#  Performance of periodic EOM-CCSD for band gaps of inorganic semiconductors
and insulators

Ethan A. Vo  Department of Chemistry, Columbia University, New York, NY 10027
USA  Xiao Wang  Department of Chemistry and Biochemistry, University of
California, Santa Cruz, CA 95064 USA  Timothy C. Berkelbach
<EMAIL_ADDRESS>Department of Chemistry, Columbia University, New
York, NY 10027 USA

###### Abstract

We calculate the band gaps of 12 inorganic semiconductors and insulators
composed of atoms from the first three rows of the periodic table using
periodic equation-of-motion coupled-cluster theory with single and double
excitations (EOM-CCSD). Our calculations are performed with atom-centered
triple-zeta basis sets and up to 64 $k$-points in the Brillouin zone. We
analyze the convergence behavior with respect to number of orbitals and number
of $k$-points sampled, using composite corrections and extrapolations to
produce our final values. When accounting for electron-phonon corrections to
experimental band gaps, we find that EOM-CCSD has a mean signed error of
$-0.12$ eV and a mean absolute error of 0.42 eV; the largest outliers are C
(error of $-0.93$ eV), BP ($-1.00$ eV), and LiH ($+0.78$ eV). Surprisingly, we
find that the more affordable partitioned EOM-MP2 theory performs as well as
EOM-CCSD.

##  I Introduction

The accurate ab initio prediction of electronic band structures and band gaps
is a major driving force behind the improvement and development of
increasingly accurate electronic structure methods. The failures of local and
semilocal density functional theory (DFT) for this task have long been
understood [1, 2, 3], motivating the use of various hybrid functionals [4, 5,
6]. In a many-body framework, the GW approximation to the self-energy [7, 8,
9] is arguably the most successful method on the basis of its good accuracy
and relatively low computational cost, with technical issues and extensions
such as self-consistency and vertex corrections continuing to be explored [10,
11, 12, 13, 14, 15].

In the last decade, wavefunction-based electronic structure methods have been
increasingly applied to solids. Seven years ago, McClain et al. [16] performed
the first calculation of band gaps of periodic three-dimensional solids using
equation-of-motion coupled-cluster theory with single and double excitations
(EOM-CCSD) for ionization potentials and electron affinities, presenting
encouraging results for diamond and silicon. Since then, the method has been
applied to select solids, including MnO, NiO [17], MoS2 [18], and ZnO [19],
and has been developed as an impurity solver for dynamical mean-field theory
[20, 21, 22]. However, the statistical performance of the method over a wide
range of solids has yet to be established. We note that EOM-CCSD has also been
used to study the neutral excitation energies of bulk solids, in periodic [23,
24, 25] and embedded cluster [26] frameworks, and defects [27, 28].

Here, we aim to conclude this first phase of exploratory work by presenting
converged band gaps of 12 simple, canonical semiconductors and insulators
composed of atoms from the first three rows of the periodic table.
Importantly, this work benefits from recent infrastructure developments,
including efficient calculation of periodic integrals and density fitting [29,
30, 31] and the development of Gaussian basis sets that can be reliably
converged to the basis set limit in closely packed solids [32]. We hope that
our results can help direct future research on the use of wavefunction based
methods for excitation energies of solids, such as applications to more
complex solids or improvements to cost or accuracy.

##  II Methods

###  II.1 Materials

We study 12 semiconductors and insulators with a range of covalent and ionic
bonding and diverse crystal structures, including diamond (C, Si), zinc blende
(SiC, BN, BP, AlP), rock salt (MgO, MgS, LiH, LiF, LiCL), and wurtzite (AlN).
Each contains two atoms per unit cell except for AlN, which contains four
atoms per unit cell. We use the experimental lattice constants for all solids,
which are given in Tab. 1. For all solids, we use DFT with the PBE functional
and a large $k$-point mesh to identify the points in the Brillouin zone where
the valence band maximum and conduction band minimum occur.

###  II.2 EOM-CCSD calculations

Periodic EOM-CCSD calculations were performed using PySCF [33] with Gaussian
density fitting of two-electron integrals [29, 30, 31]. We perform a periodic
Hartree-Fock calculation, then a ground-state CCSD calculation, and finally an
EOM-CCSD calculation for the ionization potential [IP, $E(N-1)-E(N)$] and the
electron affinity [EA, $E(N+1)-E(N)$], where $E(N)$ is (formally) the ground
state energy of a system with $N$ electrons. The band gap is the sum [IP+EA,
$E(N+1)+E(N-1)-2E(N)$]. The total cost is dominated by the ground-state CCSD
calculation, which has $O(N_{k}^{4}o^{2}v^{4})$ scaling in CPU time and
$O(N_{k}^{3}o^{2}v^{2})$ scaling in storage, where $N_{k}$ is the number of
$k$-points sampled in the Brillouin zone and $o,v$ are the number of occupied
and virtual (unoccupied) orbitals per $k$-point. These relatively high CPU and
storage costs necessitate composite corrections and extrapolations to make
predictions in the complete basis set limit and thermodynamic limit (TDL).

Figure 1:  Convergence of the IP, EA, and band gap with the number of
correlated virtual orbitals for silicon. For $N_{k}=2^{3}$ data, the error is
defined with respect to the full TZ basis value (with 54 virtual orbitals per
$k$-point). The $N_{k}=3^{3}$ and $4^{3}$ data have been shifted to align with
that of the smaller $k$-point mesh at the largest accessed number of virtual
orbitals, which is 23 for $3^{3}$ and 10 for $4^{3}$.

We use GTH pseudopotentials optimized for Hartree-Fock calculations [34, 35]
and the recently developed correlation consistent GTH-cc-pVTZ basis set [30].
Separate testing (not shown) with the QZ basis set confirm that basis set
incompleteness errors due to our use of the TZ basis set are less than 0.1 eV
in band gaps. We sample the Brillouin zone using uniform $k$-point meshes with
$N_{k}=2^{3}$, $3^{3}$, and $4^{3}$. For the larger $k$-point meshes,
prohibitive storage costs require that we freeze virtual orbitals [we always
correlate all occupied (valence) orbitals, of which there are 4 per unit cell
for all materials studied here except LiH (1 occupied orbital) and AlN (8
occupied orbitals)]. We thus use a composite correction based on calculations
with smaller $N_{k}$,

$E(N_{k,2},\mathrm{L})\approx
E(N_{k,2},\mathrm{S})+\left[E(N_{k,1},\mathrm{L})-E(N_{k,1},\mathrm{S})\right]$
(1)

where L and S indicate a large and small number of active virtual orbitals and
$N_{k,2}>N_{k,1}$. For $N_{k}=2^{3}$, we correlate the entire TZ basis set
(except for AlN), which contains between 34 and 58 (67 for truncated AlN)
orbitals per $k$-point; for $N_{k}=3^{3}$, we correlate 27 total orbitals per
$k$-point and correct based on calculations with $N_{k}=2^{3}$; for
$N_{k}=4^{3}$, we correlate 14 total orbitals per $k$-point and correct based
on calculations with $N_{k}=3^{3}$. Therefore, our calculations treat a few
hundred electrons in about 1000 total orbitals, but benefit from the savings
implied by the translational symmetry of solids.

In Fig. 1, we show the convergence of the (composite corrected) IP, EA, and
band gap as a function of the number of virtual orbitals, using silicon as an
example. We see that the shape of convergence for all $N_{k}$ is quite
similar, suggesting that the composite corrections are accurate. The IP and EA
have large, but opposite, frozen orbital errors; these errors cancel in the
band gap, which converges to within 0.2 eV when correlating only 10 virtual
orbitals per $k$-point and to within 0.1 eV when correlating 23 virtual
orbitals per $k$-point. Beyond about 10 virtual orbitals, we see that the band
gap converges from below, which is a general trend seen in most other
materials.

With these basis-set corrected estimates, we use a two-point extrapolation to
the TDL assuming finite-size errors that decay as $N_{k}^{-1/3}$,

$E(N_{k}\rightarrow\infty)\approx\frac{N_{k,2}^{1/3}E(N_{k,2})-N_{k,1}^{1/3}E(N_{k,1})}{N_{k,2}^{1/3}-N_{k,1}^{1/3}},$
(2)

where $N_{k,2}=4^{3}$ and $N_{k,1}=3^{3}$. This slow convergence with $N_{k}$
is attributable to the finite-size error of a charged unit cell and is shared
by most many-body methods [36].

In Fig. 2, we demonstrate this composite correction and extrapolation scheme
for four example solids: Si, BN, MgO, and LiH. As already mentioned, we see
that increasing the number of correlated orbitals increases the band gap. We
see that the difference in band gaps when correlating 27 or 14 orbitals is
relatively independent of $N_{k}$ (at least for $N_{k}=2^{3}$ and $3^{3}$),
indicating that our additive composite correction should be accurate. For any
given $N_{k}$, we estimate that our band gaps are basis-set converged to
within about 0.1 eV, although extrapolation can magnify these individual
errors. Moreover, the final basis-set corrected data (with $N_{k}=2^{3}$,
$3^{3}$, and $4^{3}$) do not always fall on a straight line, suggesting that
we may not have reached the limit where the finite-size error is exclusively
due to the leading-order term of $O(N_{k}^{-1/3})$.

Figure 2: Behavior of the band gap as function of $N_{k}$ when correlating the
total number of orbitals indicated (14, 27, or the full TZ basis). The
composite corrected curve (blue) is our best estimate of the basis set limit,
which is then extrapolated to the thermodynamic limit. The black dashed line
indicates the experimental band gap.

###  II.3 P-EOM-MP2 calculations

Because of the high cost of periodic EOM-CCSD calculations, it is worthwhile
to compare to more affordable theories such as MP2, i.e., the calculation of
IPs, EAs, and band gaps using the second-order self-energy with a Hartree-Fock
reference. However, MP2 has been shown to perform poorly for band gaps,
predicting negative band gaps for semiconductors like silicon [37, 38, 39].
Instead, our group recently explored the accuracy of the closely related
partitioned EOM-MP2 (P-EOM-MP2) theory [39]. In P-EOM-MP2, we make two
approximations to an EOM-CCSD calculation. First, we replace the ground-state
CCSD amplitudes by their second-order approximations. Second, we replace the
large doubles-doubles block of the similarity transformed Hamiltonian by a
diagonal matrix, keeping only the orbital energy differences. These
approximations lower the CPU and memory costs significantly.

A diagrammatic analysis in Ref. 39 showed that a P-EOM-MP2 calculation
essentially corresponds to supplementing the second-order self-energy with two
third-order diagrams. Remarkably, this minor difference resulted in
significantly improved performance. In this work, we will compare our EOM-CCSD
results to P-EOM-MP2 results, which have been recalculated here for
consistency with the pseudopotentials, basis sets, and $k$-points used in this
work. However, we find numbers in good agreement with our previous work [39],
suggesting that errors due to these latter technical details are small.

##  III Results

###  III.1 Comparing EOM-CCSD and P-EOM-MP2

Figure 3: Comparison of EOM-CCSD and P-EOM-MP2 band gaps upon extrapolation to
the thermodynamic limit. The black dashed line indicates the experimental band
gap. Figure 4: Difference between P-EOM-MP2 and EOM-CCSD for the IP, EA, and
band gap $E_{\mathrm{g}}$ of all materials studied in this work. Differences
are shown for values calculated with a $4\times 4\times 4$ $k$-point mesh and
for values extrapolated to the thermodynamic limit (TDL).

Before comparing to experimental band gaps, we first compare the performance
of EOM-CCSD and P-EOM-MP2. In Fig. 3, we compare the (composite corrected)
band gaps as a function of $N_{k}$ for four example materials. Clearly, for
many materials, the calculated band gaps are very similar: at fixed $N_{k}$,
the gaps commonly differ by less than 0.2 eV, which is a significant finding
given the difference in cost between the two methods.

However, for silicon (and a few other materials, see below), we see large
differences that are magnified on approaching the TDL. With $N_{k}=4^{3}$, the
EOM-CCSD gap is smaller by 0.4 eV, and extrapolation to the TDL magnifies this
difference to 1.0 eV. We note that our EOM-CCSD band gap in the TDL is 0.96
eV, which is slightly smaller than the previously reported value of 1.19 eV
[16]. We have confirmed that this discrepancy is due to a combination of small
errors in our composite correction and differences in our basis set,
pseudopotential, and $k$-point used for the conduction band minimum.

We next sought to identify any trend in the difference between P-EOM-MP2 and
EOM-CCSD band gaps. In Fig. 4, we show the difference of the IP, EA, and band
gap as a function of the experimental band gap; because extrapolation
sometimes alters the difference, we show differences at $N_{k}=4^{3}$ and in
the extrapolated TDL. We identify the following rough trends. The P-EOM-MP2 IP
is larger for small gap materials and smaller for large gap materials; the
P-EOM-MP2 EA is larger for most materials; the P-EOM-MP2 band gap is larger
for most materials (typically by less than 0.5 eV), but similar for large gap
materials. The similarity of the band gaps for large materials is consistent
with their more weakly correlated nature. Overall, we find that P-EOM-MP2 and
EOM-CCSD predict surprisingly similar band gaps. The largest outliers occur
for small gap materials upon extrapolation, and these are Si and BP, for which
P-EOM-MP2 predicts a band gap that is larger by about 1 eV.

###  III.2 Comparing to experiment

|  | Reference | P-EOM-MP2 | EOM-CCSD
---|---|---|---|---
Material |  $a$ ($c$) (Å) | Expt. $E_{\mathrm{g}}$ (corrected) | el-ph | $E_{\mathrm{g}}$ | Error | $E_{\mathrm{g}}$ | Error
Si | 5.431 | $1.30$ | $-0.06$ | $1.99$ | $0.69$ | $0.96$ | $-0.34$
SiC | 4.350 | $2.37$ | $-0.17$ | $2.59$ | $0.22$ | $2.54$ | $0.17$
AlP | 5.451 | $2.54\cite[cite]{[\@@bibref{Number}{Lorenz1970}{}{}]}$ | $-0.09$ | $2.93$ | $0.39$ | $2.62$ | $0.08$
BP | 4.538 | $2.65$ | $-0.25\cite[cite]{[\@@bibref{Number}{Ha2020}{}{}]}$ | $2.58$ | $-0.07$ | $1.65$ | $-1.00$
MgS | 5.191 | $4.59\cite[cite]{[\@@bibref{Number}{Ching1995}{}{}]}$ | $-$ | $5.10$ | $(0.51)$ | $5.26$ | $(0.67)$
LiH | 4.083 | $5.07\cite[cite]{[\@@bibref{Number}{Baroni1985}{}{}]}$ | $-0.08\cite[cite]{[\@@bibref{Number}{Monserrat2013}{}{}]}$ | $6.05$ | $0.98$ | $5.85$ | $0.78$
C | 3.567 | $5.81$ | $-0.33$ | $5.24$ | $-0.57$ | $4.88$ | $-0.93$
BN | 3.615 | $6.61$ | $-0.41$ | $6.16$ | $-0.45$ | $6.45$ | $-0.16$
AlN | 3.110 (4.980) | $6.62$ | $-0.42$ | $6.45$ | $-0.17$ | $6.33$ | $-0.29$
MgO | 4.213 | $8.19$ | $-0.52$ | $8.41$ | $0.22$ | $8.34$ | $0.15$
LiCl | 5.130 | $9.40\cite[cite]{[\@@bibref{Number}{Brown1970}{}{}]}$ | $-$ | $9.55$ | $(0.15)$ | $9.43$ | $(0.03)$
LiF | 4.035 | $15.09$ | $-0.59$ | $15.41$ | $0.32$ | $15.43$ | $0.34$
MSE (eV) |  |  |  |  | $0.16$ |  | $-0.12$
MAE (eV) |  |  |  |  | $0.41$ |  | $0.42$
Table 1: Lattice constants and electronic band gaps of the 12 materials
studied in this work (all energies in eV). Experimental band gaps have been
corrected for the effects of electron-phonon coupling, when such corrections
are available in literaure. Except where indicated, experimental band gaps
have been taken from the collection in Ref. 46, and electron-phonon
corrections have been taken from the collection in Ref. 39.

We now turn to a comparison between calculated and experimental band gaps. Our
basis-set corrected and TDL extrapolated results for the electronic band gap
are given in Tab. 1, where they are compared to experimental band gaps. For a
fair comparison, we have corrected the experimental band gaps for finite-
temperature and vibrational zero-point energy effects, which typically act to
reduce the purely electronic band gap. If we exclude solids without available
electron-phonon corrections, EOM-CCSD predicts band gaps with a mean signed
error (MSE) of $-0.12$ eV and a mean absolute error (MAE) of 0.42 eV. The
largest outliers are diamond (error of $-0.93$ eV), BP ($-1.00$ eV), and LiH
($+0.78$ eV). The statistical performance of P-EOM-MP2 is quite similar, with
an MSE of 0.16 eV and MAE of 0.41 eV.

The performance of both methods is shown graphically in Fig. 5, along with
that of two other Green’s function based methods, whose band gaps were
previously published: the extended second-order algebraic diagrammatic
construction [ADC(2)-X] [47] and the G0W0 approximation with a PBE reference
[46]. We choose these two for comparison because their calculations were also
performed with PySCF, using similar basis sets and pseudopotentials. We see
that EOM-CCSD and P-EOM-MP2 outperform ADC(2)-X, which predicts band gaps that
are too small, but are comparable to the G0W0 approximation.

Figure 5: Band gap error for the 12 semiconductors and insulators studied in
this work. In addition to our own EOM-CCSD and P-EOM-MP2 results, we include
ADC(2)-X results [47] and G0W0 results [46] from previous works, for
comparison; only some of the same solids were studied in these previous works.

##  IV Conclusions

Through a study of 12 simple semiconductors and insulators, we have found that
EOM-CCSD predicts band gaps with a mean absolute error of about 0.4 eV.
Perhaps unsurprisingly, this accuracy is very similar to our group’s previous
finding concerning the accuracy of EOM-CCSD for neutral excitation energies of
solids [24, 25]. Overall, we conclude that the performance of EOM-CCSD is not
measurably better than that of P-EOM-MP2, even for small gap semiconductors
that one might expect to be more strongly correlated.

Although we believe our results are converged (with respect to basis set and
number of $k$-points) to about 0.1–0.2 eV, the need for composite corrections
and extrapolation introduces uncertainties and prevents routine use of these
correlated methods. More robust methods for the reduction of basis set errors
and finite-size errors would be very valuable. Having established the
performance of band gaps of simple solids, future work should explore more
complex solids as well as core ionization energies and the charged excitations
of metals. The degree to which EOM-CCSD can replace or complement existing and
more affordable methods remains to be seen. Predicting very precise excitation
energies through the incorporation of full or perturbative triples—a manner of
systematic improvability that is not shared by most existing methods—is
perhaps the most exciting near-term goal. Addressing the cost is the
outstanding challenge.

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