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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 \ell/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 m_i, v^{qq}/v^{qc}/v^{cc} are pair potentials between quantum-quantum/quantum-classical/classical-classical particles, and v^{ext} 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/id^o text anything h0 Unique id for this Hamiltonian instance
type^o text   generic No current function
role^o text primary/extra extra Designate as Hamiltonian or not
source^o text particleset.name i Identify classical particleset
target^o text particleset.name e Identify quantum particleset
default^o 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
  skpot Unknown

shared attributes:

Name Datatype Values Default Description
type^r text See above 0 Select pairpot type
name^r text Anything any Unique name for this pairpot
source^r text particleset.name hamiltonian.target Identify interacting particles
target^r text particleset.name hamiltonian.target Identify interacting particles
units^o 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 (skpot) are not covered.
  • Available only if OHMMS_DIM==3: mpc, vhxc, pseudo.

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 :cite:`Natoli1995`) for details.

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

pairpot type=coulomb element:

parent elements: hamiltonian
child elements: None

attributes:

Name Datatype Values Default Description
type^r text coulomb   Must be coulomb
name/id^r text anything ElecElec Unique name for interaction
source^r text particleset.name hamiltonian.target Identify interacting particles
target^r text particleset.name hamiltonian.target Identify interacting particles
pbc^o boolean yes/no yes Use Ewald summation
physical^o boolean yes/no yes Hamiltonian(yes)/Observable(no)
gpu boolean yes/no depend Offload computation to GPU
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.
  • gpu: When not specified, use the gpu attribute of particleset.
<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 (\ell_{max}) 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 \Psi_T^{-1}V^{PP}\Psi_T requires additional angular integrals. These integrals are evaluated on a randomly shifted angular grid. The size of this grid is determined by \ell_{max}. See :cite:`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, \vec{s} 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
type^r text pseudo   Must be pseudo
name/id^r text anything PseudoPot No current function
source^r text particleset.name i Ion particleset name
target^r text particleset.name hamiltonian.target Electron particleset name
pbc^o boolean yes/no yes* Use Ewald summation
forces boolean yes/no no Deprecated
wavefunction^r text wavefunction.name invalid Identify wavefunction
format^r text xml/table table Select file format
algorithm^o text batched/non-batched batched Choose NLPP algorithm
DLA^o text yes/no no Use determinant localization approximation
physicalSO^o boolean yes/no yes Include the SO contribution in the local energy
spin_integrator^o text exact / simpson exact Choose which spin integration technique to use

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 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) :cite:`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.
  • spin_integrator Selects which spin integration technique to use. simpson uses a numerical integration scheme which can be inefficient but was previously the default. The exact method exploits the structure of the Slater-Jastrow wave function in order to analytically perform the spin integral.
  <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/symbol^r text groupe.name none Identify ionic species
href^r text filepath none Pseudopotential file path
format^r text xml/casino xml Specify file format
cutoff^o real     Nonlocal cutoff radius
lmax^o integer     Largest angular momentum
nrule^o integer     Integration grid order
l-local^o integer     Override local channel 
  <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
type^r text mpc   Must be MPC
name/id^r text anything MPC Unique name for interaction
source^r text particleset.name hamiltonian.target Identify interacting particles
target^r text particleset.name hamiltonian.target Identify interacting particles
physical^o 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\rightarrowDMC\rightarrowDMC). 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 :ref:`ccz-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
type^r text See above 0 Select estimator type
name^r 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 :cite:`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
type^r text chiesa   Must be chiesa
name^o text anything KEcorr Always reset to KEcorr
source^o text particleset.name e Identify quantum particles
psi^o 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

\hat{n}_r = \sum_i\delta(r-r_i)\:.

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 \Omega_c is then the average number of particles in that cell:

n_c = \int dR \left|{\Psi}\right|^2 \int_{\Omega_c}dr \sum_i\delta(r-r_i)\:.

estimator type=density element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
type^r text density   Must be density
name^r text anything any Unique name for estimator
delta^o real array(3) 0\le v_i \le 1 0.1 0.1 0.1 Grid cell spacing, unit coords
x_min^o real >0 0 Grid starting point in x (Bohr)
x_max^o real >0 | lattice[0] | Grid ending point in x (Bohr)
y_min^o real >0 0 Grid starting point in y (Bohr)
y_max^o real >0 | lattice[1] | Grid ending point in y (Bohr)
z_min^o real >0 0 Grid starting point in z (Bohr)
z_max^o real >0 | lattice[2] | Grid ending point in z (Bohr)
potential^o boolean yes/no no Accumulate local potential, deprecated
debug^o 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"\rightarrow 10\times 10\times 20 grid, delta="0.01 0.01 0.01"\rightarrow 100\times 100\times 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\times 10\times 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
type^r text spindensity   Must be spindensity
name^r text anything any Unique name for estimator
report^o boolean yes/no no Write setup details to stdout

parameters:

Name Datatype Values Default Description
grid^o integer array(3) v_i>   Grid cell count
dr^o real array(3) v_i>   Grid cell spacing (Bohr)
cell^o real array(3,3) anything   Volume grid exists in
corner^o real array(3) anything   Volume corner location
center^o real array (3) anything   Volume center/origin location
voronoi^o text particleset.name   Under development
test_moves^o 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 \times 40\times 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\times 20\times 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<u_d<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<u_d<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>

Magnetization density estimator

NOTE: This is only compatible with Spin-Orbit QMC with the batched QMC drivers. See "Spin-Orbit Calculations in QMC" for more information.

The magnetization density computes the vectorial spin per unit volume on a grid in real space. This is used with spinor-type wave functions where the spin expectation value is not exclusively aligned along the z-direction.

The formula that is implemented is the following:

\mathbf{m}_c = \int d\mathbf{X} \left|{\Psi(\mathbf{X})}\right|^2 \int_{\Omega_c}d\mathbf{r} \sum_i\delta(\mathbf{r}-\hat{\mathbf{r}}_i)\int_0^{2\pi} \frac{ds'_i}{2\pi} \frac{\Psi(\ldots \mathbf{r}_i s'_i \ldots )}{\Psi(\ldots \mathbf{r}_i s_i \ldots)}\langle s_i | \hat{\sigma} | s'_i \rangle\:.

Here, \hat{\sigma} is the vector of Pauli matrices.

estimator type=magnetizationdensity element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
type^r text magnetizationdensity   Must be magnetizationdensity
name^r text anything any Unique name for estimator
report^o boolean yes/no no Write setup details to stdout

parameters:

Name Datatype Values Default Description
grid^o integer array(3) v_i>   Grid cell count
dr^o real array(3) v_i>   Grid cell spacing (Bohr)
corner^o real array(3) anything   Volume corner location
center^o real array (3) anything   Volume center/origin location
integrator^o string simpsons/montecarlo simpsons Method to evaluate spin integral
samples^o integer anything 9 Number of points for spin integral

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="MagnetizationDensity."
  • grid: The grid sets the dimension of the histogram grid. Input like <parameter name="grid"> 40 40 40 </parameter> requests a 40 \times 40\times 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\times 20\times 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.
  • corner: The grid volume is defined as corner+\sum_{d=1}^3u_dcell_d with 0<u_d<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<u_d<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.
  • integrator: How the spin-integral is performed. By default, this is done determinstically with Simpson's rule. However, one can also Monte-Carlo sample this integral. Simpson's is preferred, but Monte-Carlo sampling might be more efficient for large systems.
  • samples: How many points are used to perform the spin integral. For Simpson's integration, this is just the number of quadrature points. For Monte-Carlo, this is literally the number of MC samples.
  • All information is dumped to hdf5. Each grid point has 3 real numbers associated with it, one for \langle \hat{\sigma_x} \rangle, \langle \hat{\sigma_y} \rangle, and \langle \hat{\sigma_z} \rangle respectively. Post-processing tools are provided in Nexus.
<estimator type="MagnetizationDensity" name="magdensity">
  <parameter name="integrator"   >  simpsons       </parameter>
  <parameter name="samples"      >  9             </parameter>
  <parameter name="center"       >  0.0 0.0 0.0    </parameter>
  <parameter name="grid"         >  10 10 10          </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 N_s is the number of particles of species s and V is the supercell volume. If s=s', then the sum is restricted so that i_s\ne j_s.

In QMCPACK, an estimate of g_{ss'}(r) is obtained as a radial histogram with a set of N_b uniform bins of width \delta r. 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, \delta r/2, 3 \delta r/2, 5 \delta r/2, \ldots.

estimator type=gofr element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
type^r text gofr   Must be gofr
name^o text anything any No current function
num_bin^r integer >1 20 # of histogram bins
rmax^o real >0 10 Histogram extent (Bohr)
dr^o real 0 0.5 No current function
debug^o boolean yes/no no No current function
target^o text particleset.name hamiltonian.target Quantum particles
source/sources^o 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 \delta r=\texttt{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 Bohr^3.
  • 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 \rho^e_{\mathbf{k}}=\sum_j e^{i \mathbf{k}\cdot\mathbf{r}_j^e} be the Fourier space electron density, with \mathbf{r}^e_j being the coordinate of the j-th electron. \mathbf{k} is a wavevector commensurate with the simulation cell. QMCPACK allows the user to accumulate the static electron structure factor S(\mathbf{k}) at all commensurate \mathbf{k} such that |\mathbf{k}| \leq (LR\_DIM\_CUTOFF) r_c. N^e is the number of electrons, LR_DIM_CUTOFF is the optimized breakup parameter, and r_c 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
type^r text sk   Must sk
name^r text anything any Unique name for estimator
hdf5^o 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 \mathbf{k} vectors commensurate with the unit cell, S(\mathbf{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 \rho(\mathbf{k}) is required. What sets SkAll apart from sk is that SkAll records \rho(\mathbf{k}) in addition to s(\mathbf{k}).

estimator type=SkAll element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
type^r text sk   Must be sk
name^r text anything any Unique name for estimator
source^r text Ion ParticleSet name None -
target^r text Electron ParticleSet name None -
hdf5^o boolean yes/no no Output to stat.h5 (yes) or scalar.dat (no)
writeionion^o boolean yes/no no Writes file rhok_IonIon.dat containing s(\mathbf{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 \mathbf{k} vectors commensurate with the unit cell, S(\mathbf{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
type^r text specieskinetic   Must be specieskinetic
name^r text anything any Unique name for estimator
hdf5^o 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
type^r text latticedeviation   Must be latticedeviation
name^r text anything any Unique name for estimator
hdf5^o boolean yes/no no Output to stat.h5 (yes)
per_xyz^o boolean yes/no no Directionally resolved (yes)
source^r text e/ion0/... no source particleset
sgroup^r text u/d/... no source particle group
target^r text e/ion0/... no target particleset
tgroup^r 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 :ref:`Listing 33 <Listing 33>` 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 :ref:`Listing 33 <Listing 33>`, 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 \hat{H} 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}
    \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}

Here, r_i and \tilde{r}_\ell represent electron and ion positions, respectively; \hat{p}_i is a single electron momentum operator; and \hat{v}^{ee}(r_i,r_j), \hat{v}^{eI}(r_i,\tilde{r}_\ell), and \hat{v}^{II}(\tilde{r}_\ell,\tilde{r}_m) 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 :cite:`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
type^r text EnergyDensity   Must be EnergyDensity
name^r text anything   Unique name for estimator
dynamic^r text particleset.name   Identify electrons
static^o text particleset.name   Identify ions
ion_points^o 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 :numref:`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
coord^r 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 :numref:`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 a_1 Cell axis 1
a2 a_2 Cell axis 2
a3 a_3 Cell axis 3
f1p a_1/2 Cell face 1+
f1m -a_1/2 Cell face 1-
f2p a_2/2 Cell face 2+
f2m -a_2/2 Cell face 2-
f3p a_3/2 Cell face 3+
f3m -a_3/2 Cell face 3-
cppp (a_1+a_2+a_3)/2 Cell corner +,+,+
cppm (a_1+a_2-a_3)/2 Cell corner +,+,-
cpmp (a_1-a_2+a_3)/2 Cell corner +,-,+
cmpp (-a_1+a_2+a_3)/2 Cell corner -,+,+
cpmm (a_1-a_2-a_3)/2 Cell corner +,-,-
cmpm (-a_1+a_2-a_3)/2 Cell corner -,+,-
cmmp (-a_1-a_2+a_3)/2 Cell corner -,-,+
cmmm (-a_1-a_2-a_3)/2 Cell corner -,-,- 
.. centered:: Table 8 Reference points available by default. Vectors :math:`a_1`, :math:`a_2`, and :math:`a_3` 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
coord^r 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
p1^r text reference_point.label   Select end point
p2^o text reference_point.label   Select end point
fraction^o 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
label^r text See below   Axis/dimension label
grid^r text   "0 1" Grid ranges/intervals
p1^r text reference_point.label   Select end point
p2^o text reference_point.label   Select end point
scale^o 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 \hat{\rho}_N=\left|{\Psi_{T}}\rangle{}\langle{\Psi_{FN}}\right| (for VMC, substitute \Psi_T for \Psi_{FN}). The one body reduced density matrix (1RDM) is obtained by tracing out all particle coordinates but one:

\hat{n}_1 = \sum_nTr_{R_n}\left|{\Psi_{T}}\rangle{}\langle{\Psi_{FN}}\right|

In this formula, the sum is over all electron indices and Tr_{R_n}(*)\equiv\int dR_n\langle{R_n}\left|{*}\right|{R_n}\rangle with R_n=[r_1,...,r_{n-1},r_{n+1},...,r_N]. 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, \{\phi_i\}:

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)\:.

The integration over r' in :eq:`eq43` is inefficient when one is also interested in obtaining matrices involving energetic quantities, such as the energy density matrix of :cite:`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 \hat{\rho}_N^\dagger (see appendix A of :cite:`Krogel2014` for more detail).

estimtor type=dm1b element:

parent elements: hamiltonian, qmc
child elements: None

attributes:

Name Datatype Values Default Description
type^r text dm1b   Must be dm1b
name^r text anything   Unique name for estimator

parameters:

Name Datatype Values Default Description
basis^r text array sposet.name(s)   Orbital basis
integrator^o text uniform_grid uniform density uniform_grid Integration method
evaluator^o text loop/matrix loop Evaluation method
scale^o real 0<scale<1 1.0 Scale integration cell
center^o real array(3) any point   Center of cell
points^o integer >0 10 Grid points in each dim
samples^o integer >0 10 MC samples
warmup^o integer >0 30 MC warmup
timestep^o real >0 0.5 MC time step
use_drift^o boolean yes/no no Use drift in VMC
check_overlap^o boolean yes/no no Print overlap matrix
check_derivatives^o boolean yes/no no Check density derivatives
acceptance_ratio^o boolean yes/no no Print accept ratio
rstats^o boolean yes/no no Print spatial stats
normalized^o boolean yes/no yes basis comes norm'ed
volume_normed^o boolean yes/no yes basis norm is volume
energy_matrix^o 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, \sum_{b\in \texttt{basis}}\left|{\phi_b}\right|^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 \times 10 \times 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 \times 200 \times 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 :cite:`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" 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" 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 \langle \Phi_0 | \Phi_0\rangle. Specifically, one multiplies each walker’s DMC mixed estimate for the observable \mathcal{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 \mathbf{R} is proportional to the total number of descendants the walker will have after a sufficiently long projection time \beta.

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
type^r text ForwardWalking   Must be "ForwardWalking"
name^r text anything any Unique name for estimator

Observable element:

parent elements: estimator, hamiltonian, qmc
child elements: None
Name Datatype Values Default Description
name^r text anything any Registered name of existing estimator on which to forward walk
max^r integer >0   Maximum projection time in steps (max=\beta/\tau)
frequency^r text \geq 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 \texttt{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>

Chiesa-Ceperley-Zhang Force Estimators

All force estimators implemented in QMCPACK are invoked with type="Force". The mode determines the specific estimator to be used. Currently, QMCPACK supports Chiesa-Ceperley-Zhang (CCZ) all-electron and Assaraf-Caffarel Zero-Variance Zero-Bias (AC) force estimators. We'll discuss the CCZ estimator in this section, and the AC estimator in the following section.

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, \mathcal{R} is a real-space cutoff of the sphere within which the bare-force estimator is smoothed, and c_\ell 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,

\chi^2 = \int_0^\mathcal{R}dr\,r^m\,[f_z(r) - \tilde{f}_z(r)]^2\:.

Here, m is the weighting exponent, f_z(r) is the unfiltered radial force density for the z force component, and \tilde{f}_z(r) is the smoothed polynomial function for the same force density.

Currently, open and periodic boundary conditions are supported but for all-electron calculations only.

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
mode^o text See above bare Select estimator type
lrmethod^o text ewald or srcoul ewald Select long-range potential breakup method
type^r text Force   Must be "Force"
name^o text Anything ForceBase Unique name for this estimator
pbc^o boolean yes/no yes Using periodic BCs or not
addionion^o boolean yes/no no Add the ion-ion force contribution to output force estimate

parameters:

Name Datatype Values Default Description
rcut^o real >0 1.0 Real-space cutoff \mathcal{R} in bohr
nbasis^o integer >0 2 Degree of smoothing polynomial M
weightexp^o integer >0 2 \chi^2 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>

Assaraf-Caffarel Force Estimators

*WARNING: The following estimator formally has infinite variance. You MUST do something to mitigate this in postprocessing or during the run before publishing.*

QMCPACK has an implementation of force estimation using the Assaraf-Caffarel Zero-Variance Zero-Bias method :cite:`Tiihonen2021`. This has the desirable property that as the trial wave function and trial wave function derivative become exact, the estimator itself becomes an exact estimate of the force and the variance of the estimator goes to ero--much like the local energy. In practice, the estimator is usually significantly more accurate and has much lower variance than the bare Hellman-Feynman estimator.

Currently, this is the only recommended way to estimate forces for systems with non-local pseudopotentials.

The zero-variance, zero-bias force estimator is given by the following expression:

\mathbf{F}^{ZVZB}_I = \mathbf{F}^{ZV}_I+\mathbf{F}^{ZB}_I = -\nabla_I E_L(\mathbf{R}) - 2 \left( E_L(\mathbf{R})-\langle E_L \rangle \right) \nabla_I \ln \Psi_T \:.

The first term is the zero-variance force estimator, given by the following.

\mathbf{F}^{ZV}_I = -\nabla_I E_L(\mathbf{R}) = \frac{-\left(\nabla_I \hat{H}\right) \Psi_T}{\Psi_T} - \frac{\left(\hat{H} - E_L\right)\nabla_I \Psi_T}{\Psi_T}\:.

The first term is the bare "Hellman-Feynman" term (denoted "hf" in output), and the second is a fluctuation cancelling zero-variance term (called "pulay" in output). This splitting allows the user to investigate the individual contributions to the force estimator, but we recommend always adding "hf" and "pulay" terms unless there is a compelling reason to do otherwise.

The second term is the "zero-bias" term:

\mathbf{F}^{ZB}_I = - 2 \left( E_L(\mathbf{R})-\langle E_L \rangle \right) \nabla_I \ln \Psi_T \:.

Because knowledge of \langle E_L \rangle is needed to compute the zero-bias term, QMCPACK returns E_L(\mathbf{R}) \ln \Psi_T (called "Ewfngrad" in output), and \ln \Psi_T (called "wfngrad" in output), which in conjunction with the local energy, allows one to construct the zero-bias term in post-processing.

There is an initial implementation of space-warp variance reduction that is accessible to the end-user, subject to the caveat that evaluation of these terms is currently slow (derivatives of local energy are computed with finite differences, rather than analytically). If the space-warp option is enabled, the following term is added to the zero-variance force estimator:

\mathbf{F}^{ZV-SW}_I = - \sum_{i=1}^{N_e} \omega_I(\mathbf{r}_i) \nabla_i E_L \:.

The variance reduction with this term is observed to be rather large. A faster, more efficient implementation of this term will be available in a future QMCPACK release.

The following term is added to the wave function gradient:

[\nabla_I \ln \Psi_T ]_{SW} = \sum_{i=1}^{N_e} \omega_I(\mathbf{r}_i) \nabla_i \ln \Psi_T + \frac{1}{2} \nabla_i\omega_I(\mathbf{r}_i) \:.

Currently, there is only one choice for damping function \omega_I(\mathbf{r}). This is given by:

\omega_I(\mathbf{r}) = \frac{F(|\mathbf{r}-\mathbf{R}_I|)}{\sum_I F(|\mathbf{r}-\mathbf{R}_I|)} \:.

We use F(r)=r^{-4} for the real space damping.

Finally, the estimator provides two methods to evaluate the necessary derivatives of the wave function and Hamiltonian. The first is a straightforward analytic differentiation of all required terms. While mathematically transparent, this algorithm has poor scaling with system size. The second utilizes the fast-derivative algorithm of Assaraf, Moroni, and Filippi :cite:`Filippi2016`, which has a smaller computational prefactor and at least an O(N) speed-up over the legacy implementation. Both of these methods are accessible with appropraite flags.

estimator type=Force element:

parent elements: hamiltonian, qmc
child elements: none

attributes:

Name Datatype Values Default Description
mode^o text acforce   Required to use ACForce estimator
type^r text Force   Must be "Force"
name^o text Anything ForceBase Unique name for this estimator
epsilon^o real >=0 0 Epsilon parameter for Pathak-Wagner regularizer.
spacewarp^o text yes/no no Add space-warp variance reduction terms
fast_derivatives^o text yes/no no Use Filippi fast derivative algorithm

Additional information:

  • Naming Convention: The unique identifier name is appended with a term label ( hf, pulay, Ewfngrad, or wfgrad) name_term_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.
  • Note: The fast force algorithm returns the total derivative of the local energy, and does not make the split between "Hellman-Feynman" and zero-variance terms like the legacy force implementation does. As such, expect name_pulay_X_Y to be zero if fast_derivatives="yes". However, this will be identical to the sum of Hellman-Feynman and zero-variance terms in the legacy implementation.

The following is an example use case.

<hamiltonian>
  <estimator name="F" type="Force" mode="acforce" fast_derivatives="yes" spacewarp="no"/>
</hamiltonian>

Stress estimators

QMCPACK takes the following parameters.

parent elements: hamiltonian

attributes:

Name Datatype Values Default Description
mode^r text stress bare Must be "stress"
type^r text Force   Must be "Force"
source^r text ion0   Name of ion particleset
name^o text Anything ForceBase Unique name for this estimator
addionion^o 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>

.. bibliography:: /bibs/hamiltonianobservable.bib