Trial wavefunction specification

Introduction

This section describes the input blocks associated with the specification of the trial wavefunction in a QMCPACK calculation. These sections are contained within the <wavefunction> \(...\) </wavefunction> xml blocks. Users are expected to rely on converters to generate the input blocks described in this section. The converters and the workflows are designed such that input blocks require minimum modifications from users. Unless the workflow requires modification of wavefunction blocks (e.g., setting the cutoff in a multideterminant calculation), only expert users should directly alter them.

The trial wavefunction in QMCPACK has a general product form:

(1)\[\Psi_T(\vec{r}) = \prod_k \Theta_k(\vec{r}) ,\]

where each \(\Theta_k(\vec{r})\) is a function of the electron coordinates (and possibly ionic coordinates and variational parameters). For problems involving electrons, the overall trial wavefunction must be antisymmetric with respect to electron exchange, so at least one of the functions in the product must be antisymmetric. Notice that, although QMCPACK allows for the construction of arbitrary trial wavefunctions based on the functions implemented in the code (e.g., slater determinants, jastrow functions), the user must make sure that a correct wavefunction is used for the problem at hand. From here on, we assume a standard trial wavefunction for an electronic structure problem

(2)\[Psi_T(\vec{r}) = \textit{A}(\vec{r}) \prod_k \textit{J}_k(\vec{r}),\]

where \(\textit{A}(\vec{r})\) is one of the antisymmetric functions: (1) slater determinant, (2) multislater determinant, or (3) pfaffian and \(\textit{J}_k\) is any of the Jastrow functions (described in Jastrow Factors). The antisymmetric functions are built from a set of single particle orbitals (sposet). QMCPACK implements four different types of sposet, described in the following section. Each sposet is designed for a different type of calculation, so their definition and generation varies accordingly.

Single determinant wavefunctons

Placing a single determinant for each spin is the most used ansatz for the antisymmetric part of a trial wavefunction. The input xml block for slaterdeterminant is given in Listing 1. A list of options is given in Table 2.

slaterdeterminant element:

Parent elements

determinantset

Child elements

determinant

Attribute:

Name

Datatype

Values

Default

Description

delay_rank

Integer

>=0

1

Number of delayed updates.

optimize

Text

yes/no

yes

Enable orbital optimization.

Table 2 Options for the slaterdeterminant xml-block.

Listing 1 Slaterdeterminant set XML element.
<slaterdeterminant delay_rank="32">
   <determinant id="updet" size="208">
     <occupation mode="ground" spindataset="0">
     </occupation>
   </determinant>
   <determinant id="downdet" size="208">
     <occupation mode="ground" spindataset="0">
     </occupation>
   </determinant>
 </slaterdeterminant>

Additional information:

  • delay_rank This option enables delayed updates of the Slater matrix inverse when particle-by-particle move is used. By default or if delay_rank=0 given in the input file, QMCPACK sets 1 for Slater matrices with a leading dimension \(<192\) and 32 otherwise. delay_rank=1 uses the Fahy’s variant [FWL90] of the Sherman-Morrison rank-1 update, which is mostly using memory bandwidth-bound BLAS-2 calls. With delay_rank>1, the delayed update algorithm [LK18][MDAzevedoL+17] turns most of the computation to compute bound BLAS-3 calls. Tuning this parameter is highly recommended to gain the best performance on medium-to-large problem sizes (\(>200\) electrons). We have seen up to an order of magnitude speedup on large problem sizes. When studying the performance of QMCPACK, a scan of this parameter is required and we recommend starting from 32. The best delay_rank giving the maximal speedup depends on the problem size. Usually the larger delay_rank corresponds to a larger problem size. On CPUs, delay_rank must be chosen as a multiple of SIMD vector length for good performance of BLAS libraries. The best delay_rank depends on the processor microarchitecture. GPU support is under development.

Single-particle orbitals

Spline basis sets

In this section we describe the use of spline basis sets to expand the sposet. Spline basis sets are designed to work seamlessly with plane wave DFT code (e.g.,Quantum ESPRESSO as a trial wavefunction generator).

In QMC algorithms, all the SPOs \(\{\phi(\vec{r})\}\) need to be updated every time a single electron moves. Evaluating SPOs takes a very large portion of computation time. In principle, PW basis set can be used to express SPOs directly in QMC, as in DFT. But it introduces an unfavorable scaling because the basis set size increases linearly as the system size. For this reason, it is efficient to use a localized basis with compact support and a good transferability from the plane wave basis.

In particular, 3D tricubic B-splines provide a basis in which only 64 elements are nonzero at any given point in [AlfeG04]. The 1D cubic B-spline is given by

(3)\[f(x) = \sum_{i'=i-1}^{i+2} b^{i'\!,3}(x)\,\, p_{i'},\]

where \(b^{i}(x)\) is the piecewise cubic polynomial basis functions and \(i = \text{floor}(\Delta^{-1} x)\) is the index of the first grid point \(\le x\). Constructing a tensor product in each Cartesian direction, we can represent a 3D orbital as

(4)\[\phi_n(x,y,z) = \!\!\!\!\sum_{i'=i-1}^{i+2} \!\! b_x^{i'\!,3}(x) \!\!\!\!\sum_{j'=j-1}^{j+2} \!\! b_y^{j'\!,3}(y) \!\!\!\!\sum_{k'=k-1}^{k+2} \!\! b_z^{k'\!,3}(z) \,\, p_{i', j', k',n}.\]

This allows the rapid evaluation of each orbital in constant time. Furthermore, this basis is systematically improvable with a single spacing parameter so that accuracy is not compromised compared with the plane wave basis.

The use of 3D tricubic B-splines greatly improves computational efficiency. The gain in computation time from a plane wave basis set to an equivalent B-spline basis set becomes increasingly large as the system size grows. On the downside, this computational efficiency comes at the expense of increased memory use, which is easily overcome, however, by the large aggregate memory available per node through OpenMP/MPI hybrid QMC.

The input xml block for the spline SPOs is given in Listing 2. A list of options is given in Table 3.

Listing 2 Determinant set XML element.
<determinantset type="bspline" source="i" href="pwscf.h5"
                tilematrix="1 1 3 1 2 -1 -2 1 0" twistnum="-1" gpu="yes" meshfactor="0.8"
                twist="0  0  0" precision="double">
  <slaterdeterminant>
    <determinant id="updet" size="208">
      <occupation mode="ground" spindataset="0">
      </occupation>
    </determinant>
    <determinant id="downdet" size="208">
      <occupation mode="ground" spindataset="0">
      </occupation>
    </determinant>
  </slaterdeterminant>
</determinantset>

determinantset element:

Parent elements

wavefunction

Child elements

slaterdeterminant

attribute:

Name

Datatype

Values

Default

Description

type

Text

Bspline

Type of sposet

href

Text

Path to hdf5 file from pw2qmcpack.x.

tilematrix

9 integers

Tiling matrix used to expand supercell.

twistnum

Integer

Index of the super twist.

twist

3 floats

Super twist.

meshfactor

Float

\(\le 1.0\)

Grid spacing ratio.

precision

Text

Single/double

Precision of spline coefficients

gpu

Text

Yes/no

GPU switch.

gpusharing

Text

Yes/no

No

Share B-spline table across GPUs.

Spline_Size_Limit_MB

Integer

Limit B-spline table size on GPU.

check_orb_norm

Text

Yes/no

Yes

Check norms of orbitals from h5 file.

save_coefs

Text

Yes/no

No

Save the spline coefficients to h5 file.

source

Text

Any

Ion0

Particle set with atomic positions.

Table 3 Options for the determinantset xml-block associated with B-spline single particle orbital sets.

Additional information:

  • precision. Only effective on CPU versions without mixed precision, “single” is always imposed with mixed precision. Using single precision not only saves memory use but also speeds up the B-spline evaluation. We recommend using single precision since we saw little chance of really compromising the accuracy of calculation.

  • meshfactor. The ratio of actual grid spacing of B-splines used in QMC calculation with respect to the original one calculated from h5. A smaller meshfactor saves memory use but reduces accuracy. The effects are similar to reducing plane wave cutoff in DFT calculations. Use with caution!

  • twistnum. If positive, it is the index. We recommend not taking this way since the indexing might show some uncertainty. If negative, the super twist is referred by twist.

  • save_coefs. If yes, dump the real-space B-spline coefficient table into an h5 file on the disk. When the orbital transformation from k space to B-spline requires more than the available amount of scratch memory on the compute nodes, users can perform this step on fat nodes and transfer back the h5 file for QMC calculations.

  • gpusharing. If enabled, spline data is shared across multiple GPUs on a given computational node. For example, on a two-GPU-per-node system, each GPU would have half of the orbitals. This enables larger overall spline tables than would normally fit in the memory of individual GPUs to be used, potentially up to the total GPU memory on a node. To obtain high performance, large electron counts or a high-performing CPU-GPU interconnect is required. To use this feature, the following needs to be done:

    • The CUDA Multi-Process Service (MPS) needs to be used (e.g., on Summit/SummitDev use “-alloc_flags gpumps” for bsub). If MPS is not detected, sharing will be disabled.

    • CUDA_VISIBLE_DEVICES needs to be properly set to control each rank’s visible CUDA devices (e.g., on OLCF Summit/SummitDev one needs to create a resource set containing all GPUs with the respective number of ranks with “jsrun –task-per-rs Ngpus -g Ngpus”).

  • Spline_Size_Limit_MB. Allows distribution of the B-spline coefficient table between the host and GPU memory. The compute kernels access host memory via zero-copy. Although the performance penalty introduced by it is significant, it allows large calculations to go through.

Gaussian basis tests

In this section we describe the use of localized basis sets to expand the sposet. The general form of a single particle orbital in this case is given by:

(5)\[\phi_i(\vec{r}) = \sum_k C_{i,k} \ \eta_k(\vec{r}),\]

where \(\{\eta_k(\vec{r})\}\) is a set of M atom-centered basis functions and \(C_{i,k}\) is a coefficient matrix. This should be used in calculations of finite systems employing an atom-centered basis set and is typically generated by the convert4qmc converter. Examples include calculations of molecules using Gaussian basis sets or Slater-type basis functions. Initial support for periodic systems is described in Periodic LCAO for Solids. Even though this section is called “Gaussian basis sets” (by far the most common atom-centered basis set), QMCPACK works with any atom-centered basis set based on either spherical harmonic angular functions or Cartesian angular expansions. The radial functions in the basis set can be expanded in either Gaussian functions, Slater-type functions, or numerical radial functions.

In this section we describe the input sections for the atom-centered basis set and the sposet for a single Slater determinant trial wavefunction. The input sections for multideterminant trial wavefunctions are described in Multideterminant wavefunctions. The basic structure for the input block of a single Slater determinant is given in Listing 3. A list of options for determinantset associated with this sposet is given in Table 4.

Listing 3 Basic input block for a single determinant trial wavefunction using a sposet expanded on an atom-centered basis set.
 <wavefunction id="psi0" target="e">
   <determinantset>
     <basisset>
       ...
     </basisset>
     <slaterdeterminant>
       ...
     </slaterdeterminant>
   </determinantset>
 </wavefunction>

The definition of the set of atom-centered basis functions is given by the basisset block, and the sposet is defined within slaterdeterminant. The basisset input block is composed from a collection of atomicBasisSet input blocks, one for each atomic species in the simulation where basis functions are centered. The general structure for basisset and atomicBasisSet are given in Listing 4, and the corresponding lists of options are given in Table 5 and Table 6.

determinantset element:

Parent elements

wavefunction

Child elements

basisset , slaterdeterminant , sposet , multideterminant

Attribute:

Name

Datatype

Values

Default

Description

name/id

Text

Any

‘’ ‘’

Name of determinant set

type

Text

See below

‘’ ‘’

Type of sposet

keyword

Text

NMO, GTO, STO

NMO

Type of orbital set generated

transform

Text

Yes/no

Yes

Transform to numerical radial functions?

source

Text

Any

Ion0

Particle set with the position of atom centers

cuspCorrection

Text

Yes/no

No

Apply cusp correction scheme to sposet?

Table 4 Options for the determinantset xml-block associated with atom-centered single particle orbital sets.

Listing 4 Basic input block for basisset.
<basisset name="LCAOBSet">
  <atomicBasisSet name="Gaussian-G2" angular="cartesian" elementType="C" normalized="no">
    <grid type="log" ri="1.e-6" rf="1.e2" npts="1001"/>
    <basisGroup rid="C00" n="0" l="0" type="Gaussian">
      <radfunc exponent="5.134400000000e-02" contraction="1.399098787100e-02"/>
      ...
    </basisGroup>
    ...
  </atomicBasisSet>
  <atomicBasisSet name="Gaussian-G2" angular="cartesian" type="Gaussian" elementType="C" normalized="no">
    ...
  </atomicBasisSet>
  ...
</basisset>

basisset element:

Parent elements

determinantset

Child elements

atomicBasisSet

Attribute:

Name

Datatype

Values

Default

Description

name / id

Text

Any

” “

Name of atom-centered basis set

Table 5 Options for the basisset xml-block associated with atom-centered single particle orbital sets.

AtomicBasisSet element:

Parent elements

basisset

Child elements

grid , basisGroup

Attribute:

Name

Datatype

Values

Default

Description

name / id

Text

Any

” “

Name of atomic basis set

angular

Text

See below

Default

Type of angular functions

expandYlm

Text

See below

Yes

Expand Ylm shells?

expM

Text

See below

Yes

Add sign for \((-1)^{m}\)?

elementType/species

Text

Any

e

Atomic species where functions are centered

normalized

Text

Yes/no

Yes

Are single particle functions normalized?

Table 6 Options for the atomicBasisSet xml-block.

basicGroup element:

Parent elements

AtomicBasisSet

Child elements

radfunc

Attribute:

Name

Datatype

Values

Default

Description

rid/id

Text

Any

‘’ ‘’

Name of the basisGroup

type

Text

Any

‘’ ‘’

Type of basisGroup

n/l/m/s

Integer

Any

0

Quantum numbers of basisGroup

Table 7 Options for the basisGroup xml-block.

Listing 5 Basic input block for slaterdeterminant with an atom-centered sposet.
  <slaterdeterminant>
  </slaterdeterminant>

element:

Parent elements:

Child elements:

Attribute:

Name

Datatype

Values

Default

Description

name/id

Text

Any

‘’ ‘’

Name of determinant set

Text

Any

‘’ ‘’

Detailed description of attributes:

In the following, we give a more detailed description of all the options presented in the various xml-blocks described in this section. Only nontrivial attributes are described. Those with simple yes/no options and whose previous description is enough to explain the intended behavior are not included.

determinantset attributes:

  • type

    Type of sposet. For atom-centered based sposets, use type=”MolecularOrbital” or type=“MO.” Other options described elsewhere in this manual are “spline,” “composite,” “pw,” “heg,” “linearopt,” etc.

  • keyword/key

    Type of basis set generated, which does not necessarily match the type of basis set on the input block. The three possible options are: NMO (numerical molecular orbitals), GTO (Gaussian-type orbitals), and STO (Slater-type orbitals). The default option is NMO. By default, QMCPACK will generate numerical orbitals from both GTO and STO types and use cubic or quintic spline interpolation to evaluate the radial functions. This is typically more efficient than evaluating the radial functions in the native basis (Gaussians or exponents) and allows for arbitrarily large contractions without any additional cost. To force use of the native expansion (not recommended), use GTO or STO for each type of input basis set.

  • transform

    Request (or avoid) a transformation of the radial functions to NMO type. The default and recommended behavior is to transform to numerical radial functions. If transform is set to yes, the option keyword is ignored.

  • cuspCorrection

    Enable (disable) use of the cusp correction algorithm (CASINO REFERENCE) for a basisset built with GTO functions. The algorithm is implemented as described in (CASINO REFERENCE) and works only with transform=”yes” and an input GTO basis set. No further input is needed.

atomicBasisSet attributes:

  • name/id

    Name of the basis set. Names should be unique.

  • angular

    Type of angular functions used in the expansion. In general, two angular basis functions are allowed: “spherical” (for spherical Ylm functions) and “Cartesian” (for functions of the type \(x^{n}y^{m}z^{l}\)).

  • expandYlm

    Determines whether each basis group is expanded across the corresponding shell of m values (for spherical type) or consistent powers (for Cartesian functions). Options:

    • “No”: Do not expand angular functions across corresponding angular shell.

    • “Gaussian”: Expand according to Gaussian03 format. This function is compatible only with angular=”spherical.” For a given input (l,m), the resulting order of the angular functions becomes (1,-1,0) for l=1 and (0,1,-1,2,-2,…,l,-l) for general l.

    • “Natural”: Expand angular functions according to (-l,-l+1,…,l-1,l).

    • “Gamess”: Expand according to Gamess’ format for Cartesian functions. Notice that this option is compatible only with angular=”Cartesian.” If angular=”Cartesian” is used, this option is not necessary.

  • expM

    Determines whether the sign of the spherical Ylm function associated with m (\(-1^{m}\)) is included in the coefficient matrix or not.

  • elementType/species

    Name of the species where basis functions are centered. Only one atomicBasisSet block is allowed per species. Additional blocks are ignored. The corresponding species must exist in the particleset given as the source option to determinantset. Basis functions for all the atoms of the corresponding species are included in the basis set, based on the order of atoms in the particleset.

basisGroup attributes:

  • type

    Type of input basis radial function. Note that this refers to the type of radial function in the input xml-block, which might not match the radial function generated internally and used in the calculation (if transform is set to “yes”). Also note that different basisGroup blocks within a given atomicBasisSet can have different types.

  • n/l/m/s

    Quantum numbers of the basis function. Note that if expandYlm is set to “yes” in atomicBasisSet, a full shell of basis functions with the appropriate values of “m” will be defined for the corresponding value of “l.” Otherwise a single basis function will be given for the specific combination of “(l,m).”

radfunc attributes for type = “Gaussian”:

  • TBDoc

slaterdeterminant attributes:

  • TBDoc

Hybrid orbital representation

The hybrid representation of the single particle orbitals combines a localized atomic basis set around atomic cores and B-splines in the interstitial regions to reduce memory use while retaining high evaluation speed and either retaining or increasing overall accuracy. Full details are provided in [LEKS18], and users of this feature are kindly requested to cite this paper. In practice, we have seen that using a meshfactor=0.5 is often possible and achieves huge memory savings. Fig. 3 illustrates how the regions are assigned.

_images/hybrid_new.jpg

Fig. 3 Regular and hybrid orbital representation. Regular B-spline representation (left panel) contains only one region and a sufficiently fine mesh to resolve orbitals near the nucleus. The hybrid orbital representation (right panel) contains near nucleus regions (A) where spherical harmonics and radial functions are used, buffers or interpolation regions (B), and an interstitial region (C) where a coarse B-spline mesh is used.

Orbitals within region A are computed as

\[\phi^A_n({\bf r})=R_{n,l,m}(r)Y_{l,m}(\hat{r})\]

Orbitals in region C are computed as the regular B-spline basis described in Spline basis sets above. The region B interpolates between A and C as

(6)\[\phi^B_n({\bf r}) = S(r) \phi^A_n({\bf r}) + (1-S(r))\phi^C_n({\bf r})\]
(7)\[(S(r) = \frac{1}{2}-\frac{1}{2} tanh \left[\alpha\left(\frac{r-r_{\rm A/B}}{r_{\rm B/C}-r_{\rm A/B}}-\frac{1}{2}\right)\right]\]

To enable hybrid orbital representation, the input XML needs to see the tag hybridrep="yes" shown in Listing 6.

Listing 6 Hybrid orbital representation input example.
<determinantset type="bspline" source="i" href="pwscf.h5"
              tilematrix="1 1 3 1 2 -1 -2 1 0" twistnum="-1" gpu="yes" meshfactor="0.8"
              twist="0  0  0" precision="single" hybridrep="yes">
  ...
</determinantset>

Second, the information describing the atomic regions is required in the particle set, shown in Listing 7.

Listing 7 particleset elements for ions with information needed by hybrid orbital representation.
<group name="Ni">
  <parameter name="charge">          18 </parameter>
  <parameter name="valence">         18 </parameter>
  <parameter name="atomicnumber" >   28 </parameter>
  <parameter name="cutoff_radius" > 1.6 </parameter>
  <parameter name="inner_cutoff" >  1.3 </parameter>
  <parameter name="lmax" >            5 </parameter>
  <parameter name="spline_radius" > 1.8 </parameter>
  <parameter name="spline_npoints">  91 </parameter>
</group>

The parameters specific to hybrid representation are listed as

attrib element

Attribute:

Name

Datatype

Values

Default

Description

cutoff_radius

Real

>=0.0

None

Cutoff radius for B/C boundary

lmax

Integer

>=0

None

Largest angular channel

inner_cutoff

Real

>=0.0

Dep.

Cutoff radius for A/B boundary

spline_radius

Real

>0.0

Dep.

Radial function radius used in spine

spline_npoints

Integer

>0

Dep.

Number of spline knots

  • cutoff_radius is required for every species. If a species is intended to not be covered by atomic regions, setting the value 0.0 will put default values for all the reset parameters. A good value is usually a bit larger than the core radius listed in the pseudopotential file. After a parametric scan, pick the one from the flat energy region with the smallest variance.

  • lmax is required if cutoff_radius \(>\) 0.0. This value usually needs to be at least the highest angular momentum plus 2.

  • inner_cutoff is optional and set as cutoff_radius \(-0.3\) by default, which is fine in most cases.

  • spline_radius and spline_npoints are optional. By default, they are calculated based on cutoff_radius and a grid displacement 0.02 bohr. If users prefer inputing them, it is required that cutoff_radius <= spline_radius \(-\) 2 \(\times\) spline_radius/(spline_npoints \(-\) 1).

In addition, the hybrid orbital representation allows extra optimization to speed up the nonlocal pseudopotential evaluation using the batched algorithm listed in Pseudopotentials.

Plane-wave basis sets

Homogeneous electron gas

The interacting Fermi liquid has its own special determinantset for filling up a Fermi surface. The shell number can be specified separately for both spin-up and spin-down. This determines how many electrons to include of each time; only closed shells are currently implemented. The shells are filled according to the rules of a square box; if other lattice vectors are used, the electrons might not fill up a complete shell.

This following example can also be used for Helium simulations by specifying the proper pair interaction in the Hamiltonian section.

Listing 8 2D Fermi liquid example: particle specification
<qmcsystem>
<simulationcell name="global">
<parameter name="rs" pol="0" condition="74">6.5</parameter>
<parameter name="bconds">p p p</parameter>
<parameter name="LR_dim_cutoff">15</parameter>
</simulationcell>
<particleset name="e" random="yes">
<group name="u" size="37">
<parameter name="charge">-1</parameter>
<parameter name="mass">1</parameter>
</group>
<group name="d" size="37">
<parameter name="charge">-1</parameter>
<parameter name="mass">1</parameter>
</group>
</particleset>
</qmcsystem>
Listing 9 2D Fermi liquid example (Slater Jastrow wavefunction)
<qmcsystem>
  <wavefunction name="psi0" target="e">
    <determinantset type="electron-gas" shell="7" shell2="7" randomize="true">
  </determinantset>
    <jastrow name="J2" type="Two-Body" function="Bspline" print="no">
      <correlation speciesA="u" speciesB="u" size="8" cusp="0">
        <coefficients id="uu" type="Array" optimize="yes">
      </correlation>
      <correlation speciesA="u" speciesB="d" size="8" cusp="0">
        <coefficients id="ud" type="Array" optimize="yes">
      </correlation>
    </jastrow>

Jastrow Factors

Jastrow factors are among the simplest and most effective ways of including dynamical correlation in the trial many body wavefunction. The resulting many body wavefunction is expressed as the product of an antisymmetric (in the case of Fermions) or symmetric (for Bosons) part and a correlating Jastrow factor like so:

(8)\[\Psi(\vec{R}) = \mathcal{A}(\vec{R}) \exp\left[J(\vec{R})\right]\]

In this section we will detail the types and forms of Jastrow factor used in QMCPACK. Note that each type of Jastrow factor needs to be specified using its own individual jastrow XML element. For this reason, we have repeated the specification of the jastrow tag in each section, with specialization for the options available for that given type of Jastrow.

One-body Jastrow functions

The one-body Jastrow factor is a form that allows for the direct inclusion of correlations between particles that are included in the wavefunction with particles that are not explicitly part of it. The most common example of this are correlations between electrons and ions.

The Jastrow function is specified within a wavefunction element and must contain one or more correlation elements specifying additional parameters as well as the actual coefficients. Example use cases gives examples of the typical nesting of jastrow, correlation, and coefficient elements.

Input Specification

Jastrow element:

name

datatype

values

defaults

description

name

text

(required)

Unique name for this Jastrow function

type

text

One-body

(required)

Define a one-body function

function

text

Bspline

(required)

BSpline Jastrow

text

pade2

Pade form

text

source

text

name

(required)

Name of attribute of classical particle set

print

text

yes / no

yes

Jastrow factor printed in external file?

elements

Correlation

Contents

(None)

To be more concrete, the one-body Jastrow factors used to describe correlations between electrons and ions take the form below:

(9)\[J1=\sum_I^{ion0}\sum_i^e u_{ab}(|r_i-R_I|)\]

where I runs over all of the ions in the calculation, i runs over the electrons and \(u_{ab}\) describes the functional form of the correlation between them. Many different forms of \(u_{ab}\) are implemented in QMCPACK. We will detail two of the most common ones below.

Spline form

The one-body spline Jastrow function is the most commonly used one-body Jastrow for solids. This form was first described and used in [EKCS12]. Here \(u_{ab}\) is an interpolating 1D B-spline (tricublc spline on a linear grid) between zero distance and \(r_{cut}\). In 3D periodic systems the default cutoff distance is the Wigner Seitz cell radius. For other periodicities, including isolated molecules, the \(r_{cut}\) must be specified. The cusp can be set. \(r_i\) and \(R_I\) are most commonly the electron and ion positions, but any particlesets that can provide the needed centers can be used.

Correlation element:

Name

Datatype

Values

Defaults

Description

ElementType

Text

Name

See below

Classical particle target

SpeciesA

Text

Name

See below

Classical particle target

SpeciesB

Text

Name

See below

Quantum species target

Size

Integer

\(> 0\)

(Required)

Number of coefficients

Rcut

Real

\(> 0\)

See below

Distance at which the correlation goes to 0

Cusp

Real

\(\ge 0\)

0

Value for use in Kato cusp condition

Spin

Text

Yes or no

No

Spin dependent Jastrow factor

Elements

Coefficients

Contents

(None)

Additional information:

  • elementType, speciesA, speciesB, spin

    For a spin-independent Jastrow factor (spin = “no”), elementType should be the name of the group of ions in the classical particleset to which the quantum particles should be correlated. For a spin-dependent Jastrow factor (spin = “yes”), set speciesA to the group name in the classical particleset and speciesB to the group name in the quantum particleset.

  • rcut

    The cutoff distance for the function in atomic units (bohr). For 3D fully periodic systems, this parameter is optional, and a default of the Wigner Seitz cell radius is used. Otherwise this parameter is required.

  • cusp

    The one-body Jastrow factor can be used to make the wavefunction satisfy the electron-ion cusp condition :cite:kato. In this case, the derivative of the Jastrow factor as the electron approaches the nucleus will be given by

(10)\[\left(\frac{\partial J}{\partial r_{iI}}\right)_{r_{iI} = 0} = -Z .\]

Note that if the antisymmetric part of the wavefunction satisfies the electron-ion cusp condition (for instance by using single-particle orbitals that respect the cusp condition) or if a nondivergent pseudopotential is used, the Jastrow should be cuspless at the nucleus and this value should be kept at its default of 0.

Coefficients element:

Name

Datatype

Values

Defaults

Description

Id

Text

(Required)

Unique identifier

Type

Text

Array

(Required)

Optimize

Text

Yes or no

Yes

if no, values are fixed in optimizations

Elements

(None)

Contents

(No name)

Real array

Zeros

Jastrow coefficients

Example use cases

Specify a spin-independent function with four parameters. Because rcut is not specified, the default cutoff of the Wigner Seitz cell radius is used; this Jastrow must be used with a 3D periodic system such as a bulk solid. The name of the particleset holding the ionic positions is “i.”

<jastrow name="J1" type="One-Body" function="Bspline" print="yes" source="i">
 <correlation elementType="C" cusp="0.0" size="4">
   <coefficients id="C" type="Array"> 0  0  0  0  </coefficients>
 </correlation>
</jastrow>

Specify a spin-dependent function with seven up-spin and seven down-spin parameters. The cutoff distance is set to 6 atomic units. Note here that the particleset holding the ions is labeled as ion0 rather than “i,” as in the other example. Also in this case, the ion is lithium with a coulomb potential, so the cusp condition is satisfied by setting cusp=”d.”

<jastrow name="J1" type="One-Body" function="Bspline" source="ion0" spin="yes">
  <correlation speciesA="Li" speciesB="u" size="7" rcut="6">
    <coefficients id="eLiu" cusp="3.0" type="Array">
    0.0 0.0 0.0 0.0 0.0 0.0 0.0
    </coefficients>
  </correlation>
  <correlation speciesA="C" speciesB="d" size="7" rcut="6">
    <coefficients id="eLid" cusp="3.0" type="Array">
    0.0 0.0 0.0 0.0 0.0 0.0 0.0
    </coefficients>
  </correlation>
</jastrow>
Pade form

Although the spline Jastrow factor is the most flexible and most commonly used form implemented in QMCPACK, there are times where its flexibility can make it difficult to optimize. As an example, a spline Jastrow with a very large cutoff can be difficult to optimize for isolated systems such as molecules because of the small number of samples present in the tail of the function. In such cases, a simpler functional form might be advantageous. The second-order Pade Jastrow factor, given in (11), is a good choice in such cases.

(11)\[u_{ab}(r) = \frac{a*r+c*r^2}{1+b*r}\]

Unlike the spline Jastrow factor, which includes a cutoff, this form has an infinite range and will be applied to every particle pair (subject to the minimum image convention). It also is a cuspless Jastrow factor, so it should be used either in combination with a single particle basis set that contains the proper cusp or with a smooth pseudopotential.

Correlation element:

Name

Datatype

Values

Defaults

Description

ElementType

Text

Name

See below

Classical particle target

Elements

Coefficients

Contents

(None)

Parameter element:

Name

Datatype

Values

Defaults

Description

Id

String

Name

(Required)

Name for variable

Name

String

A or B or C

(Required)

See (11)

Optimize

Text

Yes or no

Yes

If no, values are fixed in optimizations

Elements

(None)

Contents

(No name)

Real

Parameter value

(Required)

Jastrow coefficients

Example use case

Specify a spin-independent function with independent Jastrow factors for two different species (Li and H). The name of the particleset holding the ionic positions is “i.”

<jastrow name="J1" function="pade2" type="One-Body" print="yes" source="i">
  <correlation elementType="Li">
    <var id="LiA" name="A">  0.34 </var>
    <var id="LiB" name="B"> 12.78 </var>
    <var id="LiC" name="C">  1.62 </var>
  </correlation>
  <correlation elementType="H"">
    <var id="HA" name="A">  0.14 </var>
    <var id="HB" name="B"> 6.88 </var>
    <var id="HC" name="C"> 0.237 </var>
  </correlation>
</jastrow>
Short Range Cusp Form

The idea behind this functor is to encode nuclear cusps and other details at very short range around a nucleus in the region that the Gaussian orbitals of quantum chemistry are not capable of describing correctly. The functor is kept short ranged, because outside this small region, quantum chemistry orbital expansions are already capable of taking on the correct shapes. Unlike a pre-computed cusp correction, this optimizable functor can respond to changes in the wave function during VMC optimization. The functor’s form is

(12)\[u(r) = -\exp{\left(-r/R_0\right)} \left( A R_0 + \sum_{k=0}^{N-1} B_k \frac{ (r/R_0)^{k+2} }{ 1 + (r/R_0)^{k+2} } \right)\]

in which \(R_0\) acts as a soft cutoff radius (\(u(r)\) decays to zero quickly beyond roughly this distance) and \(A\) determines the cusp condition.

(13)\[\lim_{r \to 0} \frac{\partial u}{\partial r} = A\]

The simple exponential decay is modified by the \(N\) coefficients \(B_k\) that define an expansion in sigmoidal functions, thus adding detailed structure in a short-ranged region around a nucleus while maintaining the correct cusp condition at the nucleus. Note that sigmoidal functions are used instead of, say, a bare polynomial expansion, as they trend to unity past the soft cutoff radius and so interfere less with the exponential decay that keeps the functor short ranged. Although \(A\), \(R_0\), and the \(B_k\) coefficients can all be optimized as variational parameters, \(A\) will typically be fixed as the desired cusp condition is known.

To specify this one-body Jastrow factor, use an input section like the following.

<jastrow name="J1Cusps" type="One-Body" function="shortrangecusp" source="ion0" print="yes">
  <correlation rcut="6" cusp="3" elementType="Li">
    <var id="LiCuspR0" name="R0" optimize="yes"> 0.06 </var>
    <coefficients id="LiCuspB" type="Array" optimize="yes">
      0 0 0 0 0 0 0 0 0 0
    </coefficients>
  </correlation>
  <correlation rcut="6" cusp="1" elementType="H">
    <var id="HCuspR0" name="R0" optimize="yes"> 0.2 </var>
    <coefficients id="HCuspB" type="Array" optimize="yes">
      0 0 0 0 0 0 0 0 0 0
    </coefficients>
  </correlation>
</jastrow>

Here “rcut” is specified as the range beyond which the functor is assumed to be zero. The value of \(A\) can either be specified via the “cusp” option as shown above, in which case its optimization is disabled, or through its own “var” line as for \(R_0\), in which case it can be specified as either optimizable (“yes”) or not (“no”). The coefficients \(B_k\) are specified via the “coefficients” section, with the length \(N\) of the expansion determined automatically based on the length of the array.

Note that this one-body Jastrow form can (and probably should) be used in conjunction with a longer ranged one-body Jastrow, such as a spline form. Be sure to set the longer-ranged Jastrow to be cusp-free!

Two-body Jastrow functions

The two-body Jastrow factor is a form that allows for the explicit inclusion of dynamic correlation between two particles included in the wavefunction. It is almost always given in a spin dependent form so as to satisfy the Kato cusp condition between electrons of different spins [Kat51].

The two body Jastrow function is specified within a wavefunction element and must contain one or more correlation elements specifying additional parameters as well as the actual coefficients. Example use cases gives examples of the typical nesting of jastrow, correlation and coefficient elements.

Input Specification

Jastrow element:

name

datatype

values

defaults

description

name

text

(required)

Unique name for this Jastrow function

type

text

Two-body

(required)

Define a one-body function

function

text

Bspline

(required)

BSpline Jastrow

print

text

yes / no

yes

Jastrow factor printed in external file?

elements

Correlation

Contents

(None)

The two-body Jastrow factors used to describe correlations between electrons take the form

(14)\[J2=\sum_i^{e}\sum_{j>i}^{e} u_{ab}(|r_i-r_j|)\]

The most commonly used form of two body Jastrow factor supported by the code is a splined Jastrow factor, with many similarities to the one body spline Jastrow.

Spline form

The two-body spline Jastrow function is the most commonly used two-body Jastrow for solids. This form was first described and used in [EKCS12]. Here \(u_{ab}\) is an interpolating 1D B-spline (tricublc spline on a linear grid) between zero distance and \(r_{cut}\). In 3D periodic systems, the default cutoff distance is the Wigner Seitz cell radius. For other periodicities, including isolated molecules, the \(r_{cut}\) must be specified. \(r_i\) and \(r_j\) are typically electron positions. The cusp condition as \(r_i\) approaches \(r_j\) is set by the relative spin of the electrons.

Correlation element:

Name

Datatype

Values

Defaults

Description

SpeciesA

Text

U or d

(Required)

Quantum species target

SpeciesB

Text

U or d

(Required)

Quantum species target

Size

Integer

\(> 0\)

(Required)

Number of coefficients

Rcut

Real

\(> 0\)

See below

Distance at which the correlation goes to 0

Spin

Text

Yes or no

No

Spin-dependent Jastrow factor

Elements

Coefficients

Contents

(None)

Additional information:

  • speciesA, speciesB The scale function u(r) is defined for species pairs uu and ud. There is no need to define ud or dd since uu=dd and ud=du. The cusp condition is computed internally based on the charge of the quantum particles.

Coefficients element:

Name

Datatype

Values

Defaults

Description

Id

Text

(Required)

Unique identifier

Type

Text

Array

(Required)

Optimize

Text

Yes or no

Yes

If no, values are fixed in optimizations

Elements

(None)

Contents

(No name)

Real array

Zeros

Jastrow coefficients

Example use cases

Specify a spin-dependent function with four parameters for each channel. In this case, the cusp is set at a radius of 4.0 bohr (rather than to the default of the Wigner Seitz cell radius). Also, in this example, the coefficients are set to not be optimized during an optimization step.

<jastrow name="J2" type="Two-Body" function="Bspline" print="yes">
  <correlation speciesA="u" speciesB="u" size="8" rcut="4.0">
    <coefficients id="uu" type="Array" optimize="no"> 0.2309049836 0.1312646071 0.05464141356 0.01306231516</coefficients>
  </correlation>
  <correlation speciesA="u" speciesB="d" size="8" rcut="4.0">
    <coefficients id="ud" type="Array" optimize="no"> 0.4351561096 0.2377951747 0.1129144262 0.0356789236</coefficients>
  </correlation>
</jastrow>

User defined functional form

To aid in implementing different forms for \(u_{ab}(r)\), there is a script that uses a symbolic expression to generate the appropriate code (with spatial and parameter derivatives). The script is located in src/QMCWaveFunctions/Jastrow/codegen/user_jastrow.py. The script requires Sympy (www.sympy.org) for symbolic mathematics and code generation.

To use the script, modify it to specify the functional form and a list of variational parameters. Optionally, there may be fixed parameters - ones that are specified in the input file, but are not part of the variational optimization. Also one symbol may be specified that accepts a cusp value in order to satisfy the cusp condition. There are several example forms in the script. The default form is the simple Padé.

Once the functional form and parameters are specified in the script, run the script from the codegen directory and recompile QMCPACK. The main output of the script is the file src/QMCWaveFunctions/Jastrow/UserFunctor.h. The script also prints information to the screen, and one section is a sample XML input block containing all the parameters.

There is a unit test in src/QMCWaveFunctions/test/test_user_jastrow.cpp to perform some minimal testing of the Jastrow factor. The unit test will need updating to properly test new functional forms. Most of the changes relate to the number and name of variational parameters.

Jastrow element:

name

datatype

values

defaults

description

name

text

(required)

Unique name for this Jastrow function

type

text

One-body

(required)

Define a one-body function

Two-body

(required)

Define a two-body function

function

text

user

(required)

User-defined functor

See other parameters as appropriate for one or two-body functions

elements

Correlation

Contents

(None)

Long-ranged Jastrow factors

While short-ranged Jastrow factors capture the majority of the benefit for minimizing the total energy and the energy variance, long-ranged Jastrow factors are important to accurately reproduce the short-ranged (long wavelength) behavior of quantities such as the static structure factor, and are therefore essential for modern accurate finite size corrections in periodic systems.

Below two types of long-ranged Jastrow factors are described. The first (the k-space Jastrow) is simply an expansion of the one and/or two body correlation functions in plane waves, with the coefficients comprising the optimizable parameters. The second type have few variational parameters and use the optimized breakup method of Natoli and Ceperley [NC95] (the Yukawa and Gaskell RPA Jastrows).

Long-ranged Jastrow: k-space Jastrow

The k-space Jastrow introduces explicit long-ranged dependence commensurate with the periodic supercell. This Jastrow is to be used in periodic boundary conditions only.

The input for the k-space Jastrow fuses both one and two-body forms into a single element and so they are discussed together here. The one- and two-body terms in the k-Space Jastrow have the form:

(15)\[J_1 = \sum_{G\ne 0}b_G\rho_G^I\rho_{-G}\]
(16)\[J_2 = \sum_{G\ne 0}a_G\rho_G\rho_{-G}\]

Here \(\rho_G\) is the Fourier transform of the instantaneous electron density:

(17)\[\rho_G=\sum_{n\in electrons}e^{iG\cdot r_n}\]

and \(\rho_G^I\) has the same form, but for the fixed ions. In both cases the coefficients are restricted to be real, though in general the coefficients for the one-body term need not be. See Feature: Reciprocal-space Jastrow factors for more detail.

Input for the k-space Jastrow follows the familar nesting of jastrow-correlation-coefficients elements, with attributes unique to the k-space Jastrow at the correlation input level.

jastrow type=kSpace element:

parent elements:

wavefunction

child elements:

correlation

attributes:

Name

Datatype

Values

Default

Description

type\(^r\)

text

kSpace

must be kSpace

name\(^r\)

text

anything

0

Unique name for Jastrow

source\(^r\)

text

particleset.name

Ion particleset name

correlation element:

parent elements:

jastrow type=kSpace

child elements:

coefficients

attributes:

Name

Datatype

Values

Default

Description

type\(^r\)

text

One-body, Two-Body

Must be One-body/Two-body

kc\(^r\)

real

kc \(\ge\) 0

0.0

k-space cutoff in a.u.

symmetry\(^o\)

text

crystal,isotropic,none

crystal

symmetry of coefficients

spinDependent\(^o\)

boolean

yes,no

no

No current function

coefficients element:

parent elements:

correlation

child elements:

None

attributes:

Name

Datatype

Values

Default

Description

id\(^r\)

text

anything

cG1/cG2

Label for coeffs

type\(^r\)

text

Array

0

Must be Array

body text: The body text is a list of real values for the parameters.

Additional information:

  • It is normal to provide no coefficients as an initial guess. The number of coefficients will be automatically calculated according to the k-space cutoff + symmetry and set to zero.

  • Providing an incorrect number of parameters also results in all parameters being set to zero.

  • There is currently no way to turn optimization on/off for the k-space Jastrow. The coefficients are always optimized.

  • Spin dependence is currently not implemented for this Jastrow.

  • kc: Parameters with G vectors magnitudes less than kc are included in the Jastrow. If kc is zero, it is the same as excluding the k-space term.

  • symmetry=crystal: Impose crystal symmetry on coefficients according to the structure factor.

  • symmetry=isotropic: Impose spherical symmetry on coefficients according to G-vector magnitude.

  • symmetry=none: Impose no symmetry on the coefficients.

Listing 10 k-space Jastrow with one- and two-body terms.
<jastrow type="kSpace" name="Jk" source="ion0">
  <correlation kc="4.0" type="One-Body" symmetry="cystal">
    <coefficients id="cG1" type="Array">
    </coefficients>
  </correlation>
  <correlation kc="4.0" type="Two-Body" symmetry="crystal">
    <coefficients id="cG2" type="Array">
    </coefficients>
 </correlation>
</jastrow>
Listing 11 k-space Jastrow with one-body term only.
<jastrow type="kSpace" name="Jk" source="ion0">
   <correlation kc="4.0" type="One-Body" symmetry="crystal">
      <coefficients id="cG1" type="Array">
      </coefficients>
   </correlation>
</jastrow>
Listing 12 k-space Jastrow with two-body term only.
<jastrow type="kSpace" name="Jk" source="ion0">
   <correlation kc="4.0" type="Two-Body" symmetry="crystal">
      <coefficients id="cG2" type="Array">
      </coefficients>
   </correlation>
</jastrow>
Long-ranged Jastrows: Gaskell RPA and Yukawa forms

NOTE: The Yukawa and RPA Jastrows do not work at present and are currently being revived. Please contact the developers if you are interested in using them.

The exact Jastrow correlation functions contain terms which have a form similar to the Coulomb pair potential. In periodic systems the Coulomb potential is replaced by an Ewald summation of the bare potential over all periodic image cells. This sum is often handled by the optimized breakup method [NC95] and this same approach is applied to the long-ranged Jastrow factors in QMCPACK.

There are two main long-ranged Jastrow factors of this type implemented in QMCPACK: the Gaskell RPA [Gas61][Gas62] form and the [Cep78] form. Both of these forms were used by Ceperley in early studies of the electron gas [Cep78], but they are also appropriate starting points for general solids.

The Yukawa form is defined in real space. It’s long-range form is formally defined as

(18)\[u_Y^{PBC}(r) = \sum_{L\ne 0}\sum_{i<j}u_Y(\left|{r_i-r_j+L}\right|)\]

with \(u_Y(r)\) given by

(19)\[u_Y(r) = \frac{a}{r}\left(1-e^{-r/b}\right)\]

In QMCPACK a slightly more restricted form is used:

(20)\[u_Y(r) = \frac{r_s}{r}\left(1-e^{-r/\sqrt{r_s}}\right)\]

here “\(r_s\)” is understood to be a variational parameter.

The Gaskell RPA form—which contains correct short/long range limits and minimizes the total energy of the electron gas within the RPA—is defined directly in k-space:

(21)\[u_{RPA}(k) = -\frac{1}{2S_0(k)}+\frac{1}{2}\left(\frac{1}{S_0(k)^2}+\frac{4m_ev_k}{\hbar^2k^2}\right)^{1/2}\]

where $v_k$ is the Fourier transform of the Coulomb potential and \(S_0(k)\) is the static structure factor of the non-interacting electron gas:

\[\begin{split}S_0(k) = \left. \begin{cases} 1 & k>2k_F \\ \frac{3k}{4k_F}-\frac{1}{2}\left(\frac{k}{2k_F}\right)^3 & k<2k_F \end{cases} \right.\end{split}\]

When written in atomic units, RPA Jastrow implemented in QMCPACK has the form

(22)\[u_{RPA}(k) = \frac{1}{2N_e}\left(-\frac{1}{S_0(k)}+\left(\frac{1}{S_0(k)^2}+\frac{12}{r_s^3k^4}\right)^{1/2}\right)\]

Here “\(r_s\)” is again a variational parameter and \(k_F\equiv(\tfrac{9\pi}{4r_s^3})^{1/3}\).

For both the Yukawa and Gaskell RPA Jastrows, the default value for \(r_s\) is \(r_s=(\tfrac{3\Omega}{4\pi N_e})^{1/3}\).

jastrow type=Two-Body function=rpa/yukawa element:

parent elements:

wavefunction

child elements:

correlation

attributes:

Name

Datatype

Values

Default

Description

type\(^r\)

text

Two-body

Must be two-body

function\(^r\)

text

rpa/yukawa

Must be rpa or yukawa

name\(^r\)

text

anything

RPA_Jee

Unique name for Jastrow

longrange\(^o\)

boolean

yes/no

yes

Use long-range part

shortrange\(^o\)

boolean

yes/no

yes

Use short-range part

parameters:

Name

Datatype

Values

Default

Description

rs\(^o\)

rs

\(r_s>0\)

\(\tfrac{3\Omega}{4\pi N_e}\)

Avg. elec-elec distance

kc\(^o\)

kc

\(k_c>0\)

\(2\left(\tfrac{9\pi}{4}\right)^{1/3}\tfrac{4\pi N_e}{3\Omega}\)

k-space cutoff

Listing 13 Two body RPA Jastrow with long- and short-ranged parts.
<jastrow name=''Jee'' type=''Two-Body'' function=''rpa''>
</jastrow>

Three-body Jastrow functions

Explicit three-body correlations can be included in the wavefunction via the three-body Jastrow factor. The three-body electron-electron-ion correlation function (\(u_{\sigma\sigma'I}\)) currently used in is identical to the one proposed in [DTN04]:

(23)\[\begin{split} \begin{aligned} u_{\sigma\sigma'I}(r_{\sigma I},r_{\sigma'I},r_{\sigma\sigma'}) &= \sum_{\ell=0}^{M_{eI}}\sum_{m=0}^{M_{eI}}\sum_{n=0}^{M_{ee}}\gamma_{\ell mn} r_{\sigma I}^\ell r_{\sigma'I}^m r_{\sigma\sigma'}^n \\ &\times \left(r_{\sigma I}-\frac{r_c}{2}\right)^3 \Theta\left(r_{\sigma I}-\frac{r_c}{2}\right) \nonumber \\ &\times \left(r_{\sigma' I}-\frac{r_c}{2}\right)^3 \Theta\left(r_{\sigma' I}-\frac{r_c}{2}\right) \nonumber\end{aligned}\end{split}\]

Here \(M_{eI}\) and \(M_{ee}\) are the maximum polynomial orders of the electron-ion and electron-electron distances, respectively, \(\{\gamma_{\ell mn}\}\) are the optimizable parameters (modulo constraints), \(r_c\) is a cutoff radius, and \(r_{ab}\) are the distances between electrons or ions \(a\) and \(b\). i.e. The correlation function is only a function of the interparticle distances and not a more complex function of the particle positions, \(\mathbf{r}\). As indicated by the \(\Theta\) functions, correlations are set to zero beyond a distance of \(r_c/2\) in either of the electron-ion distances and the largest meaningful electron-electron distance is \(r_c\). This is the highest-order Jastrow correlation function currently implemented.

Today, solid state applications of QMCPACK usually utilize one and two-body B-spline Jastrow functions, with calculations on heavier elements often also using the three-body term described above.

Example use case

Here is an example of H2O molecule. After optimizing one and two body Jastrow factors, add the following block in the wavefunction. The coefficients will be filled zero automatically if not given.

<jastrow name="J3" type="eeI" function="polynomial" source="ion0" print="yes">
  <correlation ispecies="O" especies="u" isize="3" esize="3" rcut="10">
    <coefficients id="uuO" type="Array" optimize="yes"> </coefficients>
  </correlation>
  <correlation ispecies="O" especies1="u" especies2="d" isize="3" esize="3" rcut="10">
    <coefficients id="udO" type="Array" optimize="yes"> </coefficients>
  </correlation>
  <correlation ispecies="H" especies="u" isize="3" esize="3" rcut="10">
    <coefficients id="uuH" type="Array" optimize="yes"> </coefficients>
  </correlation>
  <correlation ispecies="H" especies1="u" especies2="d" isize="3" esize="3" rcut="10">
    <coefficients id="udH" type="Array" optimize="yes"> </coefficients>
  </correlation>
</jastrow>

Multideterminant wavefunctions

Multiple schemes to generate a multideterminant wavefunction are possible, from CASSF to full CI or selected CI. The QMCPACK converter can convert MCSCF multideterminant wavefunctions from GAMESS [SBB+93] and CIPSI [EG13] wavefunctions from Quantum Package [Sce17] (QP). Full details of how to run a CIPSI calculation and convert the wavefunction for QMCPACK are given in CIPSI wavefunction interface.

The script utils/determinants_tools.py can be used to generate useful information about the multideterminant wavefunction. This script takes, as a required argument, the path of an h5 file corresponding to the wavefunction. Used without optional arguments, it prints the number of determinants, the number of CSFs, and a histogram of the excitation degree.

> determinants_tools.py ./tests/molecules/C2_pp/C2.h5
Summary:
excitation degree 0 count: 1
excitation degree 1 count: 6
excitation degree 2 count: 148
excitation degree 3 count: 27
excitation degree 4 count: 20

n_det 202
n_csf 104

If the --verbose argument is used, the script will print each determinant, the associated CSF, and the excitation degree relative to the first determinant.

> determinants_tools.py -v ./tests/molecules/C2_pp/C2.h5 | head
1
alpha  1111000000000000000000000000000000000000000000000000000000
beta   1111000000000000000000000000000000000000000000000000000000
scf    2222000000000000000000000000000000000000000000000000000000
excitation degree  0

2
alpha  1011100000000000000000000000000000000000000000000000000000
beta   1011100000000000000000000000000000000000000000000000000000
scf    2022200000000000000000000000000000000000000000000000000000
excitation degree  2

Backflow Wavefunctions

One can perturb the nodal surface of a single-Slater/multi-Slater wavefunction through use of a backflow transformation. Specifically, if we have an antisymmetric function \(D(\mathbf{x}_{0\uparrow},\cdots,\mathbf{x}_{N\uparrow}, \mathbf{x}_{0\downarrow},\cdots,\mathbf{x}_{N\downarrow})\), and if \(i_\alpha\) is the \(i\)-th particle of species type \(\alpha\), then the backflow transformation works by making the coordinate transformation \(\mathbf{x}_{i_\alpha} \to \mathbf{x}'_{i_\alpha}\) and evaluating \(D\) at these new “quasiparticle” coordinates. QMCPACK currently supports quasiparticle transformations given by

(24)\[\mathbf{x}'_{i_\alpha}=\mathbf{x}_{i_\alpha}+\sum_{\alpha \leq \beta} \sum_{i_\alpha \neq j_\beta} \eta^{\alpha\beta}(|\mathbf{x}_{i_\alpha}-\mathbf{x}_{j_\beta}|)(\mathbf{x}_{i_\alpha}-\mathbf{x}_{j_\beta})\:.\]

Here, \(\eta^{\alpha\beta}(|\mathbf{x}_{i_\alpha}-\mathbf{x}_{j_\beta}|)\) is a radially symmetric backflow transformation between species \(\alpha\) and \(\beta\). In QMCPACK, particle \(i_\alpha\) is known as the “target” particle and \(j_\beta\) is known as the “source.” The main types of transformations are so-called one-body terms, which are between an electron and an ion \(\eta^{eI}(|\mathbf{x}_{i_e}-\mathbf{x}_{j_I}|)\) and two-body terms. Two-body terms are distinguished as those between like and opposite spin electrons: \(\eta^{e(\uparrow)e(\uparrow)}(|\mathbf{x}_{i_e(\uparrow)}-\mathbf{x}_{j_e(\uparrow)}|)\) and \(\eta^{e(\uparrow)e(\downarrow)}(|\mathbf{x}_{i_e(\uparrow)}-\mathbf{x}_{j_e(\downarrow)}|)\). Henceforth, we will assume that \(\eta^{e(\uparrow)e(\uparrow)}=\eta^{e(\downarrow)e(\downarrow)}\).

In the following, we explain how to describe general terms such as (24) in a QMCPACK XML file. For specificity, we will consider a particle set consisting of H and He (in that order). This ordering will be important when we build the XML file, so you can find this out either through your specific declaration of <particleset>, by looking at the hdf5 file in the case of plane waves, or by looking at the QMCPACK output file in the section labeled “Summary of QMC systems.”

Input specifications

All backflow declarations occur within a single <backflow> ... </backflow> block. Backflow transformations occur in <transformation> blocks and have the following input parameters:

Transformation element:

Name

Datatype

Values

Default

Description

name

Text

(Required)

Unique name for this Jastrow function.

type

Text

“e-I”

(Required)

Define a one-body backflow transformation.

Text

“e-e”

Define a two-body backflow transformation.

function

Text

B-spline

(Required)

B-spline type transformation (no other types supported).

source

Text

“e” if two body, ion particle set if one body.

Just like one- and two-body jastrows, parameterization of the backflow transformations are specified within the <transformation> blocks by <correlation> blocks. Please refer to Spline form for more information.

Example Use Case

Having specified the general form, we present a general example of one-body and two-body backflow transformations in a hydrogen-helium mixture. The hydrogen and helium ions have independent backflow transformations, as do the like and unlike-spin two-body terms. One caveat is in order: ionic backflow transformations must be listed in the order they appear in the particle set. If in our example, helium is listed first and hydrogen is listed second, the following example would be correct. However, switching backflow declaration to hydrogen first then helium, will result in an error. Outside of this, declaration of one-body blocks and two-body blocks are not sensitive to ordering.

<backflow>
<!--The One-Body term with independent e-He and e-H terms. IN THAT ORDER -->
<transformation name="eIonB" type="e-I" function="Bspline" source="ion0">
    <correlation cusp="0.0" size="8" type="shortrange" init="no" elementType="He" rcut="3.0">
        <coefficients id="eHeC" type="Array" optimize="yes">
            0 0 0 0 0 0 0 0
        </coefficients>
    </correlation>
    <correlation cusp="0.0" size="8" type="shortrange" init="no" elementType="H" rcut="3.0">
        <coefficients id="eHC" type="Array" optimize="yes">
            0 0 0 0 0 0 0 0
        </coefficients>
    </correlation>
</transformation>

<!--The Two-Body Term with Like and Unlike Spins -->
<transformation name="eeB" type="e-e" function="Bspline" >
    <correlation cusp="0.0" size="7" type="shortrange" init="no" speciesA="u" speciesB="u" rcut="1.2">
        <coefficients id="uuB1" type="Array" optimize="yes">
            0 0 0 0 0 0 0
        </coefficients>
    </correlation>
    <correlation cusp="0.0" size="7" type="shortrange" init="no" speciesA="d" speciesB="u" rcut="1.2">
        <coefficients id="udB1" type="Array" optimize="yes">
            0 0 0 0 0 0 0
        </coefficients>
    </correlation>
</transformation>
</backflow>

Currently, backflow works only with single-Slater determinant wavefunctions. When a backflow transformation has been declared, it should be placed within the <determinantset> block, but outside of the <slaterdeterminant> blocks, like so:

<determinantset ... >
    <!--basis set declarations go here, if there are any -->

    <backflow>
        <transformation ...>
          <!--Here is where one and two-body terms are defined -->
         </transformation>
     </backflow>

     <slaterdeterminant>
         <!--Usual determinant definitions -->
     </slaterdeterminant>
 </determinantset>

Optimization Tips

Backflow is notoriously difficult to optimize—it is extremely nonlinear in the variational parameters and moves the nodal surface around. As such, it is likely that a full Jastrow+Backflow optimization with all parameters initialized to zero might not converge in a reasonable time. If you are experiencing this problem, the following pointers are suggested (in no particular order).

Get a good starting guess for \(\Psi_T\):

  1. Try optimizing the Jastrow first without backflow.

  2. Freeze the Jastrow parameters, introduce only the e-e terms in the backflow transformation, and optimize these parameters.

  3. Freeze the e-e backflow parameters, and then optimize the e-I terms.

    • If difficulty is encountered here, try optimizing each species independently.

  4. Unfreeze all Jastrow, e-e backflow, and e-I backflow parameters, and reoptimize.

Optimizing Backflow Terms

It is possible that the previous prescription might grind to a halt in steps 2 or 3 with the inability to optimize the e-e or e-I backflow transformation independently, especially if it is initialized to zero. One way to get around this is to build a good starting guess for the e-e or e-I backflow terms iteratively as follows:

  1. Start off with a small number of knots initialized to zero. Set \(r_{cut}\) to be small (much smaller than an interatomic distance).

  2. Optimize the backflow function.

  3. If this works, slowly increase \(r_{cut}\) and/or the number of knots.

  4. Repeat steps 2 and 3 until there is no noticeable change in energy or variance of \(\Psi_T\).

Tweaking the Optimization Run

The following modifications are worth a try in the optimization block:

  • Try setting “useDrift” to “no.” This eliminates the use of wavefunction gradients and force biasing in the VMC algorithm. This could be an issue for poorly optimized wavefunctions with pathological gradients.

  • Try increasing “exp0” in the optimization block. Larger values of exp0 cause the search directions to more closely follow those predicted by steepest-descent than those by the linear method.

Note that the new adaptive shift optimizer has not yet been tried with backflow wavefunctions. It should perform better than the older optimizers, but a considered optimization process is still recommended.

Gaussian Product Wavefunction

The Gaussian Product wavefunction implements (25)

(25)\[\Psi(\vec{R}) = \prod_{i=1}^N \exp\left[ -\frac{(\vec{R}_i-\vec{R}_i^o)^2}{2\sigma_i^2} \right]\]

where \(\vec{R}_i\) is the position of the \(i^{\text{th}}\) quantum particle and \(\vec{R}_i^o\) is its center. \(\sigma_i\) is the width of the Gaussian orbital around center \(i\).

This variational wavefunction enhances single-particle density at chosen spatial locations with adjustable strengths. It is useful whenever such localization is physically relevant yet not captured by other parts of the trial wavefunction. For example, in an electron-ion simulation of a solid, the ions are localized around their crystal lattice sites. This single-particle localization is not captured by the ion-ion Jastrow. Therefore, the addition of this localization term will improve the wavefunction. The simplest use case of this wavefunction is perhaps the quantum harmonic oscillator (please see the “tests/models/sho” folder for examples).

Input specification

Gaussian Product Wavefunction (ionwf):

Name

Datatype

Values

Default

Description

Name

Text

ionwf

(Required)

Unique name for this wavefunction

Width

Floats

1.0 -1

(Required)

Widths of Gaussian orbitals

Source

Text

ion0

(Required)

Name of classical particle set

Additional information:

  • width There must be one width provided for each quantum particle. If a negative width is given, then its corresponding Gaussian orbital is removed. Negative width is useful if one wants to use Gaussian wavefunction for a subset of the quantum particles.

  • source The Gaussian centers must be specified in the form of a classical particle set. This classical particle set is likely the ion positions “ion0,” hence the name “ionwf.” However, arbitrary centers can be defined using a different particle set. Please refer to the examples in “tests/models/sho.”

Example Use Case

<qmcsystem>
  <simulationcell>
    <parameter name="bconds">
          n n n
    </parameter>
  </simulationcell>
  <particleset name="e">
    <group name="u" size="1">
      <parameter name="mass">5.0</parameter>
      <attrib name="position" datatype="posArray" condition="0">
        0.0001 -0.0001 0.0002
      </attrib>
    </group>
  </particleset>
  <particleset name="ion0" size="1">
    <group name="H">
      <attrib name="position" datatype="posArray" condition="0">
        0 0 0
      </attrib>
    </group>
  </particleset>
  <wavefunction target="e" id="psi0">
    <ionwf name="iwf" source="ion0" width="0.8165"/>
  </wavefunction>
  <hamiltonian name="h0" type="generic" target="e">
    <extpot type="HarmonicExt" mass="5.0" energy="0.3"/>
    <estimator type="latticedeviation" name="latdev"
      target="e"    tgroup="u"
      source="ion0" sgroup="H"/>
  </hamiltonian>
</qmcsystem>

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