Assessment of Approximations to the Embedding Potential in Frozen-Density Embedding Theory for the Calculation of Electric Field Gradients.

The approximations to the embedding potential in frozen-density embedding theory (FDET) have been assessed for the first time for the calculation of the electric field gradient (EFG) at a nucleus. FDET-based methods using a hierarchy of approximations are applied to evaluate the EFG at the nuclei of an HCl molecule in several noncovalently bound clusters chosen to represent potential liquid or molecular crystal systems. A detailed assessment of such approximations is made for the Hartree-Fock treatment of electron-electron correlation (both in FDET and in the reference calculations for the whole cluster). The emerging choice of the optimal set of approximations is reconfirmed in calculations in which electron-electron calculations are treated at the MP2 level. Our optimized protocol produces average errors in the complexation-induced EFG shift on the order of 25% relative to conventional quantum mechanical calculations for the whole cluster. This protocol is shown to be numerically robust and leads to enormous computational savings compared to a complete quantum mechanical treatment of the embedded species and its environment. For a cluster comprising a Na+ cation and up to 24 water molecules, the computation time is reduced by a factor of 30,000 at the expense of introducing an error in the environment-induced EFG shift of 22%.

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.

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.

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⟩.

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.

Quantifying Fluctuations of Average Solvent Environments for Embedding Calculations.

The viability and effectiveness of replacing an ensemble of embedded solute calculations by a single calculation using an average description of the solvent environment are evaluated. This work explores the fluctuations of the average description of the system obtained in two ways: from calculations on an ensemble of geometries and from an average environment constructed from the same ensemble. To this end, classical molecular dynamics simulations of a rigid acetone solute in SPCE water are performed in order to generate an ensemble of solvent environments. From this ensemble of solvent configurations, a number of different approaches for constructing an average solvent environment are employed. We perform a thorough numerical analysis of the fluctuations of the electrostatic potential experienced by the solute, as well as the resulting fluctuations of the solute's electronic density, measured through its dipole moment and fitted atomic point charges. At the same time, we inspect the accuracy of the methods used to construct average environments. Finally, the proposed method for generating the embedding potential from an average environment density is applied to estimate the solvatochromic shift of the first excitation of acetone. In order to account for quantum confinement effects, which may be important in certain cases, the fluctuations in the shift due to the interaction with the solvent are evaluated using frozen-density-embedding theory. Our results demonstrate that, for normally distributed environments, the constructed average environment is a reasonably good representation of a fluctuating molecular solvent environment. We then provide guidance for future comparisons between these theoretical treatments of solute/solvent systems to experimental measurements.

Benchmark of the Extension of Frozen-Density Embedding Theory to Nonvariational Correlated Methods: The Embedded-MP2 Case.

The extension of the frozen-density embedding theory for nonvariational methods [J. Chem. Theory Comput. 2020, 16, 6880] was utilized to evaluate intermolecular interaction energies for complexes in the Zhao-Truhlar basis set. In the applied method (FDET-MP2-FAT-LDA), the same auxiliary system is used to evaluate the correlation energy by means of the second-order Møller-Plesset perturbation theory (MP2), as in our previous work [J. Chem. Phys. 2019, 150, 121101]. Local density approximation is used for ExcTnad[ρA,ρB] in both cases. Additionally, the contribution to the energy due to the neglected correlation potential was evaluated and analyzed. The domain of applicability of the local density approximation for ExcTnad[ρA,ρB] was determined based on deviations from the interaction energies from the conventional MP2 calculations. The local density approximation for ExcTnad[ρA,ρB] performs well for hydrogen- or dipole-bound complexes. The relative errors in the interaction energy lie within 3-30%. While for charge-transfer complexes, this approximation fails consistently, and for other types of complexes, the performance of this approximation is not systematic. The sources of error are discussed in detail.

The Challenge of Accurate Computation of Two-Photon Absorption Properties of Organic Chromophores in the Condensed Phase.

Two strategies are applied to evaluate the effect of the environment on the two-photon absorption (TPA) cross sections for two characteristic excited states of C2H4 upon complexation with H2O. The supermolecular strategy provides the reference complexation-induced shifts and uses either the EOM-CCSD or ADC(2) method. The embedding strategy is based on frozen-density-embedding theory (FDET) and uses only fundamental constants. The TPA cross sections from high-level supermolecular calculations are extremely basis-set-sensitive. Literature data and the present study indicate that accuracy of the absolute TPA cross sections below 100 atomic units and their shifts below 10 atomic units remains a challenge. The obtained FDET results show a similar basis-set behavior. For the largest basis set (d-aug-cc-pVQZ), TPA cross sections obtained from these two strategies are in excellent agreement. The complexation-induced shifts have the correct sign of the effect and a small (12-33%) relative error in magnitude. The deviations of the FDET-derived shifts from the reference are of similar magnitude as the reliability threshold of the reference shifts.

Frontiers in Multiscale Modeling of Photoreceptor Proteins.

This perspective article highlights the challenges in the theoretical description of photoreceptor proteins using multiscale modeling, as discussed at the CECAM workshop in Tel Aviv, Israel. The participants have identified grand challenges and discussed the development of new tools to address them. Recent progress in understanding representative proteins such as green fluorescent protein, photoactive yellow protein, phytochrome, and rhodopsin is presented, along with methodological developments.

On the Correlation Potential in Frozen-Density Embedding Theory.

The correlation functional Ec[ρ] known in Levy's constrained search formulation of density functional theory is also one of the components of the energy functional in frozen-density embedding theory (FDET) [Wesolowski Phys. Rev. A, 77, 2008, 012504] if the embedded wave function has the single-determinant form. The relation between the FDET energy and quantities available from an auxiliary system is derived. In the auxiliary system, Ec[ρ] and its functional derivative (correlation potential) are entirely neglected. The relation is exact up to the quadratic term in density changes caused by electron-electron correlation. It is discussed in view of its practical applications in modeling an electronic structure of embedded species.

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.

The deconvolution analysis of ATR-FTIR spectra of diacetylene during UV exposure.

We performed a detailed deconvolution analysis of ATR-FTIR peaks of a common diacetylene, 10,12-tricosadiynoic acid (TRCDA) during the polymerization and the blue-to-red transition. Based on the analysis and the solvent dependence on the IR signals, we found that the triple peak from CC stretching mode that has been previously suspected as a consequence of Fermi resonance is rather associated with the macromolecular assembly of TRCDA. Besides these CC triple peaks, we found that the background in the region increased during the UV exposure due to the CC signals from polymers. In addition, the anisotropic compression during polymerization was also detected, which supports the proposed interpretation of X-ray data reported previously. These results are the benefits from the deconvolution analysis.

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.

Benchmark of Excitation Energy Shifts from Frozen-Density Embedding Theory: Introduction of a Density-Overlap-Based Applicability Threshold.

We present a thorough investigation of the errors in results obtained with the combination of frozen-density embedding theory and the algebraic diagrammatic construction scheme for the polarization propagator of second order (FDE-ADC(2)). The study was carried out on a set of 52 intermolecular complexes with varying interaction strength, each consisting of a chromophore of fundamental interest and a few small molecules in its environment. The errors emerging in frozen-density embedding theory-based methods originate from (a) the solver of the quantum many-body problem used to obtain the embedded wave function (ΨAemb), (b) the approximation for the explicit density functional for the embedding potential, and (c) the choice of the density representing the environment (ρB( r⃗)). The present work provides a comprehensive analysis of the errors in the excitation energies based on the last two factors. Furthermore, a density-overlap-based parameter is proposed to be used as an a priori criterion of applicability.

Size extensivity of elastic properties of alkane fragments.

Using MP2, CCSD, and B3LYP methods of computational chemistry, we show length dependence in the intrinsic elastic properties of short alkane fragments. For isolated alkane fragments of finite length in the gas phase and zero temperature, the intrinsic elasticity constants are found to vary with the number of carbon atoms and its parity. From extrapolation of the elasticity constants calculations to infinite chain length, and by comparing with in-situ elasticity constant of single poly(ethylene) molecule obtained with atomic force microscopy, we estimate the softening effect of environment on the extension response of the polymer.

Implementation and Application of the Frozen Density Embedding Theory with the Algebraic Diagrammatic Construction Scheme for the Polarization Propagator up to Third Order.

Implementation, benchmarking, and representative applications of the new FDE-ADC(3) method for describing environmental effects on excited states as a combination of frozen density embedding (FDE) and the algebraic-diagrammatic construction scheme for the polarization propagator of third order (ADC(3)) are presented. Results of FDE-ADC(3) calculations are validated with respect to supersystem calculations on test systems with varying molecule-environment interaction strengths from dispersion up to multiple hydrogen bonds. The overall deviation compared to the supersystem calculations is as small as 0.029 eV for excitation energies, which is even smaller than the intrinsic error of ADC(3). The dependence of the accuracy on the choice of method and functional for the calculation of the environment and the nonelectrostatic part of the system-environment interaction is evaluated. In three representative examples, the FDE-ADC method is applied to investigate larger systems and to analyze excited state properties using visualization of embedded densities and orbitals.

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.

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.