Jastrow Factors¶

Jastrow factors are correlation functions that multiply the Slater determinant wavefunction to explicitly account for electron-electron and electron-nucleus correlations. They dramatically improve the accuracy of quantum Monte Carlo calculations by capturing short-range correlation effects that are poorly described by single-particle wavefunctions.

Overview¶

The Jastrow factor \(J\) modifies the trial wavefunction:

\[\Psi_T = e^J \Psi_{SD}\]

where \(\Psi_{SD}\) is the Slater determinant wavefunction from quantum chemistry calculations.

Jastrow Components¶

The total Jastrow factor typically includes:

Electron-nucleus terms (1-body) - a parameters
- Correlation between electrons and nuclei
- Different parameters for each atom type
- Captures electron-nucleus cusp
Electron-electron terms (2-body) - b parameters
- Correlation between electron pairs (same and opposite spin)
- Single set of parameters for all electron pairs
Electron-electron-nucleus terms (3-body) - c parameters
- Three-body correlation effects
- Different parameters for each atom type
- Optional but improves accuracy

The Jastrow factor depends on the electronic (\(\mathbf{r}\)) and nuclear (\(\mathbf{R}\)) coordinates. Its defined as \(\exp(J(\mathbf{r},\mathbf{R}))\), where

\[ J = f_{en} + f_{ee} + f_{een} \]

Electron-nucleus and electron-electron: \(R={1-e^{-\kappa r} \over \kappa}\)

\[ f_{en} = \sum_{i=1}^{N_{\rm elec}} \sum_{\alpha=1}^{N_{\rm nuc}} \left( {a_1 R_{i\alpha} \over 1+a_2R_{i\alpha}} + \sum_{p=2}^{N^a_{\rm ord}} a_{p+1} R_{i\alpha}^p \right) \]

\[ f_{ee} = \sum_{i=2}^{N_{\rm elec}} \sum_{j=1}^{i-1} \left( {b_1 R_{ij} \over 1+b_2R_{ij}} + \sum_{p=2}^{N^b_{\rm ord}} b_{p+1} R_{ij}^p \right) \]

Electron-electron-nucleus: \(R=\exp\left(-\kappa r \right)\)

\[ f_{een} = \sum_{i=2}^{N_{\rm elec}} \sum_{j=1}^{i-1} \sum_{\alpha=1}^{N_{\rm nuc}} \sum_{p=2}^{N^c_{\rm ord}} \sum_{k=p-1}^0 \sum_{l=l_{\rm max}}^0 c_n R_{ij}^k (R_{i\alpha}^l+R_{j\alpha}^l) (R_{i\alpha}R_{j\alpha})^m \]

where \(m={p-k-l \over 2}\)

Typically \(N^a_{\rm ord}=N^b_{\rm ord}=5\). If \(f_{een}\) is included, \(N^c_{\rm ord}=5\).
Dependence among \(\{c_n\} \rightarrow f_{een}\) does not contribute to cusp-conditions
\(f_{en}\) and \(f_{een}\): different \(\{a_n\}\) and \(\{c_n\}\) for different atom types

File Format¶

Jastrow parameters are provided in a text file that CHAMP reads during initialization.

Basic Structure¶

jastrow_parameter   1
  5  5  0           norda,nordb,nordc
   0.60000000         scalek
   0.00000000   0.00000000  -0.41907755  -0.22916790  -0.04194614   0.08371252 (a(iparmj),iparmj=1,nparma)
   0.00000000   0.00000000  -0.09956809  -0.00598089   0.00503028   0.00600649 (a(iparmj),iparmj=1,nparma)
   0.50000000   0.36987319   0.06971895   0.00745636  -0.00306208  -0.00246314 (b(iparmj),iparmj=1,nparmb)
 (c(iparmj),iparmj=1,nparmc)
 (c(iparmj),iparmj=1,nparmc)
end

Format Specification¶

Line 1: Header

jastrow_parameter   1

The 1 indicates number of types of wavefunctions (typically always 1).

Line 2: Expansion orders

  5  5  0           norda,nordb,nordc

norda = 5: \(N^a_{\rm ord}\) Order of electron-nucleus expansion (1-body terms)

if we are using pseudopotentials (no e-n cusps), we always leave \(a_1=a_2=0\) and add \(a_3 (r_{i\alpha}^2), \ldots, a_6 (r_{i\alpha}^5)\) equal to zero, which we then optimize. We do so for each atom type.
nordb = 5: \(N^b_{\rm ord}\) Order of electron-electron expansion (2-body terms)

We set \(b_1=0.5\) (for up-down e-e cusp condition), and add \(b_3\) (\(r_{ij}^2\)), \(\ldots\), \(b_6\) (\(r_{ij}^5\)) equal to zero, which we then optimize. \(b_1\) is modified to 0.25 for up-up and down-down electrons.
nordc = 0: \(N^c_{\rm ord}\) Order of electron-electron-nucleus expansion (3-body terms, 0 = not present)

Line 3: Scaling parameter

   0.60000000         scalek

scalek scales the electron-electron distance in the Jastrow: commonly 0.5 or 0.6.

Lines 4+: Parameter sets

a-parameters (electron-nucleus): One line per unique atom type

   0.00000000   0.00000000  -0.41907755  -0.22916790  -0.04194614   0.08371252
   0.00000000   0.00000000  -0.09956809  -0.00598089   0.00503028   0.00600649

Number of a lines = number of unique atom types in geometry
Order must match atom type order in .xyz file
Number of values per line = nparmja = 2 + max(0, norda - 1)
For norda = 5: nparmja = 2 + 4 = 6 values
For norda = 3: nparmja = 2 + 2 = 4 values
For norda = 0: nparmja = 2 values
Comments in parentheses are optional documentation

b-parameters (electron-electron): Single line (if treating up and down spin electrons equally)

   0.50000000   0.36987319   0.06971895   0.00745636  -0.00306208  -0.00246314

Only one b line for entire system (if treating up and down spin electrons equally)
Number of values = nparmjb = 2 + max(0, nordb - 1)
For nordb = 5: nparmjb = 2 + 4 = 6 values
For nordb = 3: nparmjb = 2 + 2 = 4 values
For nordb = 0: nparmjb = 2 values
Applies to all electron pairs (parallel and antiparallel spins)

c-parameters (three-body): One line per unique atom type (if nordc > 0)

 (c(iparmj),iparmj=1,nparmc)
 (c(iparmj),iparmj=1,nparmc)

Number of c lines = number of unique atom types
Order must match atom type order in .xyz file
Number of values per line = nparmjc = nterms4(nordc)
If nordc = 0, c-parameter lines can be blank or omitted

Final line:

end

Examples¶

Example 1: Water Molecule (2 atom types: O, H)¶

Geometry (molecule.xyz):

3
Water molecule
  O   0.00000000   0.00000000   0.22143139
  H   0.00000000   1.43042809  -0.88572555
  H   0.00000000  -1.43042809  -0.88572555

Jastrow file (jastrow.jas):

jastrow_parameter   1
  5  5  0           norda,nordb,nordc
   0.60000000         scalek
   0.00000000   0.00000000  -0.39803661  -0.19727356  -0.04112912   0.08477303 (a for O)
   0.00000000   0.00000000  -0.25378471   0.03693657   0.02169054  -0.00707375 (a for H)
   0.50000000   0.48551602   0.09924779   0.00590014  -0.00626481  -0.00347973 (b)
end

Explanation:

Two a lines: first for O (atom type 1), second for H (atom type 2)
One b line for electron-electron correlation
nordc = 0, so no c parameters needed

Example 2: Water with 3-body Terms¶

Geometry (molecule.xyz):

3
Water molecule
  O   0.00000000   0.00000000   0.22143139
  H   0.00000000   1.43042809  -0.88572555
  H   0.00000000  -1.43042809  -0.88572555

Jastrow file (jastrow.jas):

jastrow_parameter   1
  5  5  5           norda,nordb,nordc
 0.40000  0.00000  scalek a21
 0.00000  0.00000 -0.04730 -0.39690 -0.15944  0.09709 (a(iparmj),iparmj=1,nparma)
 0.00000  0.00000 -0.01800  0.13474 -0.11232  0.00851 (a(iparmj),iparmj=1,nparma)
 0.50000  0.94655  0.14913 -0.13216  0.09053 -0.01728 (b(iparmj),iparmj=1,nparmb)
 1.69203  1.24656 -1.88214  3.91513 -0.74384 -8.64444 -3.45930  7.94537 -0.61759 -7.28154 -3.20830 10.10651  2.84683  3.11254 -1.82060 -4.12168 -2.73375  7.97867  2.63741  0.58993 -3.22734 -3.43777 -0.80525 (c(iparmj),iparmj=1,nparmc)
-0.18440 -2.24997 -1.73422  0.47251  1.54643  6.06904  3.29715  1.00667 -3.97339 -1.90822  2.48432 -1.92995 -5.13868 -1.55965 -3.12079  4.42562  3.87653 -4.16459  0.11133 -3.79462  3.68329  0.66761  1.96643 (c(iparmj),iparmj=1,nparmc)
end

Explanation:

Two atom types: O and H
norda = 5, nordb = 5, nordc = 5 (includes fully optimized 3-body terms)
Two a lines (O and H), each with 6 parameters
One b line with 6 parameters
Two c lines (O and H), each with 23 parameters (from nterms4(5))

Example 3: Initial Guess (Simple)¶

For starting VMC calculations, a minimal Jastrow:

jastrow_parameter   1
  3  3  0           norda,nordb,nordc
   0.60000000         scalek
   0.0   0.0  -0.25  -0.10 (a - adjust for each atom type)
   0.5   0.2   0.05   0.0  (b)
end

This provides a reasonable starting point for optimization.

Loading Jastrow in CHAMP Input¶

Specify the Jastrow file using the load jastrow command:

%module general
    title  'VMC with Jastrow'
    pool   './pool/'
%endmodule

load trexio          $pool/molecule.hdf5
load jastrow         jastrow.jas
load jastrow_der     jastrow.der  # Optional, for optimization

%module electrons
    nup    5
    nelec  10
%endmodule

Common Jastrow Forms¶

Standard Padé-Jastrow¶

Most common form in CHAMP:

\[J = \sum_I \sum_i \frac{a_{I}(r_{iI})}{1 + a_0 r_{iI}} + \sum_{i<j} \frac{b(r_{ij})}{1 + b_0 \cdot \text{scalek} \cdot r_{ij}}\]

where:

\(a_I(r)\) are polynomials in electron-nucleus distance \(r_{iI}\)
\(b(r)\) is a polynomial in electron-electron distance \(r_{ij}\)
The denominator ensures proper asymptotic behavior

Cusp Conditions¶

Jastrow factors must satisfy:

Electron-nucleus cusp: \(\frac{\partial J}{\partial r_{iI}}\bigg|_{r=0} = -Z_I\) for electron \(i\) at nucleus \(I\)
Electron-electron cusp: \(\frac{\partial J}{\partial r_{ij}}\bigg|_{r=0} = \pm\frac{1}{2}\) for parallel/antiparallel spins

Jastrow Derivatives - Required for wavefunction optimization
Wavefunction Optimization - Optimizing Jastrow parameters
VMC Calculations - Using optimized Jastrows
TREXIO Files - Jastrows must be supplied separately

Tips¶

Start with simple Jastrow forms before adding complexity
Check that atom type ordering matches geometry exactly
Verify parameter counts match declared orders (norda, nordb, nordc)
Use VMC to test Jastrow quality (variance should decrease with optimization)
Consult Optimization Guide for parameter tuning