qmcpack/tests/molecules/Li2_STO_ae

This test reproduces Li2 STO + Pade Jastrow runs from Umrigar, 
Nightingale, and Runge (UNR)

Reference: C. J. Umrigar et al., J. Chem. Phys. 99 2865 (1993)
DOI:       http://dx.doi.org/10.1063/1.465195

As part of the test development, the behavior of the QMCPACK Green's 
function has been compared to the UNR results as a function of 
timestep.  Here we relate the timestep behavior comparison for 
validation purposes before going into details specific to the tests. 


UNR Reference Data
==================
Reference total energies from UNR for VMC and DMC are:

VMC  -14.9472(2) Ha
DMC  -14.9890(2) Ha (extrapolated to zero timestep)

Other data from UNR include the DMC variance and autocorrelation time, 
among other quantities (UNR Table III):

sigma : sqrt(variance) (Ha)
tcorr : auto-correlation time (steps)
teff/t: ratio of effective to inputted timestep
ar    : acceptance ratio

tau    sigma tcorr  teff/t   ar  
0.010  0.408  33.4   0.963   0.968
0.050  0.390  11.5   0.836   0.858
0.100  0.378   5.1   0.751   0.786
0.150  0.371   4.5   0.710   0.753
0.200  0.370   3.6   0.689   0.740
0.250  0.368   3.3   0.679   0.734

UNR also includes plots of the DMC total energy and kinetic energy as 
a function of timestep (Figures 7 and 11, respectively).  The data 
from UNR Fig. 7 (including the statistical error bars) have been 
extracted using the "Web Plot Digitizer" tool online at 
http://arohatgi.info/WebPlotDigitizer/app/.  Systematic errors 
introduced by the extraction process are estimated to be ~1% of the 
axis scale, which translates into ~0.00002 Ha for the energy values.  
This estimate is quantitative since the x-axis values (timesteps) are 
known.  The extracted data are:

tau       E_DMC        errorbar
0.2500  -14.993939 +/- 0.000077
0.2250  -14.992622 +/- 0.000122
0.2000  -14.992026 +/- 0.000122
0.1750  -14.991245 +/- 0.000122
0.1500  -14.990342 +/- 0.000122
0.1250  -14.989867 +/- 0.000138
0.1000  -14.989378 +/- 0.000138
0.0750  -14.989224 +/- 0.000138
0.0500  -14.988888 +/- 0.000199
0.0250  -14.988781 +/- 0.000184
0.0100  -14.989469 +/- 0.000352


Comparison of QMCPACK with UNR
==============================
Current (v3.2.0) and historical (SVN rev. 6259) QMCPACK total energy 
values agree with the quoted VMC and extracted DMC values from UNR:

       VMC          DMC (tau=0.01/Ha)
6259  -14.94721(5) -14.98944(6)
v320  -14.9470(2)  -14.98949(8)
UNR   -14.9472(2)  -14.9895(4) 

The timestep study above was recreated with QMCPACK v3.2.0 (see 
UNR_timestep_comparison directory).  The resulting QMCPACK data 
are:

tau       E_DMC        errorbar
0.2500  -15.027879 +/- 0.000462
0.2250  -15.015879 +/- 0.000119
0.2000  -15.008000 +/- 0.000077
0.1750  -15.002293 +/- 0.000069
0.1500  -14.997681 +/- 0.000069
0.1250  -14.994168 +/- 0.000069
0.1000  -14.991667 +/- 0.000068
0.0750  -14.989910 +/- 0.000070
0.0500  -14.989020 +/- 0.000084
0.0250  -14.988972 +/- 0.000080
0.0100  -14.989505 +/- 0.000078
0.0050  -14.989772 +/- 0.000067

Comparison plots (with corresponding pdf image files) of the QMCPACK 
and UNR results can be generated with the script E_DMC_vs_tau.py.  
The QMCPACK input file used to generate these data is 
Li2.STO.textrap.in.xml and the raw data from the UNR Fig. 7 extraction 
is found in UNR_Fig7_E_DMC_vs_tau.csv.

The QMCPACK and UNR data agree closely for timesteps in the range 
0.025/Ha<tau<0.05/Ha with QMCPACK demonstrating a larger timestep 
error than UNR beyond 0.05/Ha.  The estimated zero timestep limit 
of UNR and QMCPACK differ slightly by about 1 mHa:

      E_DMC (t=0 quadratic extrapolation)
v320 -14.98997(7) Ha 
UNR  -14.9890(2)  Ha

A comparison between QMCPACK and UNR results for the variance 
(sigma^2), autocorrelation time, effective timestep, and acceptance 
ratio are shown below as a function of timestep.  In each case, the 
first line corresponds to UNR data (from Table III) and the second 
line is from QMCPACK v3.2.0.

tau    0.010  0.050  0.100  0.150  0.200  0.250

sigma  0.408  0.390  0.378  0.371  0.370  0.368
       0.389  0.390  0.392  0.396  0.400  0.411  

tcorr   33.4   11.5    5.1    4.5    3.6    3.3
        40.3    9.0    4.9    6.0    7.2   15.9

teff/t 0.963  0.836  0.751  0.710  0.689  0.679
       0.979  0.855  0.738  0.647  0.586  0.539

ar     0.968  0.858  0.786  0.753  0.740  0.734
       0.988  0.913  0.827  0.758  0.701  0.654


Test details
============
The tests include short and long VMC and DMC (tau=0.01/Ha) runs for a 
Li dimer. The wavefunction is of Slater-Jastrow form with the Slater 
determinant orbitals represented in a Slater-type orbital (STO) 
exponential basis. The Jastrow factor is represented with simple Pade 
functions.  All wavefunction parameters were taken from UNR.  The 
original QMCPACK input file was recovered from the informally managed 
qmcdb.  This test was likely used during early QMCPACK development and 
testing.  Reference data for the tests was produced with QMCPACK SVN 
revision 6259.

Reference data was generated with runs the same overall length 
(blocks x steps) as those included in ./qmc-ref, but using 10x more 
blocks (8000) and 10x fewer steps.  Using more blocks reduced the 
statistical error of the error bars, but resulted in output data files 
that were too large to include in the QMCPACK distribution.  The 
reference values and error bars were therefore computed from the 8000 
block runs.  The included reference runs were confirmed to be 
consistent with the original higher block count runs.

Reference data from the high block count runs is found below:

                   LocalEnergy             Variance               ratio
Li2.STO series 0  -14.947214 +/- 0.000050  0.170618 +/- 0.000103  0.0114 
Li2.STO series 1  -14.988797 +/- 0.001006  0.151149 +/- 0.000845  0.0101 
Li2.STO series 2  -14.989438 +/- 0.000064  0.151118 +/- 0.000031  0.0101 


For comparison, corresponding data for the lower block count reference 
data found in ./qmc-ref is:

>qmca -e 2 -q ev *scalar*
                   LocalEnergy             Variance               ratio
Li2.STO series 0  -14.947224 +/- 0.000051  0.170596 +/- 0.000109  0.0114 
Li2.STO series 1  -14.986769 +/- 0.001208  0.151474 +/- 0.001493  0.0101 
Li2.STO series 2  -14.989479 +/- 0.000062  0.151159 +/- 0.000035  0.0101 

Test input files were created by reducing the number of steps from the 
included reference by a factor of 10 for the "long" tests and by a 
further factor of 10 for the "short" tests.

Reference values for the mean local energy and variance were taken 
from the high block count reference runs.  Expected error bars were 
derived from the high block count data by multiplying the reference 
error bars by a factor of sqrt(10+1) and sqrt(100+1) for the long 
and short tests, respectively.  The factor of +1 accounts for the 
error bar intrinsic to the high block count runs.

The target means and error bars for long and short tests are:

                refmean   referr    longerr   shorterr
VMC  energy   -14.947214  0.000050  0.000166  0.000502
     variance   0.170618  0.000103  0.000342  0.001035
DMC  energy   -14.989438  0.000064  0.000212  0.000643
     variance   0.151118  0.000031  0.000103  0.000312

Error bars from typical long and short runs are shown below:

Typical long run results:
>qmca -e 2 -q ev *long*scalar*
                        LocalEnergy             Variance               ratio 
Li2.STO.long series 0  -14.947014 +/- 0.000152  0.170796 +/- 0.000357  0.0114 
Li2.STO.long series 1  -14.988692 +/- 0.001624  0.151845 +/- 0.000494  0.0101 
Li2.STO.long series 2  -14.989612 +/- 0.000216  0.151080 +/- 0.000094  0.0101 

Typical short run results:
>qmca -e 2 -q ev *short*scalar*
                         LocalEnergy             Variance               ratio 
Li2.STO.short series 0  -14.946531 +/- 0.000525  0.171710 +/- 0.001163  0.0115 
Li2.STO.short series 1  -14.988802 +/- 0.000887  0.152505 +/- 0.000782  0.0102 
Li2.STO.short series 2  -14.988719 +/- 0.000536  0.150840 +/- 0.000252  0.0101 

Rough timings for the tests on an old (circa 2013) AMD system are:

>grep 'Total Exe' *long*.out
  Total Execution time = 4.2856237206e+02 secs

>grep 'Total Exe' *short*.out
  Total Execution time = 4.6398911903e+01 secs