Introduce and optimize Jastrow
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
Add a simple e-e and e-n Jastrow factor
- \[N^a_{\rm ord}=5\]
Since 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.
- \[N^b_{\rm ord}=5\]
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.
The following file is your starting Jastrow factor
jastrow.start:jastrow_parameter 1 5 5 0 norda,nordb,nordc 0.60000000 scalek 0.00000000 0.00000000 0. 0. 0. 0. (a(iparmj),iparmj=1,nparma) ! e-n O 0.00000000 0.00000000 0. 0. 0. 0. (a(iparmj),iparmj=1,nparma) ! e-n H 0.50000000 1. 0. 0. 0. 0. (b(iparmj),iparmj=1,nparmb) ! e-e (c(iparmj),iparmj=1,nparmc) ! e-e-n O (c(iparmj),iparmj=1,nparmc) ! e-e-n H end
Optimize the Jastrow factor
Create the file jastrow.der:
jasderiv
4 4 5 0 0 0 0 nparma,nparmb,nparmc,nparmf
3 4 5 6 (iwjasa(iparm),iparm=1,nparma) ! e-n O
3 4 5 6 (iwjasa(iparm),iparm=1,nparma) ! e-n H
2 3 4 5 6 (iwjasb(iparm),iparm=1,nparmb) ! e-e
3 5 7 8 9 11 13 14 15 16 17 18 20 21 23 (c(iparmj),iparmj=1,nparmc)
3 5 7 8 9 11 13 14 15 16 17 18 20 21 23 (c(iparmj),iparmj=1,nparmc)
end
where you are telling CHAMP to optimize \(a_i, 3\le i \le 6\) for e-n of O and H (4 parameters for both O and H), and \(b_i, 2 \le i \le 6\) (5 parameters in total).
Now, specify the name of the info of the derivatives of the Jastrow in the input file, below the line where the jastrow.start file is specified. You also need to add a block with different options for the optimizer as follows.
load jastrow jastrow.start
load jastrow_der jastrow.der
%module optwf
ioptwf 1
ioptci 0
ioptjas 1
ioptorb 0
method 'sr_n'
nopt_iter 20
nblk_max 4000
ncore 0
nextorb 100
sr_tau 0.05
sr_eps 0.001
sr_adiag 0.01
%endmodule
Optimization doesn't require a long QMC simulation in the first SR steps. You can reduce the number of blocks in blocking_vmc to 100, and the code will slowly increase the number of blocks to nblk_max in the optwf module.
%module blocking_vmc
vmc_nstep 20
vmc_nblk 100
vmc_nblkeq 1
vmc_nconf_new 0
%endmodule
If you grep 'total E' output, you will see the optimization progressing and generating new Jastrow factors in jastrow_optimal.1.iterX.
If you grep nblk output you will see that the code automatically increases the maximum number of blocks.