Excitation Energies of Embedded Chromophores from Frozen-Density Embedding Theory Using State-Specific Electron Densities of the Environment.

Starting from the Perdew-Levy theorem on extrema of the Hohenberg-Kohn functional, the expression for the vertical excitation energy is derived within the formal framework of Frozen-Density Embedding Theory (FDET) that makes it possible to use state-specific electron densities of the environment (ρB) of an embedded species. The derived general expression involves the embedded wave functions for ground and excited states that are orthogonal and is exact up to quadratic terms in the appropriate density expansion. It can be applied in practice using various methods differing in the treatment of the electron-electron correlation for embedded electrons, the method to evaluate different contributions to the excitation energy, the method to generate state-specific ρB, and the approximation used for the non-electrostatic component of the FDET embedding potential. The derived expression is applied for 47 local excitations in 10 embedded organic chromophores. The explicit treatment of the differential polarization of ρB improves indeed the accuracy of the excitation energy as compared to the implicit treatment in which the same ρB is used for all states of embedded chromophore. For 47 local excitations in 10 embedded organic chromophores, the average absolute errors in excitation energies drop from 0.04 to 0.03 eV and their standard deviations from 0.032 to 0.025 eV, respectively. The maximal errors show similar trends.

Symmetrized non-decomposable approximations of the non-additive kinetic energy functional.

In subsystem density functional theory (DFT), the bottom-up strategy to approximate the multivariable functional of the non-additive kinetic energy (NAKE) makes it possible to impose exact properties on the corresponding NAKE potential (NAKEP). Such a construction might lead to a non-symmetric and non-homogeneous functional, which excludes the use of such approximations for the evaluation of the total energy. We propose a general formalism to construct a symmetric version based on a perturbation theory approach of the energy expression for the asymmetric part. This strategy is then applied to construct a symmetrized NAKE corresponding to the NAKEP developed recently [Polak et al., J. Chem. Phys. 156, 044103 (2022)], making it possible to evaluate consistently the energy. These functionals were used to evaluate the interaction energy in several model intermolecular complexes using the formal framework of subsystem DFT. The new symmetrized energy expression shows a superior qualitative performance over common decomposable models.

A non-decomposable approximation on the complete density function space for the non-additive kinetic potential.

A new non-decomposable approximation of the non-additive kinetic energy potential is constructed starting from the same exact property in the limit (ρA → 0 and ∫ρB = 2), as introduced in the work of Lastra et al. [J. Chem. Phys. 129, 074107 (2008)]. In order to cover the complete function space for exponentially decaying densities, the kernel of a differential operator Dγ[ρ] is introduced and analyzed in dependence of γ. The conclusive choice of γ = 1 assures that the solution functions span the complete space of molecular electron densities. As a result, the new approximant preserves the desired feature of the older approximation, which is the reciprocal singularity if the electron density decays exponentially, and eliminates artificial shallow wells (holes), which are responsible for an artificial "charge leak." Numerical considerations using the standard validation procedure introduced by Wesolowski and Weber [Chem. Phys. Lett. 248, 71-76 (1996)] demonstrate the numerical performance of the developed approximation, which increases the range of applicability of semilocal functionals.

Simulations of electric field gradient fluctuations and dynamics around sodium ions in ionic liquids.

The T1 relaxation time measured in nuclear magnetic resonance experiments contains information about electric field gradient (EFG) fluctuations around a nucleus, but computer simulations are typically required to interpret the underlying dynamics. This study uses classical molecular dynamics (MD) simulations and quantum chemical calculations, to investigate EFG fluctuations around a Na+ ion dissolved in the ionic liquid 1-ethyl 3-methylimidazolium tetrafluoroborate, [Im21][BF4], to provide a framework for future interpretation of NMR experiments. Our calculations demonstrate that the Sternheimer approximation holds for Na+ in [Im21][BF4], and the anti-shielding coefficient is comparable to its value in water. EFG correlation functions, CEFG(t), calculated using quantum mechanical methods or from force field charges are roughly equivalent after 200 fs, supporting the use of classical MD for estimating T1 times of monatomic ions in this ionic liquid. The EFG dynamics are strongly bi-modal, with 75%-90% of the de-correlation attributable to inertial solvent motion and the remainder to a highly distributed diffusional processes. Integral relaxation times, ⟨τEFG⟩, were found to deviate from hydrodynamic predictions and were non-linearly coupled to solvent viscosity. Further investigation showed that Na+ is solvated by four tetrahedrally arranged [BF4]- anions and directly coordinated by ∼6 fluorine atoms. Exchange of [BF4]- anions is rare on the 25-50 ns timescale and suggests that motion of solvent-shell [BF4]- is the primary mechanism for the EFG fluctuations. Different couplings of [BF4]- translational and rotational diffusion to viscosity are shown to be the source of the non-hydrodynamic scaling of ⟨τEFG⟩.

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.

Extension of frozen-density embedding theory for non-variational embedded wavefunctions.

In the original formulation, frozen-density embedding theory [T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, 8050-8053 (1993); T. A. Wesolowski, Phys. Rev. A 77, 012504 (2008)] concerns multi-level simulation methods in which variational methods are used to obtain the embedded NA-electron wavefunction. In this work, an implicit density functional for the total energy is constructed and used to derive a general expression for the total energy in methods in which the embedded NA electrons are treated non-variationally. The formula is exact within linear expansion in density perturbations. Illustrative numerical examples are provided.

Explicit vs. implicit electronic polarisation of environment of an embedded chromophore in frozen-density embedding theory.

Frozen-Density Embedding Theory (FDET) provides a system-independent formal framework for multi-level computational methods. Despite apparent similarity, the interaction energy components commonly used in QM/MM methods do not have their corresponding counterparts in FDET. We show how the effect of the polarisation on the electron distribution in the environment can be (or is) accounted for either explicitly or implicitly within the FDET framework. Numerical examples are provided for vertical excitation energies in four representative cases of embedded chromophores.

Looking for the Origin of Allosteric Cooperativity in Metallopolymers.

The basic concept of allosteric cooperativity used in biology, chemistry and physics states that any change in the intermolecular host-guest interactions operating in multisite receptors can be assigned to intersite interactions. Using lanthanide metals as guests and linear multi-tridentate linear oligomers of variable lengths and geometries as hosts, this work shows that the quantitative modeling of metal loadings requires the consideration of a novel phenomenon originating from solvation processes. It stepwise modulates the intrinsic affinity of each isolated site in multisite receptors, and this without resorting to allosteric cooperativity. An easy-to-handle additive model predicts a negative power law dependence of the intrinsic affinity on the length of the linear metallopolymer. Applied to lanthanidopolymers, the latter common analysis overestimates cooperativity factors by more than two orders of magnitude.

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.

Fluorescence quantum yield rationalized by the magnitude of the charge transfer in π-conjugated terpyridine derivatives.

Terpyridine derivatives are of great interest due to their unique photophysical properties when used as antennas in metallic complexes. Several experimental and theoretical studies indicate strong charge-transfer character of the lowest electronic excited state, which could be exploited for predicting fluorescence quantum yields from the magnitude of the charge separation induced by electronic transitions. Focusing on substituted 4'-phenyl-2,2':6'2''-terpyridyl, we report on two measures of the charge separation obtained from high-level calculations in ground and excited states (length of the change of the dipole moment and the electron-hole distance). Our refined model confirms that the fluorescence quantum yield shows a global S-shape dependence on the magnitude of the charge separation, which can be quantified either by the change in dipole moments between the ground and excited states or by the associated charge-hole distances. This approach provides a remarkable tool for the molecular design of a fluorescent polyaromatic antenna.

First time combination of frozen density embedding theory with the algebraic diagrammatic construction scheme for the polarization propagator of second order.

The combination of Frozen Density Embedding Theory (FDET) and the Algebraic Diagrammatic Construction (ADC) scheme for the polarization propagator for describing environmental effects on electronically excited states is presented. Two different ways of interfacing and expressing the so-called embedding operator are introduced. The resulting excited states are compared with supermolecular calculations of the total system at the ADC(2) level of theory. Molecular test systems were chosen to investigate molecule-environment interactions of varying strength from dispersion interaction up to multiple hydrogen bonds. The overall difference between the supermolecular and the FDE-ADC calculations in excitation energies is lower than 0.09 eV (max) and 0.032 eV in average, which is well below the intrinsic error of the ADC(2) method itself.

Ion Pair-π Interactions.

We report that anion-π and cation-π interactions can occur on the same aromatic surface. Interactions of this type are referred to as ion pair-π interactions. Their existence, nature, and significance are elaborated in the context of spectral tuning, ion binding in solution, and activation of cell-penetrating peptides. The origin of spectral tuning by ion pair-π interactions is unraveled with energy-minimized excited-state structures: The solvent- and pH-independent red shift of absorption and emission of push-pull fluorophores originates from antiparallel ion pair-π attraction to their polarized excited state. In contrast, the complementary parallel ion pair-π repulsion is spectroscopically irrelevant, in part because of charge neutralization by intriguing proton and electron transfers on excited push-pull surfaces. With time-resolved fluorescence measurements, very important differences between antiparallel and parallel ion pair-π interactions are identified and quantitatively dissected from interference by aggregation and ion pair dissociation. Contributions from hydrogen bonding, proton transfer, π-π interactions, chromophore twisting, ion pairing, and self-assembly are systematically addressed and eliminated by concise structural modifications. Ion-exchange studies in solution, activation of cell-penetrating peptides in vesicles, and computational analysis all imply that the situation in the ground state is complementary to spectral tuning in the excited state; i.e., parallel rather than antiparallel ion pair-π interactions are preferred, despite repulsion from the push-pull dipole. The overall quite complete picture of ion pair-π interactions provided by these remarkably coherent yet complex results is expected to attract attention throughout the multiple disciplines of chemistry involved.

Rigid urea and self-healing thiourea ethanolamine monolayers.

A series of long-tail alkyl ethanolamine analogs containing amide-, urea-, and thiourea moieties was synthesized and the behavior of the corresponding monolayers was assessed on the Langmuir-Pockels trough combined with grazing incidence X-ray diffraction experiments and complemented by computer simulations. All compounds form stable monolayers at the soft air/water interface. The phase behavior is dominated by strong intermolecular headgroup hydrogen bond networks. While the amide analog forms well-defined monolayer structures, the stronger hydrogen bonds in the urea analogs lead to the formation of small three-dimensional crystallites already during spreading due to concentration fluctuations. The hydrogen bonds in the thiourea case form a two-dimensional network, which ruptures temporarily during compression and is recovered in a self-healing process, while in the urea clusters the hydrogen bonds form a more planar framework with gliding planes keeping the structure intact during compression. Because the thiourea analogs are able to self-heal after rupture, such compounds could have interesting properties as tight, ordered, and self-healing monolayers.

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.

Spectral tuning of rhodopsin and visual cone pigments.

Retinal is the light-absorbing biochromophore responsible for the activation of vision pigments and light-driven ion pumps. Nature has evolved molecular tuning mechanisms that significantly shift the optical properties of the retinal pigments to enable their absorption of visible light. Using large-scale quantum chemical calculations at the density functional theory level combined with frozen density embedding theory, we show here how the protein environment of vision pigments tunes the absorption of retinal by electrostatically dominated interactions between the chromophore and the surrounding protein residues. The calculations accurately reproduce the experimental absorption maxima of rhodopsin and the red, green, and blue color pigments. We further identify key interactions responsible for the color-shifting effects by mutating the rhodopsin structure in silico, and we find that deprotonation of the retinyl is likely to be responsible for the blue-shifted absorption in the blue cone vision pigment.

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.

Embedding potentials for excited states of embedded species.

Frozen-Density-Embedding Theory (FDET) is a formalism to obtain the upper bound of the ground-state energy of the total system and the corresponding embedded wavefunction by means of Euler-Lagrange equations [T. A. Wesolowski, Phys. Rev. A 77(1), 012504 (2008)]. FDET provides the expression for the embedding potential as a functional of the electron density of the embedded species, electron density of the environment, and the field generated by other charges in the environment. Under certain conditions, FDET leads to the exact ground-state energy and density of the whole system. Following Perdew-Levy theorem on stationary states of the ground-state energy functional, the other-than-ground-state stationary states of the FDET energy functional correspond to excited states. In the present work, we analyze such use of other-than-ground-state embedded wavefunctions obtained in practical calculations, i.e., when the FDET embedding potential is approximated. Three computational approaches based on FDET, that assure self-consistent excitation energy and embedded wavefunction dealing with the issue of orthogonality of embedded wavefunctions for different states in a different manner, are proposed and discussed.

Atomic shell structure from the Single-Exponential Decay Detector.

The density of atomic systems is analysed via the Single-Exponential Decay Detector (SEDD). SEDD is a scalar field designed to explore mathematical, rather than physical, properties of electron density. Nevertheless, it has been shown that SEDD can serve as a descriptor of bonding patterns in molecules as well as an indicator of atomic shells [P. de Silva, J. Korchowiec, and T. A. Wesolowski, ChemPhysChem 13, 3462 (2012)]. In this work, a more detailed analysis of atomic shells is done for atoms in the Li-Xe series. Shell populations based on SEDD agree with the Aufbau principle even better than those obtained from the Electron Localization Function, which is a popular indicator of electron localization. A link between SEDD and the local wave vector is given, which provides a physical interpretation of SEDD.

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.

Anion-π and cation-π interactions on the same surface.

Herein, we address the question whether anion-π and cation-π interactions can take place simultaneously on the same aromatic surface. Covalently positioned carboxylate-guanidinium pairs on the surface of 4-amino-1,8-naphthalimides are used as an example to explore push-pull chromophores as privileged platforms for such "ion pair-π" interactions. In antiparallel orientation with respect to the push-pull dipole, a bathochromic effect is observed. A red shift of 41 nm found in the least polar solvent is in good agreement with the 70 nm expected from theoretical calculations of ground and excited states. Decreasing shifts with solvent polarity, protonation, aggregation, and parallel carboxylate-guanidinium pairs imply that the intramolecular Stark effect from antiparallel ion pair-π interactions exceeds solvatochromic effects by far. Theoretical studies indicate that carboxylate-guanidinium pairs can also interact with the surfaces of π-acidic naphthalenediimides and π-basic pyrenes.