Periodic EOM-CC Theory in Crystalline Solids

• Periodic EOM-CC theory is an extension of coupled-cluster methods to periodic systems, leveraging translational symmetry and momentum resolution to accurately describe excitations.
• It employs a similarity-transformed Hamiltonian with singles and doubles truncation to model both neutral and charged excitations, achieving precise predictions for exciton dispersions and band gaps.
• The approach supports dual implementations using localized Gaussian and plane-wave bases, enabling detailed studies of optical absorption, defect spectroscopy, and finite-size corrections in crystalline materials.

Searching arXiv for recent and foundational papers on periodic equation-of-motion coupled-cluster theory. {"query":"periodic equation-of-motion coupled-cluster theory solids EOM-CCSD arXiv", "max_results": 10} Periodic equation-of-motion coupled-cluster theory is the extension of equation-of-motion coupled-cluster (EOM-CC) methodology to crystalline solids under periodic boundary conditions, with the excitation operators, amplitudes, and Hamiltonian resolved in crystal momentum and constrained by translational symmetry. In the published solid-state formulations, the method is built on a periodic coupled-cluster ground state, uses a similarity-transformed Hamiltonian, and treats neutral excitations, charged excitations, and frequency-dependent optical response within a unified many-body framework. At the singles-and-doubles level, periodic EOM-CCSD has been applied to direct excitons, finite-momentum excitons, optical absorption spectra, point defects, and band gaps in semiconductors and insulators (Wang et al., 2020, Wang et al., 2021, Moerman et al., 30 Jan 2025).

1. Formal structure of the periodic EOM-CC ansatz

The ground state is represented by the standard coupled-cluster exponential ansatz,

$|\Psi_0\rangle = e^{\hat T} |\Phi_0\rangle,$

or equivalently $\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$, where the reference determinant is obtained from a periodic mean-field calculation and $\hat T$ is truncated at singles and doubles in the CCSD approximation. The similarity-transformed Hamiltonian is

$\bar H = e^{-\hat T}\hat H e^{\hat T},$

and the periodic CCSD amplitude equations are the usual projected coupled-cluster equations written in a momentum-resolved basis (Wang et al., 2021, Gallo et al., 2020).

For neutral excitations, the periodic EE-EOM-CCSD ansatz takes the form

$$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$

or, in the $\mathbf q$-resolved notation used for crystalline excitons,

$$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$

At the CCSD level,

$$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$

with the corresponding deexcitation operator

$\hat \Lambda=\hat \Lambda_1+\hat \Lambda_2.$

The excited-state energies follow from the non-Hermitian eigenvalue problem

$$(\bar H-E_0)\hat R_n|\Phi_0\rangle=\omega_n \hat R_n|\Phi_0\rangle.$$

In periodic systems, every amplitude carries orbital and $\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$0-point labels, and momentum conservation is enforced modulo reciprocal lattice vectors. For neutral excitations, the Hamiltonian is block-diagonal in the total crystal momentum $\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$1, so optical excitations at $\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$2 and finite-momentum excitons at $\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$3 are treated within the same formal structure (Wang et al., 2020, Wang et al., 2021).

For charged excitations, the ionization-potential and electron-affinity variants of periodic EOM-CC are used. The $\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$4 and $\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$5 states are written as

$\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$6

with IP/EA operators truncated at singles and doubles in the standard EOM-CCSD manner. The fundamental band gap is then

$\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$7

A diagnostic introduced for the solid-state band-gap problem is the single-excitation character,

$\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$8

which measures how strongly the charged state resembles a one-particle quasiparticle. In the reported calculations, values near 1 correlate with higher IP/EA-EOM-CCSD accuracy (Moerman et al., 30 Jan 2025).

2. Periodic representations and implementations

Two implementation strategies are prominent in the literature. One uses translation-adapted Gaussian atomic orbitals. A localized Gaussian $\Psi_{\mathrm{CC}}=e^{\hat T}\ket{0}$9 is Bloch-summed as

$\hat T$0

and the periodic Hartree-Fock orbitals are expanded as

$\hat T$1

This Gaussian-based framework was used for periodic EOM-CCSD calculations of excitons and absorption spectra in three-dimensional solids with PySCF, GTH pseudopotentials, DZVP and TZVP basis sets, and Gaussian density fitting with an even-tempered auxiliary basis (Wang et al., 2020).

Within that formulation, the long-range Coulomb singularity is handled through chargeless pair densities,

$\hat T$2

with

$\hat T$3

The paper states that this is equivalent to omitting the $\hat T$4 Coulomb component in a plane-wave language. Excited states are obtained by Davidson diagonalization, with an initial guess from dense diagonalization in the singles subspace (Wang et al., 2020).

A second implementation uses periodic supercells and a plane-wave/PAW basis. In that approach, the ground-state reference is obtained from Hartree-Fock calculations performed with VASP using the PAW formalism and a plane-wave cutoff of 900 eV, and the post-HF treatment is carried out at the $\hat T$5 point in $\hat T$6, $\hat T$7, and $\hat T$8 fcc supercells. The code is built on the CC4S tensor framework. The implementation was benchmarked against NWChem for finite molecular systems, with agreement to eight significant digits for both UCCSD and EOM-CCSD excitation energies on test cases such as Ne and H$\hat T$9O in aug-cc-pVDZ (Gallo et al., 2020).

These two representations emphasize complementary aspects of periodic EOM-CC theory. The Bloch/Gaussian formulation makes translational symmetry explicit at the $\bar H = e^{-\hat T}\hat H e^{\hat T},$0-point level, whereas the supercell/plane-wave formulation is well suited to localized defect excitations. Both preserve the central EOM-CC structure of a similarity-transformed Hamiltonian acting in a truncated excitation manifold.

3. Neutral excitations, excitons, and direct optical response

For neutral excitations in bulk solids, periodic EE-EOM-CCSD has been applied to eight semiconductors and insulators—diamond, Si, SiC, LiF, LiCl, MgO, BN, and AlP—with direct singlet excitation energies in the range of 3 to 15 eV. After finite-size extrapolation and basis corrections, the reported final excitation energies include 7.47 eV for diamond, 3.52 eV for Si, 6.27 eV for SiC, 13.48 eV for LiF, 9.29 eV for LiCl, 8.29 eV for MgO, 11.11 eV for BN, and 4.48 eV for AlP. The overall statistics are a mean signed error of 0.24 eV and a mean absolute error of 0.27 eV relative to experiment (Wang et al., 2020).

The same momentum-resolved formalism yields finite-$\bar H = e^{-\hat T}\hat H e^{\hat T},$1 excitons. In LiF, periodic EOM-CCSD was used to compute exciton dispersion and compare it with inelastic X-ray scattering data from Abbamonte et al. After rigid shifts aligning the $\bar H = e^{-\hat T}\hat H e^{\hat T},$2-point excitation and placing the exciton band center at 14.2 eV, the reported discrepancy is less than 0.2 eV along $\bar H = e^{-\hat T}\hat H e^{\hat T},$3. The largest disagreement occurs near $\bar H = e^{-\hat T}\hat H e^{\hat T},$4, where experimental data were unavailable and the paper notes that the mismatch may reflect limitations of the simple tight-binding model rather than the EOM-CCSD result (Wang et al., 2020).

Optical absorption spectra can also be obtained without explicit enumeration of excited states. In the frequency-domain formulation, the spectrum is written as

$\bar H = e^{-\hat T}\hat H e^{\hat T},$5

where the response vector satisfies

$\bar H = e^{-\hat T}\hat H e^{\hat T},$6

This linear-system approach gives access to an energy range of tens of eV without explicit diagonalization of the full excited-state spectrum. Using Gaussian-based periodic EOM-CCSD, absorption spectra were reported for Si, SiC, diamond, MgO, BN, and LiF. The calculated lineshapes were found to be in good agreement with experiment, but the spectra were uniformly shifted to higher energies by about 1 eV. The same study reports that CIS dramatically overestimates excitation energies by several eV, whereas EOM-CCSD captures excitonic redistribution of oscillator strength and the overall structure of the absorption edge and higher features much better than CIS (Wang et al., 2021).

The residual upward shift in the absorption spectra was tentatively attributed to a combination of omitted vibrational effects and missing higher-order correlation, specifically triple excitations and above. The paper further notes that the apparent spectral discrepancy is larger than the previously observed overestimation of the first excitation energy because peaks in a full absorption spectrum occur at higher energies than the weakly absorbing onset (Wang et al., 2021).

4. Defect spectroscopy in periodic supercells

Periodic EE-EOM-CCSD has also been applied to localized excitations associated with neutral $\bar H = e^{-\hat T}\hat H e^{\hat T},$7-centers in alkaline-earth oxides. In that setting, the oxygen vacancy traps two electrons in a cavity stabilized by the Madelung field, and the low-lying excitations are interpreted as transitions among $\bar H = e^{-\hat T}\hat H e^{\hat T},$8- and $\bar H = e^{-\hat T}\hat H e^{\hat T},$9-like defect states. The calculations use a configuration-coordinate picture along an approximate $$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$0 vibrational mode, with absorption defined as the vertical transition from the ground singlet state $$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$1 to the excited singlet state $$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$2 at the ground-state geometry, and emission modeled as decay from the excited triplet $$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$3 to the singlet ground state (Gallo et al., 2020).

For MgO, the extrapolated periodic EOM-CCSD values are 5.28 eV for absorption and 3.66 eV for emission. These are compared with experiment, quantum Monte Carlo, CASPT2, and $$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$4@LDA0-BSE. The absorption energy agrees very well with experiment and other ab initio methods, while the emission energy is significantly higher than the often-cited 2.4 eV experimental band. The paper argues that this discrepancy likely reflects the assignment of the experimental luminescence rather than a failure of the method, and proposes that the 2.4 eV band is associated with recombination processes, possibly involving photoconversion to $$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$5 centers or defect/hole recombination, rather than the direct $$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$6 transition of the neutral $$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$7-center (Gallo et al., 2020).

For CaO, the extrapolated absorption and emission energies are 3.13 eV and 1.93 eV, in close agreement with experimental absorption of about 3.0–3.1 eV and experimental emission of about 1.93–2.05 eV. For SrO, the reported values are 2.34 eV for absorption and 1.2 eV for emission; the absorption agrees with the experimental estimate of about 2.4 eV, and the emission energy is presented as a prediction because no emission measurement was available at the time of writing (Gallo et al., 2020).

This defect literature establishes that periodic EOM-CCSD is not restricted to delocalized Bloch excitons. It is also usable for localized color-center excitations in large periodic supercells, provided that orbital truncation, complete-basis-set extrapolation, and finite-size corrections are handled carefully.

5. Charged excitations and fundamental band gaps

The periodic IP/EA-EOM-CCSD framework addresses charged excitations and fundamental band gaps. In this formulation, the band gap is obtained from the sum of the lowest electron-removal and electron-addition energies,

$$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$8

and the practical difficulty is severe finite-size error in feasible simulation cells. For three-dimensional systems, the dominant asymptotic behavior is reported as

$$|\Psi_n(q)\rangle = \hat R_n(q)\, e^{\hat T} |\Phi_0\rangle,$$9

with $\mathbf q$0 the number of $\mathbf q$1-points in the equivalent mesh (Moerman et al., 30 Jan 2025).

To make thermodynamic-limit estimates practical, a hybrid EOM-CCSD+$\mathbf q$2 strategy was introduced. The key observation is that, for the systems studied, $\mathbf q$3 and EOM-CCSD exhibit essentially the same finite-size convergence behavior. The finite-size dependence of $\mathbf q$4 gaps is fitted to the asymptotic form above, and the two methods are related at fixed size through

$\mathbf q$5

so that the thermodynamic-limit estimate becomes

$\mathbf q$6

The $\mathbf q$7 calculations are performed as $\mathbf q$8HF, using the same Hartree-Fock orbitals and energies as the EOM-CCSD calculations (Moerman et al., 30 Jan 2025).

The reported comparisons show good agreement with experiment for several materials, including C, Si, BN, BP, and LiCl, and larger discrepancies for LiH, LiF, and MgO, where the EOM-CCSD gap is too large. The same study reports that $\mathbf q$9 agrees very well with zero-point-renormalization-corrected experimental gaps, and uses that agreement to support the reliability of the reference experimental values. It also notes that for some materials the extrapolated EOM-CCSD gaps change by only a few tenths of an eV when the largest cell is omitted, indicating reasonable stability of the hybrid procedure (Moerman et al., 30 Jan 2025).

A major physical conclusion of the band-gap study is that EOM-CCSD accuracy depends strongly on the single-excitation character of the charged state. In a hydrogen-chain test, agreement with the $$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$0CCSD(T) reference worsens as the chain grows and the single-excitation character drops from about 95% to about 90%. For real solids, materials with $$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$1 show small relative errors against experiment, whereas LiH, LiF, and MgO, which fall below that threshold, show much larger deviations. The paper interprets reduced single-excitation character as a signal that the band-edge state departs from a simple quasiparticle picture, making the singles-and-doubles truncation less accurate (Moerman et al., 30 Jan 2025).

6. Convergence, extrapolation, and controlled approximations

A recurrent feature of periodic EOM-CC theory is the sensitivity of excited-state observables to finite-size effects. In the absorption-spectrum work, Brillouin-zone sampling is identified as a crucial difficulty because periodic EOM-CCSD is expensive enough that only moderately dense meshes can be used, yet spectral intensities and peak positions change significantly with mesh density and with the choice of mesh shift. Meshes centered at $$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$2 were compared with randomly shifted, symmetry-breaking meshes, and the random shifts were found to converge faster to the thermodynamic limit and reduce aliasing-like artifacts. Even a $$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$3 mesh may not be fully converged for small-gap materials such as Si and SiC (Wang et al., 2021).

In that same study, thermodynamic-limit effects were estimated by extrapolating the first excitation energy as

$$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$4

and then rigidly shifting the full spectrum by the resulting finite-size correction. The correction is reported to be typically in the 0.1–0.4 eV range for the materials studied, although the effect on spectral intensities is stated to be harder to correct. Basis-set incompleteness was found to be small for the chosen pseudopotential and Gaussian basis combinations. Freezing core and high-lying virtual orbitals introduces a roughly rigid upward shift in the spectrum, typically a few tenths of an eV, whereas partitioning the doubles block shifts spectra downward by a similar amount; in production calculations these two errors often cancel partially (Wang et al., 2021).

For direct neutral excitations, a related finite-size treatment was used in the earlier exciton study: $$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$5 while LiF exciton binding energies were extrapolated as

$$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$6

because the fundamental gap inherits the slower convergence of the charged IP/EA sectors (Wang et al., 2020).

The supercell defect calculations use a different convergence strategy. Excitation energies were found to converge approximately linearly with $$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$7, where $$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$8 and $$|\Psi(\mathbf{q})\rangle=\left[R_1(\mathbf{q})+R_2(\mathbf{q})\right]e^{T}|\Phi_0\rangle.$$9 are the numbers of active virtual and occupied orbitals. For MgO defect excitations, convergence with respect to the virtual space is much slower than with respect to occupied orbitals; using only the defect occupied orbital leads to errors of about 120 meV in the lowest singlet-triplet excitation, while using more than about 25 occupied orbitals gives excitation energies converged to within a few meV. Residual finite-size errors are then removed by assuming $$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$0 convergence with supercell size. The paper also explains anomalously high $$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$1 energies in CaO and SrO by the fact that their conduction-band minima lie at the Brillouin-zone boundary, which odd-sized supercells do not include properly (Gallo et al., 2020).

These convergence studies collectively show that periodic EOM-CC theory is not limited primarily by formal correctness; it is limited by the cost of reaching the thermodynamic and basis-set limits in a many-body method whose amplitudes and response vectors are momentum resolved.

7. Scaling, parallelization, and methodological position

The computational cost of periodic EOM-CCSD is severe even after exploiting translational symmetry. In the frequency-domain absorption formalism, the iterative solve at each frequency scales as

$$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$2

for full EOM-CCSD. A partitioned EOM-CCSD approximation replaces the doubles-doubles block of the similarity-transformed Hamiltonian by a diagonal matrix of orbital energy differences and reduces the cost to

$$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$3

This reduction is the main reason larger Brillouin-zone meshes are feasible in production spectrum calculations (Wang et al., 2021).

For the underlying periodic CCSD ground state, the Gaussian/Bloch formulation reports a scaling of

$$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$4

and the largest published EOM-CCSD calculation in that study used a $$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$5 mesh, corresponding in canonical supercell terms to 768 electrons in 640 orbitals (Wang et al., 2020). A complementary distributed-memory analysis for periodic CCSD gives the naive sequential scaling as

$$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$6

in cost and

$$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$7

in memory, and introduces a momentum-aware process-grid strategy using $$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$8 processes. With that layout, the reported per-process cost becomes

$$\hat T=\hat T_1+\hat T_2,\qquad \hat R=\hat R_1+\hat R_2,$$9

and the per-process memory becomes

$\hat \Lambda=\hat \Lambda_1+\hat \Lambda_2.$0

a reduction by a factor of $\hat \Lambda=\hat \Lambda_1+\hat \Lambda_2.$1 relative to the sequential baseline. The paper also emphasizes that communication appears only in the outermost loop of the nested-loop evaluation (Yamashita et al., 2019).

Although that parallelization paper is formulated for periodic CCSD rather than EOM-CC itself, it explicitly identifies its relevance to periodic EOM-CCSD: EOM calculations require the same ground-state $\hat \Lambda=\hat \Lambda_1+\hat \Lambda_2.$2-amplitudes and similarity-transformed Hamiltonian ingredients, obey the same momentum structure, and can use the same strategy of distributing tensors by momentum pairs so that most contractions remain local (Yamashita et al., 2019).

In methodological terms, periodic EOM-CC theory occupies a position beyond CIS, TDDFT, and $\hat \Lambda=\hat \Lambda_1+\hat \Lambda_2.$3-BSE in the sense stated in the absorption-spectrum study: it provides a controlled many-body treatment, reproduces realistic excitonic lineshapes, and can be formulated either as an eigenvalue problem or as a direct frequency-domain response calculation. The published limitations are equally consistent across the literature: high computational cost, difficult Brillouin-zone convergence, residual sensitivity of spectral intensities and band gaps to finite-size effects, and missing higher-order correlation and vibrational renormalization at the EOM-CCSD level (Wang et al., 2021, Moerman et al., 30 Jan 2025).