Computational protocol: Outer-valenceElectron Spectra of Prototypical AromaticHeterocycles from an Optimally Tuned Range-Separated Hybrid Functional

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[…] All PBE calculations (within the KS scheme) and PBE0 and RSH calculations (within the GKS scheme) presented in this article were obtained within the QChem and NWChem codes, using cc-PVTZ basis functions. All geometries were optimized using the PBE functional. As the above-described tuning scheme is based on components taken from well-established density functionals (cf. eq ) and range-separation is available in many different electronic structure codes, optimal tuning of γ via eq is straightforward to perform. Moreover, the tuning strategy is computationally efficient, as it relies on a series of standard total-energy DFT calculations. We note, however, that when charging the gas-phase molecule as part of the above-explained tuning procedure, the configuration of the cation and/or anion must be identified carefully, as it may affect the results of the calculation.We illustrate the possible complications associated with different ion configurations by considering pyridine—one of the molecules of Figure , which is discussed extensively in the results section. When tuning γ using the above-described approach, it is important to ensure that the character of the HOMO (related to the left-hand term in eq ) corresponds to the “hole density,” defined as the charge-density difference of the neutral and charged states (and, consequently, related to the right-hand term in eq ). For pyridine, the self-consistent solution of the GKS equation with the RSH functional was found to lead to two different doublet configurations of the cation, depending on the initial guess used in the procedure. These two configurations correspond to two qualitatively different hole densities (see Figure , left part). As shown in the right part of Figure , the reason for the different hole densities is that the two cation configurations possess two different LUMO orbitals; i.e., the two cationic ground states represent two different ionization processes. The main difference is that the electron “loss” is, in one case, from a π orbital and, in the other case, from a σ orbital. These two cationic configurations are energetically close, which is consistent with the observation that the HOMO and the HOMO–1 of pyridine are very close in energy (vide infra). Importantly, however, only the “hole density” depicted in Figure b—which is associated with the configuration lower in energy, i.e., the true ground state predicted for the cation—corresponds to the HOMO of pyridine (see Figure c). Therefore, one has to ensure that the cationic state shown in Figure b is indeed the one entering the tuning procedure, in order to retain consistency for the orbital energies and total energies required in eq .For our analysis of the OT-RSH results, we also performed comparative GW calculations, as well as self-interaction-corrected calculations and KS-PBE0 calculations (the latter are defined and explained below). Our GW calculations are based on a standard G0W0 scheme, where quasi-particle energies are computed via a first-order correction to DFT eigenvalues, with no self-consistent update of the starting wave functions. The starting quasi-particle wave function for the G0W0 corrections was obtained from the PBE functional. The static dielectric function is computed within the random-phase approximation and extended to finite frequency via the generalized plasmon-pole (GPP) model of Hybertsen and Louie.Our G0W0 calculations were performed using the BerkeleyGW package, which employs a plane-wave basis set to compute the dielectric function and self-energy, using a PBE starting point. DFT-PBE calculations were performed within the Quantum Espresso package, which is compatible with BerkeleyGW. The nuclei and core electrons were described by Troullier–Martins relativistic norm-conserving pseudopotentials, which are part of the Quantum Espresso pseudopotential library. Here, one, four, five, and six electrons were explicitly considered as valence electrons for H, C, N, and S, respectively, with cutoff radii (in Bohr) of 1.3, 0.5, 1.0, and 1.7, respectively. We used a plane-wave basis cutoff of 80 Ry for benzene and 120 Ry for pyridine, pyrimidine, and 3N-thiol. These values lead to a total DFT energy converged to <1 meV/atom. To avoid spurious interactions with periodic images, we used a supercell with lattice vectors twice the size necessary to contain 99% of the charge density and, when computing the GW self-energy, the Coulomb interaction was truncated at distances larger than half of the unit cell size. The supercell dimensions, in atomic units, were 35 × 39 × 24; 30 × 20 × 32, 19 × 30 × 30; and 64 × 26 × 15.5 for benzene, pyridine, pyrimidine, and 3N-thiol, respectively.Our static dielectric function and self-energy were constructed from 4914, 5515, 5071, and 3598 unoccupied states, respectively, for benzene, pyridine, pyrimidine, and 3N-thiol. For the former three prototypical small molecules, this energy range corresponds to 90 eV above the vacuum energy, while for 3N-thiol, it corresponds to 50 eV above the vacuum energy. Fewer states were included for 3N-thiol due to the greater computational expense associated with this rather large system. A static remainder approach was applied to the self-energy to approximately complete the unoccupied subspace. The plane-wave cutoff for the dielectric function was 30 Ry for pyridine and pyrimidine and 24 Ry for 3N-thiol and benzene. We find that these parameters converge the HOMO energies of the prototypical small molecules to less than 0.1 eV. On the basis of the convergence behavior of these molecules and the residual differences that we find for GW and OT-RSH HOMO energies (vide infra), we extrapolate the errors associated with eigenvalues of the 3N-thiol calculation to be less than 0.2 eV.All SIC calculations were based on the seminal SIC concept of Perdew and Zunger. However, we constructed a spatially local, multiplicative exchange-correlation potential identical for all orbitals in the system, which ensures that the SIC remains within the KS realm., This is based on the generalized optimized effective potential (OEP) equation, which extends the original OEP equation to the case of unitarily variant functionals. It is solved using the generalized Krieger–Li–Iafrate (KLI) approximation., Unlike the KLI approximation to the standard OEP equation, which can introduce significant deviations for the SIC, the generalized KLI approximation used here has been shown to be an excellent approximation to the generalized OEP. The additional degree of freedom arising from the variance inherent to our procedure can be used to construct a set of orbitals that minimize the total SIC energy of the system, where we applied a complex-valued energy minimizing unitary orbital transformation. For additional insights, we also used the PBE0 functional in conjunction with a local multiplicative potential, constructed—in contrast to traditional GKS schemes—via the KLI approximation for the exact exchange part of the functional. We refer to these calculations as PBE0KS.All SIC and PBE0KS calculations were performed with the Bayreuth version of the PARSEC real-space code, where we employed a grid-spacing of 0.2 Bohr and Troullier–Martins norm-conserving pseudopotentials.Finally, for a meaningful comparison between the results of different functionals and/or computational approaches, we tested explicitly that eigenvalues obtained from different codes and basis-set expansions (Gaussian, planewave, real-space) do not differ by more than 0.1 eV for the same underlying functional. Furthermore, we verified by visual inspection, that eigenvalues calculated from different methods correspond to the same molecular orbitals. […]

Pipeline specifications

Software tools NWChem, Quantum ESPRESSO
Application Mathematical modeling
Chemicals Benzene