How to choose the frozen density in Frozen-Density Embedding Theory-based numerical simulations of local excitations?

Calculating Molecular Polarizabilities Using Exact Frozen Density Embedding with External Orthogonality

Frozen density embedding (FDE) with freeze-thaw cycles is a formally exact embedding scheme. In practice, this method is limited to systems with small density overlaps when approximate nonadditive kinetic energy functionals are used. It has been shown that the use of approximate nonadditive kinetic energy functionals can be avoided when external orthogonality (EO) is enforced, and FDE can then generate exact results even for strongly overlapping subsystems. In this work, we present an implementation of exact FDEc-EO (coupled FDE TDDFT with EO) for the calculation of polarizabilities in the Amsterdam density functional program package. EO is enforced using the level-shift projection operator method, which ensures that orbitals between fragments are orthogonal. For pure functionals, we show that only the symmetric EO contributions to the induced density matrix are needed. This leads to a simplified implementation for the calculation of polarizability that can exactly reproduce the supermolecular TDDFT results. We further discuss the limitation of exact FDEc-EO in interpreting subsystem polarizabilities due to the nonunique partitioning of the total density. We show that this limitation is due to the fact that subsystem polarizability partitioning is dependent on how the subsystems are initially polarized. As supermolecular virtual orbitals are exactly reproduced, this dependence is attributed to the description of the occupied orbitals. In contrast, for excitations of subsystems that are localized within one subsystem, we show that the excitation energies are stable with respect to the order of polarization. This observation shows that impacts from the nonunique nature of exact FDE on subsystem properties can be minimized by better fragmentation of the supermolecular systems if the property is localized. For global properties like polarizability, this is not the case, and nonuniqueness remains independent of the fragmentation used.

Multiscale Frozen Density Embedding/Molecular Mechanics Approach for Simulating Magnetic Response Properties of Solvated Systems

We present a three-layer hybrid quantum mechanical/quantum embedding/molecular mechanics approach for calculating nuclear magnetic resonance (NMR) shieldings and J-couplings of molecular systems in solution. The model is based on the frozen density embedding (FDE) and polarizable fluctuating charges (FQ) and fluctuating dipoles (FQFμ) force fields and permits the accurate ab initio description of short-range nonelectrostatic interactions by means of the FDE shell and cost-effective treatment of long-range electrostatic interactions through the polarizable force field FQ(Fμ). Our approach's accuracy and potential are demonstrated by studying NMR spectra of Brooker's merocyanine in aqueous and nonaqueous solutions.

Orbital-Free Density Functional Theory: An Attractive Electronic Structure Method for Large-Scale First-Principles Simulations

Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.

Efficient Implementation of Density Functional Theory Based Embedding for Molecular and Periodic Systems Using Gaussian Basis Functions

A practical and effective implementation of density functional theory based embedding is reported, which allows us to treat both periodic and aperiodic systems on an equal footing. Its essence is the expansion of orbitals and electron density of the periodic system using Gaussian basis functions, rather than plane-waves, which provides a unique all-electron direct-space representation, thus avoiding the need for pseudopotentials. This makes the construction of embedding potential for a molecular active subsystem due to a periodic environment quite convenient, as transformation between representations is far from trivial. The three flavors of embedding, molecule-in-molecule, molecule-in-periodic, and periodic-in-periodic embedding, are implemented using embedding potentials based on non-additive kinetic energy density functionals (approximate) and level-shift projection operator (exact). The embedding scheme is coupled with a variety of correlated wave function theory (WFT) methods, thereby providing an efficient way to study the ground and excited state properties of low-dimensional systems using high-level methods for the region of interest. Finally, an implementation of real time-time-dependent density functional embedding theory (RT-TDDFET) is presented that uses a projection operator-based embedding potential and provides accurate results compared to full RT-TDDFT for systems with uncoupled excitations. The embedding potential is calculated efficiently using a combination of density fitting and continuous fast multipole method for the Coulomb term. The applicability of (i) WFT-in-DFT embedding, in predicting the adsorption and excitation energies, and (ii) RT-TDDFET, in predicting the absorption spectra, is explored for various test systems.

N-representability of the target density in Frozen-Density Embedding Theory based methods: numerical significance and its relation to electronic polarisation

The accuracy of any observable derived from multi-scale simulations based on Frozen-Density Embedding Theory (FDET) is affected by two inseparable factors: {\it i}) the approximation for the ${E}_{xcT}^{nad}[\rho_A,\rho_B]$ <p>component of the FDET energy functional and {\it ii}) the choice of the density $\rho_B(\mathbf{r})$</p> <p>for which the FDET eigenvalue equation for the embedded wavefunction is solved.</p> <p>A procedure is proposed to estimate the relative the significance of these two factors.</p> <p>Numerical examples are given for four weakly bound intermolecular complexes.</p> <p>It is shown that the violation of the non-negativity condition is the principal source of error in the FDET energy</p> <p>if $\rho_B$ is the density of the isolated environment, i.e. is generated without taking into account the interactions with the embedded species.</p> <p>Reduction of both the magnitude of the violation of the non-negativity condition and the error in the FDET energy can be pragmatically achieved by means of the explicit treatment of the electronic polarisation of the environment.

The Seamless Connection of Local and Collective Excited States in Subsystem Time-Dependent Density Functional Theory

The theoretical understanding of photoinduced processes in multichromophoric systems requires, as an essential ingredient, the possibility of accurately describing their electronically excited states. However, the size of these systems often prohibits the usage of conventional electronic-structure methods, so that often multiscale approaches based on phenomenologically motivated models are employed. In contrast, subsystem time-dependent density functional theory (sTDDFT) allows for a subsystem-based ab initio description of multichromophoric systems and therefore allows for, in principle, an exact description of photoinduced processes. This Perspective aims to outline the theoretical foundations and commonly used practical realizations as well as to illustrate benefits of recent developments and open issues in the field of sTDDFT. Prospective, potential future applications and possible methodological developments are discussed.

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 E_xcT^nad[ρ_A,ρ_B] 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 E_xcT^nad[ρ_A,ρ_B] was determined based on deviations from the interaction energies from the conventional MP2 calculations. The local density approximation for E_xcT^nad[ρ_A,ρ_B] 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.

A polarizable three-layer frozen density embedding/molecular mechanics approach

We present a novel multilayer polarizable embedding approach in which the system is divided into three portions, two of which are treated using density functional theory and their interaction is based on frozen density embedding (FDE) theory, and both also mutually interact with a polarizable classical layer described using an atomistic model based on fluctuating charges (FQ). The efficacy of the model is demonstrated by extending the formalism to linear response properties and applying it to the simulation of the excitation energies of organic molecules in aqueous solution, where the solute and the first solvation shell are treated using FDE, while the rest of the solvent is modeled using FQ charges.

Quantum Mechanics/Extremely Localized Molecular Orbital Embedding Strategy for Excited States: Coupling to Time-Dependent Density Functional Theory and Equation-of-Motion Coupled Cluster

The QM/ELMO (quantum mechanics/extremely localized molecular orbital) method is a recently developed embedding technique in which the most important region of the system under examination is treated at fully quantum mechanical level, while the rest is described by means of transferred and frozen extremely localized molecular orbitals. In this paper, we propose the first application of the QM/ELMO approach to the investigation of excited states, and, in particular, we present the coupling of the QM/ELMO philosophy with Time-Dependent Density Functional Theory (TDDFT) and Equation-of-Motion Coupled Cluster with single and double substitutions (EOM-CCSD). The proposed TDDFT/ELMO and EOM-CCSD/ELMO strategies underwent a series of preliminary tests that were already considered for the validation of other embedding methods for excited states. The obtained results showed that the novel techniques allow the accurate description of localized excitations in large systems by only including a relatively small number of atoms in the region treated at fully quantum chemical level. Furthermore, for TDDFT/ELMO, it was also observed that (i) the method enables to avoid the presence of artificial low-lying charge-transfer states that may affect traditional TDDFT calculations, even using functionals that do not take into account long-range corrections, and (ii) the novel approach can be also successfully exploited to investigate local electronic transitions in quite large systems (e.g., reduced model of the Green Fluorescent Protein), and the accuracy of the results can be improved by including a sufficient number of chemically crucial fragments/residues in the quantum mechanical region. Finally, concerning EOM-CCSD/ELMO, it was also seen that, despite the quite crude approximation of an embedding potential given by frozen extremely localized molecular orbitals, the new strategy is able to satisfactorily account for the effects of the environment. This work paves the way to further extensions of the QM/ELMO philosophy for the study of local excitations in extended systems, suggesting the coupling of the QM/ELMO approach with other quantum chemical strategies for excited states, from the simplest ΔSCF techniques to the most advanced and computationally expensive multireferences methods.

Electronic couplings for photo-induced processes from subsystem time-dependent density-functional theory: The role of the diabatization

Subsystem time-dependent density-functional theory (sTDDFT) making use of approximate non-additive kinetic energy (NAKE) functionals is known to be capable of describing excitation energy transfer processes in a variety of applications. Here, we show that sTDDFT, especially when combined with projection-based embedding (PbE), can be employed for the entire range of photo-induced electronic couplings essential for modeling photophysical properties of complex chemical and biological systems and therefore represents a complete toolbox for this class of problems. This means that it is capable of capturing the interaction/coupling associated with local- and charge-transfer (CT) excitons. However, this requires the choice of a reasonable diabatic basis. We therefore propose different diabatization strategies of the virtual orbital space in PbE-sTDDFT and show how CT excitations can be included in sTDDFT using NAKE functionals via a phenomenological approach. Finally, these electronic couplings are compared to couplings from a multistate fragment excitation difference (FED)-fragment charge difference (FCD) diabatization procedure. We show that both procedures, multistate FED-FCD and sTDDFT (with the right diabatization procedure chosen), lead to an overall good agreement for the electronic couplings, despite differences in their general diabatization strategy. We conclude that the entire range of photo-induced electronic couplings can be obtained using sTDDFT (with the right diabatization procedure chosen) in a black-box manner.

Embedding-theory-based simulations using experimental electron densities for the environment

The basic idea of frozen-density embedding theory (FDET) is the constrained minimization of the Hohenberg-Kohn density functional EHK[ρ] performed using the auxiliary functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B], where ΨA is the embedded NA-electron wavefunction and ρB(r) is a non-negative function in real space integrating to a given number of electrons NB. This choice of independent variables in the total energy functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B] makes it possible to treat the corresponding two components of the total density using different methods in multi-level simulations. The application of FDET using ρB(r) reconstructed from X-ray diffraction data for a molecular crystal is demonstrated for the first time. For eight hydrogen-bonded clusters involving a chromophore (represented as ΨA) and the glycylglycine molecule [represented as ρB(r)], FDET is used to derive excitation energies. It is shown that experimental densities are suitable for use as ρB(r) in FDET-based simulations.

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

A comparison of strategies to account for environment polarisation in Frozen Density Embedding Theory (FDET).

Role of the Molecular Environment in Flavoprotein Color and Redox Tuning: QM Cluster versus QM/MM Modeling

We investigate the origin of the excitation energy shifts induced by the apoprotein in the active site of the bacterial photoreceptor BLUF (Blue Light sensor Using Flavin adenine dinucleotide). In order to compute the vertical excitation energies of three low-lying electronic states, including two π-π* states of flavin (S1 and S2) and a π-π* tyrosine-flavin electron-transfer state (ET), with respect to the energy of the closed-shell ground state (S0), we prepared alternative quantum mechanical (QM) cluster and quantum mechanics/molecular mechanics (QM/MM) models. We found that the excitation energies computed with both types of models correlate with the magnitude of the charge transfer character of the excitation. Accordingly, we conclude that the small charge transfer character of the light absorbing S0-S1 transition and the substantial charge transfer character of the nonabsorbing but redox active S0-ET transition explain the small color changes but substantial redox tuning in BLUF and also in other flavoproteins. Further analysis showed that redox tuning is governed by the electrostatic interaction in the QM/MM model and transfer of charge between the active site and its environment in the QM cluster. Moreover, the wave function polarization of the QM subsystem by the MM subsystem influences the magnitude of the charge transfer, resulting in the QM/MM and QM excitation energies that are not entirely consistent.

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

Besides molecular electron densities obtained within the Born–Oppenheimer approximation (ρ_B(r)) to represent the environment, the ensemble averaged density (〈ρ_B〉(r)) is also admissible in frozen-density embedding theory (FDET) [Wesolowski, Phys. Rev. A, 2008, 77, 11444].

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.