# Diffusion Monte Carlo evaluation of disiloxane linearisation barrier

Adie Tri Hanindriyo1,∗<EMAIL_ADDRESS>Amit Kumar Singh Yadav2 Tom Ichibha3
Ryo Maezono4 Kousuke Nakano4,5 Kenta Hongo6 1 School of Materials Science,
JAIST, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan  2 Department of
Electrical Engineering, Indian Institute of Technology Gandhinagar, Palaj
382355, Gujarat, India  3 Materials Science and Technology Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, USA  4 School of Information
Science, JAIST, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan  5 Scuola
Internazionale Superiore di Studi Avanzati(SISSA), via Bonomea, 265-34136
Trieste ITALY.  6 Research Center for Advanced Computing Infrastructure,
JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan

###### Abstract

The disiloxane molecule is a prime example of silicate compounds containing
the Si-O-Si bridge. The molecule is of significant interest within the field
of quantum chemistry, owing to the difficulty in theoretically predicting its
properties. Herein, the linearisation barrier of disiloxane is investigated
using a fixed-node diffusion Monte Carlo (FNDMC) approach, which is currently
the most reliable ab initio method in accounting for an electronic
correlation. Calculations utilizing the density functional theory (DFT) and
the coupled cluster method with single and double substitutions, including
noniterative triples (CCSD(T))are carried out alongside FNDMC for comparison.
Two families of basis sets are used to investigate the disiloxane
linearisation barrier - Dunning’s correlation-consistent basis sets cc-pV$x$Z
($x=$ D, T, and Q) and their core-valence correlated counterparts, cc-pCV$x$Z.
It is concluded that FNDMC successfully predicts the disiloxane linearisation
barrier and does not depend on the completeness of the basis sets as much as
DFT or CCSD(T), thus establishing its suitability.

##  1 INTRODUCTION

The simplest molecule containing the Si-O-Si bond is disiloxane or Si2H6O.
Also called disilyl ether, its structure is a single Si-O-Si bond terminated
by three H atoms at each end (H3Si-O-SiH3). There have been numerous studies
investigating the Si-O-Si bond, particularly owing to its importance in the
modelling of silica compounds, which are the most abundant constituent of the
Earth’s crust. Most importantly, silica compounds range in function from
glasses to quartz crystals, both of which comprise large sectors in industry.
In some studies, disiloxane has been used as a sealant and as a component in
cosmetics, or as a prototype region of a zeolite or clay substrate for
applications ranging from catalysis to prebiotic synthesis 1.

Experimental evaluations of the Si-O-Si bond indicate an anharmonic bending
potential with a low linearisation barrier, which makes it quite difficult to
attain sufficient accuracy in such measurements 2, 3. Despite the significant
volume of previous studies dedicated to the Si-O-Si bond 4, 1, 5, 6, 7, 8, 9,
10, the properties of Si-O-Si bond obtained in most of these studies are not
consistent with each other. In ab initio studies, in particular, multiple
calculation methods have resulted in different values for the bond angle and
length 11, 12, 9, as well as the linearisation energy 4, 5, 8 and Si-O-Si
potential energy surface 1, 9, among other factors These properties and the
bond geometry itself are shown to be sensitive to the choice of basis set and
level of theory (the former more than the latter) according to at least one
previous study 8.

To narrow down the possibilities, it would be ideal to employ the most
reliable methods at our disposal. Quantum Monte Carlo (QMC), currently the
most reliable of the many-body calculation methods, is expected to provide a
reasonable and reliable prediction. 13 We used the fixed-node diffusion Monte
Carlo (FNDMC) method, which has been widely and successfully applied to
several molecular systems 14, 15, 16, 17, 18, 19, 20, 21. Although FNDMC
results are also affected by the choice of basis sets 18, 22, we note that the
basis-set dependence is considerably different from that for an SCF-based
method, such as the density functional theory (DFT) and molecular orbital (MO)
methods. In SCF-based approaches, the choice of basis set affects both the
amplitude and the nodal positions of the corresponding many-body wavefunctions
(although the methods do not explicitly employ a many-body wavefunction
condition). With FNDMC, by contrast, the choice only affects the nodal
positions. The amplitude can be automatically adjusted such that its shape may
approach that of the exact solution as closely as possible under a restriction
with a fixed nodal position 13, 23. A typical example is the description of
electron-nucleus cusps. 24 Even when using such a poor basis set that cannot
describe the singularity of the cusp based on its analytical form, an initial
guess is applied for further numerical evolution driven by the FNDMC, making
the amplitude at the nucleus positions singular with a cusp 13, 23. Owing to
this self-healing property of the amplitude, the dependence on the choice of
basis sets in FNDMC becomes considerably weaker. That is, the bias arising
from the choice is reduced more than that for SCF-based ones, which is a
difficulty faced by the present systems. 8 Note that “cuspless” basis sets
cause a singularity of the local energy in FNDMC, thus resulting in numerical
instability. However, this singularity can be easily remedied by introducing
the cusp correction proposed by Ma et al 24.

In this study, we applied FNDMC to investigate the basis-set dependence of the
linearization energy of a disiloxane molecule in comparison with other ab
initio methods including DFT and CCSD(T), as well as empirical measurements
from earlier studies conducted on disiloxane 25, 2, 3. The FNDMC depends on
the basis set implicitly through its fixed nodal surface provided by the
Slater determinant whose orbitals are expanded in terms of the basis set,
whereas DFT and CCSD(T) strongly depend on the choice of basis set. The cc-
pV$x$Z 26, 27, 28 and cc-pCV$x$Z 29 basis sets ($x=$ D, T, Q), which are
standard correlation methods, were examined in the present study and then
applied to all the ab initio calculations to investigate the basis-set
dependence of linearisation energy evaluation.

##  2 MODEL AND METHODOLOGY

Our target property is simply a linearisation barrier of disiloxane Si2H6O
between “linear” and “non-linear” (bent) structures. As can be seen from
Figure 1, the linear structure possesses an eclipsed ${\rm D_{3h}}$ symmetry 7
and a non-linear structure is a bent conformation of the ${\rm C_{2v}}$
symmetry 30. For the two fixed structures, the barrier, $\Delta
E_{\rm{barrier}}$, is defined as the energy difference between the linear and
non-linear structures:

$\displaystyle\Delta E_{\rm barrier}={E_{\rm linear}}-{E_{\rm delinear}},$ (1)

where ${E_{\rm linear}}$ and ${E_{\rm delinear}}$ are the (electronic) total
energies of the linear and non-linear structures, respectively.

Figure 1: Linear and non-linear molecular structures of disiloxane, Si2H6O.

The accuracy of computed $\Delta E_{\rm{barrier}}$ values can be calibrated by
referring to experimentally observed $\Delta E_{\rm{barrier}}$ values. To
compare our computational results with the experimental results, however, we
note that the experimental $\Delta E_{\rm{barrier}}$ involves the zero-point
energy (ZPE) contribution, whereas our computational $\Delta E_{\rm{barrier}}$
value considers only the electronic contribution. The ZPE contributions can
also be evaluated at each level of theory, although their accuracy depends on
methods adopted and the basis sets. To investigate the basis-set dependence of
$\Delta E_{\rm{barrier}}$, we simply consider the electronic $\Delta
E_{\rm{barrier}}$. Thus, the experimental $\Delta E_{\rm{barrier}}$ values are
corrected by adopting a theoretical estimate from Koput 6. We address this
issue in more detail later.

In the present study, the above two energies ${E_{\rm linear}}$ and ${E_{\rm
delinear}}$ were computed through various combinations of (1) ab initio
methods and (2) basis sets

(1) Our methods applied are DFT with B3LYP exchange-correlation functional 31
(DFT-B3LYP), CCSD(T) 32, and FNDMC 13 with trial wavefunctions generated from
DFT-B3LYP. Within the framework of DFT, DFT-B3LYP is a standard method for
covalent systems. Within the correlated methods, CCSD(T) is known to be the
“gold standard” in quantum chemistry. A variety of methods and basis sets were
chosen in line with the previous results 8 indicating dependence of the Si-O-
Si bond description on such choice. FNDMC is known to be comparable with
CCSD(T); however, for the first time, we applied FNDMC to an evaluation of
$\Delta E_{\rm{barrier}}$.

(2) Our basis sets are a family of Dunning’s correlation-consistent basis sets
(cc-pV$x$Z; $x$ = D, T, Q) 26, 27, 28 and their core-valence correlated
variants (cc-pCV$x$Z; $x$ = D, T, Q) 29. The cc-pV$x$Z and cc-pCV$x$Z basis
sets were originally developed for correlated methods such as CCSD(T), but
have been found to be appropriate even for DFT-B3LYP and FNDMC in properly
reproducing the dynamic electron correlation. 26 In particular, the
polarisation functions included implicitly in cc-pV$x$Z are essential for
properly describing the Si-O-Si bond. To describe the correlation effects more
precisely, the present study considers the dynamic correlation between core
and valence electrons by applying the cc-pCV$x$Z basis sets, which can
reproduce the core-valence correlation by minimising the difference in the
correlation energies between all-electron and valence-only (using
pseudopotentials) calculations. 33 The number of basis functions entailed in
each basis set is shown in Table 1. As shown, accounting for the core-valence
correlation is more expensive for increasingly complete orbital and
polarisation functions, with a quadruple zeta-level basis set adding close to
130 more basis functions to the standard correlation-consistent basis set. The
number of basis functions is expected to largely correlate with the
reliability of the calculations, particularly for the core-valence variants
pertaining to all-electron calculations.

Table 1: List of basis functions Basis set | Linear basis functions
---|---
cc-pVDZ |  080
cc-pCVDZ | 102
cc-pVTZ | 182
cc-pCVTZ | 245
cc-pVQZ | 353
cc-pCVQZ | 482

In the present study we obtained the linear and delinear structures using the
second-order Møller-Plesset perturbation theory (MP2) with the cc-pVQZ basis
set (MP2/cc-pVQZ level), and previous studies conducted at the MP2/cc-pVQZ
level of theory 11, 7 have established a reliable favourability toward a
delinear structure and provide good agreement for the Si-O-Si angle in
accordance with experimental results. 30 Modelling of the Si-O-Si angle in
particular has been heavily emphasised in previous theoretical studies for
both disiloxane 8 and pyrosilisic acid 9, 10. This necessitates even larger
numbers of basis functions, which means that the cc-pCVQZ basis set would
necessitate the highest number of basis functions by far. The two optimised
structures at the MP2/cc-pVQZ level of theory were used in this work to
calculate all $\Delta E_{\rm{barrier}}$ values, i.e., common to all levels of
theory, where the linear structure has a Si-O-Si angle of 180∘ and the
delinear structure has an optimised Si-O-Si angle of 146.8∘ comparable with
the experimental value of 144.1∘ 30. The other structural parameters are given
in the Supporting Information, and are also in good agreement with
experiments.

The present study adopted no pseudopotential calculations, but rather all-
electron calculations with a total of 42 electrons for a single disiloxane
molecule. This molecular system is not so large that the CCSD(T) calculation
within the frozen core approximation is feasible, despite the computational
cost of CCSD(T) scaling as $N^{7}$, where $N$ is the number of electrons in
the system. By contrast, the DFT cost scales with $N^{3}$ (or less) and is
therefore unimportant. Similar to DFT, the FNDMC cost scales as $N^{3}$,
although the prefactor of FNDMC is much larger than that of DFT. This is
because a vast number of random sampling points are required to obtain a
sufficiently small error bar (sub-chemical accuracy of $\sim 0.1$ kcal/mol) to
calibrate a small $\Delta E_{\rm{barrier}}$ ($\sim 0.5$ kcal/mol). Recent
parallel computers, however, enable us to apply FNDMC to evaluate such a tiny
$\Delta E_{\rm{barrier}}$, because its algorithm is intrinsically parallel.

In our FNDMC calculations, we adopted Slater-Jastrow type wavefunctions as
their fixed-node trial wavefunctions 34. Molecular orbitals entering the
single Slater determinants were generated by DFT-B3LYP with various types of
basis sets. The Jastrow factor consists of one-, two-, and three-body terms 35
including 88 variational parameters in total. These parameters were optimised
through a variance minimisation scheme 36, and only the two-body term holds
the electron-electron cusp condition 37. The electron-nucleus cusp condition,
which is a short-range one-body correlation effect, is satisfied by the cusp
correction scheme applied to the Gaussian basis sets 24 instead of imposing
the cusp condition on the one-body term. Note that the Jastrow factor does not
change the (fixed) nodal surfaces and is responsible for the numerical
stability in FNDMC. However, the quality of the nodal surfaces determines the
accuracy of the FNDMC energies in terms of the fixed-node variational
principles 38. Accordingly, the FNDMC accuracy depends implicitly on the basis
set adopted, which is used to expand the molecular orbitals entering the
Slater determinant. In addition to the fixed-node error, another source of
bias in actual FNDMC calculations arises from the short-time approximation
with finite (small) timesteps 39. To remove this bias, it is common to use
multiple timesteps to make the linear regression and obtain calculation
results for $\delta t\rightarrow 0$. This regression is an approximation of
the theoretical $\delta t=0$ result. The present study considers both linear
and quadratic extrapolations.

The software package Gaussian09 40 was used to conduct the MP2 (geometry
optimisation), DFT-B3LYP, and CCSD(T) calculations, whereas the CASINO 41 code
was used to apply the FNDMC calculations. An electron-nucleus cusp correction
scheme 24 for Gaussian orbitals was utilised in the all-electron FNDMC
calculation in CASINO. In addition, cc-pCV$x$Z ($x$ = D, T, Q) basis sets for
Si and O as well as the corresponding cc-pV$x$Z versions for the H atoms
during the same calculations were obtained from the online Basis Set Exchange
library 42. In FNDMC for both structures, the number of target population was
11,520, and the number of steps in the imaginary-time evolution was set to
2,000 and 500,000 for equilibrated and accumulated phases, respectively. In
addition, we carried out FNDMC calculations with different timesteps of 0.01,
0.005, and 0.001 to remove the short-time bias (see SI for more details).

##  3 RESULTS and DISCUSSION

###  3.1 Geometry and ZPE contribution

Several studies have been conducted on the geometry of the disiloxane
molecule, although most ab initio methods tested did not manage to replicate
the available experimental results. Rather than the Si-O bond length, the Si-
O-Si bond angle and linearisation barrier have been found to be greatly
dependent on the choice of basis set used to represent the wavefunction 8.
This study focuses on the linearisation barrier of disiloxane, taking the same
optimised geometries for all calculations, partly owing to the notorious
difficulty in optimising the geometries in FNDMC.

Table 2: Linearisation barrier values obtained from theoretical calculations and experiments. Its ZPE correction and ZPE-corrected Raman value are also given. Computational values ($\Delta\epsilon_{0}$) are to be compared with a ZPE-corrected experimental value ($\Delta\epsilon_{1}-\Delta\mathrm{ZPE}$); see the text for the definition of notation, sign, etc. . Basis set |  $\Delta\epsilon_{0}$ [kcal/mol]
---|---
DFT-B3LYP | CCSD(T) |  | FNDMC
cc-pVDZ |  -0.80 |  -1.44 |  |  -0.40 $\pm$ 0.15
cc-pCVDZ |  -0.76 |  -1.42 |  |  -0.54 $\pm$ 0.12
cc-pVTZ |  -0.18 |  -0.46 |  |  -0.47 $\pm$ 0.14
cc-pCVTZ | -0.01 |  -0.32 |  |  -0.31 $\pm$ 0.11
cc-pVQZ |  -0.18 |  -0.44 |  |  -0.26 $\pm$ 0.10
cc-pCVQZ |  -0.06 |  -0.35 |  |  -0.36 $\pm$ 0.10
$\Delta\epsilon_{1}$ [kcal/mol] |  |  $\Delta\epsilon_{1}-\Delta$ZPE [kcal/mol]
Far IR spectrum 25 | IR-Raman (solid) 2 | Raman 3 |  | ZPE-corrected Raman ($\Delta$ZPE) 6
1.1-1.4 | 0.32 | 0.3 |  | 0.36 (-0.06)

The computational and experimental linearisation barriers of disiloxane,
$\Delta E_{\rm{barrier}}$, defined in Eq. (1) are listed in Table 2 and
plotted in Figure 2 as a visual guide. It has been commonly observed
experimentally that the non-linear (bent) structure of disiloxane is
energetically more favourable than a linear structure, which translates to a
positive linearisation barrier. All three experimental results seem to agree
on this point, and 25 in particular reports a higher linearisation barrier (at
1.1 to 1.4 kcal/mol) than the other two results 2, 3, which report a barrier
of approximately 0.3 kcal/mol. Both of these measurements are more recent than
the first and achieve good consilience with the DFT predictions from the
highest quality basis sets 8. Therefore, it is reasonable to infer that a
linearisation barrier of approximately 0.3 kcal/mol is a reliable value for
disiloxane.

Figure 2: Linearisation barrier of disiloxane calculated in this work.

It needs to be stressed, however, that the experimentally measured
linearisation barrier may not be comparable to the ground state values
calculated from the first principles. The ground state energy obtained through
ab initio calculations are physically unobtainable experimentally because of
the ZPE, i.e. the difference between the ground state and the lowest energy
vibrational state. Even at absolute zero temperature, the lowest energy level
achievable is a vibrational state $\epsilon_{1}$ instead of the electronic
ground state $\epsilon_{0}$:

$\displaystyle\epsilon_{1}=\epsilon_{0}+{\rm ZPE}$ (2)

Consequentially, any energetic barriers measured in the experiments is at best
the difference between the lowest vibrational states $\Delta\epsilon_{1}$,
whereas energetic barriers calculated through ab initio methods are from the
electronic ground states $\Delta\epsilon_{0}$. Therefore, a comparison between
energetic barriers obtained from the theoretical calculations and experimental
measurements must also account for the difference in ZPE. The energetic
barrier of a transition from quantum state A to state B is calculated as
follows:

$\displaystyle\epsilon^{\rm B}_{1}-\epsilon^{\rm A}_{1}$ $\displaystyle=$
$\displaystyle(\epsilon^{\rm B}_{0}+{\rm ZPE}^{\rm B})-(\epsilon^{\rm
A}_{0}+{\rm ZPE}^{\rm A})$ $\displaystyle\epsilon^{\rm B}_{1}-\epsilon^{\rm
A}_{1}$ $\displaystyle=$ $\displaystyle(\epsilon^{\rm B}_{0}-\epsilon^{\rm
A}_{0})+({\rm ZPE}^{\rm B}-{\rm ZPE}^{\rm A})$
$\displaystyle\Delta\epsilon_{1}$ $\displaystyle=$
$\displaystyle\Delta\epsilon_{0}+\Delta{\rm ZPE}$
$\displaystyle\Delta\epsilon_{0}$ $\displaystyle=$
$\displaystyle\Delta\epsilon_{1}-\Delta{\rm ZPE}$ (3)

The difference in ZPE $\Delta{\rm ZPE}$ between the initial and final states A
$\to$ B is usually considered insignificant. However, barrier height
discrepancies on the order of 0.1 kcal/mol might well be caused by this term.

The preceding theoretical study by Koput 6 estimates a $\Delta$ZPE value of
$-20$ cm${}^{-1}\approx-0.06$ kcal/mol, which in line with Equation 3.1 should
result in a higher ground state linearisation barrier than the experimentally
measured result. Therefore, this study treats the ground state linearisation
barrier of 0.36 kcal/mol as a reasonably accurate “exact” linearisation
barrier for a point of comparison with ab initio calculations. This value is
referred to as “ZPE-corrected Raman” in Table 2 and based on a dotted line in
the figures herein, as a point of comparison. ZPE correction has not been
considered in ab initio comparisons or based on experimental results, which
have cited 0.3 kcal/mol as the point of comparison. 8

###  3.2 DFT-B3LYP results

Both the cc-pV$x$Z and cc-pCV$x$Z basis sets show positive linearisation
barriers (except for cc-pCVTZ), energetically favouring the non-linear over
the linear conformer of disiloxane. The simplest basis set, i.e. the double
zeta cc-pVDZ basis set, results in the highest barrier of all (0.80 kcal/mol),
in certain agreement with the far infrared absorption spectra of gaseous
disiloxane 25. The core-valence correlated counterpart, i.e. the cc-pCVDZ
basis set, also produces a high linearisation barrier of 0.78 kcal/mol. Triple
zeta basis sets (cc-pVTZ and cc-pCVTZ) show a significant reduction in the
barrier height, eventually converging to the quadruple zeta set results. These
results reinforce the conclusions drawn in preceding studies 5, 6, 11, 7, 8
that the basis sets at the double zeta level are insufficient to properly
describe the disiloxane molecular properties, and that the triple zeta level
is the minimally reliable level for the description of the atomic orbitals.
The trends of the linearisation barrier, meanwhile, is nearly identical for
both standard and core-correlated variants (Figure 4), with higher quality
sets producing good agreement with two of the three available experimental
results 2, 3.

The results for DFT-B3LYP in this study, particularly the converged results
for the quadruple zeta basis sets cc-pVQZ and cc-pCVQZ, seem to indicate that
previous agreement between the B3LYP results and experimental data 8 is due to
coincidence instead of convergence toward the complete basis set limit.
Indeed, for the largest basis sets used in this study, the linearisation
barrier calculated by B3LYP is at best approximately 0.1 kcal/mol smaller than
the ZPE-corrected measurement. It does not seem that the B3LYP exchange-
correlation functional provides adequate accounting of the electron
correlation for properly describing a disiloxane molecule.

###  3.3 CCSD(T) results

CCSD(T) calculations were conducted with the same geometries and basis sets as
the B3LYP calculations. The results of these calculations are shown in Table
2. The trends of the linearisation barrier agree almost perfectly with B3LYP
calculations, with a universal shift in the energetic favourability toward the
non-linear conformer denoted by the larger linearisation barriers. CCSD(T)
calculations seem to converge to values closer to the ZPE-corrected Raman
linearisation barrier of 0.36 kcal/mol. The cc-pV$x$Z basis sets converge to a
linearisation barrier of approximately 0.44 kcal/mol, whereas the core-valence
correlated set cc-pCV$x$Z converges to a barrier height of approximately 0.35
kcal/mol, in excellent agreement with the benchmark value. We also achieved
good agreement with another CCSD(T) calculation in an earlier study 8
reporting a linearisation barrier of 0.48 kcal/mol using the cc-pVTZ basis
set. These results show that accounting for the electron correlation is indeed
necessary to properly model the Si-O-Si bond in disiloxane, and supports the
conclusion of earlier studies citing cc-pVTZ and CCSD(T) as the minimum
reliable level of description for the disiloxane molecule 7, 8.

###  3.4 FNDMC results

The linearisation barrier calculated by FNDMC is also shown in Table 2 and
Figure 2. As with CCSD(T), the linearisation barrier values from FNDMC
converge to a value close to the ZPE-corrected Raman benchmark adopted in this
study of 0.36 kcal/mol for the largest basis sets. Also observed is the
independence of the linearisation barrier calculated from the basis set used
to form the initial trial wavefunction using DFT-B3LYP relative to the other
two calculation methods. The influence of the different basis sets on the end
result of the FNDMC calculations directly translates into how they affect the
trial wavefunction nodal surfaces. The amplitudes of the trial wavefunction,
meanwhile, do not affect the end result at ($\tau\rightarrow\infty$), which,
in turn, limits the dependence of the end result on the basis sets used to
form the trial wavefunction.

The quality of the trial wavefunction nodal surface (how close it resembles
the true ground-state wavefunction nodal surface) is reflected in the total
energy values of the FNDMC calculation (the fixed-node variational principle)
13. Therefore, the expected values of the total energy in the all-electron
FNDMC calculations (such as those conducted in this study) are good indicators
of the quality of the trial wavefunction nodal surfaces, because all-electron
FNDMC calculations retain the variational principle with respect to the total
energy of the ground-state. These absolute values are shown in Table 3, sorted
in accord with the quality of the nodal surface (from low quality, high total
energy, to high quality, low total energy), and displayed in Figure 3. It can
be observed that the cc-pCVTZ produces a nodal surface of slightly better
quality than cc-pVQZ, indicating that the electron correlation between the
core and valence electrons can be a significant factor for improvement beyond
the triple zeta level of atomic orbital description.

Figure 3: Linearisation barrier of disiloxane calculated using cc- basis sets, sorted based on the quality of the nodal surface. Dashed lines serve to better illustrate the trends present in the data. Table 3: Total energies from FNDMC calculations, sorted from highest to lowest. Basis set |  $E_{\rm total}$ [Hartree]
---|---
Delinear | Linear
cc-pVDZ | -657.7994(2) | -657.7988(2)
cc-pCVDZ000 |  0-657.8062(1)0 |  0-657.8053(1)0
cc-pVTZ | -657.8270(2) | -657.8263(2)
cc-pVQZ | -657.8366(1) | -657.8362(1)
cc-pCVTZ | -657.8376(1) | -657.8371(1)
cc-pCVQZ | -657.8437(1) | -657.8431(1)

###  3.5 Effects of basis sets

Disiloxane linearisation barrier dependence on the basis set was observed for
all cases. In agreement with previous studies 8, this dependence is more
significant than the methodologies used in the ab initio calculations,
particularly considering the double zeta-level basis sets cc-pVDZ and cc-
pCVDZ. Triple zeta basis sets seem to offer the minimum level of description
to reliably predict the linearisation barrier, whereas quadruple zeta basis
sets result in an extremely good prediction, especially for the CCSD(T) and
FNDMC calculations. Figure 2 clearly shows very similar trends for each
calculation method with a substantial dependence on the basis set indicated,
particularly towards the smaller sets. Convergence of the disiloxane
linearisation barrier is generally observed for all three calculation methods,
albeit not necessarily converging to the same value.

Previous theoretical studies suggest that this converging trend is attributed
to the increasing addition of polarisation functions 8 within the basis sets
used. Adding polarisation functions serves to better reproduce dynamical
correlations in the system. This is in line with previous theoretical works
with semi-empirical methods implying that calculations without electron
correlation favour the linear structure, thereby resulting in negative values
of linearisation barrier. This is also reflected in preceding geometry
optimisation calculations without polarisation functions, resulting in Si-O-Si
angles close to 170∘. 5, 4. Therefore, it is expected that both cc-pV$x$Z and
cc-pCV$x$Z basis sets should converge to a reliable predicted linearisation
barrier value because polarisation functions are systematically included with
an increasing description of the atomic orbitals.

From Figure 4, a comparison can be observed between the correlation consistent
cc-pV$x$Z basis sets and its core-valence correlated counterpart cc-pCV$x$Z.
The B3LYP and CCSD(T) linearisation barrier trends closely follow one another,
with CCSD(T) cc-pCV$x$Z in particular converging to a nearly identical value
as that of the ZPE-corrected Raman. Meanwhile, the FNDMC results show a slight
difference in the trends between the two families when observing the
respective expectation values of the linearisation barriers. It should be
noted, however, that the stochastic nature of FNDMC diminishes the
significance of the energy trends (which need to be considered
probabilistically).

Figure 3 shows that FNDMC is overall less dependent on the basis set used
compared to both B3LYP and CCSD(T). Although previous studies have recommended
treating disiloxane with the cc-pVTZ basis set at a minimum, cc-pVDZ is shown
to generate a sufficiently high quality nodal surface for use in DMC
calculations, giving a linearisation barrier in good agreement with the
experimental values. Even for the smallest basis sets tested in this study,
FNDMC shows relatively more accurate values of the linearisation barrier, and
therefore less dependence on the basis set used to form the trial wavefunction
per the initial expectations.

Figure 4: Linearisation barrier of disiloxane calculated using cc-basis sets,
separated into standard and core-valence-correlated versions (left and right).
Dashed lines serve to better illustrate the trends present in the data.

###  3.6 Effects of methodologies

In line with previous studies, we find the dependence on the methods weaker
than that on the basis sets. With increasing levels of basis sets, DFT-B3LYP
and CCSD(T) values of the linearisation barriers converge to approximately 0.1
and 0.3-0.4 kcal/mol, respectively. This slight difference is in accord with
the variance in experimental measurements 25, 2, 3 and is clearly less
significant than the variance as a result of the basis sets.

Previous expectations on the trend in methodologies are derived from previous
studies 4, 43, particularly the study by Koput in 1990 4, in which the
inclusion of the electron correlation proved vital to predicting the energetic
favourability of the non-linear structure of disiloxane because the SCF
calculation produced a near-linear structure of disiloxane. This and other
studies 43 gave rise to the general expectation that the inclusion of an
electron correlation is important to properly model the structure of
disiloxane. As previously mentioned, it can be observed that the B3LYP
exchange-correlation functional is insufficient to properly recover the
electron correlation and thereby predict the disiloxane linearisation barrier.
Meanwhile, CCSD(T) and FNDMC calculations both converge towards the ZPE-
corrected Raman value, indicating a proper accounting of the dynamical
correlation.

FNDMC has the added advantage of being able to reliably predict the energetics
of disiloxane with smaller basis sets. With better scalability for application
in high performance computing (HPC) systems and the availability of
pseudopotentials, FNDMC is a promising alternative method to CCSD(T) for
describing the Si-O-Si bonds, particularly for larger systems.

##  4 CONCLUSION

The Si2H6O linearisation barrier was calculated using three separate ab initio
methods, DFT-B3LYP, CCSD(T), and FNDMC, with six different basis set choices
in line with expectations derived from previous theoretical studies on
disiloxane. Similar with previous studies, we observed that the systematic
inclusion of polarisation functions, along with an increasing level of
description for atomic orbitals, eventually result in a reliable prediction of
the disiloxane linearisation barrier. All calculation methods eventually
produced converged values with increasing level of basis sets for the
linearisation barrier, with 0.1 kcal/mol for DFT-B3LYP and 0.3-0.4 kcal/mol
for CCSD(T), respectively, in line with previous theoretical studies 8, and
similar expectation values of the barrier for FNDMC calculations. The
agreement between the experimental measurements and the DFT-B3LYP results at
0.3 kcal/mol are shown to be likely accidental. ZPE-corrected experimental
measurements are in good agreement with the CCSD(T) and FNDMC results, with
the ground state linearisation barrier taken at 0.36 kcal/mol. FNDMC is shown
to be least dependent on the choice of basis set among the three calculation
methods applied, in line with initial expectations owing to the nature of
FNDMC.

## ACKNOWLEDGEMENTS

The computations in this study were conducted using the facilities of Research
Center for Advanced Computing Infrastructure at JAIST. R.M. would like to
extend his appreciation for the financial support from MEXT-KAKENHI (19H04692
and 16KK0097), FLAGSHIP2020 (project nos. hp190169 and hp190167 at
K-computer), the Air Force Office of Scientific Research (AFOSR-
AOARD/FA2386-17-1-4049;FA2386-19-1-4015), and JSPS Bilateral Joint Projects
(with India DST). K.H. would like to thank the HPCI System Research Project
(Project ID: hp190169) and MEXT-KAKENHI (JP16H06439, JP17K17762, JP19K05029,
and JP19H05169) for the financial support.

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