# Supplemental material: Towards Quantum Monte Carlo Forces on Heavier Ions:
Scaling Properties

Juha Tiihonen  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, United States of America  Raymond C Clay III
Sandia National Laboratories, Albuquerque, NM 87185, United States of America
Jaron Krogel  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, United States of America

##  I Tail-regression estimation

To analyze heavy-tailed QMC force data, we use a tail-regression estimator
toolbox (TREAT) from Ref. [Rios2019]. Detailed theoretical foundations and
advanced techniques for tail-regression are given in the original publication.
For the purposes of this work, the task is to provide or estimate four
essential parameters for left and right tail of the form [Rios2019]

$\displaystyle P(A)=\sum_{n=0}^{N}c_{n}|A-A_{0}|^{-\mu-n\Delta}.$ (1)

The parameters corresponding to $\mu$, $\Delta$ and $N+1$ are called mu, dmu,
and nparams, respectively. A fourth parameter, mlogq, is the negative
logarithm of the quantile of input data defining the axial cutoff between
samples and the fit. With these parameters set, TREAT can perform the
regression and provide statistical analysis of the data.

Let us briefly outline three approaches or modes for assessing these
parameters: manual, automatic and median. In the manual mode, one supplies
them by hand, thus enabling enforcement of symmetries, asymptotic behavior and
other theoretical constraints to the numeric data. However, such information
is in practice unavailable or difficult to apply to QMC data that has been
reblocked to eliminate sample autocorrelation, or for other practical reasons.
Therefore, we mostly use automatic assessment of parameters, which brackets
common ranges of mu and nparams to provide as accurate regression of the tail
as possible, requiring minimal expert supervision. The automatic assessment is
used exclusively to assess isolated atoms and dimers at the equilibrium. For
dimers along the binding curve we use so-called median mode to increase
robustness of the auto-assessment. In the median mode TREAT is run three
times: first to assess median mu and dmu over the binding curve, second to
assess median mlogq and nparams, and finally fitting and evaluating the tail.
The stabilizing effects of automatic and median modes are illustrated in Fig.
1 with the VMC net force of TiO. The standard means and error bars have large
variability due to occasional spikes in the data. Tail-regression estimates in
the automatic mode are usually more uniform, but sometimes fail the assessment
due to insufficient data. The median mode has a steadier performance.

Figure 1: Sum of forces in TiO dimer using VMC with PBE0 orbitals and
different error estimation methods: standard, TREAT (automatic) and TREAT
(median). Dashed lines indicate the targeted zeros of forces.

The sample autocorrelation time $\tau$ is a measure of how many steps apart
the samples become effectively uncorrelated. Therefore, the true statistical
uncertainty, assuming standard distribution, is given by

$\sigma_{e}=\sqrt{\frac{\tau}{N-1}}\sigma.$ (2)

Estimation of $\tau$ in complex simulations can be challenging, and while
TREAT can diagnose $\tau$ in terms of the standard estimator of uncertainty,
it does not rectify the tail-regression uncertainty but assumes uncorrelated
samples. In this work we use reblocking technique to estimate the
autocorrelation. That is, starting from a relatively large set of input data
(e.g. 1-8 $\times 10^{6}$ samples with $36$ cores for VMC and 1-5 $\times
10^{5}$ samples over $1000$ walkers for DMC), we produce a series of data sets
taking averages of $R$ subsequent samples, where $R$ have been chosen to be
powers of 2. We then fit the uncertainties $\sigma_{e}$ obtained with TREAT
from the reblocked data as a function of $R$ to a simple form

$\sigma_{e}(R)=a\exp(-bR^{-1/2}),$ (3)

where $a$ and $b$ are positive coefficients. The fit converges to $a$ with
$R\rightarrow\infty$, as shown in Fig. 2, giving the statistically true
uncertainty. Estimate of the original sample auto-correlation is given by

$\tau=\frac{a}{\sigma_{e}(R=1)},$ (4)

which is used to estimate intrinsic variances from TREAT uncertainties.

Figure 2: Example of converging the VMC ZVZB-TRE uncertainties of Ti atom with
reblocking. The exponential form usually fits well the transient region
$R<\tau$, but is also resilient to large uncertainties of $\sigma_{e}$ at high
$R$.

##  II Curve fitting and integrated quantities

Curve fitting is used to study three integrated quantities based on binding
curves of dimers: equilibrium bond length $r$, vibrational frequency $\omega$
and self-consistency of the force $\Delta$, that is, the deviation from the
PES gradient. The first two are done separately by fitting either energies,
$E_{\mathrm{fit}}$ or forces, $F_{\mathrm{fit}}$. The equilibrium bond length
is where $F_{\mathrm{fit}}=0$ ($\mathrm{d}E_{\mathrm{fit}}/\mathrm{d}r=0$).
The vibrational frequency depends on the force constant $k$, which is the
derivative of $F_{\mathrm{fit}}$ or the second derivative of
$E_{\mathrm{fit}}$ at the equilibrium. The fitted range is approximately
$r_{\mathrm{expt}}\pm 20\%$ for each dimer. The range of separations used in
the calculation of $|\bar{\Delta}|$ is $r_{\mathrm{expt}}\pm 15\%$, because
the residual fitting error of
$\tfrac{\mathrm{d}E_{\mathrm{fit}}}{\mathrm{d}r}$ tends to grow near the
boundaries.

The functional form of the fit can be important and has been studied in the
past, e.g., in Refs. Badinski2008a, Zen2012, Luo2014. Choosing the optimal
fitting procedure depends on many parameters, such as the grid point density,
spatial range of the grid, and the purpose of the fit. We will use order $n=6$
polynomials to fit to the PES and $n=5$ to forces, although using a different
forms, including the Morse potential, might lead to modest but insubstantial
changes in the results.

Error estimation of the integrated quantities is done by using the resampling
technique. A number of independent random samples (1000) are generated based
on the means and errorbars at each grid point, and the fluctuation propagated
to the integrated quantities defines their statistical uncertainties.

##  III Validity of self-consistent VMC forces

Two curves of VMC ZVZB self-consistency error $\Delta$ of FeO are presented in
Fig. 3. The first curve (red) is such as elsewhere in this work, where the
Jastrow factor of the trial wavefunction is separately optimized in each
geometry. This leads to systematic inconsistencies between forces and
energies, due to missing implicit derivatives. The second curve (blue) is
based on a clamped trial wavefunction, i.e., with orbitals and Jastrow
parameters optimized at equilibrium separation and held fixed in other
geometries. The lack of systematic errors beyond statistical fluctuation
validates the implementation of the forces.

Figure 3: Self-consistency error of the VMC ZVZB forces of FeO based on
optimized (red) or clamped (blue) trial wavefunctions. The distinct values of
the forces on Fe and O are indicated by $\blacktriangleleft$ and
$\blacktriangleright$, respectively.

##  IV Numerical data for estimated bond lengths and vibrational frequencies

Table 1 contains numerical data for the estimated bond lengths $r$ and
vibrational frequencies $\omega$ based on VMC and DMC (PBE0 orbitals) energies
and ZVZB forces including hybrid estimates from this work, but also benchmark
CCSD(T) calculations from Refs. [NIST2019, Miliordos2010, Sakellaris2011] and
experimental results from Refs. [CRChandbook, Ram1999, Allen1996]. Deviations
of the bond lengths from experiment are discussed and presented in Sec. IV of
the main article. Deviations of the vibrational frequencies are presented in
Fig. 4, including median estimates from available prior works of QMC forces.
Detailed comparison of the prior QMC works is given later in the supplemental
material.

| Method | $r_{E}$ |  $r_{F}$ (left) |  $r_{F}$ (right) | $\omega_{E}$ |  $\omega_{F}$ (left) |  $\omega_{F}$ (right)
---|---|---|---|---|---|---|---
LiH | CCSD(T)[NIST2019] | 1.5963 |  |  | 1412.0 |  |
| PBE0 | 1.61106 |  |  | 1377.1 |  |
| VMC | 1.6002(2) | 1.59750(5) | 1.59746(8) | 1417(3) | 1408.0(0.3) | 1408.3(0.4)
| DMC | 1.602(2) | 1.6026(3) | 1.6003(5) | 1400(30) | 1389(2) | 1392(3)
| hybrid |  | 1.6073(6) | 1.6036(10) |  | 1366(4) | 1375(5)
| Expt[CRChandbook] | 1.5948994 |  |  | 1405.5 |  |
Li2 | CCSD(T)[NIST2019] | 2.667 |  |  | 345.0 |  |
| PBE0 | 2.75263 |  |  | 333.9 |  |
| VMC | 2.7113(3) | 2.71630(8) | 2.71522(8) | 350.5(0.4) | 340.56(6) | 341.32(7)
| DMC | 2.708(3) | 2.7145(7) | 2.7141(7) | 350(6) | 340.4(0.5) | 338.8(0.5)
| hybrid |  | 2.7123(14) | 2.7123(14) |  | 338.3(1.0) | 336.8(1.1)
| Expt[CRChandbook] | 2.6732974 |  |  | 351.41 |  |
MgO | CCSD(T)[NIST2019] | 1.7536 |  |  | 798.0 |  |
| PBE0 | 1.7308 |  |  | 812.7 |  |
| VMC | 1.7219(5) | 1.71864(11) | 1.7185(6) | 829(3) | 821.7(0.4) | 825(2)
| DMC | 1.751(2) | 1.7395(3) | 1.7179(8) | 784(13) | 765.1(0.9) | 805(4)
| hybrid |  | 1.7635(7) | 1.720(2) |  | 705(2) | 786(7)
| Expt[CRChandbook] | 1.7481687 |  |  | 785.21 |  |
CO | CCSD(T)[NIST2019] | 1.1325 |  |  | 2171.0 |  |
| PBE0 | 1.11852 |  |  | 2249.4 |  |
| VMC | 1.11340(11) | 1.11147(9) | 1.1113(2) | 2289(3) | 2292.5(1.3) | 2296(3)
| VMC-J3 | 1.11716(12) | 1.11350(10) | 1.1137(2) | 2268(4) | 2272(2) | 2270(3)
| VMC-J3+oo. | 1.11718(11) | 1.11581(9) | 1.1156(2) | 2259(3) | 2240.0(1.2) | 2241(3)
| DMC | 1.1222(5) | 1.1139(2) | 1.1111(2) | 2270(20) | 2258(3) | 2279(3)
| DMC-J3 | 1.1218(6) | 1.1143(2) | 1.1143(3) | 2190(20) | 2241(3) | 2249(4
| DMC-J3+oo. | 1.1220(6) | 1.1162(2) | 1.1157(3) | 2200(20) | 2222(3) | 2227(4)
| hybrid-J2 |  | 1.1159(4) | 1.1111(4) |  | 2230(5) | 2268(6)
| hybrid |  | 1.1159(4) | 1.1111(4) |  | 2230(5) | 2268(6)
| Expt[CRChandbook] | 1.1283199 |  |  | 2169.81 |  |
TiO | CCSD(T)[Miliordos2010] | 1.619002 |  |  | 1018.0 |  |
| PBE0 | 1.59687 |  |  | 1067.8 |  |
| VMC | 1.5874(6) | 1.5809(8) | 1.5821(7) | 1126(8) | 1117(5) | 1113(4)
| DMC | 1.611(3) | 1.598(2) | 1.5857(12) | 1040(30) | 1063(8) | 1115(6)
| hybrid |  | 1.619(4) | 1.589(3) |  | 980(20) | 1089(13)
| Expt[Ram1999] | 1.6203 |  |  | 1009.18 |  |
FeO | CCSD(T)[Sakellaris2011] | 1.611001 |  |  | 929.0 |  |
| PBE0 | 1.60126 |  |  | 903.3 |  |
| VMC | 1.6007(11) | 1.583(3) | 1.5792(12) | 882(10) | 962(13) | 950(6)
| DMC | 1.622(3) | 1.603(4) | 1.589(3) | 900(30) | 860(20) | 940(12)
| hybrid |  | 1.622(10) | 1.610(6) |  | 800(40) | 900(30)
| Expt[Allen1996] | 1.6193997 |  |  | 882.0 |  |
Table 1: Equilibrium bond lengths $r$ (Å) and vibrational frequencies $\omega$
(cm-1) calculated in this work, benchmark studies [NIST2019, Miliordos2010,
Sakellaris2011] and experiments [CRChandbook, Ram1999, Allen1996]. All of our
results are based on PBE0 trial wavefunctions with two-body Jastrow factor,
except CO contains also results with three-body Jastrow factor with (J3+oo.)
and without (J3) orbital optimization, which are discussed in later sections.
Subscript $E$ indicates that the property was obtained from the PES; otherwise
the values are fitted to ZVZB forces on left or right ions.

Figure 4: Absolute deviations from experiment of estimated vibrational
frequencies $\omega$ from VMC (top) and DMC (bottom). Bars for each system
from right to left represent QMC PES, QMC ZVZB force, and the median value out
of available QMC force values from Refs. Filippi2000, Assaraf2003,
Casalegno2003, Chiesa2005, Lee2005, Badinski2007. Solid and opaque color bars
of the DMC forces correspond respectively to the better and worse results out
of the left and right ionic forces, highlighting the net force induced. Gray
marks acceptable accuracy.

##  V Scaling of self-consistency error with $Z_{\mathrm{eff}}$

As seen and discussed earlier, the accuracy of both VMC and DMC forces suffer
from inconsistency with the PES due to Jastrow optimization without orbital
rotations. Let us boil this effect down to mean unsigned self-consistency
error

$\displaystyle|\bar{\Delta}_{\lambda}|=\frac{1}{\lambda_{2}-\lambda_{1}}\int_{\lambda_{1}}^{\lambda_{2}}|\Delta(\lambda)|\,\mathrm{d}\lambda,$
(5)

where $\Delta_{\lambda}$ is averaged over an arbitrary range, here
$r_{\mathrm{expt}}\pm 15\%$. Scaling of $|\bar{\Delta}_{\lambda}|$ with the
mean $Z_{\mathrm{eff}}$ of the selected dimers,
$\tilde{Z}_{\mathrm{eff}}=(Z^{\mathrm{left}}_{\mathrm{eff}}+Z^{\mathrm{right}}_{\mathrm{eff}})/2$,
is presented in Fig. 5. Fitting the data to $aZ_{\mathrm{eff}}^{b}$ shows that
the self-consistency error of VMC forces is characterized reasonably well by
the polynomial scaling relationship, behaving like
$\tilde{Z}_{\mathrm{eff}}^{1.6(3)}$. Similar exponents for UHF-based trial
wavefunction yields $\tilde{Z}_{\mathrm{eff}}^{2.1(10)}$, because the effect
of Jastrow factor is even higher. With DMC forces the PBE0-based self-
consistency error scales with 1.7(13).

The similarity in self-consistency scaling among VMC and DMC forces suggests a
commonality in the missing information from implicit wavefunction parameter
derivatives which cannot be captured by any of these force estimators.
Interestingly, the degree of missing information trends in a similar fashion
to the correlation energy, suggesting that the energetics of each physical
system sets the scale of the expected self-consistency error. The correlation
energy, as estimated from the energy difference between UHF and DMC at the
equilibrium geometry, is shown according to the axes on the right of Fig. 5.
The Jastrow factor is designed to take into account the many-body correlation,
and the similarity between both the self-consistency error of forces and the
correlation energy is in harmony with the understanding that the effect
originates from the Jastrow optimization.

The rapid growth of self-consistency error with $\tilde{Z}_{\mathrm{eff}}$
also highlights the added challenge systems with heavier elements are likely
to pose for ZVZB force calculations with pseudopotentials. An increased self-
consistency error in the mean-field trial wave function suggests that the
improvement needed from more optimized forms of the wave function also grows
for these systems. This gives some strong limits on how accurate forces can be
with a community standard workflow.

Figure 5: Mean unsigned self-consistency error $|\bar{\Delta}|$ is plotted as
a function of
$\tilde{Z}_{\mathrm{eff}}=(Z^{\mathrm{left}}_{\mathrm{eff}}+Z^{\mathrm{right}}_{\mathrm{eff}})/2$
for VMC ZVZB dimer forces with PBE0 trial wavefunctions. The correlation
energy, as estimated from DMC, is plotted on a different scale to allow
qualitative comparison with $|\bar{\Delta}|$.

##  VI Mitigating the force self-consistency error

It is therefore imperative to find ways of mitigating the self-consistency
error of QMC forces in ways that are both effective and computationally
tractable. In the following we shall confine ourselves to getting the most
accurate forces out of a single-determinant Slater-Jastrow wavefunction. We
focus strictly on the CO dimer, because it is the simplest of the oxides to
demonstrate all characteristics seen in the heavier ones: measurable VMC self-
consistency error, large DMC self-consistency error, and a noticeable
difference between the DMC forces on the different atoms in the dimer.

Figure 6: (left) Self-consistency error $\Delta$ (mHa/Bohr) of the left force
versus dimer separation $r$ (Å) for the CO dimer. The different colors
represent orbitals generated using different EXX fractions. (right) Mean
signed self-consistency error of VMC (red), and mean net force of DMC (blue)
plotted against respective mean energy differences from the minimum.

Figure 7: (Left) VMC self-consistency error $\Delta$ (mHa/Bohr) vs. dimer
separation $r$ (Å) for the CO dimer. This was calculated using the following
single determinant ansätze: (red) 1 and 2-body Jastrows, (green) one, 2-body,
and 3-body Jastrows, and (blue) one, two, and 3-body Jastrows with optimized
orbitals in a limited active space (56 orbitals). (Right) Averaged DMC self-
consistency error (mHa/Bohr) vs. dimer separation r (Å).

A popular way to maximize the QMC performance is to perform a parameter scan
of the mean-field orbitals. In particular, the optimal exact exchange fraction
(EXX), scaling between pure PBE (EXX=0) and Hartree–Fock (EXX=1), can be
sought to minimize the local energy for each individual system, and VMC and
DMC separately. However, as seen in Figure 6, while EXX scans can improve the
SC errors, the EXX fraction corresponding to minimal self-consistency error
does not correspond to a variational energy or discernable force variance
minimum. The left panel of Fig. 6 shows that $\Delta$ can be minimized by
varying the EXX fraction between pure UHF (EXX=1.0; pure red) and PBE (EXX=0;
pure blue). However, based on the right panel, the VMC energy minimum is near
EXX=0.4, while the minimal $\Delta$ (red circle) is around EXX=0.75. Similar
comparison of the DMC energy to a measure of self-inconsistency (net force)
leads to different EXX values. The conclusion is that EXX optimization by
itself is not an efficient way to improve the performance of ZVZB forces.

Let us instead consider improvements in the forces resulting from improving
the Jastrow factor and orbital re-optimization in the presence of the Jastrow.
We compare the CO dimer curve at both the VMC and DMC level using three
different types of trial wave functions: Slater-Jastrow with up to two body
Jastrows using PBE0 orbitals (1+2J), Slater-Jastrow with up to three-body
Jastrows using PBE0 orbitals (1+2+3J), and lastly a Slater-Jastrow with up to
three-body Jastrows using re-optimized orbitals using the PBE0 set as a
starting point (1+2+3J+OO). The aug-ccVQZ basis set for CO has 206 basis
functions, and so a full spin-unrestricted optimization of these orbitals
would introduce an additional 2030 parameters into the linear method
optimization procedure[Filippi2016, Assaraf2017, Flores2019]. In order to
reduce computational cost, we restrict orbital optimization to the 56 lowest
energy orbitals, which introduces 510 optimizable parameters. Optimization
within this reduced set does not guarantee a zero self-consistency error, but
we will show that even within this restricted active space we are able to
dramatically improve the accuracy of the VMC and DMC force estimators.

We consider first the VMC self-consistency error. The origin of this problem
has been recognized for a long time[Casalegno2003, Moroni2014]: the parameters
defining the mean-field orbitals have an implicit ion dependence yet are not
at a variational minimum of the QMC Hamiltonian. Moroni et al. [Moroni2014]
showed that the solution to this problem is orbital optimization. In the left
panel of Figure 7, we see that relative to the 1+2J wave function, the
inclusion of a three-body Jastrow does little to mitigate the self-consistency
error of VMC forces, and in fact arguably increases it at intermediate
separations. However, by performing orbital optimization on the 1+2+3J wave
function–even in the restricted active space–we can significantly reduce the
self-consistency error. While this behavior is expected, what is striking is
the incredibly small energy differences between the 1+2+3J and orbital
optimized 1+2+3J wave functions: across the range of separations, orbital
optimization recovers on the order of $0.2$ mHa, but can easily change the
self-consistency error by a factor of 2. Consistent with prior results, this
suggests that orbital optimization remains an appealing route for reduction of
self-consistency error.

We now consider the impact of the improved trial wavefunctions on DMC ZVZB
forces. In the right side of Figure 7, we show the average DMC self-
consistency error versus dimer separation. It is immediately noticeable that
the self-consistency error is larger at the DMC level than it is for VMC. The
self-consistency error for the 1+2J wave function is between between 25-30mHa,
3-5 times larger on average than it is for VMC. However, the DMC estimator
does appear to be unambiguously improved by improving the wave function. The
1+2+3J wave function reduces the SC error by 20-25% across the dimer curve
relative to 1+2J. Upon performing orbital optimization on the 1+2+3J
wavefunction, we further observe a consistent decrease of the SC error across
the range of separations by between 5-8 mHa/Bohr, which is comparable to the
improvement we saw at the VMC level. Despite these gains, further improvements
to the wavefunction are needed for the DMC forces to match the quality of the
PES when it comes to derived quantities such as the bond length. While both
the VMC and DMC forces have improved in accuracy, the DMC forces still do not
exceed the VMC forces in terms of absolute accuracy, since the deviation from
experiment for the bond lengths are $0.01251(9)$ Å for VMC forces, $0.0124(3)$
Å for mixed DMC forces, versus $0.0063(6)$ Å for the DMC PES.

In order to assess how the force imbalance depends on quality of trial wave
function, we computed the net force at each dimer separation and performed an
average over the whole dimer curve for each of the three considered wave
functions. We find that, overall, the force imbalance is a secondary source of
error compared to the overall self-consistency error. The mean self-
consistency error ranges between 5-18 mHa/Bohr with our best wavefunction,
while the mean net forces per wavefunction over the dimer separation are only
7.1(3) mHa/Bohr for 1+2J, 1.6(4) mHa/Bohr for 1+2+3J, and 3.4(4) mHa/Bohr for
1+2+3J+OO. The improvement of the Jastrow has also rectified the discrepancy
in the derived bond length from left and right DMC forces, with the two now
giving the same bond length to within $0.0001(4)$ Å. Since the DMC fixed-node
wave function is related to the trial wave function by a change of Jastrow
factor, it is expected that improvements to the Jastrow factor within the
trial wave function should substantially impact this error. Orbital
optimization, while bringing down the overall self-consistency error, actually
worsens the force imbalance a small amount, though still with a minimal impact
on imbalanced bond length as they remain in agreement to $0.0005(4)$ Å.

As expected, the Reynolds approximation introduces an extremely strong
dependence on the quality of the underlying trial wave function, which is not
particularly satisfying given the greater insensitivity of the DMC PES to the
wave function parameters than in VMC. While we were able to demonstrate a
substantial mitigation of these errors by deploying better wave functions and
performing orbital optimization, there’s a good chance that in cases where the
VMC and DMC PES’s are not extremely similar to begin with, forces obtained
from the Reynolds approximation might not mirror the true DMC PES accurately
enough to significantly improve the force estimate over the one from VMC. On a
practical side however, as long as the VMC ZVZB and DMC ZVZB with Reynolds
forces are better than those from DFT, they still have value. In Table I (main
text), we see that even for single Slater-determinant with two body-jastrow
calculations, the VMC and DMC predicted bond lengths and vibrational
frequencies are often very comparable to those derived from PBE0, and they are
likely to be improved with orbital optimization and the inclusion of a 3-body
Jastrow. For general applications where the accuracy of DMC is required,
however, significantly better approximations to the left Pulay should be
employed.

##  VII Basis set convergence and equilibrium forces

The basis set convergence is studied in terms of force values at the reference
geometry, $r_{\mathrm{expt}}$. Both PBE0 forces and the associated VMC forces
are presented in Fig. 8 with three basis sets, VTZ, VQZ, and V5Z. Only the
left forces are shown, because the VMC forces are manifestly symmetric apart
from variance. The Jastrow factors of each dimer are optimized with V5Z and
used with all the bases. Based on the data, we have chosen to use VTZ basis
for all dimers, except MgO, where VQZ is chosen.

The data including SCF and DMC forces is tabulated in Table 2 for UHF
determinants and Table 3 for PBE0 determinants. Comparison of mixed and hybrid
DMC forces at equilibrium is given in Fig. 9 with the V5Q basis set.

Figure 8: Estimated ZVZB forces at experimental bond lengths computed with SCF and VMC using UHF (above) and PBE0 (below) orbitals with aug-VTZ, aug-VDZ and aug-V5Z basis sets.  Figure 9: Estimated ZVZB forces at experimental bond lengths computed with SCF, VMC, DMC and hybrid ZVZB estimators using UHF (above) and PBE0 (below) orbitals with V5Z basis set.  | Method |  |  $F$ (left) |  |  |  $F$ (right) |
---|---|---|---|---|---|---|---
|  | VTZ | VQZ | V5Z | VTZ | VQZ | V5Z
LiH | UHF | -1.1733635 | -1.1333974 | -1.0311665 |  |  |
| VMC | -0.390(11) | -0.432(11) | -0.405(13) | 0.44(2) | 0.41(2) | 0.40(2)
| DMC | -0.99(7) | -0.99(7) | -0.50(8) | 0.62(10) | 0.50(11) | 0.12(9)
Li2 | UHF | -3.1202507 | -3.1409551 | -3.0274961 |  |  |
| VMC | -1.470(7) | -1.587(7) | -1.688(9) | 1.474(7) | 1.585(7) | 1.685(8)
| DMC | -1.37(5) | -1.43(5) | -1.48(5) | 1.45(5) | 1.40(5) | 1.40(5)
MgO | UHF | 9.5078863 | 11.5212239 | 13.0436547 |  |  |
| VMC | 0.6(0.6) | 7.16(9) | 12.25(8) | -0.7(0.5) | -7.2(0.5) | -13.2(0.5)
| DMC | -6.3(0.3) | -0.4(0.3) | 5.3(0.3) | -1.3(1.0) | -9.3(0.8) | -11.3(1.3)
CO | UHF | 78.3546928 | 81.5848802 | 81.7178423 |  |  |
| VMC | 32.8(0.4) | 36.3(0.4) | 32.0(0.4) | -35.1(0.8) | -37.3(0.8) | -32.6(0.8)
| DMC | 28.1(0.7) | 35.1(0.7) | 26.8(0.7) | -33.3(1.3) | -38.5(1.0) | -29.0(1.3)
TiO | UHF | -40.9783141 | -40.3456662 | -40.1295219 |  |  |
| VMC | -41(3) | -41(3) | -40(3) | 38(2) | 47(2) | 39(2)
| DMC | -44(5) | -47(5) | -41(5) | 32(3) | 39(4) | 33(4)
FeO | UHF | -106.2975707 | -106.0764988 | -106.0129389 |  |  |
| VMC | -110(6) | -103(6) | -98(6) | 105(3) | 100(3) | 103(3)
| DMC | -105(6) | -88(6) | -83(6) | 91(5) | 83(4) | 88(5)
Table 2: Equilibrium force values in mHa/Bohr using UHF trial wavefunctions and various basis sets. | Method |  |  $F$ (left) |  |  |  $F$ (right) |
---|---|---|---|---|---|---|---
|  | VTZ | VQZ | V5Z | VTZ | VQZ | V5Z
LiH | PBE0 | -1.7174509 | -1.6366382 | -1.5557202 |  |  |
| VMC | -0.248(10) | -0.358(11) | -0.617(14) | 0.22(2) | 0.34(2) | 0.64(2)
| DMC | -1.00(6) | -0.93(5) | -0.98(8) | 0.59(10) | 0.49(9) | 0.47(10)
Li2 | PBE0 | -2.3979264 | -2.3681756 | -2.3448929 |  |  |
| VMC | -1.254(6) | -1.154(13) | -1.460(7) | 1.241(6) | 1.142(14) | 1.472(7)
| DMC | -1.19(5) | -1.21(13) | -1.39(5) | 1.24(5) | 1.14(13) | 1.36(5)
MgO | PBE0 | 3.9364609 | 5.7371909 | 7.0522571 |  |  |
| VMC | 9.55(7) | 12.0(0.1) | 13.17(8) | -8.2(0.6) | -12.6(0.6) | -12.5(0.6)
| DMC | -0.5(0.4) | 3.0(0.4) | 4.2(0.3) | -4.5(1.1) | -11.9(1.0) | -14.7(0.9)
CO | PBE0 | 20.5586222 | 23.5869345 | 23.7307699 |  |  |
| VMC | 37.8(0.4) | 41.2(0.4) | 40.0(0.4) | -37.1(0.8) | -41.4(0.8) | -40.5(0.9)
| DMC | 33.3(1.0) | 36.6(0.9) | 29.7(0.6) | -41.6(1.0) | -44.4(1.1) | -38.7(1.4)
TiO | PBE0 | 21.3299149 | 21.8679477 | 21.9946493 |  |  |
| VMC | 39(3) | 36(3) | 44(3) | -36(3) | -40(3) | -38(2)
| DMC | 18(5) | 21(6) | 26(3) | -29(4) | -31(5) | -34(4)
FeO | PBE0 | 11.9526571 | 12.2093923 | 12.2333751 |  |  |
| VMC | 36(6) | 23(6) | 17(6) | -29(3) | -25(3) | -29(3)
| DMC | 8(7) | 3(6) | 6(5) | -22(3) | -22(2) | -21(4)
Table 3: Equilibrium force values in mHa/Bohr using PBE0 trial wavefunctions
and various basis sets.

##  VIII Energy and force curves for dimers

Binding curves (PES) for each dimer are presented in Figs. 10 with SCF, VMC
and DMC, and trial wavefunctions based on UHF and PBE0. The respective ZVZB
force curves are presented in 11, and self-consistency curves (force plus the
PES gradient) in Fig. 12. Coarse fluctuations in the self-consistency curves
are mostly due to statistical and residual errors in fitting to the PES. The
basis sets are the following: LiH (VTZ), Li2 (VTZ), CO (VQZ), MgO (VQZ), TiO
(VTZ) and FeO (VTZ).

Figure 10: Binding curves of dimers ranging $\pm 20\%$ around experimental
bond length (green marking). The solid (PBE0) and dotted (UHF) lines with
black (SCF), red (VMC) and blue (DMC) color are 6-degree polynomial fits and
markers are the QMC estimates. The absolute SCF (UHF) energy curve is often
off the chart. Figure 11: ZVZB forces for each dimer: LiH (top-left), Li2
(top-right), CO (middle-left), MgO (middle-right), TiO (bottom-left), FeO
(bottom-right). The solid (PBE0) and dotted (UHF) lines with black (SCF), red
(VMC) and blue (DMC) color are 5-degree polynomial fits and markers are the
QMC estimates. The error bars are smaller than the marker size. Green crosses
insets are located at the experimental bond lengths with slopes of the
experimental force constant. Figure 12: Self-consistency error $\Delta$ for
each dimer: LiH (top-left), Li2 (top-right), CO (middle-left), MgO (middle-
right), TiO (bottom-left), FeO (bottom-right). The solid (PBE0) and dotted
(UHF) lines with red (VMC) and blue (DMC) color are differences between
5-degree polynomial fit to forces and the derivative of 6-degree polynomial
fit to energies. The markers are original QMC forces adjusted by the energy
derivatives.

##  IX UHF bond lengths and vibrational frequencies

Data for estimated bond lengths and vibrational frequencies from SCF, VMC and
DMC calculations based on UHF orbitals are presented in Table 4.

| Method | $r_{E}$ |  $r$ (left) |  $r$ (right) | $\omega_{E}$ |  $\omega$ (left) |  $\omega$ (right)
---|---|---|---|---|---|---|---
LiH | UHF | 1.60609 |  |  | 1415.4 |  |
| VMC | 1.60333(14) | 1.59723(5) | 1.59728(7) | 1383(3) | 1406.6(0.2) | 1406.5(0.3)
| DMC | 1.603(2) | 1.6022(3) | 1.5995(5) | 1330(30) | 1385(2) | 1399(3)
Li2 | UHF | 2.77294 |  |  | 337.9 |  |
| VMC | 2.7257(3) | 2.71943(7) | 2.71920(8) | 343.6(0.4) | 342.83(6) | 342.93(6)
| DMC | 2.704(3) | 2.7161(7) | 2.7171(7) | 333(5) | 341.7(0.5) | 341.5(0.5)
MgO | UHF | 1.7125 |  |  | 800.9 |  |
| VMC | 1.7008(3) | 1.72821(13) | 1.7273(8) | 929(2) | 711.5(0.5) | 715(3)
| DMC | 1.722(2) | 1.7484(5) | 1.7292(13) | 894(14) | 673(2) | 699(5)
CO | UHF | 1.09744 |  |  | 2442.8 |  |
| VMC | 1.11157(11) | 1.11356(8) | 1.1133(2) | 2318(3) | 2301.4(1.3) | 2304(3)
| DMC | 1.1195(5) | 1.1145(2) | 1.1137(2) | 2244(14) | 2269(3) | 2281(3)
TiO | UHF | 1.66688 |  |  | 956.6 |  |
| VMC | 1.6326(8) | 1.6701(10) | 1.6669(8) | 1000(7) | 942(7) | 955(5)
| DMC | 1.661(4) | 1.677(2) | 1.6609(13) | 920(30) | 891(13) | 952(10)
FeO | UHF | 1.80768 |  |  | 598.9 |  |
| VMC | 1.737(3) | 1.802(6) | 1.799(2) | 640(20) | 670(20) | 646(8)
| DMC | 1.719(13) | 1.781(5) | 1.767(3) | 660(40) | 690(30) | 700(14)
Table 4: Equilibrium bond lengths $r$ (Å) and vibrational frequencies $\omega$
(cm-1) calculated in this work using UHF, and VMC and DMC with UHF orbitals.
Subscript $E$ indicates that the property was obtained from the PES; otherwise
the values are fitted to ZVZB forces on left or right ions.

##  X Comparison to earlier works

We present comparisons to earlier works on QMC forces [Assaraf2000,
Filippi2000, Assaraf2003, Lee2005, Chiesa2005, Lee2007, Casalegno2003,
Badinski2007], where applicable, and selected works on QMC energy surfaces
[Cleland2016, Krogel2016]. Comparison of equilibrium force values is done in
Table 5, and binding curves and vibrational frequencies in Table 6.

Our observations about the quality of VMC forces are consistent with results
from prior works [Filippi2000, Assaraf2003, Casalegno2003, Lee2005,
Badinski2007, Cleland2016] that use equivalent single reference trial
wavefunctions. Comparable prior force results are available for the LiH, Li2
and CO dimers. For LiH, we find a force derived bond length of $1.5975(5)$ Å
versus $1.5987(6)$[Badinski2007]-$1.610(4)$[Casalegno2003] Å from prior works
and $1.5948994$ Å from experiment. Our LiH force derived vibrational frequency
is $1408.3(4)$, which compares well with prior works
(1372(9)[Badinski2007]-1407(4)[Lee2005] cm-1) and experiment (1405.5 cm-1).
Li2 is a challenging system from the point of view of obtaining an accurate
bond length, largely due to the very soft nature of the bond ($351.41$ cm-1
experimentally). Small errors in energy lead to large errors in the bond
length, as seen both in our results and those from prior works. While the VMC
vibrational frequency is well described here ($340.9(5)$ cm-1) and elsewhere
($366(9)$ cm-1)[Assaraf2003], the bond length is not. Our force derived VMC
bond length overestimates the experimental value by $0.04246(6)$ Å, while
prior works overestimate by
$0.0540(11)\cite[cite]{[\@@bibref{Number}{Filippi2000}{}{}]}-0.156(15)\cite[cite]{[\@@bibref{Number}{Assaraf2003}{}{}]}$
Å. Based on the experimental force constant for Li2, errors in the PES of
about $0.02$ eV in the vicinity of the minimum are all that is required to
produce errors in the bond length both we and others observe. For CO, one
prior force based VMC estimate for the bond length and vibrational frequency
is available[Lee2005]. That work finds a both length $-0.0197(6)$ Å smaller
than experiment, similar to our underestimation by $-0.0170(1)$ Å, while
overestimating $\omega$ by $369(16)$ cm-1 compared to $125(4)$ cm-1 currently.
Other more recent studies, such as Refs. [Badinski2010, Moroni2014, Rios2019],
feature improved wave functions and methods, although results from those works
are not represented here due to lack of overlapping data.

Our results for LiH and Li2 with mixed DMC forces are more accurate than the
mixed estimates from Ref. [Assaraf2003], but not the improved estimates
obtained by hybrid extrapolation. Advanced DMC estimators, such as Ref.
[Badinski2010], show impressive accuracy with LiH but also other dimers
outside of this study.

Similar to the VMC case, the DMC bond length of Li2 is significantly
overestimated, despite its apparent simplicity. In this case the DMC fixed
node approximation is exact since the wavefunction is nodeless (a two electron
singlet following the removal of the He core via pseudization). The only
source of uncontrolled approximation in the PES enters through approximate
pseudopotential localization via the T-move scheme. We have confirmed that the
quality of the pseudopotential itself is not at issue, since the CCSD(T)
binding curve yields an essentially exact bond length. A prior pseudopotential
study shows a similar overestimation of the PES bond length, while also
demonstrating superior performance with multi-reference
wavefunctions[Cleland2016]. Mixed DMC forces result in a slight increase in
the overestimation, showing combined effects of self-consistency error and
pseudopotential localization. A pioneering all-electron DMC force study from
Ref. [Filippi2000] reproduces Li2 binding properties with great accuracy using
multi-reference trial wavefunction and correlated sampling instead of Pulay
corrections.

For the CO dimer, one previous DMC force study exists[Lee2005]. There, using
Hartree-Fock based trial wavefunctions and mixed DMC forces, the authors find
good bond length (accurate to $-0.0086(11)$ Å) and moderate quality frequency
of vibration (within $81(26)$ cm-1 of experiment). Our mixed force results are
somewhat worse, with the bond length and vibration frequency deviating from
experiment by $-0.0161(2)$ Å and $100(2)$ cm-1, but still within the range of
variability seen for other systems.

No earlier DMC force data is available for the metal oxide dimers. Compared to
earlier benchmark studies [Cleland2016, Krogel2016] of the DMC PES, this
present work obtains a similar accuracy for the derived bond length and
vibration frequencies, but generally with larger error bars due the the
sensitivities of fitting to polynomials instead of the Morse potential.

Dimer | Reference | Method | Orbitals |  $F$ (left) |  $F$ (right)
---|---|---|---|---|---
LiH | This work | VMC | SD-PBE0 | -0.617(14) | 0.64(2)
|  | DMC |  | -0.98(8) | 0.47(10)
| Ref. [Assaraf2000] | VMC | SD-HF | -26.3(2) |
|  | DMC |  | -12.5(9) |
|  | hybrid |  | -1.3(1.1) |
| Ref. [Casalegno2003] | VMC | SD-HF | -1.9(4) |
|  | DMC |  | -0.2(2) |
Li2 | This work | VMC | SD-PBE0 | -1.460(7) | 1.472(7)
|  | DMC |  | -1.39(5) | 1.36(5)
| Ref. [Assaraf2000] | VMC | SD-HF | -196(2) |
|  | DMC |  | -96(3) |
|  | hybrid |  | -4(4) |
| Ref. [Assaraf2003] | VMC | MD-VB | 6.92(9) |
|  | DMC |  | 3.58(19) |
|  | hybrid |  | 0.24(39) |
CO | This work | VMC | SD-PBE0 | 40.0(0.4) | -40.5(0.9)
|  | DMC |  | 29.7(0.6) | -38.7(1.4)
| Ref. [Lee2007] | VMC | SD-HF | 28(5) |
|  | DMC |  | 16(8) |
Table 5: Comparison of this work (PBE0 with V5Z) and earlier QMC studies on left and right forces (mHa/Bohr), when applicable, at experimental equilibrium geometry.  Dimer | Reference | Method | Orbitals |  $r$ (PES) |  $r$ (left) |  $r$ (right) |  $\omega$ (PES) |  $\omega$ (left) |  $\omega$ (right)
---|---|---|---|---|---|---|---|---|---
LiH | Expt |  |  | 1.5948994 |  |  | 1405.5 |  |
| This work | VMC | SD-PBE0 | 1.6002(2) | 1.59750(5) | 1.59746(8) | 1417(3) | 1408.3(0.2) | 1408.3(0.4)
|  | DMC |  | 1.602(2) | 1.6026(3) | 1.6003(5) | 1400(30) | 1389(2) | 1392(3)
|  | hybrid |  |  | 1.6073(6) | 1.6036(10) |  | 1366(4) | 1375(5)
|  Ref. [Assaraf2003] | VMC | MD-VB |  | 1.646(9) |  |  | 1559(40) |
|  | DMC |  |  | 1.617(4) |  |  | 1549(6) |
|  | hybrid |  |  | 1.588(8) |  |  | 1519(31) |
|  Ref. [Casalegno2003] | VMC | SD-HF | 1.595(14) | 1.610(4) |  |  |  |
|  | DMC |  | 1.609(3) | 1.601(3) |  |  |  |
|  Ref. [Chiesa2005] | DMC | SD-HF |  | 1.592(4) |  |  | 1402(4) |
|  Ref. [Lee2005] | VMC | SD-HF |  | 1.6076(6) |  |  | 1407(4) |
|  | DMC |  |  | 1.5981(6) |  |  | 1445(20) |
|  Ref. [Badinski2007] | VMC | SD-HF | 1.5952(4) | 1.6063(8) | 1.5911(8) | 1416(15) | 1392(13) | 1352(12)
|  | DMC |  | 1.6000(3) | 1.5995(8) | 1.5970(12) | 1395(20) | 1424(24) | 1398(22)
|  Ref. [Cleland2016] | VMC | SD-B3LYP | 1.6007(10) |  |  |  |  |
|  | VMC | MD-MCSCF | 1.6116(9) |  |  |  |  |
|  | DMC | SD-PBE | 1.6047(10) |  |  |  |  |
|  | DMC | MD-MCSCF | 1.6077(10) |  |  |  |  |
Li2 | Expt |  |  | 2.6732974 |  |  | 351.41 |  |
| This work | VMC | SD-PBE0 | 2.7113(3) | 2.71630(8) | 2.71522(8) | 350.5(0.4) | 340.56(6) | 341.32(7)
|  | DMC |  | 2.708(3) | 2.7145(7) | 2.7141(7) | 350(6) | 340.4(0.5) | 338.8(0.5)
|  | hybrid |  |  | 2.7123(14) | 2.7123(14) |  | 338.3(1.0) | 336.8(1.1)
|  Ref. [Filippi2000] | VMC | MD-MCSCF |  | 2.7273(11) |  |  |  |
|  | DMC |  |  | 2.663(2) |  |  |  |
|  Ref. [Assaraf2003] | VMC | MD-VB |  | 2.829(15) |  |  | 366(9) |
|  | DMC |  |  | 2.752(9) |  |  | 373(5) |
|  | hybrid |  |  | 2.682(15) |  |  | 387(8) |
|  Ref. [Cleland2016] | VMC | SD-B3LYP | 2.6849(40) |  |  |  |  |
|  | VMC | MD-MCSCF | 2.6930(8) |  |  |  |  |
|  | DMC | SD-PBE | 2.722(6) |  |  |  |  |
|  | DMC | MD-MCSCF | 2.6819(20) |  |  |  |  |
MgO | Expt |  |  | 1.7481687 |  |  | 785.21 |  |
| This work | VMC | SD-PBE0 | 1.7219(5) | 1.71864(11) | 1.7185(6) | 829(3) | 821.7(0.4) | 825(2)
|  | DMC |  | 1.751(2) | 1.7395(3) | 1.7185(6) | 784(13) | 765.1(0.9) | 805(4)
|  | hybrid |  |  | 1.7635(7) | 1.720(2) |  | 705(2) | 786(7)
|  Ref. [Cleland2016] | VMC | SD-B3LYP | 1.705(3) |  |  |  |  |
|  | VMC | MD-MCSCF | 1.753(2) |  |  |  |  |
|  | DMC | SD-HF | 1.708(2) |  |  |  |  |
|  | DMC | MD-MCSCF | 1.749(2) |  |  |  |  |
CO | Expt |  |  | 1.1283199 |  |  | 2169.81 |  |
| This work | VMC | SD-PBE0 | 1.11343(11) | 1.11122(9) | 1.1115(2) | 2266(3) | 2296.9(1.3) | 2293(3)
|  | DMC |  | 1.1225(5) | 1.1140(2) | 1.1105(2) | 2190(20) | 2252(3) | 2288(3)
|  | hybrid |  |  | 1.1164(4) | 1.1097(4) |  | 2215(5) | 2282(6)
|  Ref. [Lee2005] | VMC | SD-HF |  | 1.1086(6) |  |  | 2539(16) |
|  | DMC |  |  | 1.1197(11) |  |  | 2251(26) |
|  Ref. [Cleland2016] | VMC | SD-B3LYP | 1.11642(50) |  |  |  |  |
|  | VMC | MD-MCSCF | 1.12672(40) |  |  |  |  |
|  | DMC | SD-HF | 1.12122(40) |  |  |  |  |
|  | DMC | MD-MCSCF | 1.12572(30) |  |  |  |  |
TiO | Expt |  |  | 1.6203 |  |  | 1009.18 |  |
| This work | VMC | SD-PBE0 | 1.5874(6) | 1.5809(8) | 1.5821(7) | 1126(7) | 1117(5) | 1113(4)
|  | DMC |  | 1.611(3) | 1.598(2) | 1.5857(12) | 1040(30) | 1063(8) | 1115(6)
|  | hybrid |  |  | 1.619(4) | 1.589(3) |  | 980(20) | 1089(13)
|  Ref. [Krogel2016] | DMC | SD-LDA | 1.614(1) |  |  | 1019(6) |  |
|  | DMC | SD-LDA |  |  |  | 1010(6) |  |
FeO | Expt |  |  | 1.6193997 |  |  | 882.0 |  |
| This work | VMC | SD-PBE0 | 1.6007(11) | 1.583(3) | 1.5792(12) | 882(10) | 962(13) | 950(6)
|  | DMC |  | 1.622(3) | 1.603(4) | 1.589(3) | 900(40) | 860(20) | 940(12)
|  | hybrid |  |  | 1.622(10) | 1.610(6) |  | 800(40) | 900(30)
|  Ref. [Krogel2016] | DMC | SD-LDA | 1.618(1) |  |  | 894(5) |  |
|  | DMC | SD-LDA |  |  |  | 881(6) |  |
Table 6: Comparison of this work (PBE0) and selected values from other QMC
studies on bond lengths $r$ (Å) and vibrational frequencies $\omega$ (cm-1)
based on left or right force, or the PES.

##  XI Intrinsic variances and their scaling relations with $Z_{\mathrm{eff}}$

The estimated intrinsic variances of energies and ZVZB-TRE forces are
presented in Table 7 for VMC and DMC simulations of isolated atoms. The
intrinsic variances of ZVZB-TRE and ZVZB-SW-TRE forces are given in Tables 8
and 9 based on VMC and DMC simulations of homonuclear dimers, respectively.

Scaling relations of form $aZ_{\mathrm{eff}}^{b}$ have studied using Theil-Sen
fits of the energies (E) and ZVZB-TRE forces (F) in VMC and DMC simulation:
intrinsic variances $\sigma^{2}$, autocorrelation times $\tau$, and the
autocorrelation renormalized variance $\sigma^{2}\tau$. The fitted
coefficients are presented in Table 10, and Figs. 13, 14, and 15.

Atom | Core | $Z_{\mathrm{eff}}$ | E (VMC) | E (DMC) | F (VMC) | F (DMC)
---|---|---|---|---|---|---
Li | He | 1 | 0.00018 | 0.0002 | 0.00013 | 0.00014
Be |  | 2 | 0.016 | 0.014 | 0.06 | 0.056
B |  | 3 | 0.041 | 0.045 | 1.1 | 2.3
C |  | 4 | 0.069 | 0.072 | 3.5 | 7.2
O |  | 6 | 0.25 | 0.26 | 35 | 94
Ne |  | 8 | 0.65 | 0.66 | 150 | 410
S |  | 14 | 2.9 | 3.3 | 2600 | 5000
Ar |  | 16 | 4.9 | 5.1 | 6300 | 14000
Na | Ne | 1 | 0.00015 | 0.00014 | 0.0002 | 0.0003
Mg |  | 2 | 0.0095 | 0.009 | 0.052 | 0.024
Al |  | 3 | 0.015 | 0.016 | 0.11 | 0.11
Si |  | 4 | 0.03 | 0.029 | 0.25 | 0.41
S |  | 6 | 0.095 | 0.098 | 1.9 | 2.1
K |  | 9 | 0.36 | 0.38 | 20 | 26
Ca |  | 10 | 0.5 | 0.49 | 38 | 49
Ti |  | 12 | 0.85 | 0.86 | 110 | 200
Fe |  | 16 | 2.2 | 2.4 | 930 | 1900
Zn |  | 20 | 4.5 | 4.7 | 3900 | 6800
Table 7: Intrinsic variances of energies (E) and ZVZB-TRE forces (F) of isolated atoms with VMC and DMC Atom | Core | $Z_{\mathrm{eff}}$ | ZVZB-TRE (z) | ZVZB-SW-TRE (x,y) | ZVZB-TRE (x,y) | ZVZB-SW-TRE (x,y)
---|---|---|---|---|---|---
B | He | 3 | 1.9 | 0.61 | 1.9 | 0.056
C |  | 4 | 13 | 4.5 | 8 | 0.55
N |  | 5 | 33 | 9 | 20 | 0.9
O |  | 6 | 60 | 12 | 60 | 1.1
Si |  | 12 | 2000 | 270 | 2000 | 0.53
S |  | 14 | 5100 | 700 | 4400 | 1.9
Table 8: Intrinsic variances of ZVZB-TRE and ZVZB-SW-TRE forces of homonuclear dimers with Helium core from VMC. Atom | Core | $Z_{\mathrm{eff}}$ | ZVZB-TRE (z) | ZVZB-SW-TRE (x,y) | ZVZB-TRE (x,y) | ZVZB-SW-TRE (x,y)
---|---|---|---|---|---|---
B | He | 3 | 3.8 | 0.77 | 3.5 | 0.095
C |  | 4 | 24 | 5.3 | 23 | 0.85
N |  | 5 | 66 | 12 | 47 | 1.4
O |  | 6 | 130 | 52 | 140 | 1.9
Si |  | 12 | 3400 | 320 | 3200 | 0.79
S |  | 14 | 8900 | 870 | 7700 | 1800
Table 9: Intrinsic variances of ZVZB-TRE and ZVZB-SW-TRE forces of homonuclear dimers with Helium core from DMC. Quantity | Core | Method | $a$ | $b$
---|---|---|---|---
$\sigma^{2}_{E}$ | He | VMC | 0.0012 | 3.0(5)
| He | DMC | 0.0011 | 3.0(4)
| Ne | VMC | 0.00036 | 3.1(2)
| Ne | DMC | 0.00038 | 3.1(2)
$\tau_{E}$ | He | VMC | 0.57 | 0.5(4)
| He | DMC | 240 | -1.1(0.2)
| Ne | VMC | 1.0 | 0.1(1)
| Ne | DMC | 430 | -1.1(2)
$\sigma^{2}_{E}\tau_{E}$ | He | VMC | 0.00040 | 3.8(7)
| He | DMC | 0.35 | 1.7(5)
| Ne | VMC | 0.00024 | 3.3(3)
| Ne | DMC | 0.19 | 1.9(2)
$\sigma^{2}_{F}$ | He | VMC | 0.0017 | 5.5(6)
| He | DMC | 0.0022 | 5.9(9)
| Ne | VMC | 0.00010 | 5.5(4)
| Ne | DMC | 0.000096 | 5.6(4)
$\tau_{F}$ | He | VMC | 1.5 | 0.5(5)
| He | DMC | 95 | -1.6(8)
| Ne | VMC | 0.47 | 0.5(3)
| Ne | DMC | 180 | -1.1(2)
$\sigma^{2}_{F}\tau_{F}$ | He | VMC | 0.0020 | 6.2(1.0)
| He | DMC | 0.068 | 4.7(6)
| Ne | VMC | 0.000073 | 5.7(5)
| Ne | DMC | 0.026 | 4.3(3)
Table 10: Fitting parameters to $aZ_{\mathrm{eff}}^{b}$ of intrinsic variances
of local energy $\sigma^{2}_{E}$ and ZVZB-TRE forces from VMC and DMC
simulations of isolated atoms with variable pseudized cores. 90% confidence
intervals of the exponents $b$ from Theil-Sen fits are given in parentheses.
The prefactors $a$ are in the units of Ha2 for energies and (Ha/Bohr)2 for
forces.  Figure 13: Autocorrection normalized variances $\sigma^{2}\tau$ of
DMC and VMC energies and ZVZB-TRE forces. Figure 14: Sample autocorrelation
times $\tau$ of DMC energies and ZVZB-TRE forces. Figure 15: Autocorrection
normalized variances $\sigma^{2}\tau$ of DMC and VMC ZVZB-SW-TRE and ZVZB-TRE
forces.

##  XII System sizes with equivalent statistical cost

System sizes with equivalent statistical costs to 64, 126 and 768 Carbon atoms
are presented based on ZVZB-TRE forces on isolated atoms from VMC (Table 11)
or DMC (Table 12) simulations. Similar data is presented based on comparison
of ZVZB-TRE forces and ZVZB-SW-TRE forces from VMC (Table 13) and DMC (Table
14). The cost isocontours for VMC are presented and discussed in Sec. V of the
main body, and similar presentation for DMC is given in Fig. 16.

| C64 |  | C216 |  | C768 |
---|---|---|---|---|---|---
$Z_{\mathrm{eff}}$ | He | Ne | He | Ne | He | Ne
1 | 3150 | 2656 | 10634 | 8967 | 37810 | 31883
2 | 410 | 536 | 1385 | 1810 | 4926 | 6436
3 | 101 | 258 | 342 | 872 | 1216 | 3103
4 | 63 | 168 | 215 | 568 | 767 | 2022
5 | 38 | 106 | 131 | 358 | 466 | 1275
6 | 27 | 73 | 91 | 249 | 326 | 888
7 | 20 | 51 | 68 | 173 | 241 | 618
8 | 15 | 37 | 52 | 127 | 186 | 453
9 | 12 | 29 | 41 | 100 | 146 | 356
10 | 9 | 22 | 32 | 76 | 117 | 273
11 | 7 | 16 | 26 | 56 | 95 | 201
12 | 6 | 13 | 22 | 44 | 80 | 156
13 | 5 | 10 | 19 | 34 | 68 | 121
14 | 4 | 8 | 16 | 27 | 59 | 98
15 | 3 | 6 | 12 | 23 | 46 | 82
16 | 3 | 5 | 10 | 20 | 37 | 71
17 | - | 5 | - | 17 | - | 63
18 | - | 4 | - | 15 | - | 56
19 | - | 4 | - | 14 | - | 50
20 | - | 3 | - | 12 | - | 46
Table 11: Numbers of Helium (He) and Neon (Ne) core atoms with variable $Z_{\mathrm{eff}}$ that can be computed with computational cost equivalent of 64, 216 or 768 Carbon (He) atoms, using ZVZB-TRE force estimator for VMC, according to data of isolated atoms. | C64 |  | C216 |  | C768 |
---|---|---|---|---|---|---
$Z_{\mathrm{eff}}$ | He | Ne | He | Ne | He | Ne
1 | 1124 | 1191 | 3796 | 4020 | 13499 | 14295
2 | 206 | 285 | 695 | 963 | 2472 | 3425
3 | 88 | 146 | 299 | 495 | 1065 | 1761
4 | 63 | 92 | 215 | 312 | 767 | 1111
5 | 43 | 60 | 145 | 205 | 517 | 730
6 | 30 | 44 | 101 | 148 | 362 | 529
7 | 17 | 33 | 60 | 113 | 215 | 404
8 | 13 | 26 | 45 | 90 | 161 | 320
9 | 11 | 22 | 38 | 74 | 138 | 265
10 | 10 | 18 | 34 | 62 | 123 | 221
11 | 9 | 15 | 31 | 52 | 111 | 185
12 | 8 | 12 | 28 | 43 | 101 | 154
13 | 7 | 10 | 25 | 34 | 91 | 122
14 | 6 | 8 | 23 | 27 | 81 | 98
15 | 4 | 6 | 16 | 23 | 59 | 83
16 | 3 | 6 | 13 | 20 | 46 | 73
17 | - | 5 | - | 18 | - | 65
18 | - | 4 | - | 16 | - | 59
19 | - | 4 | - | 15 | - | 54
20 | - | 4 | - | 14 | - | 50
Table 12: Numbers of Helium (He) and Neon (Ne) core atoms with variable $Z_{\mathrm{eff}}$ that can be computed with computational cost equivalent of 64, 216 or 768 Carbon (He) atoms, using ZVZB-TRE force estimator for DMC, according to data of isolated atoms. | C64 |  | C216 |  | C768 |
---|---|---|---|---|---|---
$Z_{\mathrm{eff}}$ | ZVZB-TRE | ZVZB-SW-TRE | ZVZB-TRE | ZVZB-SW-TRE | ZVZB-TRE | ZVZB-SW-TRE
3 | 90 | 118 | 305 | 399 | 1084 | 1418
4 | 38 | 63 | 130 | 215 | 464 | 767
5 | 28 | 44 | 94 | 151 | 336 | 539
6 | 19 | 34 | 64 | 116 | 229 | 413
7 | 13 | 25 | 45 | 86 | 162 | 307
8 | 10 | 19 | 34 | 65 | 123 | 234
9 | 8 | 15 | 27 | 52 | 98 | 185
10 | 6 | 12 | 22 | 42 | 81 | 151
11 | 5 | 10 | 19 | 35 | 69 | 127
12 | 5 | 9 | 16 | 30 | 60 | 109
13 | 4 | 7 | 14 | 23 | 50 | 84
14 | 3 | 5 | 12 | 19 | 43 | 68
Table 13: Numbers of Helium-core atoms with variable $Z_{\mathrm{eff}}$ that can be computed with computational cost equivalent of 64, 216 or 768 Carbon atoms, using ZVZB-TRE or ZVZB-TRE-SW force estimators for VMC, according to data of homonuclear dimers. | C64 |  | C216 |  | C768 |
---|---|---|---|---|---|---
$Z_{\mathrm{eff}}$ | ZVZB-TRE | ZVZB-SW-TRE | ZVZB-TRE | ZVZB-SW-TRE | ZVZB-TRE | ZVZB-SW-TRE
3 | 97 | 107 | 328 | 362 | 1167 | 1287
4 | 57 | 63 | 194 | 215 | 689 | 767
5 | 39 | 46 | 131 | 156 | 469 | 557
6 | 29 | 38 | 98 | 129 | 349 | 460
7 | 20 | 31 | 68 | 105 | 243 | 374
8 | 15 | 25 | 50 | 85 | 181 | 305
9 | 12 | 21 | 40 | 71 | 145 | 253
10 | 10 | 17 | 34 | 60 | 122 | 214
11 | 8 | 15 | 30 | 51 | 107 | 184
12 | 8 | 13 | 27 | 45 | 97 | 160
13 | 7 | 11 | 25 | 38 | 89 | 136
14 | 6 | 9 | 23 | 32 | 83 | 116
Table 14: Numbers of Helium-core atoms with variable $Z_{\mathrm{eff}}$ that
can be computed with computational cost equivalent of 64, 216 or 768 Carbon
atoms, using ZVZB-TRE or ZVZB-TRE-SW force estimators for DMC, according to
data of homonuclear dimers. Figure 16: Cost isocontours for DMC simulations
with variable system sizes $N$ and $Z_{\mathrm{eff}}$ based on ZVZB-TRE forces
of isolated atoms (left) and ZVZB-TRE and ZVZB-SW-TRE forces on homonuclear
dimers (right)

##  XIII Mapping between bond stiffness and force uncertainty

A mapping between the bond stiffness (force constant) of the dimers in this
study and force uncertainty to reach the precision of $\sim 0.01$ Å is
presented in Fig. 17.

Figure 17: Mapping between stiffness $k$ and the maximum force uncertainty
required for spatial accuracy of $\sim 0.01$ Å.

##  XIV Implementation Details of Forces for Non-Local Pseudopotentials

QMCPACK uses the following form of the non-local pseudopotential, following
the normalization and quadrature rules of Mitas and Ceperley[Mitas1991]:

$\displaystyle\frac{\hat{V}_{NL}\Psi_{T}}{\Psi_{T}}$ $\displaystyle=$
$\displaystyle\sum_{I}\sum_{i}\sum_{\ell}\left((2\ell+1)v_{l}(r_{iI})\sum_{\alpha}\left[w_{\alpha}P_{\ell}\left(\cos\theta^{\alpha}_{iI}\right)\frac{\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}\right]\right)$
(6) $\displaystyle\mathbf{r}_{iI}$ $\displaystyle=$
$\displaystyle\mathbf{r}_{i}-\mathbf{R}_{I}$ (7) $\displaystyle r_{iI}$
$\displaystyle=$ $\displaystyle|\mathbf{r}_{i}-\mathbf{R}_{I}|$ (8)
$\displaystyle\mathbf{q}^{\alpha}_{iI}$ $\displaystyle=$
$\displaystyle\mathbf{R}_{I}+r_{iI}\mathbf{u}_{\alpha}$ (9)
$\displaystyle\cos\theta^{\alpha}_{iI}$ $\displaystyle=$
$\displaystyle(\mathbf{r}_{iI}\cdot\mathbf{u}_{\alpha})r_{iI}^{-1}$ (10)

Lower case vectors and indices denote electron coordinates, upper case vectors
and indices denote ionic coordinates. $P_{\ell}(x)$ is a Legendre polynomial,
$w_{\alpha}$ is a weight given by whichever quadrature rule is being used, and
$v_{\ell}(r)$ the radial potential for the $\ell$ angular momentum channel.

We now consider the derivative of this expression w.r.t the ion displacement
$R_{J,\gamma}$. Formally,

$\displaystyle\scriptstyle\frac{\partial}{\partial
R_{J,\gamma}}\frac{\hat{V}_{NL}\Psi_{T}}{\Psi_{T}}$ $\displaystyle=$
$\displaystyle\scriptstyle\sum_{I}\sum_{i}\sum_{\ell}(2\ell+1)\frac{\partial}{\partial
R_{J,\gamma}}v_{l}(r_{iI})\sum_{\alpha}\left[w_{\alpha}P_{\ell}\left(\cos\theta^{\alpha}_{iI}\right)\frac{\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}\right]$
(11) $\displaystyle+$
$\displaystyle\scriptstyle(2\ell+1)v_{l}(r_{iI})\sum_{\alpha}\left[w_{\alpha}\frac{\partial}{\partial
R_{J,\gamma}}P_{\ell}\left(\cos\theta^{\alpha}_{iI}\right)\frac{\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}\right]$
(12) $\displaystyle+$
$\displaystyle\scriptstyle(2\ell+1)v_{l}(r_{iI})\sum_{\alpha}\left[w_{\alpha}P_{\ell}\left(\cos\theta^{\alpha}_{iI}\right)\frac{\partial}{\partial
R_{J,\gamma}}\frac{\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}\right]$
(13)

First, note that the first two terms vanish unless $I=J$. The last term can
have Pulay like contributions coming from explicit $\Psi_{T}$ dependence on
$i_{J,\gamma}$. Before crunching on evaluating this whole expression, lets
take some intermediate derivatives. Using Einstein notation, with the big
caveat that capital indices are not summed over (e.g. $I$ and $J$):

$\frac{\partial}{\partial
R_{J,\gamma}}r_{{iI},\mu}=-\delta_{IJ}\delta_{\gamma\mu}$ (15)
$\frac{\partial}{\partial
R_{J,\gamma}}r_{iI}=-\delta_{IJ}\frac{r_{iI,\gamma}}{r_{iI}}$ (16)
$\frac{\partial}{\partial
R_{J,\gamma}}r_{iI,\mu}u^{\alpha}_{\mu}=-\delta_{IJ}\delta_{\gamma\mu}u^{\alpha}_{\mu}=-\delta_{IJ}u^{\alpha}_{\gamma}$
(17)

These allow us to evaluate the derivatives of the quadrature points and
angles:

$\frac{\partial}{\partial
R_{J,\gamma}}q^{\alpha}_{iI,\mu}=\delta_{IJ}\left[\delta_{\gamma\mu}-\frac{r_{iI,\gamma}u^{\alpha}_{\mu}}{r_{iI}}\right]$
(18) $\frac{\partial}{\partial
R_{J,\gamma}}\cos\theta^{\alpha}_{iI}=\frac{\partial}{\partial
R_{J,\gamma}}\left(r_{iI,\mu}u^{\alpha}_{\mu}r_{iI}^{-1}\right)=-\delta_{IJ}r^{-1}_{iI}\left[u^{\alpha}_{\gamma}-\cos\theta^{\alpha}_{iI}\frac{r_{iI,\gamma}}{r_{iI}}\right]$
(19)

Now let’s consider what happens to the wavefunction derivative:

$\frac{\partial}{\partial
R_{J,\gamma}}\frac{\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}=\frac{\frac{\partial}{\partial
R_{J,\gamma}}\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}-\frac{\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}\frac{\frac{\partial}{\partial
R_{J,\gamma}}\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}$
(20)

The ion derivative in the second term is a standard Pulay term and is
evaluated as such. The first term is complicated by the potential explicit
dependence of $\Psi_{T}$ on the ion coordinates (as in the second term), but
also by the ion dependence of the quadrature points. The total derivative of
the numerator is:

$\displaystyle\scriptstyle\frac{d}{dR_{J,\gamma}}\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)$
$\displaystyle=$ $\displaystyle\scriptstyle\frac{\partial}{\partial
R_{J,\gamma}}\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)+\partial_{i,\mu}\Psi_{T}(\ldots
q^{\alpha}_{iI,\mu}\ldots)\frac{\partial q^{\alpha}_{iI,\mu}}{\partial
R_{J,\gamma}}$ (21) $\displaystyle=$
$\displaystyle\scriptstyle\frac{\partial}{\partial
R_{J,\gamma}}\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iI}\ldots)+\delta_{IJ}\partial_{i,\mu}\Psi_{T}(\ldots
q^{\alpha}_{iI,\mu}\ldots)\left[\delta_{\gamma\mu}-\frac{r_{iI,\gamma}u^{\alpha}_{\mu}}{r_{iI}}\right]$
(22)

Now for the radial potential derivative:

$\frac{\partial}{\partial
R_{J,\gamma}}v_{l}(r_{iI})=-\delta_{IJ}v^{\prime}_{l}(r_{iI})\frac{r_{iI,\gamma}}{r_{iI}}$
(23)

The legendre polynomial…

$\displaystyle\frac{\partial}{\partial
R_{J,\gamma}}P_{\ell}(\cos\theta^{\alpha}_{iI})$ $\displaystyle=$
$\displaystyle
P^{\prime}_{\ell}(\cos\theta^{\alpha}_{iI})\frac{\partial}{\partial
R_{J,\gamma}}\cos\theta^{\alpha}_{iI}$ (24) $\displaystyle=$
$\displaystyle-\delta_{IJ}P^{\prime}_{\ell}(\cos\theta^{\alpha}_{iI})r^{-1}_{iI}\left[u^{\alpha}_{\gamma}-\cos\theta^{\alpha}_{iI}\frac{r_{iI,\gamma}}{r_{iI}}\right]$
(25)

The Legendre Polynomials in QMCPACK are computed via the recursion relation:

$(n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x)$ (26)

$P_{0}(x)=1$, and $P_{1}(x)=x$.

We can take the derivative of the above equation to obtain a recursion
relation for the derivatives of the Legendre polynomials:

$(n+1)P^{\prime}_{n+1}(x)=(2n+1)\left[xP^{\prime}_{n}(x)+P_{n}(x)\right]-nP^{\prime}_{n-1}(x)$
(27)

With of course, $P^{\prime}_{0}(x)=0$, and $P^{\prime}_{1}(x)=1$.

We now take all these results and plug them in:

$\displaystyle\scriptstyle\frac{\partial}{\partial
R_{J,\gamma}}\frac{\hat{V}_{NL}\Psi_{T}}{\Psi_{T}}$ $\displaystyle=$
$\displaystyle\scriptstyle-\sum_{i}\sum_{\ell}(2\ell+1)\underbrace{\scriptstyle
v^{\prime}_{l}(r_{iJ})\frac{r_{iJ,\gamma}}{r_{iJ}}}\sum_{\alpha}\left[w_{\alpha}P_{\ell}\left(\cos\theta^{\alpha}_{iJ}\right)\frac{\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iJ}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}\right]$
(28) $\displaystyle-$
$\displaystyle\scriptstyle\sum_{i}\sum_{\ell}(2\ell+1)v_{l}(r_{iJ})\sum_{\alpha}\left[w_{\alpha}\underbrace{\scriptstyle
P^{\prime}_{\ell}(\cos\theta^{\alpha}_{iJ})r^{-1}_{iJ}\left(u^{\alpha}_{\gamma}-\cos\theta^{\alpha}_{iJ}\frac{r_{iJ,\gamma}}{r_{iJ}}\right)}\frac{\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iJ}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}\right]$
(29) $\displaystyle+$
$\displaystyle\scriptstyle\sum_{i}\sum_{\ell}(2\ell+1)v_{l}(r_{iJ})\sum_{\alpha}\left[w_{\alpha}P_{\ell}\left(\cos\theta^{\alpha}_{iJ}\right)\underbrace{\scriptstyle\frac{\partial_{i,\mu}\Psi_{T}(\ldots\mathbf{q}^{\alpha}_{iJ}\ldots)}{\Psi_{T}(\ldots\mathbf{r}_{i}\ldots)}\left(\delta_{\gamma\mu}-\frac{r_{iJ,\gamma}u^{\alpha}_{\mu}}{r_{iJ}}\right)}\right]$
(30) $\displaystyle+$
$\displaystyle\scriptstyle\frac{\hat{V}_{NL}\frac{\partial\Psi_{T}}{\partial
R_{J,\gamma}}}{\Psi_{T}}-\left(\frac{\hat{V}_{NL}\Psi_{T}}{\Psi_{T}}\right)\frac{\frac{\partial\Psi_{T}}{\partial
R_{J,\gamma}}}{\Psi_{T}}$ (31)

I’ve put braces under the terms that need to be computed beyond what’s already
computed for the nonlocal pseudopotential energy evaluation. In the last line,
we have first what’s essentially a Pulay term, and the last is a ZVZB like
term.

##  XV Acknowledgements

This research has been provided by the US Department of Energy, Office of
Science, Basic Energy Sciences, Materials Sciences and Engineering Division,
as part of the Computational Materials Sciences Program and Center for
Predictive Simulation of Functional Materials. This research used resources of
the Compute and Data Environment for Science (CADES) at the Oak Ridge National
Laboratory, which is supported by the Office of Science of the U.S. Department
of Energy under Contract No. DE-AC05-00OR22725.

This manuscript has been authored in part by UT-Battelle, LLC, under contract
DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government
retains and the publisher, by accepting the article for publication,
acknowledges that the US government retains a nonexclusive, paid-up,
irrevocable, worldwide license to publish or reproduce the published form of
this manuscript, or allow others to do so, for US government purposes. DOE
will provide public access to these results of federally sponsored research in
accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-
public-access-plan).

Sandia National Laboratories is a multimission laboratory managed and operated
by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned
subsidiary of Honeywell International Inc., for the U.S. Department of
Energy’s National Nuclear Security Administration under contract DE-NA0003525.
This paper describes objective technical results and analysis. Any subjective
views or opinions that might be expressed in the paper do not necessarily
represent the views of the U.S. Department of Energy or the United States
Government.

The data that support the findings of this study are openly available in The
Materials Data Facility at http://doi.org/[doi], reference number [reference
number].

##  XVI References