Excitons in Solids from Periodic Equation-Of-Motion Coupled-Cluster

Excitons in solids from periodic

equation-of-motion coupled-cluster theory

Xiao Wang† and Timothy C. Berkelbach∗,†,‡

†Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010 USA ‡Department of Chemistry, Columbia University, New York, New York 10027 USA

E-mail: [email protected]

Abstract

We present an ab initio study of electronically excited states of three-dimensional solids

using Gaussian-based periodic equation-of-motion coupled-cluster theory with single and dou-

ble excitations (EOM-CCSD). The explicit use of translational symmetry, as implemented via

Brillouin zone sampling and momentum conservation, is responsible for a large reduction in

cost. Our largest system studied, which samples the Brillouin zone using 64 k-points (a 4×4×4

mesh) corresponds to a canonical EOM-CCSD calculation of 768 electrons in 640 orbitals. We

study eight simple main-group semiconductors and insulators, with direct singlet excitation en-

ergies in the range of 3 to 15 eV. Our predicted excitation energies exhibit a mean signed error

of 0.24 eV and a mean absolute error of 0.27 eV when compared to experiment. Although

this error is similar to that found for EOM-CCSD applied to molecules, it may also reﬂect the

role of vibrational eﬀects, which are neglected in the calculations. Our results support recently

proposed revisions of experimental optical gaps for AlP and cubic BN. We furthermore cal-

culate the energy of excitons with nonzero momentum and compare the exciton dispersion of

LiF with experimental data from inelastic X-ray scattering. By calculating excitation energies

under strain, we extract hydrostatic deformation potentials in order to quantify the strength of

1 interactions between excitons and acoustic phonons. Our results indicate that coupled-cluster

theory is a promising method for the accurate study of a variety of exciton phenomena in solids.

1 Introduction

Density functional theory (DFT) is the workhorse of computational materials science and commonly predicts ground-state properties with high accuracy.1–8 Extending this success to the prediction of excited-state properties is an ongoing challenge. Time-dependent DFT (TDDFT)9 strug- gles to treat neutral excitations in the condensed phase and reasonable accuracy requires the use of hybrid functionals and/or frequency-dependent exchange-correlation kernels.10–20 Instead, the community typically builds upon DFT through Green’s function-based many-body perturbation theory.13 For weakly-correlated materials, including simple metals, semiconductors, and insulators, the GW approximation to the self-energy21–26 combined with the same approximation to the Bethe-Salpeter equation27–31 yields excited state properties that are typically in good agreement with experiment. Strongly-correlated materials are more commonly treated via dynamical mean- field theory (DMFT),32,33 either in the DFT+DMFT framework34–36 or the GW+DMFT framework.37,38 Alternatively, quantum Monte Carlo methods have been adapted for the calculation of excitation energies, including variational39 and diffusion Monte Carlo40–42 and auxiliary-field quantum Monte Carlo.43 In recent years, wavefunction-based techniques from the quantum chemistry community have been adapted for periodic boundary conditions and applied to condensed-phase systems. For ground-state properties, we mention the application of Møller-Plesset perturbation theory,44–47 coupled-cluster theory,48–52 and full configuration interaction quantum Monte Carlo.53–55 The same methods have also been used for finite-temperature properties.56–58 Charged excitation energies, as quantified through the one-particle spectral function or the band structure, have been studied by periodic equation-of-motion coupled-cluster theory for the uniform electron gas,59 for the sim-

52 60 ple semiconductors silicon and diamond, for two-dimensional MoS2, and for transition metal

2 oxides.61 Neutral excitation energies have been primarily studied by configuration interaction with single excitations (CIS),62,63 which is equivalent to the use of the Hartree-Fock self-energy in the Bethe-Salpeter equation (with the Tamm-Dancoff approximation). Due to the unscreened nature of the Coulomb interaction, periodic CIS typically predicts excitation energies which are much too large.64 Recently, periodic equation-of-motion coupled-cluster theory with single and double excitations (EOM-CCSD) was applied to one-dimensional polyethylene in a small single-particle basis set,50 producing results in reasonable agreement with experiment. For three-dimensional condensed-phase systems, our group has recently applied EOM-CCSD to the neutral excited-state properties of the uniform electron gas65 as well as the absorption and pump-probe spectroscopy of the naphthalene crystal at the Γ point.66 Here, we continue this endeavor and present the results of a new EOM-CCSD implementation with translational symmetry and Brillouin zone sampling for three-dimensional atomistic solids. The layout of the article is as follows. In section2, we describe the theory underlying our implementation, including details about our symmetry-adapted Gaussian basis sets and periodic EOM-CCSD. In section3, we present computational details, including details about the materials studied, the basis sets used, and integral evaluation. In section4, we provide EOM-CCSD results on the excitation energies (including a comparison with CIS and a discussion of finite-size effects), the exciton binding energy, the dispersion of excitons with nonzero momentum, and the exciton- phonon interaction. In section5, we summarize our results and conclude with future directions.

2 Theory

Our periodic calculations are performed using a translational-symmetry-adapted single-particle basis,

X ik·T φµk(r) = e φ˜µ(r − T ), (1) T

3 where φ˜µ(r) is an atom-centered Gaussian orbital, T is a lattice translation vector, and k is a crystal momentum vector sampled from the ﬁrst Brillouin zone. A Hartree-Fock calculation in this basis produces crystalline orbitals (COs)

X ψpk(r) = Cµp(k)φµk(r), (2) µ

where p is the band index and Cµp(k) are the CO coeﬃcients. The periodic CCSD energy and cluster amplitudes are determined by the usual equations,49,67–71

E0 = hΦ0|H¯ |Φ0i, (3)

0 = hΦaka |H¯ |Φ i, (4) iki 0 0 = hΦakabkb |H¯ |Φ i, (5) iki jk j 0 where Φaka and Φakabkb are Slater determinants with one and two electron-hole pairs, indices i, j, k, l iki iki jk j denote occupied orbitals, and a, b, c, d denote virtual orbitals. The similarity transformed Hamil-

−T T tonian is given by H¯ ≡ e He , where T = T1 + T2 is a momentum-conserving cluster operator with

X X0 aka † T = t a a k , (6) 1 iki aka i i ai kaki X X0 1 akabkb † † T2 = t a a a jk aik . (7) 4 iki jk j aka bkb j i abi j kakbkik j

The primed summations indicate momentum conservation. Excited states are accessed in coupled-cluster theory using the equation-of-motion (EOM) formalism,71–76 which amounts to diagonalizing the eﬀective Hamiltonian H¯ in a truncated space of excitations. For neutral excitations considered in this work, we use electronic excitation (EE) EOM-CCSD, where the diagonalization is performed in the basis of 1-particle+1-hole and 2- particle+2-hole states. We study excitations with zero and nonzero momentum q by using a basis

4 of determinants with the corresponding momentum. The (right-hand) excited state is therefore given by

T |Ψ(q)i = [R1(q) + R2(q)] e |Φ0i (8) with

X X0 aka † R (q) = r a a k , (9) 1 iki aka i i ai kaki X X0 1 akabkb † † R2(q) = r a a a jk aik , (10) 4 iki jk j aka bkb j i abi j kakbkik j

where the primed summations indicate that the momenta sum to q. The use of translational sym-

2 4 4 metry in CCSD leads to a computational cost that scales like o v Nk , where o is the number of

occupied orbitals per unit cell, v is the number of virtual orbitals per unit cell, and Nk is the number of k-points sampled. Further details about periodic Gaussian-based Hartree-Fock and CCSD can be found in Ref. 52.

3 Computational Details

We study eight semiconducting and insulating materials featuring the diamond/zinc-blende crystal structure and the rock salt crystal structure. The materials have a wide range of both direct and indirect band gaps and a variety of ionic and covalent bonding. The eight materials and the lattice constants used are given in Tab.1. In all calculations, the Brillouin zone was sampled with a

77 uniform Monkhorst-Pack mesh of Nk k-points that includes the Γ point. Our calculations are performed with GTH pseudopotentials,78,79 although we perform some all- electron calculations for comparison. For pseudopotential calculations, we use the corresponding polarized double- and triple-zeta basis sets DZVP and TZVP.80 For all-electron calculations, we use a modification of the cc-pVDZ basis set presented in Ref. 64, denoted AE-PVDZ. The finite-size errors of periodic calculations are influenced by the treatment of two-electron repulsion integrals (ERIs). Many of these integrals are formally divergent, due to the long-range

5 nature of the Coulomb interaction; however, these divergent ERIs enter into expressions for ob- servables as integrable divergences, producing well-deﬁned results in the thermodynamic limit. In order to avoid divergent ERIs, we calculate all atomic orbital ERIs as

Z Z kµkν kκkλ −1 ρµν (r1)ρκλ (r2) (µkµ, νkν|κkκ, λkλ) = Nk dr1 dr2 , (11) |r1 − r2|

where the orbital-pair densities have had their net charge removed,

ρkµkν (r) = φ∗ (r)φ (r) − ρ , (12a) µν µkµ νkν µν 1 Z ρ = drφ∗ (r)φ (r), (12b) µν µkµ νkν NkΩ

and Ω is the volume of the unit cell. The ERIs are then calculated using periodic Gaussian density ﬁtting with an even-tempered auxiliary basis as described in Ref. 81. We note that the use of chargeless pair densities is equivalent to neglecting the G = 0 component of the ERIs when calculated in a plane-wave basis. At the HF level, this treatment of ERIs produces an energy that

−1/3 converges to the thermodynamic limit as Nk , due to the exchange energy; this can be corrected −1 82,83 with a Madelung constant, leading to Nk convergence. This particular ﬁnite-size behavior is also present in the occupied orbital energies εi(k). We use a closed-shell implementation of periodic EOM-CCSD and study singlet excitations in this work, which are calculated by Davidson diagonalization.84,85 The initial guess used in the iterative diagonalization is obtained from dense diagonalization of the eﬀective Hamiltonian in the single excitation subspace. All calculations were performed with the open-source PySCF software package.86 Our implementation of EOM-CCSD with k-point symmetry is not yet parallelized; for the largest systems studied here, these serial calculations take a few days on a single core.

6 4 Results and Discussion

4.1 Direct optical excitation energy

In this section, we study the performance of periodic EOM-CCSD on the lowest-lying direct singlet excitation energies for the eight selected solids. Such states are relevant for absorption spectroscopy where no momentum is transferred during excitation. In periodic electronic structure calculations, it is desirable to achieve convergence with respect to the single-particle basis set, the level of correlation, and Brillioun zone sampling. The first two categories (extensively characterized in molecular systems) have been studied to some extent in solids53,87,88 and the convergence behavior of the third one is well understood at the mean-field level.83,89,90 For ground-state energies, the finite-size behavior has been extensively discussed in the quantum Monte Carlo community91,92 and only more recently from the quantum chemistry perspective.51,53,93–97 The finite-size behavior of excitation energies has received considerably less attention.41,42,52 We hope our preliminary results and discussion presented below encourage future work in this direction.

4.1.1 CIS

As a warm-up to EOM-CCSD, we first present results for periodic configuration interaction with single excitations (CIS), which forms a minimal theory for electronic excited states in the condensed phase. Importantly, the relatively low cost of CIS allows us to study the convergence with respect to Brillouin zone sampling up to relatively large k-point meshes (either 73 or 83). The finite-size error of the CIS excitation energy can be analyzed approximately by considering it as the difference between the energy of two determinants, the HF one |Φ0i and a single excitation

ak |Φik i. As discussed above, the energy of a single determinant calculated using our handling of ERIs −1/3 exhibits a ﬁnite-size error decaying like Nk , but which can be removed by a Madelung constant that depends only on the number of electrons in the unit cell. Therefore, the correction is identical for both states, and this leading-order error cancels in the energy diﬀerence. We thus posit that the

7 Figure 1: Excitation energy of LiF calculated with CIS using the DZVP basis set. The dashed line is a ﬁt to results obtained with a number of k-points between 33 and 73. The dotted line includes only 23, 33, and 43 k-points. Both ﬁts use the functional form given in Eq. 13.

−1 CIS energy converges at least as fast as Nk . In Fig.1, we show the excitation energy predicted by CIS for the LiF crystal as a function of

−1 −1 Nk . Clearly, the ﬁnite-size error decays at least as fast as Nk and so we use the three-parameter ﬁtting function

−1 −2 E(Nk) = E∞ + a Nk + b Nk , (13)

in order to extrapolate to the thermodynamic limit (Nk → ∞). This ﬁt is shown by the dashed line in Fig.1, which includes results between 3 3 and 73 k-points. Using an all-electron double-zeta basis set, we performed CIS calculations on a subset of our eight materials, in order to compare to previously published CIS results from Lorenz et al.64 By extrapolation of our results obtained at large k-point meshes using the above form, we obtain converged excitation energies (the “AE-PVDZ/Conv” column of Tab.1) that are within 0.1 eV of Ref. 64 for all materials studied. However, we emphasize that the CIS excitation energies are larger than experiment by 2 eV or more. Before moving on to our EOM-CCSD results, we use CIS to assess some of the future approximations we will have to make. In particular, we will only access k-point meshes up to 4 × 4 × 4 and

8 we will use GTH pseudopotentials and corresponding DZVP basis sets. First, considering finite- size errors, we re-fit the CIS data using the same form but excluding all k-point meshes larger than 4 × 4 × 4; for LiF, the result of this fit is shown as the dotted line in Fig.1. Clearly, without larger k-point meshes, this extrapolation predicts an excitation energy which is too high by about 0.2 eV. These limited extrapolation results are shown for all materials in the “AE-PVDZ/234-Extrap” column of Tab.1 and exhibit errors of about ±0.5 eV. Second, considering pseudopotential errors, in the last two columns of Tab.1, we show the excitation energies calculated with GTH pseudopotentials. In many cases, the pseudopotential error is less than 0.1 eV; naturally, materials containing heavier second-row atoms such as Si or Mg exhibit the largest errors, which are about 0.3 eV.

4.1.2 EOM-CCSD

We now move on to our results from periodic EOM-CCSD. Again, due to the high cost of these calculations, we utilized GTH pseudopotentials, sampled the Brillouin zone with meshes up to 4×4×4, and used basis set corrections. In particular, our primary calculations were based on a HF reference obtained with the full DZVP basis set; subsequent CCSD and EOM-CCSD calculations then employed frozen virtual orbitals, typically correlating 4 lowest virtual bands. The results of these calculations were used to extrapolate to the thermodynamic limit using the same empirical

Table 1: Singlet excitation energies (eV) from all-electron and pseudopotential-based periodic CIS

System a (Å) AE-PVDZ DZVP 234-Extrapa Convb Ref. 64 234-Extrapa Convb Diamond 3.567 10.99 11.76 11.72 11.07 11.81 Si 5.431 5.84 6.14 6.05 5.43 5.71 SiC 4.360 9.41 9.83 9.74 9.12 9.47 LiF 3.990 16.02 15.85 15.84 16.06 15.87 LiCl 5.130 11.04 10.89 MgO 4.213 12.00 11.91 11.94 11.69 11.66 BN 3.615 14.17 14.32 AlP 5.451 6.59 6.76 a Extrapolation based on results obtained with 23, 33, and 43 k-point meshes b Extrapolation based on results obtained with 33 up to 73 k-point meshes

9 Table 2: Singlet excitation energies (eV) from periodic EOM-CCSD

a b c System 234-Extrap ∆frz ∆TZ Final Expt. Diamond 7.70 −0.18 −0.05 7.47 7.398 Si 3.96 −0.37 −0.07 3.52 3.499 SiC 6.53 −0.19 −0.08 6.27 6.0100 LiF 14.29 −0.82 +0.01d 13.48 12.7,101 13.68102 LiCl 9.62 −0.27 −0.07 9.29 8.9103 MgO 8.55 −0.19 −0.07 8.29 7.5,104 7.6105 BN 11.48 −0.35 −0.02 11.11 11106 AlP 4.97 −0.42 −0.07 4.48 4.6107,108 MSE 0.24 MAE 0.27 a Extrapolation based on frozen-virtual DZVP results obtained with 23, 33, and 43 k-point meshes b ∆frz is the energy difference between complete DZVP and frozen- virtual DZVP on a 3×3×3 k-point grid c ∆TZ is the energy difference between TZVP and DZVP on a 2×2×2 k-point grid d For LiF, the TZVP basis set has severe linear dependencies and was modified by doubling the exponents of the two most diffuse s- and p-type primitive Gaussian functions of Li (all basis functions of F are unchanged) formula as described above (Eq. 13). These results are given in the “234-Extrap” column of Tab.2.

To these values, we then added two basis set corrections, ∆frz and ∆TZ; ∆frz is the energy diﬀerence between complete and frozen-orbital DZVP calculations with a 3 × 3 × 3 k-point mesh; ∆TZ is the energy diﬀerence between TZVP and DZVP calculations with a 2 × 2 × 2 k-point mesh. Whereas

∆frz is typically between 0.2 and 0.8 eV, ∆TZ is less than 0.1 eV.

−1 Basis-set corrected excitation energies as a function of Nk are shown in Figure2 for diamond, Si, LiF, and MgO. Our ﬁnal values for all eight materials are given in the “Final” column of Tab.2 and compared to experiment. Unsurprisingly, EOM-CCSD is a massive improvement over CIS; for the eight solids studied here, EOM-CCSD predicts excitation energies with a mean signed error (MSE) of 0.24 eV and a mean absolute error (MAE) of 0.27 eV. A few results in Tab.2 are noteworthy. First, we note that the excitation energy of cubic BN is frequently reported as 6.4 eV, which is almost 5 eV lower than the EOM-CCSD prediction. How-

10 ever, a GW/BSE calculation reported in 2004109 also found a value of around 11 eV and proposed a reinterpretation of the experimental data. Indeed, a joint theory-experiment paper published in 2018106 attributed the lower energy absorption features to a combination of defects and domains of hexagonal BN, further supporting a direct excitation energy of about 11 eV. Second, the excitation energy of AlP has frequently been reported as 3.6 eV, about 1 eV below the EOM-CCSD prediction. A 2019 publication reporting the results of TDDFT and GW/BSE calculations108 sug- gested that the 3.6 eV feature seen in experimental spectra is due to the indirect transition of AlP. Experimental spectra show a much stronger peak at around 4.6 eV,107 which is the likely value of the first direct excitation energy. The two materials with the largest error are LiF and MgO. Whereas absorption spectra of LiF typically show a narrow peak at about 12.7 eV101 (leading to an overprediction of 0.8 eV), inelastic X-ray scattering data is consistent with a value of 13.68 eV.102 For MgO, EOM-CCSD overpredicts the excitation energy by about 0.7 eV. Dittmer et al.110 very recently applied a flavor of similarity-transformed EOM-CCSD (which we emphasize is different from EOM-CCSD111) to the calculation of band gaps and optical gaps of solids using a cluster-based approach, to be contrasted with the periodic approach taken here. Interestingly, the two solids that are shared between our works are these same two outliers, MgO and LiF, for which that work predicts optical gaps of 6.32 and 11.41 eV, respectively. Based on the experimental values identified in Tab.2, these values underestimate the experimental ones by about 1.3 eV. The tendency of vanilla EOM-CCSD to overestimate excitation energies is the same as the one typically observed in molecules and could potentially be addressed via inclusion of triple excitations. However, we also emphasize that our calculations include no information about exciton- phonon effects, which are expected to reduce the excitation energy from its purely electronic value.112–116 For this reason, our electronic optical gaps might actually be more accurate than sug- gested here.

11 Figure 2: Excitation energy of diamond, Si, LiF, and MgO calculated with CIS and EOM-CCSD. Results are extrapolated to Nk → ∞ using the functional form given in Eq. 13. The dashed lines are ﬁtted based on results obtained with a number of k-points between 33 and 73. The dotted lines include only 23, 33, and 43 k-points. A variety of experimental results are indicated by solid lines.

4.2 Exciton binding energy

We now consider the exciton binding energy, defined as the difference between the fundamental band gap and the first neutral excitation energy (i.e. the optical gap). Within periodic coupled- cluster theory, the band gap is obtained from the calculation of the ionization potential (IP-EOM- CCSD) and the electron affinity (EA-EOM-CCSD), as described in Ref. 52. Here, our IP/EA- EOM-CCSD calculations are basis-set-corrected in the same way as described for our EOM-CCSD calculations. We focus on LiF, which is a direct-gap ionic insulator with a concomitantly large exciton binding energy. The minimum band gap occurs at the Γ point, which is where we calculate the IP and EA. In Fig.3(a), we show the fundamental (IP /EA) gap and the optical (EE) gap, as a function of the number of k-points sampled in the Brillouin zone. Clearly the fundamental gap is larger than

12 Figure 3: Fundamental band gap and optical gap (a) and exciton binding energy (b) of LiF from periodic EOM-CCSD. As discussed in the text, the optical gap is ﬁt to the functional form given in Eq. 13. Due to the behavior of IP/EA-EOM-CCSD, the fundamental band gap (red squares) and exciton binding energy (brown diamonds) are extrapolated to Nk → ∞ assuming a ﬁnite-size −1/3 scaling of the form ENk = E∞ + a Nk .

the optical gap such that the exciton binding energy is positive, as expected. The exciton binding energy is plotted in Fig.3(b), and seen to be around 0.8 eV with a 4 × 4 × 4 k-point mesh. Unlike neutral excitation energies, which conserve particle number, the IP and EA are charged excitation energies corresponding to a change in particle number. In particular, the same approximation considered above for CIS, i.e. the use of a single determinant, leads to Koopmans’ approximation to

the ionization potential, IP≈ −εi; as discussed above, occupied orbital energies converge slowly

with Nk when the ERIs are handled as described in section3. Therefore, as shown in Fig.3(b), we ﬁt the exciton binding energies calculated with 3 × 3 × 3 and 4 × 4 × 4 grids to the form

−1/3 E(Nk) = E∞ + aNk , in order to extrapolate to the thermodynamic limit. Doing so gives 1.47 eV, which is in good agreement with the experimental value of 1.6 eV.105 We note that if we sepa- rately extrapolate the band gap and the optical gap and take the diﬀerence, we get a larger value of 2.74 eV.

13 4.3 Exciton dispersion

Although optical absorption spectroscopy and modern theoretical approaches have primarily focused on excitons with q = 0, it is important to also consider excitons that carry a finite momentum q. For example, electron-hole pairs with finite momentum are realized in many indirect semiconductors and are also important for a quantitative modeling of the exciton-phonon interaction, excitonic dynamics, and radiative lifetimes. The simulation of excitons with finite-momentum is straightforward in EOM-CCSD, and sim- ply requires that the involved crystal momenta sum to q in Eqs.9 and 10. The EOM Hamiltonian is block-diagonal with respect to the exciton momentum and thus all accessible momenta can be studied independentaly and calculated in parallel. Because the exciton momentum q corresponds to a momentum difference, a periodic calculation only has access to values q = k − k0, where k are momenta from the k-point mesh. Therefore, different k-point meshes can be utilized in order to access different values of the exciton momentum q, albeit with an impact on the finite-size error. Again we focus on LiF, for which we show the EOM-CCSD exciton dispersion in Fig.4. Our results are compared to inelestic X-ray scattering (IXS) spectroscopy measurements performed by Abbamonte et al.102 (open circles). We utilize 3 × 3 × 3 k-point mesh (up-pointing triangles) and 4×4×4 k-point mesh (down-pointing triangles) in order to access more momenta q in the Brillouin

Figure 4: Exciton dispersion of LiF. Periodic EOM-CCSD data are obtained with diﬀerent k-point meshes in order to access more values of the momentum transfer q. Rigid shifts were applied, as described in the text. Also shown are the experimental inelastic X-ray scattering (IXS) data from Ref. 102 and the tight-binding (TB) model ﬁtted to that data.

14 Figure 5: Ground- and excited-state potential energy surface of diamond as a function of the unit cell volume (a) and the behavior of the excitation energy as a function of hydrostatic strain (b). Results are obtained from CCSD and EOM-CCSD with the DZVP basis set and a 3 × 3 × 3 sampling of the Brillouin zone.

zone. The 3 × 3 × 3 values were rigidly shifted in order to achieve agreement at the Γ point and subsequently, both dispersion data were rigidly shifted in order to place the exciton band center at 14.2 eV, as was observed experimentally. From their experimental data along the Γ − X line, those authors paramaterized a tight-binding model (with band center 14.2 eV and nearest-neighbor transfer integral −0.065 eV), which we have extended to the entire Brillouin zone for comparison (dashed line). We can see that the EOM-CCSD results are in good agreement with the IXS data, with an error less than 0.2 eV along the Γ → X direction. Our largest discrepancy is at the L point, although we emphasize that experimental data is unavailable at that momentum and the disagreement may indicate a failure of the simple tight-binding model.

4.4 Exciton-phonon interaction

Finally, as an additional demonstration of the solid-state problems that can be studied by EOM- CCSD, we consider the behavior of excitation energies as a function of lattice strain. As an example, in Fig.5(a) we show the ground-state and ﬁrst excited-state potential energy surfaces of diamond, associated with hydrostatic strain, i.e. isotropic variation of the unit cell. While the

15 ground-state energy minimum occurs at a lattice constant of 3.567 Å (which is fortuitously the exact experimental value5,117), the excited state has a minimum which is shifted to a larger lattice constant of 3.674 Å. The behavior of the excitation energy as a function of lattice strain can be used to determine properties of the exciton-phonon interaction. In particular, the interaction with acoustic phonons can be modeled within the deformation potential approximation for the change in the excitation

118,119 energy, ∆EX = 3Dε, where D is the hydrostatic deformation potential, 3ε is the trace of the

strain tensor, ε = (a − a0)/a0 is the relative strain, a is the strained lattice constant, and a0 is the unstrained lattice constant. Deﬁned in this way, our calculations predict the excitonic hydrostatic deformation potential of diamond to be D = −2.8 eV. Repeating the same procedure for MgO, we predict a larger D = −11.7 eV, indicating a stronger coupling to hydrostatic strain than in diamond. For comparison, the use of the direct HF band gap at Γ yields D = −5.0 eV and D = −14.4 eV, indicating the importance of electron correlation. Interestingly the analogous use of the LDA band gap yields D = −2.9 eV and D = −10.0 eV, in better agreement with the EOM-CCSD result. Experimentally determined deformation potentials for exciton-phonon interactions are sporadic in the literature and we consider a direct comparison on a wider variety of materials to be a topic for future work, along with the extension to optical phonons and phonons away from the Γ point.

5 Conclusions

To summarize, we have presented the results of periodic EOM-CCSD for various neutral excited- state properties of semiconductors and insulators. The method has shown promising results for optical excitation energies, exciton binding energies, exciton dispersion relations, and exciton-phonon interaction energies. Collectively, these results demonstrate that EOM-CCSD is a promising and tractable approach for the study of excited-state properties of solids. While we have attempted to address ﬁnite-size errors, arising from incomplete sampling of the Brillouin zone, the high cost of EOM-CCSD precludes a deﬁnitive extrapolation. Future work

16 will be focused on both analytical and numerical exploration of finite-size errors and strategies for amelioration, such as those that have been developed for ground-state CCSD.51,94,95 In a similar direction, we are exploring the use of approximations to EOM-CCSD with reduced scaling,120 which will enable the study of simple solids with more k-points or the study of solids with more complex unit cells. Additional future work is targeted at the study of simple metals (where finite- order perturbation theory breaks down88), the study of exciton-phonon interactions beyond the deformation potential approximation, and the efficient calculation of optical spectra.

Acknowledgments

X.W. thanks Dr. Varun Rishi for helpful discussions. This work was supported in part by the National Science Foundation under Grant No. CHE-1848369. All calculations were performed using resources provided by the Flatiron Institute. The Flatiron Institute is a division of the Simons Foundation.

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