Correction: Benchmark and performance of long-range corrected time-dependent density functional tight binding (LC-TD-DFTB) on rhodopsins and light-harvesting complexes.

Correction for 'Benchmark and performance of long-range corrected time-dependent density functional tight binding (LC-TD-DFTB) on rhodopsins and light-harvesting complexes' by Beatrix M. Bold et al., Phys. Chem. Chem. Phys., 2020, 22, 10500-10518, https://doi.org/10.1039/C9CP05753F.

Pushing the limits: Efficient wavefunction methods for excited states in complex systems using frozen-density embedding.

Frozen density embedding (FDE) is an embedding method for complex environments that is simple for users to set up. It reduces the computation time by dividing the total system into small subsystems and approximating the interaction by a functional of their densities. Its combination with wavefunction methods is, however, limited to small- or medium-sized molecules because of the steep scaling in computation time of these methods. To mitigate this limitation, we present a combination of the FDE approach with pair natural orbitals (PNOs) in the TURBOMOLE software package. It combines the uncoupled FDE (FDEu) approach for excitation energy calculations with efficient implementations of second-order correlation methods in the ricc2 and pnoccsd programs. The performance of this combination is tested for tetraazaperopyrene (TAPP) molecular crystals. It is shown that the PNO truncation error on environment-induced shifts is significantly smaller than the shifts themselves and, thus, that the local approximations of PNO-based wavefunction methods can without the loss of relevant digits be combined with the FDE method. Computational wall times are presented for two TAPP systems. The scaling of the wall times is compared to conventional supermolecular calculations and demonstrates large computational savings for the combination of FDE- and PNO-based methods. Additionally, the behavior of excitation energies with the system size is investigated. It is found that the excitation energies converge quickly with the size of the embedding environment for the TAPPs investigated in the current study.

Wavefunction frozen-density embedding with one-dimensional periodicity: Electronic polarization effects from local perturbations.

We report an approach to treat polarization effects in a one-dimensional (1D) environment using frozen-density embedding (FDE), suitable to compute response to electron loss or attachment as occurring in organic semiconductors during charge migration. The present work provides two key developments: (a) Local perturbations are computed avoiding an infinite repetition thereof and (b) a first-order equation-of-motion ansatz is used to compute polarization effects due to electron loss and attachment, ensuring an efficient calculation by avoiding open-shell calculations. In a first step, an unperturbed 1D molecular chain is equilibrated using FDE by translation of the center molecule. In a subsequent second step, long-range contributions are frozen and a local perturbation is introduced in the center subsystem. Freeze-thaw iterations are used to relax the electronic wavefunction of both the center subsystem and subsystems in an active region around the center subsystem, avoiding the need to translate the perturbation. The proposed scheme proves to be very efficient and allows for the calculation of charged tetraazaperopyrenes in 1D chains. Due to its efficiency, the new method is capable of providing wavefunction-based reference data relevant for electronic couplings in complex environments.

Understanding excited state properties of host materials in OLEDs: simulation of absorption spectrum of amorphous 4,4-bis(carbazol-9-yl)-2,2-biphenyl (CBP).

4,4-Bis(carbazol-9-yl)-2,2-biphenyl (CBP) is widely used as a host material in phosphorescent organic light-emitting diodes (PhOLEDs). In the present study, we simulate the absorption spectra of CBP in gas and condensed phases, respectively, using the efficient time-dependent long-range corrected tight-binding density functional theory (TD-LC-DFTB). The accuracy of the condensed-phase absorption spectra computed using the structures obtained from classical molecular dynamics (MD) and quantum mechanical/molecular mechanical (QM/MM) simulations is examined by comparison with the experimental absorption spectrum. It is found that the TD-LC-DFTB gas-phase spectrum is in good agreement with the GW-BSE spectrum, indicating TD-LC-DFTB is an accurate and robust method in calculating the excitation energies of CBP. For the condensed-phase spectrum, we find that the electrostatic embedding has a minor influence on the excitation energy. Computing accurate absorption spectra is a particular challenge since static and dynamic disorders have to be taken into account. The static disorder results from the molecular packing in the material, which leads to molecule deformations. Since these structural changes sensitively impact the excitation energies of the individual molecules, a proper representation of this static disorder indicates that a reasonable structural model of the material has been generated. The good agreement between computed and experimental absorption spectra is therefore an indicator for the structural model developed. Concerning dynamic disorder, we find that molecular changes occur on long timescales in the ns-regime, which requires the use of fast computation approaches to reach convergence. The structural models derived in this work are the basis for future studies of charge and exciton transfer in CBP and related materials, also concerning the degradation mechanisms of CBP-based PhOLEDs.

Employing Pseudopotentials to Tackle Excited-State Electron Spill-Out in Frozen Density Embedding Calculations.

In frozen density embedding (FDE), the properties of a target molecule are computed in the presence of an effective embedding potential, which accounts for the attractive and repulsive contributions of the environment. The formally exact embedding potential, however, is in practice calculated using explicit kinetic-energy functionals for which the resulting potentials are in many cases not repulsive enough to account fully for Pauli repulsion by the electrons of the environment and to compensate thereby the strong electron-nuclear attraction. For the excited states on the target molecule, this leads to charge spill-out when diffuse basis functions are included, which allow that valence electrons are excited to those regions of the environment where the strong nuclear attraction is not sufficiently compensated by repulsive contributions. To reduce this insufficiency, we propose in the present work the inclusion of atomic all-electron pseudopotentials for all environment atoms on top of the conventional embedding potential. In the current work, the pseudopotentials are applied for computing vertical excitation energies of local excited states in complex systems employing the second-order algebraic diagrammatic construction (ADC(2)) scheme. The proposed approach leads to significantly reduced charge spill-out and an improved agreement of FDE and supermolecular calculations in the frozen solvent approximation. In particular, when diffuse functions are employed, the mean absolute deviation (MAD) is reduced from 0.27 to 0.05 eV for the investigated cases.

Geometry dependence of excitonic couplings and the consequences for configuration-space sampling.

Excitonic coupling plays a key role for the understanding of excitonic energy transport (EET) in, for example, organic photovoltaics. However, the calculation of realistic systems is often beyond the applicability range of accurate wavefunction methods so that lower-scaling semi-empirical methods are used to model EET events. In the present work, the distance and angle dependence of excitonic couplings of dimers of selected organic molecules are evaluated for the semi-empirical long-range corrected density functional based tight binding (LC-DFTB) method and spin opposite scaled second order approximate coupled cluster singles and doubles (SOS-CC2). While semi-empirically scaled methods can lead to slightly increased deviations for excitation energies, the excitonic couplings and their dependence on the dimer geometry are reproduced. LC-DFTB yields a similar accuracy range as density-functional theory (DFT) employing the ωB97X functional while the computation time is reduced by several orders of magnitude. The dependence of the exchange contributions to the excitonic couplings on the dimer geometry is analyzed assessing the calculation of Coulombic excitonic couplings from monomer local excited states only, which reduces the computational effort significantly. The present work is a necessary first step toward the simulation of excitonic energy transport using semi-empirical methods.

Analytical Time-Dependent Long-Range Corrected Density Functional Tight Binding (TD-LC-DFTB) Gradients in DFTB+: Implementation and Benchmark for Excited-State Geometries and Transition Energies.

The absorption and emission of light is a ubiquitous process in chemical and biological processes, making a theoretical description inevitable for understanding and predicting such properties. Although ab initio and DFT methods are capable of describing excited states with good accuracy in many cases, the investigation of dynamical processes and the need to sample the phase space in complex systems often requires methods with reduced computational costs but still sufficient accuracy. In the present work, we report the derivation and implementation of analytical nuclear gradients for time-dependent long-range corrected density functional tight binding (TD-LC-DFTB) in the DFTB+ program. The accuracy of the TD-LC-DFTB potential-energy surfaces is benchmarked for excited-state geometries and adiabatic as well as vertical transition energies. The benchmark set consists of more than 100 organic molecules taken as subsets from available benchmark sets. The reported method yields a mean deviation of 0.31 eV for adiabatic excitation energies with respect to CC2. In order to study more subtle effects, seminumerical second derivatives based on the analytical gradients are employed to simulate vibrationally resolved UV/vis spectra. This extensive test exhibits few problematic cases, which can be traced back to the parametrization of the repulsive potential.

Combining wavefunction frozen-density embedding with one-dimensional periodicity.

We present the combination of wavefunction frozen-density embedding (FDE) with a periodic repetition in one dimension (1D) for molecular systems in the KOALA program. In this periodic orbital-uncoupled FDE ansatz, no wavefunction overlap is taken into account, and only the electron density of the active subsystem is computed explicitly. This density is relaxed in the presence of the environment potential, which is obtained by translating the updated active subsystem density, yielding a fully self-consistent solution at convergence. Treating only one subsystem explicitly, the method allows for the calculation of local properties in condensed molecular systems, while no orbital band structure is obtained preventing the application, e.g., to systems with metallic bonding. In order to illustrate possible applications of the new implementation, selected case studies are presented, ranging from ground-state dipole moments using configuration interaction methods via excitation energies using time-dependent density-functional theory to ionization potentials obtained from equation-of-motion correlation methods. Different levels of approximations are assessed, revealing that an active subsystem consisting of two or three molecules leads to results that are converged with respect to the environment contributions.

A Fock-operator complete active space self-consistent field (CAS-SCF) method combined with frozen-density embedding.

We report the implementation of a Fock-operator complete-active space self-consistent field (CAS-SCF) method combined with frozen-density embedding (FDE) into the KOALA quantum-chemistry program. The implementation is based on configuration interaction from an unrestricted reference determinant and is able to treat electronic configurations such as singlet, triplet, or quintet states embedded in a molecular environment. In order to account for possible spin polarization effects, the FDE contribution is extended to the unrestricted case. We assess the convergence obtained with the implementation at the example of a stretched lithium dimer with significant multi-reference character. The efficiency of the implementation enables the orbital optimization for 25 states in a state-average SA[S0-S10,T1-T12,Q1-Q2]-CAS(10,10)-SCF calculation for the retinal molecule using a def2-TZVP basis. The FDE ansatz leads to orbitals localized by definition on the target system, thus facilitating the orbital selection required for CAS methods in complex environments.

Ultrafast Singlet Fission and Intersystem Crossing in Halogenated Tetraazaperopyrenes.

Charge carrier multiplication via singlet fission into two triplet states has the potential to increase efficiencies of photovoltaics by one-third due to the reduction of thermalization losses. In the present work, we investigate tetraazaperopyrenes, a class of N-heteropolycyles, as suitable singlet fission candidates. Using a combined experimental and theoretical approach, fundamentally different mechanisms for triplet formation in solution and thin film are identified. In solution, an ultrafast intersystem crossing process is observed, which is accelerated for heavier halide substituents not only due to enhanced spin-orbit coupling but also due to the energy tuning between the S1 and T2 states. In thin films, a correlated triplet pair is formed coherently upon photoexcitation. Subsequently, an excimer formation is observed, which competes with the electronic decorrelation of the triplet pair. The comparison with peropyrene shows that aza-substitutions within the aromatic core can be a powerful strategy for tuning the energy levels of the states important to singlet fission.

Analytical nuclear gradients for electron-attached and electron-detached states for the second-order algebraic diagrammatic construction scheme combined with frozen-density embedding.

In the present work, we report the derivation and implementation of vertical ionization potentials (IPs) and electron affinities (EAs) for embedded wavefunction methods as well as the corresponding analytical nuclear gradients. Vertical transitions have been implemented for CIS(D∞), the second-order algebraic diagrammatic construction [ADC(2)] scheme, and the second-order approximate coupled-cluster singles and doubles method. For all methods, density fitting is applied to facilitate reduced memory and disk storage requirements. Analytical nuclear gradients have been derived and implemented for CIS(D∞) and ADC(2) both with and without frozen-density embedding (FDE). The objective of the reported method is to study the properties of organic semiconductors in which charge is transported along molecular stacks in molecular crystals. The accuracy of the implemented methods is, therefore, assessed using stacked dimers of small model systems. Albeit second-order methods can yield noticeable errors with respect to reference methods in terms of absolute IP and EA values, they show a significantly improved accuracy for the shift of the IP and EA values at different intermolecular distances relative to the monomers. Besides reducing the computational costs, the FDE ansatz introduces furthermore a significant conceptual difference as it enables control over which subsystem is ionized, allowing for the calculation of transfer integrals for the interacting (embedded) systems. The new implementation is finally applied to tetraazaperopyrenes, used as organic semiconductors, to study charge-localization and long-range polarization in particular.

Perhalogenated Tetraazaperopyrenes and Their Corresponding Mono- and Dianions.

Chlorination and bromination of 2,9-perfluoropropyl-substituted tetraazaperopyrenes (TAPPs) under forcing conditions resulted in fully core-halogenated TAPP derivatives, devoid of hydrogen atoms at the polycyclic aromatic core. The octahalogenation stabilized the reduced mono- and dianionic compounds sufficiently to allow for their characterization. The additional ortho-chlorination led to an improvement of the electron mobility compared to the bay-substituted tetrachloro-TAPP when employed as an n-channel semiconductor in thin-film transistors.

Benchmark and performance of long-range corrected time-dependent density functional tight binding (LC-TD-DFTB) on rhodopsins and light-harvesting complexes.

The chromophores of rhodopsins (Rh) and light-harvesting (LH) complexes still represent a major challenge for a quantum chemical description due to their size and complex electronic structure. Since gradient corrected and hybrid density functional approaches have been shown to fail for these systems, only range-separated functionals seem to be a promising alternative to the more time consuming post-Hartree-Fock approaches. For extended sampling of optical properties, however, even more approximate approaches are required. Recently, a long-range corrected (LC) functional has been implemented into the efficient density functional tight binding (DFTB) method, allowing to sample the excited states properties of chromophores embedded into proteins using quantum mechanical/molecular mechanical (QM/MM) with the time-dependent (TD) DFTB approach. In the present study, we assess the accuracy of LC-TD-DFT and LC-TD-DFTB for rhodopsins (bacteriorhodopsin (bR) and pharaonis phoborhodopsin (ppR)) and LH complexes (light-harvesting complex II (LH2) and Fenna-Matthews-Olson (FMO) complex). This benchmark study shows the improved description of the color tuning parameters compared to standard DFT functionals. In general, LC-TD-DFTB can exhibit a similar performance as the corresponding LC functionals, allowing a reliable description of excited states properties at significantly reduced cost. The two chromophores investigated here pose complementary challenges: while huge sensitivity to external field perturbation (color tuning) and charge transfer excitations are characteristic for the retinal chromophore, the multi-chromophoric character of the LH complexes emphasizes a correct description of inter-chromophore couplings, giving less importance to color tuning. None of the investigated functionals masters both systems simultaneously with satisfactory accuracy. LC-TD-DFTB, at the current stage, although showing a systematic improvement compared to TD-DFTB cannot be recommended for studying color tuning in retinal proteins, similar to some of the LC-DFT functionals, because the response to external fields is still too weak. For sampling of LH-spectra, however, LC-TD-DFTB is a viable tool, allowing to efficiently sample absorption energies, as shown for three different LH complexes. As the calculations indicate, geometry optimization may overestimate the importance of local minima, which may be averaged over when using trajectories. Fast quantum chemical approaches therefore may allow for a direct sampling of spectra in the near future.

Twisting the TAPPs: Bay-Substituted Non-planar Tetraazapero-pyrenes and their Reduced Anions.

A new synthesis of tetraazaperopyrenes (TAPPs) starting from a halogenated perylene derivative 3,4,9,10- tetrabromo-1,6,7,12-tetrachloroperylene (1) gave access to bay-substituted TAPPs for the first time. Selective lithiation of the bromine-positions and subsequent addition of tosyl azide led to the formation of the tetraazidotetrachloroperylene (2), which was subsequently reduced by addition of sodium borohydride to the corresponding tetraaminotetrachloroperylene (3). Oxidation to its semiquinoidal form 4 and subsequent cyclization with acid chlorides gave rise to a series of bay-chlorinated TAPPs. Whereas the aromatic core of the previously studied ortho-substituted TAPPs was found to be planar, the steric pressure of the two chlorine substituents on each side leads to the twist of the peropyrene core of approximately 30 degrees, a structural feature also observed in other bay-substituted perylene derivatives. An experimental and computational analysis reveals that introducing chloride substituents at these positions leads to slightly increased electron affinities (EA) enabling the selective generation and characterization of the reduced mono-anionic radicals and closed shell di-anionic species. These anions were isolated and characterized by UV/Vis spectroscopy and EPR or NMR, respectively. Processing of the bay-chlorinated TAPPs in n-channel organic TFTs revealed electron mobilities of 0.001 to 0.003 cm2 V-1 s-1 . These reduced electron mobilities compared to the ortho-halogenated TAPPs are thought to be rooted in the less densely packed solid-state structures.

Origin of the Solvatochromism in Organic Fluorophores with Flexible Side Chains: A Case Study of Flugi-2.

The emission band for Flugi-2 solvated in dimethyl sulfoxide (DMSO) is obtained from the combined quantum-classical simulations in which the quantum mechanics/molecular mechanics excitation energies are evaluated at the equilibrated segment of the classical molecular dynamics trajectory on the lowest-excited-state potential energy surface. The classical force-field parameters were obtained and validated specifically for the purpose of the present work. The calculated gas-phase to DMSO solvatochromic shift amounts to -0.21 eV, which is in line with the experimentally determined difference between the maxima of the emission bands for Flugi-2 in decane and in DMSO (-0.26 eV). The used model describes rather well the effect of DMSO on the broadening of the emission band. The solvatochromic shift in DMSO originates from two competing effects. The structural deformation of Flugi-2 due to the interaction with DMSO, which results in a positive contribution, and the negative contribution of a larger magnitude due to favorable specific interactions with the solvent. The latter is dominated by a single hydrogen bond between the oxygen atom of a DMSO molecule and the N3 hydrogen atom of the Flugi-2 molecule in which the proton of N3 acts as the donor.

The extended explicitly-correlated second-order approximate coupled-cluster singles and doubles ansatz suitable for response theory.

We report the extended explicitly correlated approximate coupled-cluster singles and doubles CC2(F12*)-XSP method suitable for response properties. Equations are derived using an automated approach and have subsequently been hand-coded into the computer program KOALA, in which for all two-electron integrals, density fitting is employed. Numerical results are presented for the lowest two vertical singlet excitation energies of a set of selected molecules. The results show that the CC2(F12*)-XSP method provides the correct basis-set limit with no bias to the ground state, and an excellent agreement with reference CC2 values using large basis sets is found. Using Dunning's aug-cc-pVTZ basis, the CC2(F12*)-XSP method yields excitation energies which are converged within 1 mEh to the basis-set limit for valence excitations.

Similarities and differences of the Lagrange formalism and the intermediate state representation in the treatment of molecular properties.

When dealing with approximate wave functions, molecular properties can be computed either as expectation values or as derivatives of the energy with respect to a corresponding perturbation. In this work, the intermediate state representation (ISR) formalism for the computation of expectation values is compared to the Lagrange formalism following a derivative ansatz, which are two alternative approaches of which neither one can be considered superior in general. Within the ISR formalism, terms are included up to a given order of perturbation theory only, while in the Lagrange formalism, all terms are accounted for arising through the differentiation. Similarities and differences of the Lagrange and ISR formalism are illustrated using explicit working equations for selected methods and analyzing numerical results for a range of coupled-cluster as well as algebraic-diagrammatic construction (ADC) methods for excited states. The analysis explains why the ADC(3/2) method is able to yield a large amount of the orbital-relaxation effects for p-h states in contrast to ADC(2) although the same second-order ISR is used to represent the corresponding operator.

Understanding UV-Vis Spectra of Halogenated Tetraazaperopyrenes (TAPPs): A Computational Study.

The UV-vis absorption and emission spectra of halogenated tetraazaperopyrenes (TAPPs) have been investigated employing second-order approximate coupled cluster (CC2) and (time-dependent) density functional theory (DFT). We have found that the qualitative estimates of (vertical) absorption and excitation energies are possible within a single particle picture based on frontier orbitals, but the single particle picture is not sufficient to achieve quantitative accuracy. Going from the single-particle picture to the many-particle picture improves the agreement with experimental results, but still no satisfying correlation of theory and experiment is obtained. The comparison of CC2- and DFT-based methods reveals that deviations from the experimental results cannot be explained by deficiencies of the electronic-structure methods but rather stem from neglecting vibrational effects. An agreement of theoretical results and experimental spectra is found for adiabatic excitation energies, which are given as energy differences of vibronic states, which are directly accessible using both theoretical and experimental methods. The most pronounced vibronic influence is found for the Stokes shifts, which are significantly overestimated by computing the vertical electronic transitions only. Based on the vibronic contributions, the small Stokes shift of the TAPP compounds can be explained by the temperature dependence of the vibrationally resolved UV-vis spectra.

Analytical Nuclear Excited-State Gradients for the Second-Order Approximate Coupled-Cluster Singles and Doubles (CC2) Method Employing Uncoupled Frozen-Density Embedding.

We report the derivation and implementation of analytical orbital-relaxed properties and nuclear gradients for excited states using the second-order approximate coupled-cluster singles and doubles (CC2) model combined with uncoupled frozen-density embedding (FDEu). An implementation of the algebraic diagrammatic construction through second-order ADC(2), which arises from simplification of RICC2 FDEu, is also presented. In order to ensure a RICC2 FDEu Lagrange functional that is linear in the Lagrange multipliers, the Hartree-Fock density is employed for the target subsystem in the embedding contributions. The accuracy of the new scheme is assessed using the carbon monoxide molecule, 4-aminophthalimide, and a benzonitrile dimer, revealing that the obtained errors are below the method error of RICC2. Using density functional theory for the environment, the efficiency of the new method is illustrated by computing the perturbed excited-state dipole moment of a chromophore in a biological environment. For this system, comprising 32 molecules consisting of 366 atoms in total, the computation requires only a couple of days on a standard compute node. RICC2 FDEu thus enables large-scale calculations of ab initio wave functions for molecules in complex environments as routine applications.

Communication: Biological applications of coupled-cluster frozen-density embedding.

We report the implementation of the Laplace-transform scaled opposite-spin (LT-SOS) resolution-of-the-identity second-order approximate coupled-cluster singles and doubles (RICC2) combined with frozen-density embedding for excitation energies and molecular properties. In the present work, we furthermore employ the Hartree-Fock density for the interaction energy leading to a simplified Lagrangian which is linear in the Lagrangian multipliers. This approximation has the key advantage of a decoupling of the coupled-cluster amplitude and multipliers, leading also to a significant reduction in computation time. Using the new simplified Lagrangian in combination with efficient wavefunction models such as RICC2 or LT-SOS-RICC2 and density-functional theory (DFT) for the environment molecules (CC2-in-DFT) enables the efficient study of biological applications such as the rhodopsin and visual cone pigments using ab initio methods as routine applications.