)
30
Number of steps over which to ramp up step size
TJF_2Body_eta
real
\(> 0\)
0.01
Step size for two body Jastrow parameters
TJF_1Body_eta
real
\(> 0\)
0.01
Step size for one body Jastrow parameters
F_eta
real
\(> 0\)
0.001
Step size for number counting Jastrow F matrix parameters
Gauss_eta
real
\(> 0\)
0.001
Step size for number counting Jastrow gaussian basis parameters
CI_eta
real
\(> 0\)
0.01
Step size for CI parameters
Orb_eta
real
\(> 0\)
0.001
Step size for orbital parameters
collection_step
real
\(> 0\)
0.01
Step number to start collecting samples for final averages
compute_step
real
\(> 0\)
0.001
Step number to start computing averaged from stored history
print_derivs
real
yes, no
no
Whether to print parameter derivatives
These descent algortihms have been extended to the optimization of the same excited state functional as the adaptive LM. [LON20] This also allows the hybrid optimizer discussed below to be applied to excited states. The relevant parameters are the same as for targeting excited states with the adaptive optimizer above.
Additional information and recommendations:
It is generally advantageous to set different step sizes for different types of parameters. More nonlinear parameters such as those for number counting Jastrow factors or orbitals typically require smaller steps sizes than those for CI coefficients or traditional Jastrow parameters. There are defaults for several parameter types and a default of .001 has been chosen for all other parameters.
The ability to gradually ramp up step sizes to their input values is useful for avoiding spikes in the average local energy during early iterations of descent optimization. This initial rise in the energy occurs as a memory of past gradients is being built up and it may be possible for the energy to recover without ramping if there are enough iterations in the optimization.
The step sizes chosen can have a substantial influence on the quality of the optimization and the final variational energy achieved. Larger step sizes may be helpful if there is reason to think the descent optimization is not reaching the minimum energy. There are also additional hyperparameters in the descent algorithms with default values. [ON19] They seem to have limited influence on the effectiveness of the optimization compared to step sizes, but users can adjust them within the source code of the descent engine if they wish.
The sampling effort for individual descent steps can be small compared that for linear method iterations as shown in the example input below. Something in the range of 10,000 to 30,000 seems sufficient for molecules with tens of electrons. However, descent optimizations may require anywhere from a few hundred to a few thousand iterations.
For reporting quantities such as a final energy and associated uncertainty, an average over many descent steps can be taken. The parameters for
collection_step
andcompute_step
help automate this task.After the descent iteration specified by
collection_step
, a history of local energy values will be kept for determining a final error and average, which will be computed and given in the output once the iteration specified bycompute_step
is reached. For reasonable results, this procedure should use descent steps near the end of the optimization when the wave function parameters are essentially no longer changing.
In cases where a descent optimization struggles to reach the minimum and a linear method optimization is not possible or unsatisfactory, it may be useful to try the hybrid optimization approach described in the next subsection.
<loop max="2000">
<qmc method="linear" move="pbyp" checkpoint="1" gpu="no">
<! VMC inputs >
<parameter name="blocks">2000</parameter>
<parameter name="steps">1</parameter>
<parameter name="samples">20000</parameter>
<parameter name="warmupsteps">100</parameter>
<parameter name="timestep">0.05</parameter>
<parameter name="MinMethod">descent</parameter>
<estimator name="LocalEnergy" hdf5="no"/>
<parameter name="usebuffer">yes</parameter>
<estimator name="LocalEnergy" hdf5="no"/>
<! Descent Inputs >
<parameter name="flavor">RMSprop</parameter>
<parameter name="Ramp_eta">no</parameter>
<parameter name="Ramp_num">30</parameter>
<parameter name="TJF_2Body_eta">.02</parameter>
<parameter name="TJF_1Body_eta">.02</parameter>
<parameter name="F_eta">.001</parameter>
<parameter name="Gauss_eta">.001</parameter>
<parameter name="CI_eta">.1</parameter>
<parameter name="Orb_eta">.0001</parameter>
<parameter name="collection_step">500</parameter>
<parameter name="compute_step">998</parameter>
<parameter name="targetExcited"> yes </parameter>
<parameter name="targetExcited"> 11.4 </parameter>
<parameter name="print_derivs">no</parameter>
</qmc>
</loop>
Hybrid Optimizer¶
Another optimization option is to use a hybrid combination of accelerated descent and blocked linear method.
It provides a means to retain the advantages of both individual methods while scaling to large numbers of parameters beyond the traditional 10,000 parameter limit of the linear method. [ON19]
In a hybrid optimization, alternating sections of descent and BLM optimization are used.
Gradient descent is used to identify the previous important directions in parameter space used by the BLM, the number of which is set by the nold
input for the BLM.
Over the course of a section of descent, vectors of parameter differences are stored and then passed to the linear method engine after the optimization changes to the BLM.
One motivation for including sections of descent is to counteract noise in linear method updates due to uncertainties in its step direction and allow for a smoother movement to the minimum.
There are two additional parameters used in the hybrid optimization and it requires a slightly different format of input to specify the constituent methods as shown below in the example.
descent
method:
parameters:
Name
Datatype
Values
Default
Description
num_updates
integer
\(> 0\)
Number of steps for a method
Stored_Vectors
integer
\(> 0\)
5
Number of vectors to transfer to BLM
<loop max="203">
<qmc method="linear" move="pbyp" checkpoint="1" gpu="no">
<parameter name="Minmethod"> hybrid </parameter>
<optimizer num_updates="100">
<parameter name="blocks">1000</parameter>
<parameter name="steps">1</parameter>
<parameter name="samples">20000</parameter>
<parameter name="warmupsteps">1000</parameter>
<parameter name="timestep">0.05</parameter>
<estimator name="LocalEnergy" hdf5="no"/>
<parameter name="Minmethod"> descent </parameter>
<parameter name="Stored_Vectors">5</parameter>
<parameter name="flavor">RMSprop</parameter>
<parameter name="TJF_2Body_eta">.01</parameter>
<parameter name="TJF_1Body_eta">.01</parameter>
<parameter name="CI_eta">.1</parameter>
<parameter name="Ramp_eta">no</parameter>
<parameter name="Ramp_num">10</parameter>
</optimizer>
<optimizer num_updates="3">
<parameter name="blocks">2000</parameter>
<parameter name="steps">1</parameter>
<parameter name="samples">1000000</parameter>
<parameter name="warmupsteps">1000</parameter>
<parameter name="timestep">0.05</parameter>
<estimator name="LocalEnergy" hdf5="no"/>
<parameter name="Minmethod"> adaptive </parameter>
<parameter name="max_relative_cost_change">10.0</parameter>
<parameter name="max_param_change">3</parameter>
<parameter name="shift_i">0.01</parameter>
<parameter name="shift_s">1.00</parameter>
<parameter name="block_lm">yes</parameter>
<parameter name="nblocks">2</parameter>
<parameter name="nolds">5</parameter>
<parameter name="nkept">5</parameter>
</optimizer>
</qmc>
</loop>
Additional information and recommendations:
In the example above, the input for
loop
gives the total number of steps for the full optimization while the inputs fornum_updates
specify the number of steps in the constituent methods. For this case, the optimization would begin with 100 steps of descent using the parameters in the firstoptimizer
block and then switch to the BLM for 3 steps before switching back to descent for the final 100 iterations of the total of 203.The design of the hybrid method allows for more than two
optimizer
blocks to be used and the optimization will cycle through the individual methods. However, the effectiveness of this in terms of the quality of optimization results is unexplored.It can be useful to follow a hybrid optimization with a section of pure descent optimization and take an average energy over the last few hundred iterations as the final variational energy. This approach can achieve a lower statistical uncertainty on the energy for less overall sampling effort compared to what a pure linear method optimization would require. The
collection_step
andcompute_step
parameters discussed earlier for descent are useful for setting up the descent engine to do this averaging on its own.
Quartic Optimizer¶
This is an older optimizer method retained for compatibility. We recommend starting with the newest OneShiftOnly or adaptive optimizers. The quartic optimizer fits a quartic polynomial to 7 values of the cost function obtained using reweighting along the chosen direction and determines the optimal move. This optimizer is very robust but is a bit conservative when accepting new steps, especially when large parameters changes are proposed.
linear
method:
parameters:
Name
Datatype
Values
Default
Description
bigchange
real
\(> 0\)
50.0
Largest parameter change allowed
alloweddifference
real
\(> 0\)
1e4
Allowed increase in energy
exp0
real
any value
16.0
Initial value for stabilizer
stabilizerscale
real
\(> 0\)
2.0
Increase in value of
exp0
between iterations
nstabilizers
integer
\(> 0\)
3
Number of stabilizers to try
max_its
integer
\(> 0\)
1
Number of inner loops with same samples
Additional information:
exp0
This is the initial value for stabilizer (shift to diagonal of H). The actual value of stabilizer is \(10^{\textrm{exp0}}\).
Recommendations:
For hard cases (e.g., simultaneous optimization of long MSD and 3Body J), set
exp0
to 0 and do a single inner iteration (max its=1) per sample of configurations.
<! Specify the optimizer options >
<parameter name="MinMethod">quartic</parameter>
<parameter name="exp0">6</parameter>
<parameter name="alloweddifference"> 1.0e4 </parameter>
<parameter name="nstabilizers"> 1 </parameter>
<parameter name="bigchange">15.0</parameter>
General Recommendations¶
All electron wavefunctions are typically more difficult to optimize than pseudopotential wavefunctions because of the importance of the wavefunction near the nucleus.
Twobody Jastrow contributes the largest portion of correlation energy from bare Slater determinants. Consequently, the recommended order for optimizing wavefunction components is twobody, onebody, threebody Jastrow factors and MSD coefficients.
For twobody spline Jastrows, always start from a reasonable one. The lack of physically motivated constraints in the functional form at large distances can cause slow convergence if starting from zero.
Onebody spline Jastrow from old calculations can be a good starting point.
Threebody polynomial Jastrow can start from zero. It is beneficial to first optimize onebody and twobody Jastrow factors without adding threebody terms in the calculation and then add the threebody Jastrow and optimize all the three components together.
Optimization of CI coefficients¶
When storing a CI wavefunction in HDF5 format, the CI coefficients and the \(\alpha\) and \(\beta\) components of each CI are not in the XML input file. When optimizing the CI coefficients, they will be stored in HDF5 format. The optimization header block will have to specify that the new CI coefficients will be saved to HDF5 format. If the tag is not added coefficients will not be saved.
<qmc method="linear" move="pbyp" gpu="no" hdf5="yes">
The rest of the optimization block remains the same.
When running the optimization, the new coefficients will be stored in a *.sXXX.opt.h5 file, where XXX coressponds to the series number. The H5 file contains only the optimized coefficients. The corresponding *.sXXX.opt.xml will be updated for each optimization block as follows:
<detlist size="1487" type="DETS" nca="0" ncb="0" nea="2" neb="2" nstates="85" cutoff="1e2" href="../LiH.orbs.h5" opt_coeffs="LiH.s001.opt.h5"/>
The opt_coeffs tag will then reference where the new CI coefficients are stored.
When restarting the run with the new optimized coeffs, you need to specify the previous hdf5 containing the basis set, orbitals, and MSD, as well as the new optimized coefficients. The code will read the previous data but will rewrite the coefficients that were optimized with the values found in the *.sXXX.opt.h5 file. Be careful to keep the pair of optimized CI coefficients and Jastrow coefficients together to avoid inconsistencies.
Diffusion Monte Carlo¶
Main input parameters are given in Table 9, additional in Table 10.
dmc
method:
parameters:
Name 
Datatype 
Values 
Default 
Description 


integer 
\(> 0\) 
dep. 
Overall total number of walkers 

integer 
\(\geq 0\) 
1 
Number of blocks 

integer 
\(\geq 0\) 
1 
Number of steps per block 

integer 
\(\geq 0\) 
0 
Number of steps for warming up 

real 
\(> 0\) 
0.1 
Time step for each electron move 

string 
yes, no, v0, v1, v3 
no 
Run with Tmoves 

string 
classic/DRV/ZSGMA/YL 
classic 
Branch cutoff scheme 

real 
\(\geq 0\) 
3.6e5 
Maximum allowed walltime in seconds 

integer 
\(\geq 0\) 
dep. 
Wavefunction recompute frequency 

text 
yes,no 
no 
Whether or not to sample the electron spins 

real 
\(> 0\) 
1.0 
Effective mass for spin sampling 
Table 9 Main DMC input parameters.
Name 
Datatype 
Values 
Default 
Description 


integer 
\(\geq 0\) 
0 
Trial energy update interval 

real 
all values 
dep. 
Reference energy in atomic units 

double 
\(\geq 0\) 
1.0 
Population feedback on the trial energy 

10 
\(\geq 0\) 
10 
Parameter to cutoff large weights 

string 
yes/other 
no 
Kill or reject walkers that cross nodes 

option 
yes,no 
0 
Warm up with a fixed population 

string 
yes/pure/other 
no 
Fixed population technique 

integer 
\(\geq 0\) 
1 
Branching interval 

integer 
\(\geq 0\) 
1 
Branching interval 

double 
\(\geq 0\) 
10 
Kill persistent walkers 

double 
\(\geq 0\) 
2 
Limit population growth 

real 
all values 
1 
Maximum particle move 

string 
yes/other 
yes 
Scale weights (CUDA only) 

integer 
\(\geq 0\) 
100 
Number of steps between walker updates 

text 
yes/other 
yes 
Fast gradients 

integer 
all values 
0 
Store configurations 

string 
yes/no 
yes 
Using nonblocking send/recv 

string 
yes/no 
no 
Disable branching for debugging without correctness guarantee 
Table 10 Additional DMC input parameters.
Additional information:
targetwalkers
: A DMC run can be considered a restart run or a new run. A restart run is considered to be any method block beyond the first one, such as when a DMC method block follows a VMC block. Alternatively, a user reading in configurations from disk would also considered a restart run. In the case of a restart run, the DMC driver will use the configurations from the previous run, and this variable will not be used. For a new run, if the number of walkers is less than the number of threads, then the number of walkers will be set equal to the number of threads.blocks
: This is the number of blocks run during a DMC method block. A block consists of a number of DMC steps (steps), after which all the statistics accumulated in the block are written to disk.steps
: This is the number of DMC steps in a block.warmupsteps
: These are the steps at the beginning of a DMC run in which the instantaneous average energy is used to update the trial energy. During regular steps, E\(_{ref}\) is used.timestep
: Thetimestep
determines the accuracy of the imaginary time propagator. Generally, multiple time steps are used to extrapolate to the infinite time step limit. A good range of time steps in which to perform time step extrapolation will typically have a minimum of 99% acceptance probability for each step.checkproperties
: When using a particlebyparticle driver, this variable specifies how often to reset all the variables kept in the buffer.maxcpusecs
: The default is 100 hours. Once the specified time has elapsed, the program will finalize the simulation even if all blocks are not completed.spinMoves
Determines whether or not the spin variables are sampled following [MZG+16] and [MBM16]. If a relativistic calculation is desired using pseudopotentials, spin variable sampling is required.spinMass
If spin sampling is on usingspinMoves
== yes, the spin mass determines the rate of spin sampling, resulting in an effective spin timestep \(\tau_s = \frac{\tau}{\mu_s}\) where \(\tau\) is the normal spatial timestep and \(\mu_s\) is the value of the spin mass.energyUpdateInterval
: The default is to update the trial energy at every step. Otherwise the trial energy is updated everyenergyUpdateInterval
step.
where \(N\) is the current population.
refEnergy
: The default reference energy is taken from the VMC run that precedes the DMC run. This value is updated to the current mean whenever branching happens.feedback
: This variable is used to determine how strong to react to population fluctuations when doing population control. See the equation in energyUpdateInterval for more details.useBareTau
: The same time step is used whether or not a move is rejected. The default is to use an effective time step when a move is rejected.warmupByReconfiguration
: Warmup DMC is done with a fixed population.sigmaBound
: This determines the branch cutoff to limit wild weights based on the sigma andsigmaBound
.killnode
: When running fixednode, if a walker attempts to cross a node, the move will normally be rejected. Ifkillnode
= “yes,” then walkers are destroyed when they cross a node.reconfiguration
: Ifreconfiguration
is “yes,” then run with a fixed walker population using the reconfiguration technique.branchInterval
: This is the number of steps between branching. The total number of DMC steps in a block will beBranchInterval
*Steps.substeps
: This is the same asBranchInterval
.nonlocalmoves
: Evaluate pseudopotentials using one of the nonlocal move algorithms such as Tmoves.no(default): Imposes the locality approximation.
yes/v0: Implements the algorithm in the 2006 Casula paper [Cas06].
v1: Implements the v1 algorithm in the 2010 Casula paper [CMSF10].
v2: Is not implemented and is skipped to avoid any confusion with the v2 algorithm in the 2010 Casula paper [CMSF10].
v3: (Experimental) Implements an algorithm similar to v1 but is much faster. v1 computes the transition probability before each single electron Tmove selection because of the acceptance of previous Tmoves. v3 mostly reuses the transition probability computed during the evaluation of nonlocal pseudopotentials for the local energy, namely before accepting any Tmoves, and only recomputes the transition probability of the electrons within the same pseudopotential region of any electrons touched by Tmoves. This is an approximation to v1 and results in a slightly different time step error, but it significantly reduces the computational cost. v1 and v3 agree at zero time step. This faster algorithm is the topic of a paper in preparation.
The v1 and v3 algorithms are sizeconsistent and are important advances over the previous v0 nonsizeconsistent algorithm. We highly recommend investigating the importance of sizeconsistency.
scaleweight
: This is the scaling weight per Umrigar/Nightengale. CUDA only.MaxAge
: Set the weight of a walker to min(currentweight,0.5) after a walker has not moved forMaxAge
steps. Needed if persistent walkers appear during the course of a run.MaxCopy
: When determining the number of copies of a walker to branch, set the number of copies equal to min(Multiplicity,MaxCopy).fastgrad
: This calculates gradients with either the fast version or the fullratio version.maxDisplSq
: When running a DMC calculation with particle by particle, this sets the maximum displacement allowed for a single particle move. All distance displacements larger than the max are rejected. If initialized to a negative value, it becomes equal to Lattice(LR/rc).sigmaBound
: This determines the branch cutoff to limit wild weights based on the sigma andsigmaBound
.storeconfigs
: Ifstoreconfigs
is set to a nonzero value, then electron configurations during the DMC run will be saved. This option is disabled for the OpenMP version of DMC.blocks_between_recompute
: See details in Variational Monte Carlo.branching_cutoff_scheme:
Modifies how the branching factor is computed so as to avoid divergences and stability problems near nodal surfaces.classic (default): The implementation found in QMCPACK v3.0.0 and earlier. \(E_{\rm cut}=\mathrm{min}(\mathrm{max}(\sigma^2 \times \mathrm{sigmaBound},\mathrm{maxSigma}),2.5/\tau)\), where \(\sigma^2\) is the variance and \(\mathrm{maxSigma}\) is set to 50 during warmup (equilibration) and 10 thereafter. \(\mathrm{sigmaBound}\) is default to 10.
DRV: Implements the algorithm of DePasquale et al., Eq. 3 in [DRV88] or Eq. 9 of [UNR93]. \(E_{\rm cut}=2.0/\sqrt{\tau}\).
ZSGMA: Implements the “ZSGMA” algorithm of [ZSG+16] with \(\alpha=0.2\). The cutoff energy is modified by a factor including the electron count, \(E_{\rm cut}=\alpha \sqrt{N/\tau}\), which greatly improves size consistency over Eq. 39 of [UNR93]. See Eq. 6 in [ZSG+16] and for an application to molecular crystals [ZBKlimevs+18].
YL: An unpublished algorithm due to Ye Luo. \(E_{\rm cut}=\sigma\times\mathrm{min}(\mathrm{sigmaBound},\sqrt{1/\tau})\). This option takes into account both size consistency and wavefunction quality via the term \(\sigma\). \(\mathrm{sigmaBound}\) is default to 10.
<qmc method="dmc" move="pbyp" target="e">
<parameter name="blocks">100</parameter>
<parameter name="steps">400</parameter>
<parameter name="timestep">0.010</parameter>
<parameter name="warmupsteps">100</parameter>
</qmc>
The time step should be individually adjusted for each problem. Please refer to the theory section on diffusion Monte Carlo.
<qmc method="dmc" move="pbyp" checkpoint="0">
<parameter name="timestep"> 0.004 </parameter>
<parameter name="blocks"> 100 </parameter>
<parameter name="steps"> 400 </parameter>
</qmc>
The checkpoint flag instructs QMCPACK to output walker configurations.
This also works in VMC. This will output an h5 file with the name
projectid.runnumber.config.h5
. Check that this file exists before
attempting a restart. To read in this file for a continuation run,
specify the following:
<mcwalkerset fileroot="BH.s002" version="0 6" collected="yes"/>
BH is the project id, and s002 is the calculation number to read in the walkers from the previous run.
Combining VMC and DMC in a single run (wavefunction optimization can be combined in this way too) is the standard way in which QMCPACK is typically run. There is no need to run two separate jobs since method sections can be stacked and walkers are transferred between them.
<qmc method="vmc" move="pbyp" target="e">
<parameter name="blocks">100</parameter>
<parameter name="steps">4000</parameter>
<parameter name="warmupsteps">100</parameter>
<parameter name="samples">1920</parameter>
<parameter name="walkers">1</parameter>
<parameter name="timestep">0.5</parameter>
</qmc>
<qmc method="dmc" move="pbyp" target="e">
<parameter name="blocks">100</parameter>
<parameter name="steps">400</parameter>
<parameter name="timestep">0.010</parameter>
<parameter name="warmupsteps">100</parameter>
</qmc>
<qmc method="dmc" move="pbyp" target="e">
<parameter name="warmupsteps">500</parameter>
<parameter name="blocks">50</parameter>
<parameter name="steps">100</parameter>
<parameter name="timestep">0.005</parameter>
</qmc>
Reptation Monte Carlo¶
Like DMC, RMC is a projectorbased method that allows sampling of the fixednode wavefunciton. However, by exploiting the pathintegral formulation of Schrödinger’s equation, the RMC algorithm can offer some advantages over traditional DMC, such as sampling both the mixed and pure fixednode distributions in polynomial time, as well as not having population fluctuations and biases. The current implementation does not work with Tmoves.
There are two adjustable parameters that affect the quality of the RMC projection: imaginary projection time \(\beta\) of the sampling path (commonly called a “reptile”) and the Trotter time step \(\tau\). \(\beta\) must be chosen to be large enough such that \(e^{\beta \hat{H}}\Psi_T\rangle \approx \Phi_0\rangle\) for mixed observables, and \(e^{\frac{\beta}{2} \hat{H}}\Psi_T\rangle \approx \Phi_0\rangle\) for pure observables. The reptile is discretized into \(M=\beta/\tau\) beads at the cost of an \(\mathcal{O}(\tau)\) timestep error for observables arising from the TrotterSuzuki breakup of the shorttime propagator.
The following table lists some of the more practical
vmc
method:
parameters:
Name
Datatype
Values
Default
Description
beta
real
\(> 0\)
dep.
Reptile project time \(\beta\)
timestep
real
\(> 0\)
0.1
Trotter time step \(\tau\) for each electron move
beads
int
\(> 0\)
1
Number of reptile beads \(M=\beta/\tau\)
blocks
integer
\(> 0\)
1
Number of blocks
steps
integer
\(\geq 0\)
1
Number of steps per block
vmcpresteps
integer
\(\geq 0\)
0
Propagates reptile using VMC for given number of steps
warmupsteps
integer
\(\geq 0\)
0
Number of steps for warming up
maxAge
integer
\(\geq 0\)
0
Force accept for stuck reptile if age exceeds
maxAge
Additional information:
Because of the sampling differences between DMC ensembles of walkers and
RMC reptiles, the RMC block should contain the following estimator
declaration to ensure correct sampling:
<estimator name="RMC" hdf5="no">
.
beta
orbeads
? One or the other can be specified, and from the Trotter time step, the code will construct an appropriately sized reptile. If both are given,beta
overridesbeads
.Mixed vs. pure observables? Configurations sampled by the endpoints of the reptile are distributed according to the mixed distribution \(f(\mathbf{R})=\Psi_T(\mathbf{R})\Phi_0(\mathbf{R})\). Any observable that is computable within DMC and is dumped to the
scalar.dat
file will likewise be found in thescalar.dat
file generated by RMC, except there will be an appended_m
to alert the user that the observable was computed on the mixed distribution. For pure observables, care must be taken in the interpretation. If the observable is diagonal in the position basis (in layman’s terms, if it is entirely computable from a single electron configuration \(\mathbf{R}\), like the potential energy), and if the observable does not have an explicit dependence on the trial wavefunction (e.g., the local energy has an explicit dependence on the trial wavefunction from the kinetic energy term), then pure estimates will be correctly computed. These observables will be found in either thescalar.dat
, where they will be appended with a_p
suffix, or in thestat.h5
file. No mixed estimators will be dumped to the h5 file.Sampling: For pure estimators, the traces of both pure and mixed estimates should be checked. Ergodicity is a known problem in RMC. Because we use the bounce algorithm, it is possible for the reptile to bounce back and forth without changing the electron coordinates of the central beads. This might not easily show up with mixed estimators, since these are accumulated at constantly regrown ends, but pure estimates are accumulated on these central beads and so can exhibit strong autocorrelations in pure estimate traces.
Propagator: Our implementation of RMC uses Moroni’s DMC link action (symmetrized), with Umrigar’s scaled drift near nodes. In this regard, the propagator is identical to the one QMCPACK uses in DMC.
Sampling: We use Ceperley’s bounce algorithm.
MaxAge
is used in case the reptile gets stuck, at which point the code forces move acceptance, stops accumulating statistics, and requilibrates the reptile. Very rarely will this be required. For move proposals, we use particlebyparticle VMC a total of \(N_e\) times to generate a new allelectron configuration, at which point the action is computed and the move is either accepted or rejected.
 Cas06
Michele Casula. Beyond the locality approximation in the standard diffusion Monte Carlo method. Physical Review B  Condensed Matter and Materials Physics, 74:1–4, 2006. doi:10.1103/PhysRevB.74.161102.
 CMSF10(1,2)
Michele Casula, Saverio Moroni, Sandro Sorella, and Claudia Filippi. Sizeconsistent variational approaches to nonlocal pseudopotentials: Standard and lattice regularized diffusion Monte Carlo methods revisited. Journal of Chemical Physics, 2010. arXiv:1002.0356, doi:10.1063/1.3380831.
 DRV88
Michael F. DePasquale, Stuart M. Rothstein, and Jan Vrbik. Reliable diffusion quantum monte carlo. The Journal of Chemical Physics, 89(6):3629–3637, September 1988. doi:10.1063/1.454883.
 LON20
Leon Otis, Isabel Craig and Eric Neuscamman. A hybrid approach to excitedstatespecific variational monte carlo and doubly excited states. arXiv preprint arXiv:2008:03586, 2020. URL: https://arxiv.org/abs/2008.03586.
 MBM16(1,2)
Cody A. Melton, M. Chandler Bennett, and Lubos Mitas. Quantum monte carlo with variable spins. The Journal of Chemical Physics, 144(24):244113, 2016. doi:10.1063/1.4954726.
 MZG+16(1,2)
Cody A. Melton, Minyi Zhu, Shi Guo, Alberto Ambrosetti, Francesco Pederiva, and Lubos Mitas. Spinorbit interactions in electronic structure quantum monte carlo methods. Phys. Rev. A, 93:042502, Apr 2016. doi:10.1103/PhysRevA.93.042502.
 ON19(1,2,3)
Leon Otis and Eric Neuscamman. Complementary first and second derivative methods for ansatz optimization in variational monte carlo. Phys. Chem. Chem. Phys., 21:14491, 2019. doi:10.1039/C9CP02269D.
 UNR93(1,2)
C J Umrigar, M P Nightingale, and K J Runge. A diffusion Monte Carlo algorithm with very small timestep errors A diffusion Monte Carlo algorithm with very small timestep errors. The Journal of Chemical Physics, 99(4):2865, 1993. doi:10.1063/1.465195.
 ZBKlimevs+18
Andrea Zen, Jan Gerit Brandenburg, Jiř’ı Klimeš, Alexandre Tkatchenko, Dario Alfè, and Angelos Michaelides. Fast and accurate quantum monte carlo for molecular crystals. Proceedings of the National Academy of Sciences, 115(8):1724–1729, February 2018. doi:10.1073/pnas.1715434115.
 ZSG+16(1,2)
Andrea Zen, Sandro Sorella, Michael J. Gillan, Angelos Michaelides, and Dario Alfè. Boosting the accuracy and speed of quantum monte carlo: size consistency and time step. Physical Review B, June 2016. doi:10.1103/physrevb.93.241118.
 ZN16
Luning Zhao and Eric Neuscamman. An efficient variational principle for the direct optimization of excited states. J. Chem. Theory. Comput., 12:3436, 2016. doi:10.1021/acs.jctc.6b00508.
 ZN17
Luning Zhao and Eric Neuscamman. A blocked linear method for optimizing large parameter sets in variational monte carlo. J. Chem. Theory. Comput., 2017. doi:10.1021/acs.jctc.7b00119.