We calculate bandgaps 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 the 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 bandgaps, 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.

The accurate ab initio prediction of electronic band structures and bandgaps 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,1–3 motivating the use of various hybrid functionals.4–6 In a many-body framework, the GW approximation to the self-energy7–9 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.10–15 

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 bandgaps 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 It has also been developed as an impurity solver for dynamical mean-field theory,20–22 and its diagrammatic content and relation to the GW approximation have been explored.23–25 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 neutral excitation energies of bulk solids, in periodic26–28 and embedded cluster29 frameworks, and defects.30,31

Here, we aim to conclude this first phase of exploratory work by presenting converged bandgaps 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 fitting32–34 and the development of Gaussian basis sets that can be reliably converged to the basis set limit in closely packed solids.35 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.

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, and AlP), rock salt (MgO, MgS, LiH, LiF, and LiCl), and wurtzite (AlN). Each contains two atoms per primitive unit cell except for AlN, which contains four atoms per primitive unit cell. We use the experimental lattice constants for all solids, which are given in Table I. 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.

TABLE I.

Lattice constants and electronic bandgaps of the 12 materials studied in this work (all energies are in eV). Experimental bandgaps have been corrected for the effects of electron–phonon coupling, when such corrections are available in literature. Except where indicated, experimental bandgaps have been taken from the collection in Ref. 46, and electron–phonon corrections have been collected from Refs. 47–50. G0W0 bandgaps have been taken from Refs. 46 and 51, except for the bandgap of LiH, which is taken from Ref. 52 (where PW91 was used as the reference).

MaterialReferencesG0W0@PBEP-EOM-MP2EOM-CCSD
a(c) (Å)Expt. Eg (corrected)El–phEgErrorEgErrorEgError
Si 5.431 1.30 −0.06 1.08 −0.22 1.99 0.69 0.96 −0.34 
SiC 4.350 2.37 −0.17 2.44 0.07 2.59 0.22 2.54 0.17 
AlP 5.451 2.5442  −0.09 2.41 −0.13 2.93 0.39 2.62 0.08 
BP 4.538 2.65 −0.25 2.15 −0.50 2.58 −0.07 1.65 −1.00 
MgS 5.191 4.5943  ⋯ 4.80 0.21 5.10 (0.51) 5.26 (0.67) 
LiH 4.083 5.0744  −0.08 4.75 −0.32 6.05 0.98 5.85 0.78 
3.567 5.81 −0.33 5.52 −0.29 5.24 −0.57 4.88 −0.93 
BN 3.615 6.61 −0.41 6.41 −0.20 6.16 −0.45 6.45 −0.16 
AlN 3.110 (4.980) 6.62 −0.42 5.89 −0.73 6.45 −0.17 6.33 −0.29 
MgO 4.213 8.19 −0.52 7.43 −0.76 8.41 0.22 8.34 0.15 
LiCl 5.130 9.4045  ⋯ 9.69 0.29 9.55 (0.15) 9.43 (0.03) 
LiF 4.035 15.09 −0.59 14.55 −0.54 15.41 0.32 15.43 0.34 
MSE (eV)     −0.36  0.16  −0.12 
MAE (eV)     0.38  0.41  0.42 
MaterialReferencesG0W0@PBEP-EOM-MP2EOM-CCSD
a(c) (Å)Expt. Eg (corrected)El–phEgErrorEgErrorEgError
Si 5.431 1.30 −0.06 1.08 −0.22 1.99 0.69 0.96 −0.34 
SiC 4.350 2.37 −0.17 2.44 0.07 2.59 0.22 2.54 0.17 
AlP 5.451 2.5442  −0.09 2.41 −0.13 2.93 0.39 2.62 0.08 
BP 4.538 2.65 −0.25 2.15 −0.50 2.58 −0.07 1.65 −1.00 
MgS 5.191 4.5943  ⋯ 4.80 0.21 5.10 (0.51) 5.26 (0.67) 
LiH 4.083 5.0744  −0.08 4.75 −0.32 6.05 0.98 5.85 0.78 
3.567 5.81 −0.33 5.52 −0.29 5.24 −0.57 4.88 −0.93 
BN 3.615 6.61 −0.41 6.41 −0.20 6.16 −0.45 6.45 −0.16 
AlN 3.110 (4.980) 6.62 −0.42 5.89 −0.73 6.45 −0.17 6.33 −0.29 
MgO 4.213 8.19 −0.52 7.43 −0.76 8.41 0.22 8.34 0.15 
LiCl 5.130 9.4045  ⋯ 9.69 0.29 9.55 (0.15) 9.43 (0.03) 
LiF 4.035 15.09 −0.59 14.55 −0.54 15.41 0.32 15.43 0.34 
MSE (eV)     −0.36  0.16  −0.12 
MAE (eV)     0.38  0.41  0.42 

Periodic EOM-CCSD calculations were performed using PySCF36 with Gaussian density fitting of two-electron integrals.32–34 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 bandgap 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(Nk4n6) scaling in computation time and O(Nk3n4) scaling in storage, where Nk is the number of k-points sampled in the Brillouin zone and n is the number of atoms in the unit cell. The subsequent IP and EA calculations are cheaper, with computation times that scale as O(Nk3n5) per k-point in the desired band structure. 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).

We use GTH pseudopotentials optimized for Hartree–Fock calculations37,38 and the recently developed correlation consistent GTH-cc-pVTZ basis set.33 Separate testing (not shown) with the QZ basis set confirms that basis set incompleteness errors due to our use of the TZ basis set are less than 0.1 eV in bandgaps. We sample the Brillouin zone using uniform k-point meshes with Nk = 23, 33, and 43. We perform separate IP and EA calculations using meshes that are shifted to include the k-point where the valence band maximum and conduction band minimum occur (as determined by a DFT band structure calculation). 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 four per unit cell for all materials studied here except LiH (one occupied orbital) and AlN (eight occupied orbitals)]. We thus use a composite correction based on calculations with smaller Nk,
E(Nk,2,L)E(Nk,2,S)+E(Nk,1,L)E(Nk,1,S),
(1)
where L and S indicate a large and small number of active virtual orbitals and Nk,2 > Nk,1. For Nk = 23, 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 Nk = 33, we correlate 27 total orbitals per k-point and correct based on calculations with Nk = 23; and for Nk = 43, we correlate 14 total orbitals per k-point and correct based on calculations with Nk = 33. Therefore, our calculations simulate an equivalent supercell with 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 bandgap as a function of the number of virtual orbitals using silicon as an example. We see that the shape of convergence for all Nk 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 bandgap, which converges to within 0.2 eV when correlating only ten virtual orbitals per k-point and to within 0.1 eV when correlating 23 virtual orbitals per k-point. Beyond about ten virtual orbitals, we see that the bandgap converges from below, which is a general trend seen in most other materials.

FIG. 1.

Convergence of the IP, EA, and bandgap with the number of correlated virtual orbitals for silicon. For Nk = 23 data, the error is defined with respect to the full TZ basis value (with 54 virtual orbitals per k-point). The Nk = 33 and 43 data have been shifted to align with those of the smaller k-point mesh at the largest accessed number of virtual orbitals, which is 23 for 33 and 10 for 43.

FIG. 1.

Convergence of the IP, EA, and bandgap with the number of correlated virtual orbitals for silicon. For Nk = 23 data, the error is defined with respect to the full TZ basis value (with 54 virtual orbitals per k-point). The Nk = 33 and 43 data have been shifted to align with those of the smaller k-point mesh at the largest accessed number of virtual orbitals, which is 23 for 33 and 10 for 43.

Close modal
With these basis-set corrected estimates, we use a two-point extrapolation to the TDL assuming finite-size errors that decay as Nk1/3,
E(Nk)Nk,21/3E(Nk,2)Nk,11/3E(Nk,1)Nk,21/3Nk,11/3,
(2)
where Nk,2 = 43 and Nk,1 = 33. This slow convergence with Nk is attributable to the finite-size error of a charged unit cell and is shared by most many-body methods,39 even when including Madelung constant corrections.

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 bandgap. We see that the difference in bandgaps when correlating 27 or 14 orbitals is relatively independent of Nk (at least for Nk = 23 and 33), indicating that our additive composite correction should be accurate. For any given Nk, we estimate that our bandgaps 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 Nk = 23, 33, and 43) 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(Nk1/3).

FIG. 2.

Behavior of the bandgap as a function of Nk 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 bandgap.

FIG. 2.

Behavior of the bandgap as a function of Nk 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 bandgap.

Close modal

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 bandgaps using the second-order self-energy with a Hartree–Fock reference. However, MP2 has been shown to perform poorly for bandgaps, predicting negative bandgaps for semiconductors such as silicon.24,40,41 Instead, our group has recently explored the accuracy of the closely related partitioned EOM-MP2 (P-EOM-MP2) theory.24 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 computational timing and memory costs significantly. Specifically, the timing scales as O(Nk2n4) per k-point in the desired band structure due to iterative matrix-vector products (there is a non-iterative step with higher scaling that rarely dominates the timing).

A diagrammatic analysis in Ref. 24 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,24 suggesting that errors due to these latter technical details are small.

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

FIG. 3.

Comparison of EOM-CCSD and P-EOM-MP2 bandgaps upon extrapolation to the thermodynamic limit. The black dashed line indicates the experimental bandgap.

FIG. 3.

Comparison of EOM-CCSD and P-EOM-MP2 bandgaps upon extrapolation to the thermodynamic limit. The black dashed line indicates the experimental bandgap.

Close modal

However, for silicon (and a few other materials, see below), we see large differences that are magnified on approaching the TDL. With Nk = 43, 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 bandgap 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 bandgaps. In Fig. 4, we show the difference of the IP, EA, and bandgap as a function of the experimental bandgap; because extrapolation sometimes alters the difference, we show differences at Nk = 43 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 bandgap is larger for most materials (typically by less than 0.5 eV), but similar for large gap materials. The similarity of the bandgaps for large materials is consistent with their more weakly correlated nature. Overall, we find that P-EOM-MP2 and EOM-CCSD predict surprisingly similar bandgaps. The largest outliers occur for small gap materials upon extrapolation, and these are Si and BP, for which P-EOM-MP2 predicts a bandgap that is larger by about 1 eV.

FIG. 4.

Difference between P-EOM-MP2 and EOM-CCSD for the IP, EA, and bandgap Eg of all materials studied in this work. Differences are shown for values calculated with a 4 × 4 × 4 k-point mesh and for values extrapolated to the thermodynamic limit (TDL).

FIG. 4.

Difference between P-EOM-MP2 and EOM-CCSD for the IP, EA, and bandgap Eg of all materials studied in this work. Differences are shown for values calculated with a 4 × 4 × 4 k-point mesh and for values extrapolated to the thermodynamic limit (TDL).

Close modal

We now turn to a comparison between calculated and experimental bandgaps. Our basis-set corrected and TDL extrapolated results for the electronic bandgap are given in Table I, where they are compared to experimental bandgaps and G0W0@PBE bandgaps. For a fair comparison, we have corrected the experimental bandgaps for finite-temperature and vibrational zero-point energy effects, which typically act to reduce the purely electronic bandgap. These zero-point renormalizations are collected from Refs. 47–50, most of them using many-body perturbation theory in the Allen–Heine–Cardona framework, but we note that developments in the first-principles incorporation of electron–phonon interactions are still ongoing. If we exclude solids without available electron–phonon corrections, EOM-CCSD predicts bandgaps 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 a MSE of 0.16 eV and a MAE of 0.41 eV.

The performance of EOM-CCSD and P-EOM-MP2 is shown graphically in Fig. 5, along with that of two other Green’s function based methods, whose bandgaps were previously published: the extended second-order algebraic diagrammatic construction [ADC(2)-X]53 and the G0W0 approximation with a PBE reference.46,51,52 We see that EOM-CCSD and P-EOM-MP2 outperform ADC(2)-X, which predicts bandgaps that are too small, but are comparable to the G0W0 approximation. As given in Table I, the MAE of G0W0@PBE is 0.38 eV, which is similar to that of EOM-CCSD and P-EOM-MP2. However, G0W0@PBE systematically underestimates bandgaps, with a MSE of −0.36 eV, which is 2–3 times larger in magnitude than that of EOM-CCSD or P-EOM-MP2. For comparison, standard GW calculations have a computational timing that scales as O(Nk2n4), which is the same as that of P-EOM-MP2 but less than that of EOM-CCSD.

FIG. 5.

Bandgap 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 results53 and G0W0 results46,51,52 from previous works; for ADC(2)-X, only some of the same solids have been previously studied.

FIG. 5.

Bandgap 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 results53 and G0W0 results46,51,52 from previous works; for ADC(2)-X, only some of the same solids have been previously studied.

Close modal

Through a study of 12 simple semiconductors and insulators, we have found that EOM-CCSD predicts bandgaps 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.27,28 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 exhibit significant dynamical correlation.

Although we believe our results are converged (with respect to the 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 bandgaps 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 many existing methods—is perhaps the most exciting near-term goal. Addressing the cost is the outstanding challenge.

This work was supported by the National Science Foundation under Grant No. CHE-2023568 and by the Air Force Office of Scientific Research under Grant No. FA9550-21-1-0400. We acknowledge computing resources from Columbia University’s Shared Research Computing Facility project, which is supported by NIH Research Facility Improvement Grant 1G20RR030893-01, and associated funds from the New York State Empire State Development, Division of Science Technology and Innovation (NYSTAR), Contract C090171, both awarded on April 15, 2010.

The authors have no conflicts to disclose.

Ethan A. Vo: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Xiao Wang: Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Timothy C. Berkelbach: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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