Frozen-density Embedding Research Articles

Effect of metallation, substituents and inter/intra-molecular polarization on electronic couplings for hole transport in stacked porphyrin dyads.

We carried out a systematic study of the hole transport properties for a series of symmetrically stacked porphyrin dimers. In the first part of this study, we evaluated the sensitivity of electronic couplings to orbital relaxation due to molecular ionization and intermolecular interactions for a series of halogenated porphyrins. The effect of polarization was estimated by comparing electronic couplings from fragment orbital density functional theory (FODFT) and frozen density embedding electron transfer (FDE-CT). For the dimers considered, the effect of polarization was estimated to be less than 20%, in line with previous studies on different molecular dimers. Thus, we decided to employ a computationally cheaper FODFT method to continue our study of the effect of metals and substituents on the electronic couplings for hole transfer. We find that, compared to the non-metallated porphyrins, Ni, Fe and Pt significantly reduce the coupling, while Zn, Ti, Cd and Pd increase it. The effect of substituents was studied on a series of meso-substituted porphyrins (meso-tetrapyridineporphyrin, meso-tetraphenylporphyrin and derivatives) for which we could relate a reduction of the coupling to steric effects that reduce the overlap between the frontier orbitals of the monomers.

Computing UV/vis spectra using a combined molecular dynamics and quantum chemistry approach: bis-triazin-pyridine (BTP) ligands studied in solution.

We report a combined computational and experimental study to investigate the UV/vis spectra of 2,6-bis(5,6-dialkyl-1,2,4-triazin-3-yl)pyridine (BTP) ligands in solution. In order to study molecules in solution using theoretical methods, force-field parameters for the ligand-water interaction are adjusted to ab initio quantum chemical calculations. Based on these parameters, molecular dynamics (MD) simulations are carried out from which snapshots are extracted as input to quantum chemical excitation-energy calculations to obtain UV/vis spectra of BTP ligands in solution using time-dependent density functional theory (TDDFT) employing the Tamm-Dancoff approximation (TDA). The range-separated CAM-B3LYP functional is used to avoid large errors for charge-transfer states occurring in the electronic spectra. In order to study environment effects with theoretical methods, the frozen-density embedding scheme is applied. This computational procedure allows to obtain electronic spectra calculated at the (range-separated) DFT level of theory in solution, revealing solvatochromic shifts upon solvation of up to about 0.6 eV. Comparison to experimental data shows a significantly improved agreement compared to vacuum calculations and enables the analysis of relevant excitations for the line shape in solution.

Frozen-density embedding theory with average solvent charge densities from explicit atomistic simulations.

Besides molecular electron densities obtained within the Born-Oppenheimer approximation (ρB(r)) to represent the environment, the ensemble averaged density (〈ρB〉(r)) is also admissible in frozen-density embedding theory (FDET) [Wesolowski, Phys. Rev. A, 2008, 77, 11444]. This makes it possible to introduce an approximation in the evaluation of the solvent effect on quantum mechanical observables consisting of replacing the ensemble averaged observable by the observable evaluated at ensemble averaged ρB(r). This approximation is shown to affect negligibly the solvatochromic shift in the absorption of hydrated acetone. The proposed model provides a continuum type of representation of the solvent, which reflects nevertheless its local structure, and it is to be applied as a post-simulation analysis tool in atomistic level simulations.

Homogeneity properties of the embedding potential in frozen-density embedding theory

ABSTRACTIn numerical simulations, based on frozen-density embedding theory, the independent variables describing the total system are the embedded wave function (ΨA) and the density representing the environment. Due to inhomogeneity of the non-electrostatic component of the total energy: ), the expectation value of the embedding potential is not equal to the corresponding component of the total energy. The differences are evaluated using local and semi-local approximations for the functional EnadxcT[ρA, ρB] in two model systems representing embedded species weakly interacting with the environment. It is found that ΔnadxcT is typically one order of magnitude smaller than EnadxcT[ρA, ρB] and decreases with the overlap between and . The kinetic- and exchange-correlation contributions to ΔnadxcT cancel partially reducing its magnitude to mHartrees. Compared to local approximation for EnadxcT[ρA, ρB], the inhomogeneity is more pronounced in semi-local functionals.

Orthogonality of embedded wave functions for different states in frozen-density embedding theory.

Other than lowest-energy stationary embedded wave functions obtained in Frozen-Density Embedding Theory (FDET) [T. A. Wesolowski, Phys. Rev. A 77, 012504 (2008)] can be associated with electronic excited states but they can be mutually non-orthogonal. Although this does not violate any physical principles--embedded wave functions are only auxiliary objects used to obtain stationary densities--working with orthogonal functions has many practical advantages. In the present work, we show numerically that excitation energies obtained using conventional FDET calculations (allowing for non-orthogonality) can be obtained using embedded wave functions which are strictly orthogonal. The used method preserves the mathematical structure of FDET and self-consistency between energy, embedded wave function, and the embedding potential (they are connected through the Euler-Lagrange equations). The orthogonality is built-in through the linearization in the embedded density of the relevant components of the total energy functional. Moreover, we show formally that the differences between the expectation values of the embedded Hamiltonian are equal to the excitation energies, which is the exact result within linearized FDET. Linearized FDET is shown to be a robust approximation for a large class of reference densities.

A Local Variant of the Conductor-Like Screening Model for Fragment-Based Electronic-Structure Methods.

Due to steadily rising computational power and sophisticated modeling approaches, increasingly larger molecular systems can be modeled with ab initio methods. An especially promising approach is subsystem methods, where the total system is broken down into smaller subunits that can be treated individually. If an implicit solvent environment such as the conductor-like screening model (COSMO) is included in the description, then additional environmental effects can be incorporated at relatively low cost. For very large systems described with subsystem methods, however, the solution of the COSMO equations can actually become the bottleneck of the calculation. A prominent example in this area is biomolecular systems such as proteins, which can, for instance, be described with frozen density embedding (FDE), especially the related 3-FDE approach. In this article, we present an alternative COSMO variant, which exploits the subsystem nature of the underlying electronic description and has been implemented in a development version of the Amsterdam Density Functional program suite (Adf). We show that the computational cost for the solvent model can be reduced dramatically while retaining the accuracy of the regular description. We compare several schemes for density and surface charge updates and assess the effect of the single tuning parameter.

Frozen-Density Embedding Potentials and Chiroptical Properties.

The efficacy of the frozen density embedding (FDE) approach to the simulation of solvent effects is examined for two key chiroptical properties-specific rotation and circular dichroism spectra. In particular, we have investigated the performance of a wave function-theory-in-density-functional-theory (WFT-in-DFT) FDE approach for computing such properties for the small, rigid chiral compound (P)-dimethylallene interacting with up to three water molecules. Although the solvent potential is obtained through DFT, the optical response is computed using coupled cluster linear response theory for mixed electric and magnetic field perturbations. We find that the FDE potential generally yields too small a shift from the isolated molecule as compared to that introduced by the explicit solvent. In one case, the FDE potential fails to reproduce a change in sign of the ORD in which the solute interacts with two solvent molecules. The source of these errors is due primarily to the lack of solvent response to the external field and is analyzed in terms of solvent-solute charge transfer excitations.

Exact kinetic energy enables accurate evaluation of weak interactions by the FDE-vdW method.

The correlation energy of interaction is an elusive and sought-after interaction between molecular systems. By partitioning the response function of the system into subsystem contributions, the Frozen Density Embedding (FDE)-vdW method provides a computationally amenable nonlocal correlation functional based on the adiabatic connection fluctuation dissipation theorem applied to subsystem density functional theory. In reproducing potential energy surfaces of weakly interacting dimers, we show that FDE-vdW, either employing semilocal or exact nonadditive kinetic energy functionals, is in quantitative agreement with high-accuracy coupled cluster calculations (overall mean unsigned error of 0.5 kcal/mol). When employing the exact kinetic energy (which we term the Kohn-Sham (KS)-vdW method), the binding energies are generally closer to the benchmark, and the energy surfaces are also smoother.

Frozen Density Embedding with External Orthogonality in Delocalized Covalent Systems.

Frozen density embedding (FDE) has become a popular subsystem density functional theory (DFT) method for systems with weakly overlapping charge densities. The failure of this method for strongly interacting and covalent systems is due to the approximate kinetic energy density functional (KEDF), although the need for approximate KEDFs may be eliminated if each subsystem's Kohn-Sham (KS) orbitals are orthogonal to the other, termed external orthogonality (EO). We present an implementation of EO into the FDE framework within the Amsterdam density functional program package, using the level-shift projection operator method. We generalize this method to remove the need for orbital localization schemes and to include multiple subsystems, and we show that the exact KS-DFT energies and densities may be reproduced through iterative freeze-and-thaw cycles for a number of systems, including a charge delocalized benzene molecule starting from atomic subsystems. Finally, we examine the possibility of a truncated basis for systems with and without charge delocalization, and found that subsystems require a basis that allows them to correctly describe the supermolecular delocalized orbitals.

Excitation energies from frozen-density embedding with accurate embedding potentials.

We present calculations of excitation energies within the time-dependent density functional theory (TDDFT) extension of frozen-density embedding (FDE) using reconstructed accurate embedding potentials. Previous applications of FDE showed significant deviations from supermolecular calculations; our current approach eliminates one potential error source and yields excitation energies of generally much better agreement with Kohn-Sham-TDDFT. Our results demonstrate that the embedding potentials represent the main error source in FDE-TDDFT calculations using standard approximate kinetic-energy functionals for excitations localized within one subsystem.

Frozen-Density Embedding Strategy for Multilevel Simulations of Electronic Structure.

Modeling properties of chemical species and chemical reactions requires usually the quantum-mechanical level of description. Methods from the ever-growing toolbox of quantum chemistry1,2 are used for this purpose. Due to unfavorable scaling of quantum chemistry methods, a compromise must be made between the accuracy of the numerical results and the size of the system described at the wave function level. The same strategy can be extended for separating molecular fragments or molecules. Nonlocal embedding operators based on either transferrable pseudopotentials or frozen orbitals obtained from localization procedures have been developed in many groups. The essential feature of all such local potentials is that they comprise a component which takes into account the intermolecular Pauli repulsion.

Subsystem real-time time dependent density functional theory.

We present the extension of Frozen Density Embedding (FDE) formulation of subsystem Density Functional Theory (DFT) to real-time Time Dependent Density Functional Theory (rt-TDDFT). FDE is a DFT-in-DFT embedding method that allows to partition a larger Kohn-Sham system into a set of smaller, coupled Kohn-Sham systems. Additional to the computational advantage, FDE provides physical insight into the properties of embedded systems and the coupling interactions between them. The extension to rt-TDDFT is done straightforwardly by evolving the Kohn-Sham subsystems in time simultaneously, while updating the embedding potential between the systems at every time step. Two main applications are presented: the explicit excitation energy transfer in real time between subsystems is demonstrated for the case of the Na4 cluster and the effect of the embedding on optical spectra of coupled chromophores. In particular, the importance of including the full dynamic response in the embedding potential is demonstrated.

Performance of Frozen Density Embedding for Modeling Hole Transfer Reactions.

We have carried out a thorough benchmark of the frozen density-embedding (FDE) method for calculating hole transfer couplings. We have considered 10 exchange-correlation functionals, 3 nonadditive kinetic energy functionals, and 3 basis sets. Overall, we conclude that with a 7% mean relative unsigned error, the PBE and PW91 functionals coupled with the PW91k nonadditive kinetic energy functional and a TZP basis set constitute the most stable and accurate levels of theory for hole transfer coupling calculations. The FDE-ET method is found to be an excellent tool for computing diabatic couplings for hole transfer reactions.

Solvatochromic shifts of Br₂ and I₂ in water cages of type 5¹², 5¹²6², 5¹²6³, and 5¹²6⁴.

Despite the relatively small size of molecular bromine and iodine, the physicochemical behavior in different solvents is not yet fully understood, in particular when excited-state properties are sought. In this work, we investigate isolated halogen molecules trapped in clathrate hydrate cages. Relativistic supermolecular calculations reveal that the environment shift to the excitation energies of the (nondegenerate) states 3Πu and 1Πu lie within a spread of 0.05 eV, respectively, suggesting that environment shifts can be estimated with scalar-relativistic treatments. As even scalar-relativistic calculations are problematic for excited-state calculations for clathrates with growing size and basis sets, we have applied the subsystem-based scheme frozen-density embedding, which avoids a supermolecular treatment. This allows for the calculation of excited states for extended clusters with coupled-cluster methods and basis sets of triple-zeta quality with additional diffuse functions mandatory for excited-state properties, as well as a facile treatment at scalar-relativistic exact two-component level of theory for the heavy atoms bromine and iodine. This simple approach yields scalar-relativistic estimates for solvatochromic shifts introduced by the clathrate cages.

On exchange coupling and bonding in the Gd2@C80 and Gd2@C79N endohedral dimetallo-fullerenes

A series of computational experiments performed with various methods belonging to wave-function and density functional theories approaches the issue of bonding regime and exchange coupling in the title compounds. Gd2@C80 is computed with a very weak exchange coupling, the sign depending on the method, while Gd2@C79N has resulted with a strong coupling and ferromagnetic ground state, irrespective of the computational approach. The multi-configuration calculation and broken symmetry estimation are yielding closely coincident coupling constants, of about J ∼ 400 cm−1. No experimental estimation exists, but the ferromagnetic ground state of Gd2@C79N is confirmed from paramagnetic resonance data. The different behaviour is due to particularities of electron accommodation in the orbital scheme. The exchange effects localised on atom lead to preference for parallel alignment of the electrons placed in the 4f and 5d lanthanide shells, determining also a ferromagnetic inter-centre coupling. The structural insight is completed with a ligand field analysis of the density functional theory results in the context of frozen density embedding. The energy decomposition analysis of bonding effects is also discussed. Finally, with the help of home-made codes (named Xatom+Xsphere), a model for the atom encapsulated in a cage is designed, the exemplified numeric experiments showing relevance for the considered endohedral metallo-fullerene issues.

Self-consistent embedding of density-matrix renormalization group wavefunctions in a density functional environment.

We present the first implementation of a density matrix renormalization group algorithm embedded in an environment described by density functional theory. The frozen density embedding scheme is used with a freeze-and-thaw strategy for a self-consistent polarization of the orbital-optimized wavefunction and the environmental densities with respect to each other.

Features of chemical bonds based on the overlap polarizabilities: diatomic and solid-state systems with the frozen-density embedding approach.

The chemical bond overlap properties were obtained for alkali halides NaY (Y = F, Cl, Br), alkaline-earth chalcogenides MX (M = Ca, Mg and X = O, S, Se) and alkali and alkali-earth metals (Li, Na, and Mg) in diatomic and solid-state systems using an embedding approach based on the frozen density functional theory to simulate the crystalline effects. The computational protocol established provides errors for bond distances smaller than 1%. The results indicate that larger chemical bond covalency leads to larger absorption or scattering by the overlap region. The ionic specific valence and overlap polarizability are closely related to the valence orbital compactness measured by the sum of Mulliken electronegativities. The embedding approach used in this work makes it possible to quantify the effects of the crystalline environment on the chemical bond overlap properties. In the solid-state, the bond overlap charges are less polarizable, in cases of well-known ionic systems (provided by electronegativity differences), leading to smaller chemical bond covalency in solids than in diatomics. The spectroscopic properties of the polarizability of the electron density in the overlap region of a chemical bond could be measured in the 1-20 eV spectral region and could be used to characterize some bands in several spectra whose assignments are ambiguous or not available.

Periodic subsystem density-functional theory

By partitioning the electron density into subsystem contributions, the Frozen Density Embedding (FDE) formulation of subsystem Density Functional Theory (DFT) has recently emerged as a powerful tool for reducing the computational scaling of Kohn-Sham DFT. To date, however, FDE has been employed to molecular systems only. Periodic systems, such as metals, semiconductors, and other crystalline solids have been outside the applicability of FDE, mostly because of the lack of a periodic FDE implementation. To fill this gap, in this work we aim at extending FDE to treat subsystems of molecular and periodic character. This goal is achieved by a dual approach. On one side, the development of a theoretical framework for periodic subsystem DFT. On the other, the realization of the method into a parallel computer code. We find that periodic FDE is capable of reproducing total electron densities and (to a lesser extent) also interaction energies of molecular systems weakly interacting with metallic surfaces. In the pilot calculations considered, we find that FDE fails in those cases where there is appreciable density overlap between the subsystems. Conversely, we find FDE to be in semiquantitative agreement with Kohn-Sham DFT when the inter-subsystem density overlap is low. We also conclude that to make FDE a suitable method for describing molecular adsorption at surfaces, kinetic energy density functionals that go beyond the GGA level must be employed.

Non-uniform Continuum Model for Solvated Species Based on Frozen-Density Embedding Theory: The Study Case of Solvatochromism of Coumarin 153.

Recent application of the Frozen-Density Embedding Theory based continuum model of the solvent, which is used for calculating solvatochromic shifts in the UV/Vis range, are reviewed. In this model, the solvent is represented as a non-uniform continuum taking into account both the statistical nature of the solvent and specific solute-solvent interactions. It offers, therefore, a computationally attractive alternative to methods in which the solvent is described at atomistic level. The evaluation of the solvatochromic shift involves only two calculations of excitation energy instead of at least hundreds needed to account for inhomogeneous broadening. The present review provides a detailed graphical analysis of the key quantities of this model: the average charge density of the solvent (<ρB>) and the corresponding Frozen-Density Embedding Theory derived embedding potential for coumarin 153.

Nonuniform continuum model for solvatochromism based on frozen-density embedding theory.

Frozen-density embedding theory (FDET) provides the formal framework for multilevel numerical simulations, such that a selected subsystem is described at the quantum mechanical level, whereas its environment is described by means of the electron density (frozen density; ${\rho _{\rm{B}} (\vec r)}$). The frozen density ${\rho _{\rm{B}} (\vec r)}$ is usually obtained from some lower-level quantum mechanical methods applied to the environment, but FDET is not limited to such choices for ${\rho _{\rm{B}} (\vec r)}$. The present work concerns the application of FDET, in which ${\rho _{\rm{B}} (\vec r)}$ is the statistically averaged electron density of the solvent ${\left\langle {\rho _{\rm{B}} (\vec r)} \right\rangle }$. The specific solute-solvent interactions are represented in a statistical manner in ${\left\langle {\rho _{\rm{B}} (\vec r)} \right\rangle }$. A full self-consistent treatment of solvated chromophore, thus involves a single geometry of the chromophore in a given state and the corresponding ${\left\langle {\rho _{\rm{B}} (\vec r)} \right\rangle }$. We show that the coupling between the two descriptors might be made in an approximate manner that is applicable for both absorption and emission. The proposed protocol leads to accurate (error in the range of 0.05 eV) descriptions of the solvatochromic shifts in both absorption and emission.