Stephan N. Steinmann

Comprehensive Benchmarking of a Density-Dependent Dispersion Correction

Standard density functional approximations cannot accurately describe interactions between nonoverlapping densities. A simple remedy consists in correcting for the missing interactions a posteriori, adding an attractive energy term summed over all atom pairs. The density-dependent energy correction, dDsC, presented herein, is constructed from dispersion coefficients computed on the basis of a generalized gradient approximation to Becke and Johnsons exchange-hole dipole moment formalism. dDsC also relies on an extended Tang and Toennies damping function accounting for charge-overlap effects. The comprehensive benchmarking on 341 diverse reaction energies divided into 18 illustrative test sets validates the robust performance and general accuracy of dDsC for describing various intra- and intermolecular interactions. With a total MAD of 1.3 kcal mol(-1), B97-dDsC slightly improves the results of M06-2X and B2PLYP-D3 (MAD = 1.4 kcal mol(-1) for both) at a lower computational cost. The density dependence of both the dispersion coefficients and the damping function makes the approach especially valuable for modeling redox reactions and charged species in general.

Why are the Interaction Energies of Charge-Transfer Complexes Challenging for DFT?

The description of ground state charge-transfer complexes is highly challenging. Illustrative examples include large overestimations of charge-transfer by local and semilocal density functional approximations as well as inaccurate binding energies. It is demonstrated here that standard density functionals fail to accurately describe interaction energies of charge-transfer complexes not only because of the missing long-range exchange as generally assumed but also as a result of the neglect of weak interactions. Thus, accounting for the missing van der Waals interactions is of key importance. These assertions, based on the evaluation of the extent of stabilization due to dispersion using both DFT coupled with our recent density-dependent dispersion correction (dDsC) and high-level ab initio computations, reflect the imperfect error-cancellation between the overestimation of charge-transfer and the missing long-range interactions. An in-depth energy decomposition analysis of an illustrative series of four small ambidentate molecules (HCN, HNC, HF, and ClF) bound together with NF3 provides the main conclusions, which are validated on a prototypical organic charge-transfer complex (i.e., tetrathiafulvalene-tetracyanoquinodimethane, TTF-TCNQ). We establish that the interaction energies for charge-transfer complexes can only be properly described when using well-balanced functionals such as PBE0-dDsC, M06-2X, and LC-BOP-LRD.

A System-Dependent Density-Based Dispersion Correction.

Density functional approximations fail to provide a consistent description of weak molecular interactions arising from small electron density overlaps. A simple remedy to correct for the missing interactions is to add a posteriori an attractive energy term summed over all atom pairs in the system. The density-dependent energy correction, presented herein, is applicable to all elements of the periodic table and is easily combined with any electronic structure method, which lacks the accurate treatment of weak interactions. Dispersion coefficients are computed according to Becke and Johnsons exchange-hole dipole moment (XDM) formalism, thereby depending on the chemical environment of an atom (density, oxidation state). The long-range ∼R(-6) potential is supplemented with higher-order correction terms (∼R(-8) and ∼R(-10)) through the universal damping function of Tang and Toennies. A genuine damping factor depending on (iterative) Hirshfeld (overlap) populations, atomic ionization energies, and two adjustable parameters specifically fitted to a given DFT functional is also introduced. The proposed correction, dDXDM, dramatically improves the performance of popular density functionals. The analysis of 30 (dispersion corrected) density functionals on 145 systems reveals that dDXDM largely reduces the errors of the parent functionals for both inter- and intramolecular interactions. With mean absolute deviations (MADs) of 0.74-0.84 kcal mol(-1), PBE-dDXDM, PBE0-dDXDM, and B3LYP-dDXDM outperform the computationally more demanding and most recent functionals such as M06-2X and B2PLYP-D (MAD of 1.93 and 1.06 kcal mol(-1), respectively).

A generalized-gradient approximation exchange hole model for dispersion coefficients.

A simple method for computing accurate density-dependent dispersion coefficients is presented. The dispersion coefficients are modeled by a generalized gradient-type approximation to Becke and Johnsons exchange hole dipole moment formalism. Our most cost-effective variant, based on a disjoint description of atoms in a molecule, gives mean absolute errors in the C(6) coefficients for 90 complexes below 10%. The inclusion of the missing long-range van der Waals interactions in density functionals using the derived coefficients in a pair wise correction leads to highly accurate typical noncovalent interaction energies.

Unified Inter- and Intramolecular Dispersion Correction Formula for Generalized Gradient Approximation Density Functional Theory.

Density functionals fail to provide a consistent description of weak intra- (i.e., short-range) and inter- (i.e., long-range) molecular interactions arising from nonoverlapping electron densities. An efficient way to correct the long-range errors is to add an empirical atom pair wise interaction-correction, inspired by the Lennard-Jones potential (R(-6) dependence). We show that the universal damping function of Tang and Toennies (TT) that includes higher-order correction terms (R(-8) and R(-10) dependent) reduces the intramolecular errors more efficiently, without altering the long-range correction. For general applicability, the TT damping function is augmented by a Fermi damping to prevent corrections at covalent distances. The performance of the new dD10 correction was tested in combination with three nonempirical GGAs (PBE, PBEsol, RGE2) on 64 illustrative reaction energies featuring both intra- and intermolecular interactions. With only two empirical parameters, PBE-dD10 outperforms the computationally more demanding and most recent functionals such as M06-2X or B2PLYP-D (MAD = 3.78 and 1.95 kcal mol(-1), respectively). At the cc-pVTZ level, PBE-dD10 (MAD = 1.24 kcal mol(-1)) considerably reduces common DFT errors successfully balancing intra- (short-range) and inter- (long-range) molecular interactions. While REG2-dD10 performs closely to PBE-dD10 (MAD = 1.48 kcal mol(-1)), the overall MAD of PBEsol-dD10 is again slightly higher (MAD = 1.76 kcal mol(-1)).

Hierarchically Structured Microfibers of “Single Stack” Perylene Bisimide and Quaterthiophene Nanowires

Organic nanowires and microfibers are excellent model systems for charge transport in organic semiconductors under nanoscopic confinement and may be relevant for future nanoelectronic devices. For this purpose, however, the preparation of well-ordered organic nanowires with uniform lateral dimensions remains a challenge to achieve. Here, we used the self-assembly of oligopeptide-substituted perylene bisimides and quaterthiophenes to obtain well-ordered nanofibrils. The individual nanofibrils were investigated by spectroscopic and imaging methods, and the preparation of hierarchically structured microfibers of aligned nanofibrils allowed for a comprehensive structural characterization on all length scales with molecular level precision. Thus, we showed that the molecular chirality resulted in supramolecular helicity, which supposedly serves to suppress lateral aggregation. We also proved that, as a result, the individual nanofibrils comprised a single stack of the π-conjugated molecules at their core. Moreover, the conformational flexibility between the hydrogen-bonded oligopeptides and the π-π stacked chromophores gave rise to synergistically enhanced strong π-π interactions and hydrogen-bonding. The result is a remarkably tight π-π stacking inside the nanofibrils, irrespective of the electronic nature of the employed chromophores, which may render them suitable nanowire models to investigate one-dimensional charge transport along defined π-π stacks of p-type or n-type semiconductors.

How do electron localization functions describe π-electron delocalization?

Scalar fields provide an intuitive picture of chemical bonding. In particular, the electron localization function (ELF) has proven to be highly valuable in interpreting a broad range of bonding patterns. The discrimination between enhanced or reduced electron (de)localization within cyclic π-conjugated systems remains, however, challenging for ELF. In order to clearly distinguish between the local properties of ten highly and weakly π-(de)localized prototype systems, we compare the ELFs of both the canonical wave functions and electron-localized states (diabatic) with those of two closely related scalar fields: the electron localizability indicator (ELI-D) and the localized orbital locator (LOL). The simplest LOL function distinguishes enhanced from weak π-(de)localization in an insightful and reliable manner. LOL offers the finest contrast between annulenes with 4n/4n + 2 π electrons and their inorganic analogues as well as between hyperconjugated cyclopentadiene derivatives. LOL(π) also gives an appealing and intuitive picture of the π-bond. In contrast, the most popular ELF fails to capture subtle contrasting local electronic properties and suffers from the arbitrariness of the σ/π dissection. The orbital separation of the most recent ELI-D is clear-cut but the interpretations sometime less straightforward in the present context.

Fluorescence sensing of caffeine in water with polysulfonated pyrenes.

The commercially available fluorescence dye 8-hydroxypyrene-1,3,6-trisulfonate (HPTS) can be used as a selective molecular probe for the detection of caffeine in aqueous solution.

Dispersion-Corrected Energy Decomposition Analysis for Intermolecular Interactions Based on the BLW and dDXDM Methods

As the simplest variant of the valence bond (VB) theory, the block-localized wave function (BLW) method defines the intermediate electron-localized state self-consistently at the DFT level and can be used to explore the nature of intermolecular interactions in terms of several physically intuitive energy components. Yet, it is unclear how the dispersion interaction affects such a kind of energy decomposition analysis (EDA) as standard density functional approximations neglect the long-range dispersion attractive interactions. Three electron densities corresponding to the initial electron-localized state, optimal electron-localized state, and final electron-delocalized state are involved in the BLW-ED approach; a density-dependent dispersion correction, such as the recently proposed dDXDM approach, can thus uniquely probe the impact of the long-range dispersion effect on EDA results computed at the DFT level. In this paper, we incorporate the dDXDM dispersion corrections into the BLW-ED approach and investigate a range of representative systems such as hydrogen-bonding systems, acid-base pairs, and van der Waals complexes. Results show that both the polarization and charge-transfer energies are little affected by the inclusion of the long-range dispersion effect, which thus can be regarded as an independent energy component in EDA.

Benchmark tests and spin adaptation for the particle-particle random phase approximation.

The particle-particle random phase approximation (pp-RPA) provides an approximation to the correlation energy in density functional theory via the adiabatic connection [H. van Aggelen, Y. Yang, and W. Yang, Phys. Rev. A 88, 030501 (2013)]. It has virtually no delocalization error nor static correlation error for single-bond systems. However, with its formal O(N(6)) scaling, the pp-RPA is computationally expensive. In this paper, we implement a spin-separated and spin-adapted pp-RPA algorithm, which reduces the computational cost by a substantial factor. We then perform benchmark tests on the G2/97 enthalpies of formation database, DBH24 reaction barrier database, and four test sets for non-bonded interactions (HB6/04, CT7/04, DI6/04, and WI9/04). For the G2/97 database, the pp-RPA gives a significantly smaller mean absolute error (8.3 kcal/mol) than the direct particle-hole RPA (ph-RPA) (22.7 kcal/mol). Furthermore, the error in the pp-RPA is nearly constant with the number of atoms in a molecule, while the error in the ph-RPA increases. For chemical reactions involving typical organic closed-shell molecules, pp- and ph-RPA both give accurate reaction energies. Similarly, both RPAs perform well for reaction barriers and nonbonded interactions. These results suggest that the pp-RPA gives reliable energies in chemical applications. The adiabatic connection formalism based on pairing matrix fluctuation is therefore expected to lead to widely applicable and accurate density functionals.