Density-Based Basis-Set Incompleteness Correction for GW Methods

A density-fitting implementation of the density-based basis-set correction method

This work reports an efficient density-fitting implementation of the density-based basis-set correction (DBBSC) method in the MOLPRO software. This method consists in correcting the energy calculated by a wave-function method with a given basis set by an adapted basis-set correction density functional incorporating the short-range electron correlation effects missing in the basis set, resulting in an accelerated convergence to the complete-basis-set limit. Different basis-set correction density-functional approximations are explored and the complementary-auxiliary-basis-set single-excitation correction is added. The method is tested on a benchmark set of reaction energies at the second-order Møller-Plesset (MP2) level and a comparison with the explicitly correlated MP2-F12 method is provided. The results show that the DBBSC method greatly accelerates the basis convergence of MP2 reaction energies, without reaching the accuracy of the MP2-F12 method but with a lower computational cost.

Noncovalently bound excited-state dimers: a perspective on current time-dependent density functional theory approaches applied to aromatic excimer models

Excimers are supramolecular systems whose binding strength is influenced by many factors that are ongoing challenges for computational methods, such as charge transfer, exciton coupling, and London dispersion interactions. Treating the various intricacies of excimer binding at an adequate level is expected to be particularly challenging for time-dependent Density Functional Theory (TD-DFT) methods. In addition to well-known limitations for some TD-DFT methods in the description of charge transfer or exciton coupling, the inherent London dispersion problem from ground-state DFT translates to TD-DFT. While techniques to appropriately treat dispersion in DFT are well-developed for electronic ground states, these dispersion corrections remain largely untested for excited states. Herein, we aim to shed light on current TD-DFT methods, including some of the newest developments. The binding of four model excimers is studied across nine density functionals with and without the application of additive dispersion corrections against a wave function reference of SCS-CC2/CBS(3,4) quality, which approximates select CCSDR(3)/CBS data adequately. To our knowledge, this is the first study that presents single-reference wave function dissociation curves at the complete basis set level for the assessed model systems. It is also the first time range-separated double-hybrid density functionals are applied to excimers. In fact, those functionals turn out to be the most promising for the description of excimer binding followed by global double hybrids. Range-separated and global hybrids-particularly with large fractions of Fock exchange-are outperformed by double hybrids and yield worse dissociation energies and inter-molecular equilibrium distances. The deviation between each assessed functional and reference increases with system size, most likely due to missing dispersion interactions. Additive dispersion corrections of the DFT-D3(BJ) and DFT-D4 types reduce the average errors for TD-DFT methods but do so inconsistently and therefore do not offer a black-box solution in their ground-state parametrised form. The lack of appropriate description of dispersion effects for TD-DFT methods is likely hindering the practical application of the herein identified more efficient methods. Dispersion corrections parametrised for excited states appear to be an important next step to improve the applicability of TD-DFT methods and we hope that our work assists with the future development of such corrections.

Reference Vertical Excitation Energies for Transition Metal Compounds

To enrich and enhance the diversity of the quest database of highly accurate excitation energies [Véril, M.; et al. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2021, 11, e1517], we report vertical transition energies in transition metal compounds. Eleven diatomic molecules with a singlet or doublet ground state containing a fourth-row transition metal (CuCl, CuF, CuH, ScF, ScH, ScO, ScS, TiN, ZnH, ZnO, and ZnS) are considered, and the corresponding excitation energies are computed using high-level coupled-cluster (CC) methods, namely, CC3, CCSDT, CC4, and CCSDTQ, as well as multiconfigurational methods such as CASPT2 and NEVPT2. In many cases, to provide more comprehensive benchmark data, we also provide full configuration interaction estimates computed with the configuration interaction using a perturbative selection made iteratively (CIPSI) method. Based on these calculations, theoretical best estimates of the transition energies are established in both the aug-cc-pVDZ and aug-cc-pVTZ basis sets. This allows us to accurately assess the performance of the CC and multiconfigurational methods for this specific set of challenging transitions. Furthermore, comparisons with experimental data and previous theoretical results are also reported.

The three channels of many-body perturbation theory: GW, particle-particle, and electron-hole T-matrix self-energies

We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known GW self-energy, as well as the particle-particle and electron-hole T-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory. The performance of these schemes on strongly correlated systems (B2 and C2) is also discussed.

Quasiparticle Self-Consistent GW-Bethe-Salpeter Equation Calculations for Large Chromophoric Systems

The GW-Bethe–Salpeter equation (BSE) method is promising for calculating the low-lying excitonic states of molecular systems. However, so far it has only been applied to rather small molecules and in the commonly implemented diagonal approximations to the electronic self-energy, it depends on a mean-field starting point. We describe here an implementation of the self-consistent and starting-point-independent quasiparticle self-consistent (qsGW)-BSE approach, which is suitable for calculations on large molecules. We herein show that eigenvalue-only self-consistency can lead to an unfaithful description of some excitonic states for chlorophyll dimers while the qsGW-BSE vertical excitation energies (VEEs) are in excellent agreement with spectroscopic experiments for chlorophyll monomers and dimers measured in the gas phase. Furthermore, VEEs from time-dependent density functional theory calculations tend to disagree with experimental values and using different range-separated hybrid (RSH) kernels does change the VEEs by up to 0.5 eV. We use the new qsGW-BSE implementation to calculate the lowest excitation energies of the six chromophores of the photosystem II (PSII) reaction center (RC) with nearly 2000 correlated electrons. Using more than 11,000 (6000) basis functions, the calculation could be completed in less than 5 (2) days on a single modern compute node. In agreement with previous TD-DFT calculations using RSH kernels on models that also do not include environmental effects, our qsGW-BSE calculations only yield states with local characters in the low-energy spectrum of the hexameric complex. Earlier works with RSH kernels have demonstrated that the protein environment facilitates the experimentally observed interchromophoric charge transfer. Therefore, future research will need to combine correlation effects beyond TD-DFT with an explicit treatment of environmental electrostatics.

Basis Set Selection for Molecular Core-Level GW Calculations

The GW approximation has been recently gaining popularity among the methods for simulating molecular core-level X-ray photoemission spectra. Traditionally, Gaussian-type orbital GW core-level binding energies have been computed using either the cc-pVnZ or def2-nZVP basis set families, extrapolating the obtained results to the complete basis set limit, followed by an element-specific relativistic correction. Despite achieving rather good accuracy, it has been previously stated that these binding energies are chronically underestimated. In the present work, we show that those previous studies obtained results that were not well-converged with respect to the basis set size and that, once basis set convergence is achieved, there seems to be no such underestimation. Standard techniques known to offer a good cost-accuracy ratio in other theories demonstrate that the cc-pVnZ and def2-nZVP families exhibit contraction errors and might lead to unreliable complete basis set extrapolations for absolute binding energies, often deviating about 200-500 meV from the putative basis set limit found in this work. On the other hand, uncontracted versions of these basis sets offer vastly improved convergence. Even faster convergence can be obtained using core-rich property-optimized basis set families like pcSseg-n, pcJ-n, and ccX-nZ. Finally, we also show that the improvement observed for core properties using these specialized basis sets does not degrade their description of valence excitations: vertical ionization potentials and electron affinities computed with these basis sets converge as fast as the ones obtained with the aug-cc-pVnZ family, thus offering a balanced description of both core and valence regions.

Basis-set correction for coupled-cluster estimation of dipole moments

The present work proposes an approach to obtain a basis-set correction based on density-functional theory (DFT) for the computation of molecular properties in wave-function theory (WFT). This approach allows one to accelerate the basis-set convergence of any energy derivative of a non-variational WFT method, generalizing previous works on the DFT-based basis-set correction where either only ground-state energies could be computed with non-variational wave functions [Loos et al., J. Phys. Chem. Lett. 10, 2931 (2019)] or properties could be computed as expectation values over variational wave functions [Giner et al., J. Chem. Phys. 155, 044109 (2021)]. This work focuses on the basis-set correction of dipole moments in coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)], which is numerically tested on a set of 14 molecules with dipole moments covering two orders of magnitude. As the basis-set correction relies only on Hartree–Fock densities, its computational cost is marginal with respect to the one of the CCSD(T) calculations. Statistical analysis of the numerical results shows a clear improvement of the basis convergence of the dipole moment with respect to the usual CCSD(T) calculations.

Noncovalently bound excited-state dimers: a perspective on current time-dependent density functional theory approaches applied to aromatic excimer models

Excimers are supramolecular systems whose binding strength is influenced by many factors that are ongoing challenges for computational methods, such as charge transfer, exciton coupling, and London dispersion interactions. Treating the various intricacies of excimer binding at an adequate level is expected to be particularly challenging for Time-Dependent Density Functional Theory (TD-DFT) methods. In addition to well-known limitations for some TD-DFT methods in the description of charge transfer or exciton coupling, the inherent London dispersion problem from ground-state DFT translates to TD-DFT. While techniques to appropriately treat dispersion in DFT are well-developed for electronic ground states, these dispersion corrections remain largely untested for excited states. Herein, we aim to shed light on current TD-DFT methods, including some of the newest developments. The binding of four model excimers is studied across nine density functionals with and without the application of additive dispersion corrections against a wave function reference of SCS-CC2/CBS(3,4) quality, which approximates select CCSDR(3)/CBS data adequately. To our knowledge, this is the first study that presents single-reference wave function dissociation curves at the complete basis set level for the assessed model systems. It is also the first time range-separated double-hybrid density functionals are applied to excimers. In fact, those functionals turn out to be the most promising for the description of excimer binding followed by global double hybrids. Range-separated and global hybrids-particularly with large fractions of Fock exchange-are outperformed by double hybrids and yield worse dissociation energies and inter-molecular equilibrium distances. The deviation between each assessed functional and reference increases with system size, most likely due to missing dispersion interactions. Additive dispersion corrections of the DFT-D3(BJ) and DFT-D4 types reduce the average errors for TD-DFT methods but do so inconsistently and therefore do not offer a black-box solution in their ground-state parametrised form. The lack of appropriate description of dispersion effects for TD-DFT methods is likely hindering the practical application of the herein identified more efficient methods. Dispersion corrections parametrised for excited states appear to be an important next step to improve the applicability of TD-DFT methods and we hope that our work assists with the future development of such corrections.

GW100: A Slater-Type Orbital Perspective

We calculate complete basis set (CBS) limit-extrapolated ionization potentials (IPs) and electron affinities (EA) with Slater-type basis sets for the molecules in the GW100 database. To this end, we present two new Slater-type orbital (STO) basis sets of triple-(TZ) and quadruple-ζ (QZ) quality, whose polarization is adequate for correlated-electron methods and which contain extra diffuse functions to be able to correctly calculate EAs of molecules with a positive lowest unoccupied molecular orbital (LUMO). We demonstrate that going from TZ to QZ quality consistently reduces the basis set error of our computed IPs and EAs, and we conclude that a good estimate of these quantities at the CBS limit can be obtained by extrapolation. With mean absolute deviations (MAD) from 70 to 85 meV, our CBS limit-extrapolated IP are in good agreement with results from FHI-AIMS, TURBOMOLE, VASP, and WEST, while they differ by more than 130 meV on average from nanoGW. With a MAD of 160 meV, our EA are also in good agreement with the WEST code. Especially for systems with positive LUMOs, the agreement is excellent. With respect to other codes, the STO-type basis sets generally underestimate EAs of small molecules with strongly bound LUMOs. With 62 meV for IPs and 93 meV for EAs, we find much better agreement with CBS limit-extrapolated results from FHI-AIMS for a set of 250 medium to large organic molecules.

Self-consistent density-based basis-set correction: How much do we lower total energies and improve dipole moments?

This work provides a self-consistent extension of the recently proposed density-based basis-set correction method for wave function electronic-structure calculations [E. Giner et al., J. Chem. Phys. 149, 194301 (2018)]. In contrast to the previously used approximation where the basis-set correction density functional was a posteriori added to the energy from a wave-function calculation, here the energy minimization is performed including the basis-set correction. Compared to the non-self-consistent approximation, this allows one to lower the total energy and change the wave function under the effect of the basis-set correction. This work addresses two main questions: (i) What is the change in total energy compared to the non-self-consistent approximation and (ii) can we obtain better properties, namely, dipole moments, with the basis-set corrected wave functions. We implement the present formalism with two different basis-set correction functionals and test it on different molecular systems. The main results of the study are that (i) the total energy lowering obtained by the self-consistent approach is extremely small, which justifies the use of the non-self-consistent approximation, and (ii) the dipole moments obtained from the basis-set corrected wave functions are improved, being already close to their complete basis-set values with triple-zeta basis sets. Thus, the present study further confirms the soundness of the density-based basis-set correction scheme.

Assessing the G0W0Γ0(1) Approach: Beyond G0W0 with Hedin's Full Second-Order Self-Energy Contribution

We present and benchmark a self-energy approach for quasiparticle energy calculations that goes beyond Hedin's GW approximation by adding the full second-order self-energy (FSOS-W) contribution. The FSOS-W diagram involves two screened Coulomb interaction (W) lines, and adding the FSOS-W to the GW self-energy can be interpreted as first-order vertex correction to GW (GWΓ⁽¹⁾). Our FSOS-W implementation is based on the resolution-of-identity technique and exhibits better than O(N⁵) scaling with system size for small- to medium-sized molecules. We then present one-shot GWΓ⁽¹⁾ (G₀W₀Γ₀⁽¹⁾) benchmarks for the GW100 test set and a set of 24 acceptor molecules. For semilocal or hybrid density functional theory starting points, G₀W₀Γ₀⁽¹⁾ systematically outperforms G₀W₀ for the first vertical ionization potentials and electron affinities of both test sets. Finally, we demonstrate that a static FSOS-W self-energy significantly underestimates the quasiparticle energies.

Spin-Conserved and Spin-Flip Optical Excitations from the Bethe-Salpeter Equation Formalism

Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe–Salpeter equation (BSE) formalism of many-body perturbation theory, in its static approximation, is “blind” to double (and higher) excitations, which are ubiquitous, for example, in conjugated molecules like polyenes. Here, we apply the spin-flip ansatz (which considers the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism in order to access, in particular, double excitations. The present scheme is based on a spin-unrestricted version of the GW approximation employed to compute the charged excitations and screened Coulomb potential required for the BSE calculations. Dynamical corrections to the static BSE optical excitations are taken into account via an unrestricted generalization of our recently developed (renormalized) perturbative treatment. The performance of the present spin-flip BSE formalism is illustrated by computing excited-state energies of the beryllium atom, the hydrogen molecule at various bond lengths, and cyclobutadiene in its rectangular and square-planar geometries.

Cubic-Scaling All-Electron GW Calculations with a Separable Density-Fitting Space-Time Approach

We present an implementation of the GW space-time approach that allows cubic-scaling all-electron calculations with standard Gaussian basis sets without exploiting any localization or sparsity considerations. The independent-electron susceptibility is constructed in a time representation over a nonuniform distribution of real-space locations {r_k} optimized within a separable resolution-of-the-identity framework to reproduce standard Coulomb-fitting calculations with meV accuracy. The compactness of the obtained {r_k} distribution leads to a crossover with the standard Coulomb-fitting scheme for system sizes below a few hundred electrons. The needed analytic continuation follows a recent approach that requires the continuation of the screened Coulomb potential rather than the much more structured self-energy. The present scheme is benchmarked over large molecular sets, and scaling properties are demonstrated on a family of defected hexagonal boron-nitride flakes containing up to 6000 electrons.

Low-Order Scaling G0W0 by Pair Atomic Density Fitting

We derive a low-scaling G0W0 algorithm for molecules using pair atomic density fitting (PADF) and an imaginary time representation of the Green's function and describe its implementation in the Slater type orbital (STO)-based Amsterdam density functional (ADF) electronic structure code. We demonstrate the scalability of our algorithm on a series of water clusters with up to 432 atoms and 7776 basis functions and observe asymptotic quadratic scaling with realistic threshold qualities controlling distance effects and basis sets of triple-ζ (TZ) plus double polarization quality. Also owing to a very small prefactor, a G0W0 calculation for the largest of these clusters takes only 240 CPU hours with these settings. We assess the accuracy of our algorithm for HOMO and LUMO energies in the GW100 database. With errors of 0.24 eV for HOMO energies on the quadruple-ζ level, our implementation is less accurate than canonical all-electron implementations using the larger def2-QZVP GTO-type basis set. Apart from basis set errors, this is related to the well-known shortcomings of the GW space-time method using analytical continuation techniques as well as to numerical issues of the PADF approach of accurately representing diffuse atomic orbital (AO) products. We speculate that these difficulties might be overcome by using optimized auxiliary fit sets with more diffuse functions of higher angular momenta. Despite these shortcomings, for subsets of medium and large molecules from the GW5000 database, the error of our approach using basis sets of TZ and augmented double-ζ (DZ) quality is decreasing with system size. On the augmented DZ level, we reproduce canonical, complete basis set limit extrapolated reference values with an accuracy of 80 meV on average for a set of 20 large organic molecules. We anticipate our algorithm, in its current form, to be very useful in the study of single-particle properties of large organic systems such as chromophores and acceptor molecules.

Extrapolating Unconverged GW Energies up to the Complete Basis Set Limit with Linear Regression

The GW approximation to the electronic self-energy is now a well-recognized approach to obtain the electron quasiparticle energies of molecules and, in particular, their ionization potential and electron affinity. Though much faster than the corresponding wavefunction methods, the GW energies are still affected by slow convergence with respect to the basis completeness. This limitation hinders a wider application of the GW approach. Here, we show that we can reach the complete basis set limit for the cumbersome GW calculations solely based on fast preliminary calculations with an unconverged basis set. We introduce a linear model that correlates the molecular orbital characteristics and the basis convergence error for a large database of approximately 600 states in 104 organic molecules that contain H, C, O, N, F, P, S, and Cl. The model employs molecular-orbital-based non-linear descriptors that encode efficiently the chemical space offering outstanding transferability. Using a low number of descriptors (17) the performance of this extrapolation procedure is superior to that of the earlier more physically motivated approaches. The predictive power of the method is finally demonstrated for a selection of large acceptor molecules.