# 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

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

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

• 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.

Listing 20 All electron Hamiltonian XML element.
<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>

Listing 21 Pseudopotential Hamiltonian XML element.
<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

• 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:

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

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

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

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 [NC95]) for details.

For information on how to set the boundary conditions, consult Specifying the system to be simulated.

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

forces

boolean

yes/no

no

Deprecated

• 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.

Listing 22 QMCPXML element for Coulomb interaction between electrons.
<pairpot name="ElecElec" type="coulomb" source="e" target="e"/>

Listing 23 QMCPXML element for Coulomb interaction between electrons and ions (all-electron only).
<pairpot name="ElecIon"  type="coulomb" source="i" target="e"/>

Listing 24 QMCPXML element for Coulomb interaction between ions.
<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:

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

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 [MSC91] 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

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

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

• 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) 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.

Listing 25 QMCPXML element for pseudopotential electron-ion interaction (psf files).
  <pairpot name="PseudoPot" type="pseudo"  source="i" wavefunction="psi0" format="psf"/>

Listing 26 QMCPXML element for pseudopotential electron-ion interaction (xml files). If SOC terms present in xml, they are added to local energy
  <pairpot name="PseudoPot" type="pseudo"  source="i" wavefunction="psi0" format="xml">
<pseudo elementType="Li" href="Li.xml"/>
<pseudo elementType="H" href="H.xml"/>
</pairpot>

Listing 27 QMCPXML element for pseudopotential to accumulate the spin-orbit energy, but do not include in local energy
  <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

lmax$$^o$$

integer

Largest angular momentum

nrule$$^o$$

integer

Integration grid order

l-local$$^o$$

integer

Override local channel

Listing 28 QMCPXML element for pseudopotential of single ionic species.
  <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.

Listing 29 MPC for finite-size postcorrection.
  <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$$\rightarrow$$DMC$$\rightarrow$$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 Chiesa-Ceperley-Zhang Force Estimators) 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 [CCMH06]. 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

Listing 30 “Chiesa” kinetic energy finite-size postcorrection.
   <estimator name="KEcorr" type="chiesa" source="e" psi="psi0"/>


### Density estimator

The particle number density operator is given by

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

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:

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

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

• 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 $$L$$).

Listing 31 QMCPXML,caption=Density estimator (uniform grid).
   <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

• 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.

Listing 32 Spin density estimator (uniform grid).
<estimator type="spindensity" name="SpinDensity" report="yes">
<parameter name="grid"> 40 40 40 </parameter>
</estimator>

Listing 33 Spin density estimator (uniform grid centered about origin).
<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:

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

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

• 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.

Listing 34 Magnetization density estimator (uniform grid).
<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

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

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

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

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

• 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.

Listing 35 Pair correlation function estimator element.
<estimator type="gofr" name="gofr" num_bin="200" rmax="3.0" />

Listing 36 Pair correlation function estimator element with additional electron-ion correlations.
<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:

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

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)

• 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.

Listing 37 Static structure factor estimator element.
  <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

• 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.

Listing 38 SkAll estimator element.
  <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)

Listing 39 Species kinetic energy estimator element.
  <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

• 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 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 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.

Listing 40 Lattice deviation estimator element.
<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

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

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

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

with each component given by

(39)\begin{split}\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{split}

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 [KYKC13].

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

• 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.

Listing 41 Energy density estimator accumulated on a $$20 \times 10 \times 10$$ grid over the simulation cell.
<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>

Listing 42 Energy density estimator accumulated within spheres of radius 6.9 Bohr centered on the first and second atoms in the ion0 particleset.
<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>

Listing 43 Energy density estimator accumulated within Voronoi polyhedra centered on the ions.
<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

coord$$^r$$

text

Cartesian/cell

Specify coordinate system

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

• 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

$$a_1$$

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 -,-,-

Table 8 Reference points available by default. Vectors $$a_1$$, $$a_2$$, and $$a_3$$ 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.

• 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

• 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:

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

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

(41)\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, $$\{\phi_i\}$$:

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

The integration over $$r'$$ in (42) is inefficient when one is also interested in obtaining matrices involving energetic quantities, such as the energy density matrix of [KKR14] or the related (and more well known) generalized Fock matrix. For this reason, an approximation is introduced as follows:

(43)\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 [KKR14] 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

• 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 [KKR14].

Listing 44 One body density matrix with uniform grid integration.
<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>

Listing 45 One body density matrix with uniform sampling.
<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>

Listing 46 One body density matrix with density sampling.
<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>

Listing 47 Example sposet initialization for density matrix use. Occupied and virtual orbital sets are created separately, then joined (basis="spo_u spo_uv").
<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>

Listing 48 Example sposet initialization for density matrix use. Density matrix orbital basis created separately (basis="dm_basis").
<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

• 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.

Listing 49 Forward-walking estimator element.
<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:

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

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,

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

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 :mathm

• 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>
<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 [TCK21]. 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:

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

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

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

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:

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

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:

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

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.

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

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

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

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 [FAM16], 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

fast_derivatives$$^o$$

text

yes/no

no

Use Filippi fast derivative algorithm

• 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

• 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>


CCMH06

Simone Chiesa, David M. Ceperley, Richard M. Martin, and Markus Holzmann. Finite-size error in many-body simulations with long-range interactions. Phys. Rev. Lett., 97:076404, August 2006. doi:10.1103/PhysRevLett.97.076404.

FAM16

Claudia Filippi, Roland Assaraf, and Saverio Moroni. Simple formalism for efficient derivatives and multi-determinant expansions in quantum monte carlo. The Journal of Chemical Physics, 144(19):194105, 2016. URL: https://doi.org/10.1063/1.4948778, arXiv:https://doi.org/10.1063/1.4948778, doi:10.1063/1.4948778.

KKR14(1,2,3)

Jaron T. Krogel, Jeongnim Kim, and Fernando A. Reboredo. Energy density matrix formalism for interacting quantum systems: quantum monte carlo study. Phys. Rev. B, 90:035125, July 2014. doi:10.1103/PhysRevB.90.035125.

KYKC13

Jaron T. Krogel, Min Yu, Jeongnim Kim, and David M. Ceperley. Quantum energy density: improved efficiency for quantum monte carlo calculations. Phys. Rev. B, 88:035137, July 2013. doi:10.1103/PhysRevB.88.035137.

MSC91

Lubos Mitas, Eric L. Shirley, and David M. Ceperley. Nonlocal pseudopotentials and diffusion monte carlo. The Journal of Chemical Physics, 95(5):3467–3475, 1991. doi:10.1063/1.460849.

NC95

Vincent Natoli and David M. Ceperley. An optimized method for treating long-range potentials. Journal of Computational Physics, 117(1):171–178, 1995. URL: http://www.sciencedirect.com/science/article/pii/S0021999185710546, doi:10.1006/jcph.1995.1054.

TCK21

Juha Tiihonen, Raymond C. Clay, and Jaron T. Krogel. Toward quantum monte carlo forces on heavier ions: scaling properties. The Journal of Chemical Physics, 154(20):204111, 2021. URL: https://doi.org/10.1063/5.0052266, arXiv:https://doi.org/10.1063/5.0052266, doi:10.1063/5.0052266.

ZBMAlfe19

Andrea Zen, Jan Gerit Brandenburg, Angelos Michaelides, and Dario Alfè. A new scheme for fixed node diffusion quantum monte carlo with pseudopotentials: improving reproducibility and reducing the trial-wave-function bias. The Journal of Chemical Physics, 151(13):134105, October 2019. doi:10.1063/1.5119729.