2.3. CC Method

Similar to the ADC scheme, the CC method is a systematically improvable approach for solving the Schrödinger equation. Methods based on CC theory are becoming standard tools in applications of electronic structure theory for problems in chemistry and physics.^80,81 Recent works using the CC method have the reproduced X-ray absorption spectra in good agreement with experiments.³¹⁻³⁵

We use the e^T package, version 1.0.7, to compute X-ray spectra.^34,82 The equation-of-motion method with the CVS technique is used with three coupled-cluster approximations: CC2 (CC singles and perturbative doubles),⁸³ CCSD (CC singles and doubles),⁸⁴ and CC3 (CC singles and doubles and perturbative triples).^34,85 CC2 is an approximation to CCSD, while CC3 is an approximation to CCSDT (CC singles, doubles, and triples); the accuracy of the methods can thereby be formally classified as CC2 (least accurate) < CCSD < CC3 (most accurate). While the CC methods become more accurate in increasing rank, with systems with n electrons being described exactly with CC theory that includes up to n-fold substitutions, the computational cost also undergoes a steep increase at every step of the level.

We studied the basis set convergence of the CVS-CC2, CVS-CCSD, and CVS-CC3 spectra with the un-aug-pc-n basis sets (n = 0, 1, 2, and 3); the values of the corresponding K edges are given in the Supporting Information. The difference between the un-aug-pc-2 and un-aug-pc-3 K edges is of the order of 0.2 eV for all molecules, suggesting that the un-aug-pc-3 spectrum is sufficiently converged. Spectra for the un-aug-pc-3 basis are therefore used in this work. The CC3 calculations for SO₂ failed to converge due to degenerate eigenstates. Therefore, for SO₂, we only show spectra for the CC2 and CCSD methods, while for H₂S, OCS, CS, and also CC3 data are included.

The convergence of the CC spectra with respect to the number of excited states was examined. Five excited states were sufficient for a converged spectrum for SO₂; six states were necessary for CS and H₂S, while seven excited states were used for OCS. The CC stick spectra are broadened with the approach given in Section 2.

Because relativistic corrections are not included in e^T at the moment, we employ a semiempirical shift obtained at the X2C level⁶² for Hartree–Fock using the Psi4 program,⁸⁶ version 1.3.2. The resulting orbital energy shifts for the sulfur 1s are shown in Table 2. Based on these data, the CC excitation energies are shifted by + 7.9 eV to account for relativistic effects.

e^T employs a pivoted Cholesky decomposition of the atomic-orbital basis analogously to refs (68) and (69), allowing the use of overcomplete basis sets. As the basis set convergence of the CC methods is known to be similar, the importance of further augmentation of the basis, that is, the effect of Rydberg states was studied with the CVS-CC2 method. In agreement with the CVS-ADC calculations described in Section 2.1, multiple augmentation was found to have negligible effects on the CVS-CC spectra with the chosen number of excited states. The CVS-CC2/un-aug-pc-3 and CVS-CC2/un-daug-pc-3 stick spectra are given in the Supporting Information.