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Hamiltonian and Observables

QMCPACK is capable of the simultaneous measurement of the Hamiltonian and many other quantum operators. The Hamiltonian attains a special status among the available operators (also referred to as observables) because it ultimately generates all available information regarding the quantum system. This is evident from an algorithmic standpoint as well since the Hamiltonian (embodied in the projector) generates the imaginary time dynamics of the walkers in DMC and reptation Monte Carlo (RMC).

This section covers how the Hamiltonian can be specified, component by component, by the user in the XML format native to qmcpack. It also covers the input structure of statistical estimators corresponding to quantum observables such as the density, static structure factor, and forces.

The Hamiltonian

The many-body Hamiltonian in Hartree units is given by

$$\hat{H} = -\sum_i\frac{1}{2m_i}\nabla_i^2 + \sum_iv^{ext}(r_i) + \sum_{i<j}v^{qq}(r_i,r_j) + \sum_{i\ell}v^{qc}(r_i,r_\ell) + \sum_{\ell<m}v^{cc}(r_\ell,r_m)\:.$$

Here, the sums indexed by i/j are over quantum particles, while ℓ/m are reserved for classical particles. Often the quantum particles are electrons, and the classical particles are ions, though is not limited in this way. The mass of each quantum particle is denoted mi, vqq/vqc/vcc are pair potentials between quantum-quantum/quantum-classical/classical-classical particles, and vext denotes a purely external potential.

QMCPACK is designed modularly so that any potential can be supported with minimal additions to the code base. Potentials currently supported include Coulomb interactions in open and periodic boundary conditions, the MPC potential, nonlocal pseudopotentials, helium pair potentials, and various model potentials such as hard sphere, Gaussian, and modified Poschl-Teller.

Reference information and examples for the <hamiltonian/> XML element are provided subsequently. Detailed descriptions of the input for individual potentials is given in the sections that follow.

hamiltonian element:

parent elements: simulation, qmcsystem
child elements: pairpot extpot estimator constant (deprecated)

attributes:

Name Datatype Values Default Description
name/ido text anything h0 Unique id for this Hamiltonian instance
typeo text generic No current function
roleo text primary/extra extra Designate as Hamiltonian or not
sourceo text particleset.name i Identify classical particleset
targeto text particleset.name e Identify quantum particleset
defaulto boolean yes/no yes Include kinetic energy term implicitly

Additional information:

  • target: Must be set to the name of the quantum particleset. The default value is typically sufficient. In normal usage, no other attributes are provided.
<hamiltonian target="e">
  <pairpot name="ElecElec" type="coulomb" source="e" target="e"/>
  <pairpot name="ElecIon"  type="coulomb" source="i" target="e"/>
  <pairpot name="IonIon"   type="coulomb" source="i" target="i"/>
</hamiltonian>
<hamiltonian target="e">
  <pairpot name="ElecElec"  type="coulomb" source="e" target="e"/>
  <pairpot name="PseudoPot" type="pseudo"  source="i" wavefunction="psi0" format="xml">
    <pseudo elementType="Li" href="Li.xml"/>
    <pseudo elementType="H" href="H.xml"/>
  </pairpot>
  <pairpot name="IonIon"    type="coulomb" source="i" target="i"/>
</hamiltonian>

Pair potentials

Many pair potentials are supported. Though only the most commonly used pair potentials are covered in detail in this section, all currently available potentials are listed subsequently. If a potential you desire is not listed, or is not present at all, feel free to contact the developers.

pairpot factory element:

parent elements: hamiltonian
child elements: type attribute
type options coulomb Coulomb/Ewald potential
pseudo Semilocal pseudopotential
mpc Model periodic Coulomb interaction/correction
cpp Core polarization potential
skpot Unknown

shared attributes:

Name Datatype Values Default Description
typer text See above 0 Select pairpot type
namer text Anything any Unique name for this pairpot
sourcer text particleset.name hamiltonian.target Identify interacting particles
targetr text particleset.name hamiltonian.target Identify interacting particles
unitso text hartree No current function

Additional information:

  • type: Used to select the desired pair potential. Must be selected from the list of type options.
  • name: A unique name used to identify this pair potential. Block averaged output data will appear under this name in scalar.dat and/or stat.h5 files.
  • source/target: These specify the particles involved in a pair interaction. If an interaction is between classical (e.g., ions) and quantum (e.g., electrons), source/target should be the name of the classical/quantum particleset.
  • Only Coulomb, pseudo, and mpc are described in detail in the following subsections. The older or less-used types (cpp, skpot) are not covered.
  • Available only if QMC_CUDA is not defined: skpot.
  • Available only if OHMMS_DIM==3: mpc, vhxc, pseudo.
  • Available only if OHMMS_DIM==3 and QMC_CUDA is not defined: cpp.

Coulomb potentials

The bare Coulomb potential is used in open boundary conditions:

$$V_c^{open} = \sum_{i<j}\frac{q_iq_j}{\left|{r_i-r_j}\right|}\:.$$

When periodic boundary conditions are selected, Ewald summation is used automatically:

$$V_c^{pbc} = \sum_{i<j}\frac{q_iq_j}{\left|{r_i-r_j}\right|} + \frac{1}{2}\sum_{L\ne0}\sum_{i,j}\frac{q_iq_j}{\left|{r_i-r_j+L}\right|}\:.$$

The sum indexed by L is over all nonzero simulation cell lattice vectors. In practice, the Ewald sum is broken into short- and long-range parts in a manner optimized for efficiency (see Natoli1995) for details.

For information on how to set the boundary conditions, consult simulationcell.

pairpot type=coulomb element:

parent elements: hamiltonian
child elements: None

attributes:

Name Datatype Values Default Description
typer text coulomb Must be coulomb
name/idr text anything ElecElec Unique name for interaction
sourcer text particleset.name hamiltonian.target Identify interacting particles
targetr text particleset.name hamiltonian.target Identify interacting particles
pbco boolean yes/no yes Use Ewald summation
physicalo boolean yes/no yes Hamiltonian(yes)/Observable(no)
forces boolean yes/no no Deprecated

Additional information:

  • type/source/target: See description for the previous generic pairpot factory element.
  • name: Traditional user-specified names for electron-electron, electron-ion, and ion-ion terms are ElecElec, ElecIon, and IonIon, respectively. Although any choice can be used, the data analysis tools expect to find columns in *.scalar.dat with these names.
  • pbc: Ewald summation will not be performed if simulationcell.bconds== n n n, regardless of the value of pbc. Similarly, the pbc attribute can only be used to turn off Ewald summation if simulationcell.bconds!= n n n. The default value is recommended.
  • physical: If physical==yes, this pair potential is included in the Hamiltonian and will factor into the LocalEnergy reported by QMCPACK and also in the DMC branching weight. If physical==no, then the pair potential is treated as a passive observable but not as part of the Hamiltonian itself. As such it does not contribute to the outputted LocalEnergy. Regardless of the value of physical output data will appear in scalar.dat in a column headed by name.
<pairpot name="ElecElec" type="coulomb" source="e" target="e"/>
<pairpot name="ElecIon"  type="coulomb" source="i" target="e"/>
<pairpot name="IonIon"   type="coulomb" source="i" target="i"/>

Pseudopotentials

QMCPACK supports pseudopotentials in semilocal form, which is local in the radial coordinate and nonlocal in angular coordinates. When all angular momentum channels above a certain threshold (ℓmax) are well approximated by the same potential ($V_{\bar{\ell}}\equiv V_{loc}$), the pseudopotential separates into a fully local channel and an angularly nonlocal component:

$$V^{PP} = \sum_{ij}\Big(V_{\bar{\ell}}(\left|{r_i-\tilde{r}_j}\right|) + \sum_{\ell\ne\bar{\ell}}^{\ell_{max}}\sum_{m=-\ell}^\ell |{Y_{\ell m}}\rangle{\big[V_\ell(\left|{r_i-\tilde{r}_j}\right|) - V_{\bar{\ell}}(\left|{r_i-\tilde{r}_j}\right|) \big]}\langle{Y_{\ell m}}| \Big)\:.$$

Here the electron/ion index is i/j, and only one type of ion is shown for simplicity.

Evaluation of the localized pseudopotential energy ΨT − 1VPPΨT requires additional angular integrals. These integrals are evaluated on a randomly shifted angular grid. The size of this grid is determined by ℓmax. See Mitas1991 for further detail.

uses the FSAtom pseudopotential file format associated with the “Free Software Project for Atomic-scale Simulations” initiated in 2002. See http://www.tddft.org/fsatom/manifest.php for more information. The FSAtom format uses XML for structured data. Files in this format do not use a specific identifying file extension; instead they are simply suffixed with “.xml.” The tabular data format of CASINO is also supported.

In addition to the semilocal pseudopotential above, spin-orbit interactions can also be included through the use of spin-orbit pseudopotentials. The spin-orbit contribution can be written as

$$V^{\rm SO} = \sum_{ij} \left(\sum_{\ell = 1}^{\ell_{max}-1} \frac{2}{2\ell+1} V^{\rm SO}_\ell \left( \left|r_i - \tilde{r}_j \right| \right) \sum_{m,m'=-\ell}^{\ell} | Y_{\ell m} \rangle \langle Y_{\ell m} | \vec{\ell} \cdot \vec{s} | Y_{\ell m'}\rangle\langle Y_{\ell m'}|\right)\:.$$

Here, s⃗ is the spin operator. For each atom with a spin-orbit contribution, the radial functions $V_{\ell}^{\rm SO}$ can be included in the pseudopotential “.xml” file.

pairpot type=pseudo element:

parent elements: hamiltonian
child elements: pseudo

attributes:

Name Datatype Values Default Description
typer text pseudo Must be pseudo
name/idr text anything PseudoPot No current function
sourcer text particleset.name i Ion particleset name
targetr text particleset.name hamiltonian.target Electron particleset name
pbco boolean yes/no yes* Use Ewald summation
forces boolean yes/no no Deprecated
wavefunctionr text wavefunction.name invalid Identify wavefunction
formatr text xml/table table Select file format
algorithmo text batched/non-batched depends Choose NLPP algorithm
DLAo text yes/no no Use determinant localization approximation
physicalSOo boolean yes/no yes Include the SO contribution in the local energy

Additional information:

  • type/source/target See description for the generic pairpot factory element.
  • name: Ignored. Instead, default names will be present in *scalar.dat output files when pseudopotentials are used. The field LocalECP refers to the local part of the pseudopotential. If nonlocal channels are present, a NonLocalECP field will be added that contains the nonlocal energy summed over all angular momentum channels.
  • pbc: Ewald summation will not be performed if simulationcell.bconds== n n n, regardless of the value of pbc. Similarly, the pbc attribute can only be used to turn off Ewald summation if simulationcell.bconds!= n n n.
  • format: If format==table, QMCPACK looks for *.psf files containing pseudopotential data in a tabular format. The files must be named after the ionic species provided in particleset (e.g., Li.psf and H.psf). If format==xml, additional pseudo child XML elements must be provided (see the following). These elements specify individual file names and formats (both the FSAtom XML and CASINO tabular data formats are supported).
  • algorithm The default value is batched when OpenMP offload is enabled andnon-batched otherwise. The non-batched algorithm evaluates the ratios of wavefunction components together for each quadrature point and then one point after another. The batched algorithm evaluates the ratios of quadrature points together for each wavefunction component and then one component after another. Internally, it uses VirtualParticleSet for quadrature points. Hybrid orbital representation has an extra optimization enabled when using the batched algorithm. When OpenMP offload build is enabled, the default value is batched. Otherwise, non-batched is the default.
  • DLA Determinant localization approximation (DLA) Zen2019DLA uses only the fermionic part of the wavefunction when calculating NLPP.
  • physicalSO If the spin-orbit components are included in the .xml file, this flag allows control over whether the SO contribution is included in the local energy.
<pairpot name="PseudoPot" type="pseudo"  source="i" wavefunction="psi0" format="psf"/>
<pairpot name="PseudoPot" type="pseudo"  source="i" wavefunction="psi0" format="xml">
  <pseudo elementType="Li" href="Li.xml"/>
  <pseudo elementType="H" href="H.xml"/>
</pairpot>
<pairpot name="PseudoPot" type="pseudo" source="i" wavefunction="psi0" format="xml" physicalSO="no">
  <pseudo elementType="Pb" href="Pb.xml"/>
</pairpot>

Details of <pseudo/> input elements are shown in the following. It is possible to include (or construct) a full pseudopotential directly in the input file without providing an external file via href. The full XML format for pseudopotentials is not yet covered.

pseudo element:

parent elements: pairpot type=pseudo
child elements: header local grid

attributes:

Name Datatype Values Default Description
elementType/symbolr text groupe.name none Identify ionic species
hrefr text filepath none Pseudopotential file path
formatr text xml/casino xml Specify file format
cutoffo real Nonlocal cutoff radius
lmaxo integer Largest angular momentum
nruleo integer Integration grid order 
<pseudo elementType="Li" href="Li.xml"/>

MPC Interaction/correction

The MPC interaction is an alternative to direct Ewald summation. The MPC corrects the exchange correlation hole to more closely match its thermodynamic limit. Because of this, the MPC exhibits smaller finite-size errors than the bare Ewald interaction, though a few alternative and competitive finite-size correction schemes now exist. The MPC is itself often used just as a finite-size correction in post-processing (set physical=false in the input).

pairpot type=mpc element:

parent elements: hamiltonian
child elements: None

attributes:

Name Datatype Values Default Description
typer text mpc Must be MPC
name/idr text anything MPC Unique name for interaction
sourcer text particleset.name hamiltonian.target Identify interacting particles
targetr text particleset.name hamiltonian.target Identify interacting particles
physicalo boolean yes/no no Hamiltonian(yes)/observable(no)
cutoff real  > 0 30.0 Kinetic energy cutoff

Remarks:

  • physical: Typically set to no, meaning the standard Ewald interaction will be used during sampling and MPC will be measured as an observable for finite-size post-correction. If physical is yes, the MPC interaction will be used during sampling. In this case an electron-electron Coulomb pairpot element should not be supplied.
  • Developer note: Currently the name attribute for the MPC interaction is ignored. The name is always reset to MPC.
<pairpot type="MPC" name="MPC" source="e" target="e" ecut="60.0" physical="no"/>

General estimators

A broad range of estimators for physical observables are available in QMCPACK. The following sections contain input details for the total number density (density), number density resolved by particle spin (spindensity), spherically averaged pair correlation function (gofr), static structure factor (sk), static structure factor (skall), energy density (energydensity), one body reduced density matrix (dm1b), S(k) based kinetic energy correction (chiesa), forward walking (ForwardWalking), and force (Force) estimators. Other estimators are not yet covered.

When an <estimator/> element appears in <hamiltonian/>, it is evaluated for all applicable chained QMC runs (e.g., VMC→DMC→DMC). Estimators are generally not accumulated during wavefunction optimization sections. If an <estimator/> element is instead provided in a particular <qmc/> element, that estimator is only evaluated for that specific section (e.g., during VMC only).

estimator factory element:

parent elements: hamiltonian, qmc
type selector: type attribute
type options density Density on a grid
spindensity Spin density on a grid
gofr Pair correlation function (quantum species)
sk Static structure factor
SkAll Static structure factor needed for finite size correction
structurefactor Species resolved structure factor
species kinetic Species resolved kinetic energy
latticedeviation Spatial deviation between two particlesets
momentum Momentum distribution
energydensity Energy density on uniform or Voronoi grid
dm1b One body density matrix in arbitrary basis
chiesa Chiesa-Ceperley-Martin-Holzmann kinetic energy correction
Force Family of "force" estimators (see force-est)
ForwardWalking Forward walking values for existing estimators
orbitalimages Create image files for orbitals, then exit
flux Checks sampling of kinetic energy
localmoment Atomic spin polarization within cutoff radius
Pressure No current function

shared attributes:

Name Datatype Values Default Description
typer text See above 0 Select estimator type
namer text anything any Unique name for this estimator

Chiesa-Ceperley-Martin-Holzmann kinetic energy correction

This estimator calculates a finite-size correction to the kinetic energy following the formalism laid out in Chiesa2006. The total energy can be corrected for finite-size effects by using this estimator in conjunction with the MPC correction.

estimator type=chiesa element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text chiesa Must be chiesa
name:math:^o text anything KEcorr Always reset to KEcorr
sourceo text particleset.name e Identify quantum particles
psio text wavefunction.name psi0 Identify wavefunction 
<estimator name="KEcorr" type="chiesa" source="e" psi="psi0"/>

Density estimator

The particle number density operator is given by


n̂r = ∑iδ(r − ri) .

The density estimator accumulates the number density on a uniform histogram grid over the simulation cell. The value obtained for a grid cell c with volume Ωc is then the average number of particles in that cell:


nc = ∫dR|Ψ|2∫Ωcdr∑iδ(r − ri) .

estimator type=density element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text density Must be density
namer text anything any Unique name for estimator
deltao real array(3) 0 ≤ vi ≤ 1 0.1 0.1 0.1 Grid cell spacing, unit coords
x_mino real  > 0 0 Grid starting point in x (Bohr)
x_maxo real  > 0 | lattice[0] | Grid ending point in x (Bohr)
y_mino real  > 0 0 Grid starting point in y (Bohr)
y_maxo real  > 0 | lattice[1] | Grid ending point in y (Bohr)
z_mino real  > 0 0 Grid starting point in z (Bohr)
z_maxo real  > 0 | lattice[2] | Grid ending point in z (Bohr)
potentialo boolean yes/no no Accumulate local potential, deprecated
debugo boolean yes/no no No current function

Additional information:

  • name: The name provided will be used as a label in the stat.h5 file for the blocked output data. Postprocessing tools expect name="Density."
  • delta: This sets the histogram grid size used to accumulate the density: delta="0.1 0.1 0.05" → 10 × 10 × 20 grid, delta="0.01 0.01 0.01" → 100 × 100 × 100 grid. The density grid is written to a stat.h5 file at the end of each MC block. If you request many blocks in a <qmc/> element, or select a large grid, the resulting stat.h5 file could be many gigabytes in size.
  • *_min/*_max: Can be used to select a subset of the simulation cell for the density histogram grid. For example if a (cubic) simulation cell is 20 Bohr on a side, setting *_min=5.0 and *_max=15.0 will result in a density histogram grid spanning a 10 × 10 × 10 Bohr cube about the center of the box. Use of x_min, x_max, y_min, y_max, z_min, z_max is only appropriate for orthorhombic simulation cells with open boundary conditions.
  • When open boundary conditions are used, a <simulationcell/> element must be explicitly provided as the first subelement of <qmcsystem/> for the density estimator to work. In this case the molecule should be centered around the middle of the simulation cell (L/2) and not the origin (0 since the space within the cell, and hence the density grid, is defined from 0 to L).
<estimator name="Density" type="density" delta="0.05 0.05 0.05"/>

Spin density estimator

The spin density is similar to the total density described previously. In this case, the sum over particles is performed independently for each spin component.

estimator type=spindensity element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text spindensity Must be spindensity
namer text anything any Unique name for estimator
reporto boolean yes/no no Write setup details to stdout

parameters:

Name Datatype Values Default Description
grido integer array(3) vi> Grid cell count
dro real array(3) vi> Grid cell spacing (Bohr)
cello real array(3,3) anything Volume grid exists in
cornero real array(3) anything Volume corner location
centero real array (3) anything Volume center/origin location
voronoio text particleset.name Under development
test_moveso integer  >  = 0 0 Test estimator with random moves

Additional information:

  • name: The name provided will be used as a label in the stat.h5 file for the blocked output data. Postprocessing tools expect name="SpinDensity."
  • grid: The grid sets the dimension of the histogram grid. Input like <parameter name="grid"> 40 40 40 </parameter> requests a 40 × 40 × 40 grid. The shape of individual grid cells is commensurate with the supercell shape.
  • dr: The dr sets the real-space dimensions of grid cell edges (Bohr units). Input like <parameter name="dr"> 0.5 0.5 0.5 </parameter> in a supercell with axes of length 10 Bohr each (but of arbitrary shape) will produce a 20 × 20 × 20 grid. The inputted dr values are rounded to produce an integer number of grid cells along each supercell axis. Either grid or dr must be provided, but not both.
  • cell: When cell is provided, a user-defined grid volume is used instead of the global supercell. This must be provided if open boundary conditions are used. Additionally, if cell is provided, the user must specify where the volume is located in space in addition to its size/shape (cell) using either the corner or center parameters.
  • corner: The grid volume is defined as $corner+\sum_{d=1}^3u_dcell_d$ with 0 < ud < 1 (“cell” refers to either the supercell or user-provided cell).
  • center: The grid volume is defined as $center+\sum_{d=1}^3u_dcell_d$ with  − 1/2 < ud < 1/2 (“cell” refers to either the supercell or user-provided cell). corner/center can be used to shift the grid even if cell is not specified. Simultaneous use of corner and center will cause QMCPACK to abort.
<estimator type="spindensity" name="SpinDensity" report="yes">
  <parameter name="grid"> 40 40 40 </parameter>
</estimator>
<estimator type="spindensity" name="SpinDensity" report="yes">
  <parameter name="grid">
    20 20 20
  </parameter>
  <parameter name="center">
    0.0 0.0 0.0
  </parameter>
  <parameter name="cell">
    10.0  0.0  0.0
     0.0 10.0  0.0
     0.0  0.0 10.0
  </parameter>
</estimator>

Pair correlation function, g(r)

The functional form of the species-resolved radial pair correlation function operator is

$$g_{ss'}(r) = \frac{V}{4\pi r^2N_sN_{s'}}\sum_{i_s=1}^{N_s}\sum_{j_{s'}=1}^{N_{s'}}\delta(r-|r_{i_s}-r_{j_{s'}}|)\:,$$

where Ns is the number of particles of species s and V is the supercell volume. If s = s′, then the sum is restricted so that is ≠ js.

In QMCPACK, an estimate of gss′(r) is obtained as a radial histogram with a set of Nb uniform bins of width δr. This can be expressed analytically as

$$\tilde{g}_{ss'}(r) = \frac{V}{4\pi r^2N_sN_{s'}}\sum_{i=1}^{N_s}\sum_{j=1}^{N_{s'}}\frac{1}{\delta r}\int_{r-\delta r/2}^{r+\delta r/2}dr'\delta(r'-|r_{si}-r_{s'j}|)\:,$$

where the radial coordinate r is restricted to reside at the bin centers, δr/2, 3δr/2, 5δr/2, ….

estimator type=gofr element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text gofr Must be gofr
nameo text anything any No current function
num_binr integer  > 1 20 # of histogram bins
rmaxo real  > 0 10 Histogram extent (Bohr)
dro real 0 0.5 No current function
debugo boolean yes/no no No current function
targeto text particleset.name hamiltonian.target Quantum particles
source/sourceso text array particleset.name hamiltonian.target Classical particles

Additional information:

  • num_bin: This is the number of bins in each species pair radial histogram.
  • rmax: This is the maximum pair distance included in the histogram. The uniform bin width is δr = rmax/num_bin. If periodic boundary conditions are used for any dimension of the simulation cell, then the default value of rmax is the simulation cell radius instead of 10 Bohr. For open boundary conditions, the volume (V) used is 1.0 Bohr3.
  • source/sources: If unspecified, only pair correlations between each species of quantum particle will be measured. For each classical particleset specified by source/sources, additional pair correlations between each quantum and classical species will be measured. Typically there is only one classical particleset (e.g., source="ion0"), but there can be several in principle (e.g., sources="ion0 ion1 ion2").
  • target: The default value is the preferred usage (i.e., target does not need to be provided).
  • Data is output to the stat.h5 for each QMC subrun. Individual histograms are named according to the quantum particleset and index of the pair. For example, if the quantum particleset is named “e" and there are two species (up and down electrons, say), then there will be three sets of histogram data in each stat.h5 file named gofr_e_0_0, gofr_e_0_1, and gofr_e_1_1 for up-up, up-down, and down-down correlations, respectively.
<estimator type="gofr" name="gofr" num_bin="200" rmax="3.0" />
<estimator type="gofr" name="gofr" num_bin="200" rmax="3.0" source="ion0" />

Static structure factor, S(k)

Let ρke = ∑jeik ⋅ rje be the Fourier space electron density, with rje being the coordinate of the j-th electron. k is a wavevector commensurate with the simulation cell. QMCPACK allows the user to accumulate the static electron structure factor S(k) at all commensurate k such that |k| ≤ (LR_DIM_CUTOFF)rc. Ne is the number of electrons, LR_DIM_CUTOFF is the optimized breakup parameter, and rc is the Wigner-Seitz radius. It is defined as follows:

$$S(\mathbf{k}) = \frac{1}{N^e}\langle \rho^e_{-\mathbf{k}} \rho^e_{\mathbf{k}} \rangle\:.$$

estimator type=sk element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text sk Must sk
namer text anything any Unique name for estimator
hdf5o boolean yes/no no

Output to stat.h5 (yes) or scalar.dat (no)

Additional information:

  • name: This is the unique name for estimator instance. A data structure of the same name will appear in stat.h5 output files.
  • hdf5: If hdf5==yes, output data for S(k) is directed to the stat.h5 file (recommended usage). If hdf5==no, the data is instead routed to the scalar.dat file, resulting in many columns of data with headings prefixed by name and postfixed by the k-point index (e.g., sk_0 sk_1 …sk_1037 …).
  • This estimator only works in periodic boundary conditions. Its presence in the input file is ignored otherwise.
  • This is not a species-resolved structure factor. Additionally, for k vectors commensurate with the unit cell, S(k) will include contributions from the static electronic density, thus meaning it will not accurately measure the electron-electron density response.
<estimator type="sk" name="sk" hdf5="yes"/>

Static structure factor, SkAll

In order to compute the finite size correction to the potential energy, records of ρ(k) is required. What sets SkAll apart from sk is that SkAll records ρ(k) in addition to s(k).

estimator type=SkAll element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text sk Must be sk
namer text anything any Unique name for estimator
sourcer text Ion ParticleSet name None -
targetr text Electron ParticleSet name None -
hdf5o boolean yes/no no Output to stat.h5 (yes) or scalar.dat (no)
writeioniono boolean yes/no no Writes file rhok_IonIon.dat containing s(k) for the ions

Additional information:

  • name: This is the unique name for estimator instance. A data structure of the same name will appear in stat.h5 output files.
  • hdf5: If hdf5==yes, output data is directed to the stat.h5 file (recommended usage). If hdf5==no, the data is instead routed to the scalar.dat file, resulting in many columns of data with headings prefixed by rhok and postfixed by the k-point index.
  • This estimator only works in periodic boundary conditions. Its presence in the input file is ignored otherwise.
  • This is not a species-resolved structure factor. Additionally, for k vectors commensurate with the unit cell, S(k) will include contributions from the static electronic density, thus meaning it wil not accurately measure the electron-electron density response.
<estimator type="skall" name="SkAll" source="ion0" target="e" hdf5="yes"/>

Species kinetic energy

Record species-resolved kinetic energy instead of the total kinetic energy in the Kinetic column of scalar.dat. SpeciesKineticEnergy is arguably the simplest estimator in QMCPACK. The implementation of this estimator is detailed in manual/estimator/estimator_implementation.pdf.

estimator type=specieskinetic element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text specieskinetic Must be specieskinetic
namer text anything any Unique name for estimator
hdf5o boolean yes/no no Output to stat.h5 (yes) 
<estimator type="specieskinetic" name="skinetic" hdf5="no"/>

Lattice deviation estimator

Record deviation of a group of particles in one particle set (target) from a group of particles in another particle set (source).

estimator type=latticedeviation element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text latticedeviation Must be latticedeviation
namer text anything any Unique name for estimator
hdf5o boolean yes/no no Output to stat.h5 (yes)
per_xyzo boolean yes/no no Directionally resolved (yes)
sourcer text e/ion0/... no source particleset
sgroupr text u/d/... no source particle group
targetr text e/ion0/... no target particleset
tgroupr text u/d/... no target particle group

Additional information:

  • source: The “reference” particleset to measure distances from; actual reference points are determined together with sgroup.
  • sgroup: The “reference” particle group to measure distances from.
  • source: The “target” particleset to measure distances to.
  • sgroup: The “target” particle group to measure distances to. For example, in Listing 32 <Listing 32> the distance from the up electron (“u”) to the origin of the coordinate system is recorded.
  • per_xyz: Used to record direction-resolved distance. In Listing 32 <Listing 32>, the x,y,z coordinates of the up electron will be recorded separately if per_xyz=yes.
  • hdf5: Used to record particle-resolved distances in the h5 file if gdf5=yes.
<particleset name="e" random="yes">
  <group name="u" size="1" mass="1.0">
     <parameter name="charge"              >    -1                    </parameter>
     <parameter name="mass"                >    1.0                   </parameter>
  </group>
  <group name="d" size="1" mass="1.0">
     <parameter name="charge"              >    -1                    </parameter>
     <parameter name="mass"                >    1.0                   </parameter>
  </group>
</particleset>

<particleset name="wf_center">
  <group name="origin" size="1">
    <attrib name="position" datatype="posArray" condition="0">
             0.00000000        0.00000000        0.00000000
    </attrib>
  </group>
</particleset>

<estimator type="latticedeviation" name="latdev" hdf5="yes" per_xyz="yes"
  source="wf_center" sgroup="origin" target="e" tgroup="u"/>

Energy density estimator

An energy density operator, $\hat{\mathcal{E}}_r$, satisfies

$$\int dr \hat{\mathcal{E}}_r = \hat{H},$$

where the integral is over all space and Ĥ is the Hamiltonian. In QMCPACK, the energy density is split into kinetic and potential components

$$\hat{\mathcal{E}}_r = \hat{\mathcal{T}}_r + \hat{\mathcal{V}}_r\:,$$

with each component given by

$$\begin{aligned} \begin{aligned} \hat{\mathcal{T}}_r &= \frac{1}{2}\sum_i\delta(r-r_i)\hat{p}_i^2 \\\ \hat{\mathcal{V}}_r &= \sum_{i<j}\frac{\delta(r-r_i)+\delta(r-r_j)}{2}\hat{v}^{ee}(r_i,r_j) + \sum_{i\ell}\frac{\delta(r-r_i)+\delta(r-\tilde{r}_\ell)}{2}\hat{v}^{eI}(r_i,\tilde{r}_\ell) \nonumber\\\ &\qquad + \sum_{\ell< m}\frac{\delta(r-\tilde{r}_\ell)+\delta(r-\tilde{r}_m)}{2}\hat{v}^{II}(\tilde{r}_\ell,\tilde{r}_m)\:.\nonumber\end{aligned} \end{aligned}$$

Here, ri and r̃ℓ represent electron and ion positions, respectively; p̂i is a single electron momentum operator; and v̂ee(ri, rj), v̂eI(ri, r̃ℓ), and v̂II(r̃ℓ, r̃m) are the electron-electron, electron-ion, and ion-ion pair potential operators (including nonlocal pseudopotentials, if present). This form of the energy density is size consistent; that is, the partially integrated energy density operators of well-separated atoms gives the isolated Hamiltonians of the respective atoms. For periodic systems with twist-averaged boundary conditions, the energy density is formally correct only for either a set of supercell k-points that correspond to real-valued wavefunctions or a k-point set that has inversion symmetry around a k-point having a real-valued wavefunction. For more information about the energy density, see Krogel2013.

In QMCPACK, the energy density can be accumulated on piecewise uniform 3D grids in generalized Cartesian, cylindrical, or spherical coordinates. The energy density integrated within Voronoi volumes centered on ion positions is also available. The total particle number density is also accumulated on the same grids by the energy density estimator for convenience so that related quantities, such as the regional energy per particle, can be computed easily.

estimator type=EnergyDensity element:

parent elements: hamiltonian, qmc
child elements: reference_points, spacegrid

attributes:

Name Datatype Values Default Description
typer text EnergyDensity Must be EnergyDensity
namer text anything Unique name for estimator
dynamicr text particleset.name Identify electrons
statico text particleset.name Identify ions
ion_pointso text yes/no

no

 Separate ion energy density onto point field

Additional information:

  • name: Must be unique. A dataset with blocked statistical data for the energy density will appear in the stat.h5 files labeled as name.
  • Important: in order for the estimator to work, a traces XML input element (<traces array="yes" write="no"/>) must appear following the <qmcsystem/> element and prior to any <qmc/> element.
<estimator type="EnergyDensity" name="EDcell" dynamic="e" static="ion0">
   <spacegrid coord="cartesian">
     <origin p1="zero"/>
     <axis p1="a1" scale=".5" label="x" grid="-1 (.05) 1"/>
     <axis p1="a2" scale=".5" label="y" grid="-1 (.1) 1"/>
     <axis p1="a3" scale=".5" label="z" grid="-1 (.1) 1"/>
   </spacegrid>
</estimator>
<estimator type="EnergyDensity" name="EDatom" dynamic="e" static="ion0">
  <reference_points coord="cartesian">
    r1 1 0 0
    r2 0 1 0
    r3 0 0 1
  </reference_points>
  <spacegrid coord="spherical">
    <origin p1="ion01"/>
    <axis p1="r1" scale="6.9" label="r"     grid="0 1"/>
    <axis p1="r2" scale="6.9" label="phi"   grid="0 1"/>
    <axis p1="r3" scale="6.9" label="theta" grid="0 1"/>
  </spacegrid>
  <spacegrid coord="spherical">
    <origin p1="ion02"/>
    <axis p1="r1" scale="6.9" label="r"     grid="0 1"/>
    <axis p1="r2" scale="6.9" label="phi"   grid="0 1"/>
    <axis p1="r3" scale="6.9" label="theta" grid="0 1"/>
  </spacegrid>
</estimator>
<estimator type="EnergyDensity" name="EDvoronoi" dynamic="e" static="ion0">
  <spacegrid coord="voronoi"/>
</estimator>

The <reference_points/> element provides a set of points for later use in specifying the origin and coordinate axes needed to construct a spatial histogramming grid. Several reference points on the surface of the simulation cell (see table8), as well as the positions of the ions (see the energydensity.static attribute), are made available by default. The reference points can be used, for example, to construct a cylindrical grid along a bond with the origin on the bond center.

reference_points element:

parent elements: estimator type=EnergyDensity
child elements: None

attributes:

Name Datatype Values Default Description
coordr text Cartesian/cell Specify coordinate system

body text: The body text is a line formatted list of points with labels

Additional information:

  • coord: If coord=cartesian, labeled points are in Cartesian (x,y,z) format in units of Bohr. If coord=cell, then labeled points are in units of the simulation cell axes.

  • body text: The list of points provided in the body text are line formatted, with four entries per line (label coor1 coor2 coor3). A set of points referenced to the simulation cell is available by default (see table8). If energydensity.static is provided, the location of each individual ion is also available (e.g., if energydensity.static=ion0, then the location of the first atom is available with label ion01, the second with ion02, etc.). All points can be used by label when constructing spatial histogramming grids (see the following spacegrid element) used to collect energy densities.

    label point description
    zero 0 0 0 Cell center
    a1 a1 Cell axis 1
    a2 a2 Cell axis 2
    a3 a3 Cell axis 3
    f1p a1/2 Cell face 1+
    f1m -a1/2 Cell face 1-
    f2p a2/2 Cell face 2+
    f2m -a2/2 Cell face 2-
    f3p a3/2 Cell face 3+
    f3m -a3/2 Cell face 3-
    cppp (a1 + a2 + a3)/2 Cell corner +,+,+
    cppm (a1 + a2 − a3)/2 Cell corner +,+,-
    cpmp (a1 − a2 + a3)/2 Cell corner +,-,+
    cmpp ( − a1 + a2 + a3)/2 Cell corner -,+,+
    cpmm (a1 − a2 − a3)/2 Cell corner +,-,-
    cmpm ( − a1 + a2 − a3)/2 Cell corner -,+,-
    cmmp ( − a1 − a2 + a3)/2 Cell corner -,-,+
    cmmm ( − a1 − a2 − a3)/2 Cell corner -,-,-

Table 8 Reference points available by default. Vectors a1, a2, and a3 refer to the simulation cell axes. The representation of the cell is centered around zero.

The <spacegrid/> element is used to specify a spatial histogramming grid for the energy density. Grids are constructed based on a set of, potentially nonorthogonal, user-provided coordinate axes. The axes are based on information available from reference_points. Voronoi grids are based only on nearest neighbor distances between electrons and ions. Any number of space grids can be provided to a single energy density estimator.

spacegrid element:

parent elements: estimator type=EnergyDensity
child elements: origin, axis

attributes:

Name Datatype Values Default Description
coordr text Cartesian Specify coordinate system
cylindrical
spherical
Voronoi

The <origin/> element gives the location of the origin for a non-Voronoi grid.

Additional information:

  • p1/p2/fraction: The location of the origin is set to p1+fraction*(p2-p1). If only p1 is provided, the origin is at p1.

origin element:

parent elements: spacegrid
child elements: None

attributes:

Name Datatype Values Default Description
p1r text reference_point.label Select end point
p2o text reference_point.label Select end point
fractiono real 0 Interpolation fraction

The <axis/> element represents a coordinate axis used to construct the, possibly curved, coordinate system for the histogramming grid. Three <axis/> elements must be provided to a non-Voronoi <spacegrid/> element.

axis element:

parent elements: spacegrid
child elements: None

attributes:

Name Datatype Values Default Description
labelr text See below Axis/dimension label
gridr text "0 1" Grid ranges/intervals
p1r text reference_point.label Select end point
p2o text reference_point.label Select end point
scaleo real Interpolation fraction

Additional information:

  • label: The allowed set of axis labels depends on the coordinate system (i.e., spacegrid.coord). Labels are x/y/z for coord=cartesian, r/phi/z for coord=cylindrical, r/phi/theta for coord=spherical.
  • p1/p2/scale: The axis vector is set to p1+scale*(p2-p1). If only p1 is provided, the axis vector is p1.
  • grid: The grid specifies the histogram grid along the direction specified by label. The allowed grid points fall in the range [-1,1] for label=x/y/z or [0,1] for r/phi/theta. A grid of 10 evenly spaced points between 0 and 1 can be requested equivalently by grid="0 (0.1) 1" or grid="0 (10) 1." Piecewise uniform grids covering portions of the range are supported, e.g., grid="-0.7 (10) 0.0 (20) 0.5."
  • Note that grid specifies the histogram grid along the (curved) coordinate given by label. The axis specified by p1/p2/scale does not correspond one-to-one with label unless label=x/y/z, but the full set of axes provided defines the (sheared) space on top of which the curved (e.g., spherical) coordinate system is built.

One body density matrix

The N-body density matrix in DMC is ρ̂N = |ΨT⟩⟨ΨFN| (for VMC, substitute ΨT for ΨFN). The one body reduced density matrix (1RDM) is obtained by tracing out all particle coordinates but one:


n̂1 = ∑nTrRn|ΨT⟩⟨ΨFN|

In this formula, the sum is over all electron indices and TrRn( * ) ≡ ∫dRn⟨Rn|*|Rn⟩ with Rn = [r1, ..., rn − 1, rn + 1, ..., rN]. When the sum is restricted over spin-up or spin-down electrons, one obtains a density matrix for each spin species. The 1RDM computed by is partitioned in this way.

In real space, the matrix elements of the 1RDM are

$$\begin{aligned} n_1(r,r') &= \langle{r}\left|{\hat{n}_1}\right|{r'}\rangle = \sum_n\int dR_n \Psi_T(r,R_n)\Psi_{FN}^*(r',R_n)\:. \end{aligned}$$

A more efficient and compact representation of the 1RDM is obtained by expanding in the SPOs obtained from a Hartree-Fock or DFT calculation, {ϕi}:

$$\begin{aligned} n_1(i,j) &= \langle{\phi_i}\left|{\hat{n}_1}\right|{\phi_j}\rangle \nonumber \\\ &= \int dR \Psi_{FN}^*(R)\Psi_{T}(R) \sum_n\int dr'_n \frac{\Psi_T(r_n',R_n)}{\Psi_T(r_n,R_n)}\phi_i(r_n')^* \phi_j(r_n)\:. \end{aligned}$$

The integration over r′ in eq42 is inefficient when one is also interested in obtaining matrices involving energetic quantities, such as the energy density matrix of Krogel2014 or the related (and more well known) generalized Fock matrix. For this reason, an approximation is introduced as follows:

$$\begin{aligned} n_1(i,j) \approx \int dR \Psi_{FN}(R)^*\Psi_T(R) \sum_n \int dr_n' \frac{\Psi_T(r_n',R_n)^*}{\Psi_T(r_n,R_n)^*}\phi_i(r_n)^* \phi_j(r_n')\:. \end{aligned}$$

For VMC, FN-DMC, FP-DMC, and RN-DMC this formula represents an exact sampling of the 1RDM corresponding to ρ̂N† (see appendix A of Krogel2014 for more detail).

estimtor type=dm1b element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
typer text dm1b Must be dm1b
namer text anything Unique name for estimator

parameters:

Name Datatype Values Default Description
basisr text array sposet.name(s) Orbital basis
integratoro text uniform_grid uniform density uniform_grid Integration method
evaluatoro text loop/matrix loop Evaluation method
scaleo real 0 < scale < 1 1.0 Scale integration cell
centero real array(3) any point Center of cell
pointso integer  > 0 10 Grid points in each dim
sampleso integer  > 0 10 MC samples
warmupo integer  > 0 30 MC warmup
timestepo real  > 0 0.5 MC time step
use_drifto boolean yes/no no Use drift in VMC
check_overlapo boolean yes/no no Print overlap matrix
check_derivativeso boolean yes/no no Check density derivatives
acceptance_ratioo boolean yes/no no Print accept ratio
rstatso boolean yes/no no Print spatial stats
normalizedo boolean yes/no yes basis comes norm'ed
volume_normed:math:^o boolean yes/no yes basis norm is volume
energy_matrixo boolean yes/no no Energy density matrix

Additional information:

  • name: Density matrix results appear in stat.h5 files labeled according to name.
  • basis: List sposet.name’s. The total set of orbitals contained in all sposet’s comprises the basis (subspace) onto which the one body density matrix is projected. This set of orbitals generally includes many virtual orbitals that are not occupied in a single reference Slater determinant.
  • integrator: Select the method used to perform the additional single particle integration. Options are uniform_grid (uniform grid of points over the cell), uniform (uniform random sampling over the cell), and density (Metropolis sampling of approximate density, ∑b ∈ basis|ϕb|2, is not well tested, please check results carefully!). Depending on the integrator selected, different subsets of the other input parameters are active.
  • evaluator: Select for-loop or matrix multiply implementations. Matrix is preferred for speed. Both implementations should give the same results, but please check as this has not been exhaustively tested.
  • scale: Resize the simulation cell by scale for use as an integration volume (active for integrator=uniform/uniform_grid).
  • center: Translate the integration volume to center at this point (active for integrator=uniform/ uniform_grid). If center is not provided, the scaled simulation cell is used as is.
  • points: Number of grid points in each dimension for integrator=uniform_grid. For example, points=10 results in a uniform 10 × 10 × 10 grid over the cell.
  • samples: Sets the number of MC samples collected for each step (active for integrator=uniform/ density).
  • warmup: Number of warmup Metropolis steps at the start of the run before data collection (active for integrator=density).
  • timestep: Drift-diffusion time step used in Metropolis sampling (active for integrator=density).
  • use_drift: Enable drift in Metropolis sampling (active for integrator=density).
  • check_overlap: Print the overlap matrix (computed via simple Riemann sums) to the log, then abort. Note that subsequent analysis based on the 1RDM is simplest if the input orbitals are orthogonal.
  • check_derivatives: Print analytic and numerical derivatives of the approximate (sampled) density for several sample points, then abort.
  • acceptance_ratio: Print the acceptance ratio of the density sampling to the log for each step.
  • rstats: Print statistical information about the spatial motion of the sampled points to the log for each step.
  • normalized: Declare whether the inputted orbitals are normalized or not. If normalized=no, direct Riemann integration over a 200 × 200 × 200 grid will be used to compute the normalizations before use.
  • volume_normed: Declare whether the inputted orbitals are normalized to the cell volume (default) or not (a norm of 1.0 is assumed in this case). Currently, B-spline orbitals coming from QE and HEG planewave orbitals native to QMCPACK are known to be volume normalized.
  • energy_matrix: Accumulate the one body reduced energy density matrix, and write it to stat.h5. This matrix is not covered in any detail here; the interested reader is referred to Krogel2014.
<estimator type="dm1b" name="DensityMatrices">
  <parameter name="basis"        >  spo_u spo_uv  </parameter>
  <parameter name="evaluator"    >  matrix        </parameter>
  <parameter name="integrator"   >  uniform_grid  </parameter>
  <parameter name="points"       >  4             </parameter>
  <parameter name="scale"        >  1.0           </parameter>
  <parameter name="center"       >  0 0 0         </parameter>
</estimator>
<estimator type="dm1b" name="DensityMatrices">
  <parameter name="basis"        >  spo_u spo_uv  </parameter>
  <parameter name="evaluator"    >  matrix        </parameter>
  <parameter name="integrator"   >  uniform       </parameter>
  <parameter name="samples"      >  64            </parameter>
  <parameter name="scale"        >  1.0           </parameter>
  <parameter name="center"       >  0 0 0         </parameter>
</estimator>
<estimator type="dm1b" name="DensityMatrices">
  <parameter name="basis"        >  spo_u spo_uv  </parameter>
  <parameter name="evaluator"    >  matrix        </parameter>
  <parameter name="integrator"   >  density       </parameter>
  <parameter name="samples"      >  64            </parameter>
  <parameter name="timestep"     >  0.5           </parameter>
  <parameter name="use_drift"    >  no            </parameter>
</estimator>
<sposet_builder type="bspline" href="../dft/pwscf_output/pwscf.pwscf.h5" tilematrix="1 0 0 0 1 0 0 0 1" twistnum="0" meshfactor="1.0" gpu="no" precision="single">
  <sposet type="bspline" name="spo_u"  group="0" size="4"/>
  <sposet type="bspline" name="spo_d"  group="0" size="2"/>
  <sposet type="bspline" name="spo_uv" group="0" index_min="4" index_max="10"/>
</sposet_builder>
<sposet_builder type="bspline" href="../dft/pwscf_output/pwscf.pwscf.h5" tilematrix="1 0 0 0 1 0 0 0 1" twistnum="0" meshfactor="1.0" gpu="no" precision="single">
  <sposet type="bspline" name="spo_u"  group="0" size="4"/>
  <sposet type="bspline" name="spo_d"  group="0" size="2"/>
  <sposet type="bspline" name="dm_basis" size="50" spindataset="0"/>
</sposet_builder>

Forward-Walking Estimators

Forward walking is a method for sampling the pure fixed-node distribution ⟨Φ0|Φ0⟩. Specifically, one multiplies each walker’s DMC mixed estimate for the observable 𝒪, $\frac{\mathcal{O}(\mathbf{R})\Psi_T(\mathbf{R})}{\Psi_T(\mathbf{R})}$, by the weighting factor $\frac{\Phi_0(\mathbf{R})}{\Psi_T(\mathbf{R})}$. As it turns out, this weighting factor for any walker R is proportional to the total number of descendants the walker will have after a sufficiently long projection time β.

To forward walk on an observable, declare a generic forward-walking estimator within a <hamiltonian> block, and then specify the observables to forward walk on and the forward-walking parameters. Here is a summary.

estimator type=ForwardWalking element:

parent elements: hamiltonian, qmc
child elements: Observable

attributes:

Name Datatype Values Default Description
typer text ForwardWalking Must be "ForwardWalking"
namer text anything any Unique name for estimator

Observable element:

parent elements: estimator, hamiltonian, qmc
child elements: None
Name Datatype Values Default Description
namer text anything any Registered name of existing estimator on which to forward walk
maxr integer  > 0 Maximum projection time in steps (max = β/τ)
frequencyr text  ≥ 1 Dump data only for every frequency-th to scalar.dat file

Additional information:

  • Cost: Because histories of observables up to max time steps have to be stored, the memory cost of storing the nonforward-walked observables variables should be multiplied by max. Although this is not an issue for items such as potential energy, it could be prohibitive for observables such as density, forces, etc.
  • Naming Convention: Forward-walked observables are automatically named FWE_name_i, where i is the forward-walked expectation value at time step i, and name is whatever name appears in the <Observable> block. This is also how it will appear in the scalar.dat file.

In the following example case, QMCPACK forward walks on the potential energy for 300 time steps and dumps the forward-walked value at every time step.

<estimator name="fw" type="ForwardWalking">
    <Observable name="LocalPotential" max="300" frequency="1"/>
     <!--- Additional Observable blocks go here -->
 </estimator>

"Force" estimators

QMCPACK supports force estimation by use of the Chiesa-Ceperly-Zhang (CCZ) estimator. Currently, open and periodic boundary conditions are supported but for all-electron calculations only.

Without loss of generality, the CCZ estimator for the z-component of the force on an ion centered at the origin is given by the following expression:

$$F_z = -Z \sum_{i=1}^{N_e}\frac{z_i}{r_i^3}[\theta(r_i-\mathcal{R}) + \theta(\mathcal{R}-r_i)\sum_{\ell=1}^{M}c_\ell r_i^\ell]\:.$$

Z is the ionic charge, M is the degree of the smoothing polynomial, ℛ is a real-space cutoff of the sphere within which the bare-force estimator is smoothed, and cℓ are predetermined coefficients. These coefficients are chosen to minimize the weighted mean square error between the bare force estimate and the s-wave filtered estimator. Specifically,


χ2 = ∫0ℛdr rm [fz(r) − f̃z(r)]2 .

Here, m is the weighting exponent, fz(r) is the unfiltered radial force density for the z force component, and f̃z(r) is the smoothed polynomial function for the same force density. The reader is invited to refer to the original paper for a more thorough explanation of the methodology, but with the notation in hand, QMCPACK takes the following parameters.

estimator type=Force element:

parent elements: hamiltonian, qmc
child elements: parameter

attributes:

Name Datatype Values Default Description
modeo text See above bare Select estimator type
lrmethodo text ewald or srcoul ewald Select long-range potential breakup method
typer text Force Must be "Force"
name:math:^o text Anything ForceBase Unique name for this estimator
pbco boolean yes/no yes Using periodic BCs or not
addioniono boolean yes/no no Add the ion-ion force contribution to output force estimate

parameters:

Name Datatype Values Default Description
rcuto real  > 0 1.0 Real-space cutoff ℛ in bohr
nbasiso integer  > 0 2 Degree of smoothing polynomial M
weightexpo integer  > 0 2 χ2 weighting exponent :math`m`

Additional information:

  • Naming Convention: The unique identifier name is appended with name_X_Y in the scalar.dat file, where X is the ion ID number and Y is the component ID (an integer with x=0, y=1, z=2). All force components for all ions are computed and dumped to the scalar.dat file.
  • Long-range breakup: With periodic boundary conditions, it is important to converge the lattice sum when calculating Coulomb contribution to the forces. As a quick test, increase the LR_dim_cutoff parameter until ion-ion forces are converged. The Ewald method converges more slowly than optimized method, but the optimized method can break down in edge cases, eg. too large LR_dim_cutoff.
  • Miscellaneous: Usually, the default choice of weightexp is sufficient. Different combinations of rcut and nbasis should be tested though to minimize variance and bias. There is, of course, a tradeoff, with larger nbasis and smaller rcut leading to smaller biases and larger variances.

The following is an example use case.

<simulationcell>
  ...
  <parameter name="LR_handler">  opt_breakup_original  </parameter>
  <parameter name="LR_dim_cutoff">  20  </parameter>
</simulationcell>
<hamiltonian>
  <estimator name="F" type="Force" mode="cep" addionion="yes">
    <parameter name="rcut">0.1</parameter>
    <parameter name="nbasis">4</parameter>
    <parameter name="weightexp">2</parameter>
  </estimator>
</hamiltonian>

Stress estimators

QMCPACK takes the following parameters.

parent elements: hamiltonian

attributes:

Name Datatype Values Default Description
moder text stress bare Must be "stress"
typer text Force Must be "Force"
sourcer text ion0 Name of ion particleset
nameo text Anything ForceBase Unique name for this estimator
addioniono boolean yes/no no Add the ion-ion stress contribution to output

Additional information:

  • Naming Convention: The unique identifier name is appended with name_X_Y in the scalar.dat file, where X and Y are the component IDs (an integer with x=0, y=1, z=2).
  • Long-range breakup: With periodic boundary conditions, it is important to converge the lattice sum when calculating Coulomb contribution to the forces. As a quick test, increase the LR_dim_cutoff parameter until ion-ion stresses are converged. Check using QE "Ewald contribution", for example. The stress estimator is implemented only with the Ewald method.

The following is an example use case.

<simulationcell>
  ...
  <parameter name="LR_handler">  ewald  </parameter>
  <parameter name="LR_dim_cutoff">  45  </parameter>
</simulationcell>
<hamiltonian>
  <estimator name="S" type="Force" mode="stress" source="ion0"/>
</hamiltonian>

/bibs/hamiltonianobservable.bib