Quantum Monte Carlo Methods

qmc factory element:

Parent elements	simulation, loop
type selector	method attribute

type 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 particle-by-particle move. In this method, only one electron is moved for acceptance or rejection. The other method is the all-electron 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 CPU-only 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.

Listing 43 The following is an example of running a simulation that can be restarted.

<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.run-number.config.h5. Check that this file exists before attempting a restart.

To continue a run, specify the mcwalkerset element before your VMC/DMC block:

Listing 44 Restart (read walkers from previous run).

<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 fewer walkers than threads are requested.

• blocks This parameter is universal for all the QMC methods. The MC processes are divided into a number of blocks, 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 of blocks to perform statistical analysis.

• warmupsteps - warmupsteps are used only for equilibration. Property measurements are not performed during warm-up 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 particle-by-particle or an all-electron 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 no-drift 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, having stepsbetweensamples larger than 1 can be used to reduces that correlation. In practice, using larger substeps is cheaper than using stepsbetweensamples to decorrelate samples.

• storeconfigs If storeconfigs 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

[MZG+16] and [MBM16]. If a relativistic calculation is desired using pseudopotentials, spin variable sampling is required.

• spinMass If spin sampling is on using spinMoves == 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 real-space 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 state-of-the-art 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 larger substeps to control the correlation between samples.

• 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.

Correlated sampling and cost function

After generating the samples with VMC, the derivatives of the wavefunction with respect to the parameters are computed for proposing a new set of parameters by optimizers. And later, a correlated sampling calculation is performed to quickly evaluate values of the cost function on the old set of parameters and the new set for further decisions. The input parameters are listed in the following table.

linear method:

parameters:

Name	Datatype	Values	Default	Description
nonlocalpp	text	yes, no	no	include non-local PP energy in the cost function
minwalkers	real	0–1	0.3	Lower bound of the effective weight
maxWeight	real	\(> 1\)	1e6	Maximum weight allowed in reweighting

Additional information:

• maxWeight The default should be good.

• nonlocalpp The nonlocalpp contribution to the local energy depends on the wavefunction. When a new set of parameters is proposed, this contribution needs to be updated if the cost function consists of local energy. Fortunately, nonlocal contribution is chosen small when making a PP for small locality error. We can ignore its change and avoid the expensive computational cost. An implementation issue with GPU code is that a large amount of memory is consumed with this option.

• minwalkers This is a critical parameter. When the ratio of effective samples to actual number of samples in a reweighting step goes lower than minwalkers, the proposed set of parameters is invalid.

The cost function consists of three components: energy, unreweighted variance, and reweighted variance.

<cost name="energy">                   0.95 </cost>
<cost name="unreweightedvariance">     0.00 </cost>
<cost name="reweightedvariance">       0.05 </cost>

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 auto-adjusted 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">           1e-4 </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 memory-efficient eigenproblem in the basis of these supposedly important block-wise 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 overlap-based 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 overlap-based 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:

\[\Omega[\Psi]=\frac{\left<\Psi|\omega-H|\Psi\right>}{\left<\Psi|{\left(\omega-H\right)}^2|\Psi\right>}\:.\]

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 and compute_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 by compute_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 for num_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 first optimizer 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 and compute_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	\(> 0\)	1e-4	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 3-Body 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"> 1.0e-4 </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.

• Two-body Jastrow contributes the largest portion of correlation energy from bare Slater determinants. Consequently, the recommended order for optimizing wavefunction components is two-body, one-body, three-body Jastrow factors and MSD coefficients.

• For two-body 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.

• One-body spline Jastrow from old calculations can be a good starting point.

• Three-body polynomial Jastrow can start from zero. It is beneficial to first optimize one-body and two-body Jastrow factors without adding three-body terms in the calculation and then add the three-body 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="1e-2" 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 table9, additional in table10.

dmc method:

parameters:

Name	Datatype	Values	Default	Description
targetwalkers	integer	\(> 0\)	dep.	Overall total number of walkers
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
timestep	real	\(> 0\)	0.1	Time step for each electron move
nonlocalmoves	string	yes, no, v0, v1, v3	no	Run with T-moves
branching_cutoff_scheme	string	classic/DRV/ZSGMA/YL	classic	Branch cutoff scheme
maxcpusecs	real	\(\geq 0\)	3.6e5	Maximum allowed walltime in seconds
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

Table 9 Main DMC input parameters.

Name	Datatype	Values	Default	Description
energyUpdateInterval	integer	\(\geq 0\)	0	Trial energy update interval
refEnergy	real	all values	dep.	Reference energy in atomic units
feedback	double	\(\geq 0\)	1.0	Population feedback on the trial energy
sigmaBound	10	\(\geq 0\)	10	Parameter to cutoff large weights
killnode	string	yes/other	no	Kill or reject walkers that cross nodes
warmupByReconfiguration	option	yes,no	0	Warm up with a fixed population
reconfiguration	string	yes/pure/other	no	Fixed population technique
branchInterval	integer	\(\geq 0\)	1	Branching interval
substeps	integer	\(\geq 0\)	1	Branching interval
MaxAge	double	\(\geq 0\)	10	Kill persistent walkers
MaxCopy	double	\(\geq 0\)	2	Limit population growth
maxDisplSq	real	all values	-1	Maximum particle move
scaleweight	string	yes/other	yes	Scale weights (CUDA only)
checkproperties	integer	\(\geq 0\)	100	Number of steps between walker updates
fastgrad	text	yes/other	yes	Fast gradients
storeconfigs	integer	all values	0	Store configurations
use_nonblocking	string	yes/no	yes	Using nonblocking send/recv
debug_disable_branching	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: The timestep 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 particle-by-particle 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 using spinMoves == 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 every energyUpdateInterval step.

\[E_{\text{trial}}= \textrm{refEnergy}+\textrm{feedback}\cdot(\ln\texttt{targetWalkers}-\ln N)\:,\]

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 and sigmaBound.

• killnode: When running fixed-node, if a walker attempts to cross a node, the move will normally be rejected. If killnode = “yes,” then walkers are destroyed when they cross a node.

• reconfiguration: If reconfiguration 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 be BranchInterval*Steps.

• substeps: This is the same as BranchInterval.

• nonlocalmoves: Evaluate pseudopotentials using one of the nonlocal move algorithms such as T-moves.

  • 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 T-move selection because of the acceptance of previous T-moves. v3 mostly reuses the transition probability computed during the evaluation of nonlocal pseudopotentials for the local energy, namely before accepting any T-moves, and only recomputes the transition probability of the electrons within the same pseudopotential region of any electrons touched by T-moves. 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 size-consistent and are important advances over the previous v0 non-size-consistent algorithm. We highly recommend investigating the importance of size-consistency.

• 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 for MaxAge 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 full-ratio 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 and sigmaBound.

• storeconfigs: If storeconfigs 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.

Listing 45 The following is an example of a very simple DMC section.

<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.

Listing 46 The following is an example of running a simulation that can be restarted.

<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.run-number.config.h5. Check that this file exists before attempting a restart. To read in this file for a continuation run, specify the following:

Listing 47 Restart (read walkers from previous run).

<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.

Listing 48 Combined VMC and DMC run.

<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 projector-based method that allows sampling of the fixed-node wavefunciton. However, by exploiting the path-integral formulation of Schrödinger’s equation, the RMC algorithm can offer some advantages over traditional DMC, such as sampling both the mixed and pure fixed-node distributions in polynomial time, as well as not having population fluctuations and biases. The current implementation does not work with T-moves.

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 \(\mathcal{O}(\tau)\) time-step error for observables arising from the Trotter-Suzuki breakup of the short-time 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 or beads? 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 overrides beads.

• 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 the scalar.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 the scalar.dat, where they will be appended with a _p suffix, or in the stat.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 particle-by-particle VMC a total of \(N_e\) times to generate a new all-electron 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. Size-consistent 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 excited-state-specific 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. Spin-orbit 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 time-step 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.