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%.

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 (ρ_B) 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 ρ_B, 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 ρ_B improves indeed the accuracy of the excitation energy as compared to the implicit treatment in which the same ρ_B 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.

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 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.

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.