Frozen-density Embedding Research Articles

Coupled-cluster frozen-density embedding using resolution of the identity methods.

Frozen-density embedding (FDE) is combined with resolution of the identity (RI) Hartree–Fock and a RI-variant of a second-order approximate coupled-cluster singles and doubles (RI-CC2) to determine solvatochromic shifts for the lowest excitation energy of acetone and pyridazine, respectively, each solvated in different environments with total system sizes of about 2.5 nm diameter. The combination of FDE and RI-CC2 increases efficiency and enables the calculation of numerous snapshots with 100 to 300 molecules, also allowing for larger basis sets as well as diffuse functions needed for an accurate treatment of properties. The maximum errors in the solvatochromic shifts amount up to 0.2 eV, which are similar to other approximated studies in the literature.

Subsystem density‐functional theory

Subsystem density‐functional theory (subsystem DFT) has developed into a powerful alternative to Kohn–Sham DFT for quantum chemical calculations of complex systems. It exploits the idea of representing the total electron density as a sum of subsystem densities. The optimum total density is found by minimizing the total energy with respect to each of the subsystem densities, which breaks down the electronic‐structure problem into effective subsystem problems. This enables calculations on large molecular aggregates and even (bio‐)polymers without system‐specific parameterizations. We provide a concise review of the underlying theory, typical approximations, and embedding approaches related to subsystem DFT such as frozen‐density embedding (FDE). Moreover, we discuss extensions and applications of subsystem DFT and FDE to molecular property calculations, excited states, and wave function in DFT embedding methods. Furthermore, we outline recent developments for reconstruction techniques of embedding potentials arising in subsystem DFT, and for using subsystem DFT to incorporate constraints into DFT calculations.This article is categorized under:Structure and Mechanism > Computational Materials Science

Frozen density embedding with non-integer subsystems’ particle numbers

We extend the frozen density embedding theory to non-integer subsystems' particles numbers. Different features of this formulation are discussed, with special concern for approximate embedding calculations. In particular, we highlight the relation between the non-integer particle-number partition scheme and the resulting embedding errors. Finally, we provide a discussion of the implications of the present theory for the derivative discontinuity issue and the calculation of chemical reactivity descriptors.

Calculation of nuclear spin-spin coupling constants using frozen density embedding

We present a method for a subsystem-based calculation of indirect nuclear spin-spin coupling tensors within the framework of current-spin-density-functional theory. Our approach is based on the frozen-density embedding scheme within density-functional theory and extends a previously reported subsystem-based approach for the calculation of nuclear magnetic resonance shielding tensors to magnetic fields which couple not only to orbital but also spin degrees of freedom. This leads to a formulation in which the electron density, the induced paramagnetic current, and the induced spin-magnetization density are calculated separately for the individual subsystems. This is particularly useful for the inclusion of environmental effects in the calculation of nuclear spin-spin coupling constants. Neglecting the induced paramagnetic current and spin-magnetization density in the environment due to the magnetic moments of the coupled nuclei leads to a very efficient method in which the computationally expensive response calculation has to be performed only for the subsystem of interest. We show that this approach leads to very good results for the calculation of solvent-induced shifts of nuclear spin-spin coupling constants in hydrogen-bonded systems. Also for systems with stronger interactions, frozen-density embedding performs remarkably well, given the approximate nature of currently available functionals for the non-additive kinetic energy. As an example we show results for methylmercury halides which exhibit an exceptionally large shift of the one-bond coupling constants between (199)Hg and (13)C upon coordination of dimethylsulfoxide solvent molecules.

Modeling environment effects on pigment site energies: Frozen density embedding with fully quantum-chemical protein densities

The recently developed 3-FDE method (Jacob and Visscher, 2008) combines ideas from the Molecular Fractionation with Conjugate Caps (MFCC) method (Zhang and Zhang, 2003) with Frozen Density Embedding (FDE) (Wesołowski and Warshel, 1993) and is thus able to produce fully quantum-chemical electron densities of entire proteins. In contrast to the original FDE method, 3-FDE facilitates a fragmentation into covalently bound subunits. We apply the method for the first time to the calculation of excitation energies, where we use the Fenna–Matthews–Olson (FMO) pigment–protein complex as a test case. Several technical and conceptual parameters for the preparation stages are tested and a robust protocol for this type of embedding is established. We present calculations of excitation energies of the individual pigments (site energies) under the influence of full protein densities obtained with different settings and compare them to the results from a simple point charge model. Our results indicate that 3-FDE is a well-suited method for the description of excitation energies within density-based embedding.

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.

A comparison between QM/MM and QM/QM based fitting of condensed-phase atomic polarizabilities.

Recently we reported a combined QM/MM approach to estimate condensed-phase values of atomic polarizabilities for use in (bio)molecular simulation. The setup relies on a MM treatment of the solvent when determining atomic polarizabilities to describe the response of a QM described solute to its external electric field. In this work, we study the effect of using alternative descriptions of the solvent molecules when evaluating atomic polarizabilities of a methanol solute. In a first step, we show that solute polarizabilities are not significantly affected upon substantially increasing the MM dipole moments towards values that are typically reported in literature for water solvent molecules. Subsequently, solute polarization is evaluated in the presence of a QM described solvent (using the frozen-density embedding method). In the latter case, lower oxygen polarizabilities were obtained than when using MM point charges to describe the solvent, due to introduction of Pauli-repulsion effects.

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

According to Frozen-Density Embedding Theory, any observable evaluated for the embedded species is a functional of the frozen density (ρ B —the density associated with the environment). The environment-induced shifts in the energies of local excitations in organic chromophores embedded in hydrogen-bonded environments are analyzed. The excitation energies obtained for ρ B , which is derived from ground-state calculations for the whole environment applying medium quality basis sets (STO–DZP) or larger, vary in a narrow range (about 0.02 eV which is at least one order of magnitude less than the magnitude of the shift). At the same time, the ground-state dipole moment of the environment varies significantly. The lack of correlation between the calculated shift and the dipole moment of the environment reflects the fact that, in Frozen-Density Embedding Theory, the partitioning of the total density is not unique. As a consequence, such concepts as “environment polarization” are not well defined within Frozen-Density Embedding Theory. Other strategies to generate ρ B (superposition of densities of atoms/molecules in the environment) are shown to be less robust for simulating excitation energy shifts for chromophores in environments comprising hydrogen-bonded molecules.

Calculating Hyperfine Couplings in Large Ionic Crystals Containing Hundreds of QM Atoms: Subsystem DFT Is the Key

We present an application of the linear scaling frozen density embedding (FDE) formulation of subsystem DFT to the calculation of isotropic hyperfine coupling constants (hfcc's) of atoms belonging to a guanine radical cation embedded in a guanine hydrochloride monohydrate crystal. The model systems range from an isolated guanine to a 15,000 atom QM/MM cluster where the QM region is comprised of 36 protonated guanine cations, 36 chlorine anions, and 42 water molecules. Our calculations show that the embedding effects of the surrounding crystal cannot be reproduced by small model systems nor by a pure QM/MM procedure. Instead, a large QM region is needed to fully capture the complicated nature of the embedding effects in this system. The unprecedented system size for a relativistic all-electron isotropic hfcc calculation can be approached in this work because the local nature of the electronic structure of the organic crystals considered is fully captured by the FDE approach.

Solvatochromic shifts from coupled-cluster theory embedded in density functional theory

Building on the framework recently reported for determining general response properties for frozen-density embedding [S. Höfener, A. S. P. Gomes, and L. Visscher, J. Chem. Phys. 136, 044104 (2012)], in this work we report a first implementation of an embedded coupled-cluster in density-functional theory (CC-in-DFT) scheme for electronic excitations, where only the response of the active subsystem is taken into account. The formalism is applied to the calculation of coupled-cluster excitation energies of water and uracil in aqueous solution. We find that the CC-in-DFT results are in good agreement with reference calculations and experimental results. The accuracy of calculations is mainly sensitive to factors influencing the correlation treatment (basis set quality, truncation of the cluster operator) and to the embedding treatment of the ground-state (choice of density functionals). This allows for efficient approximations at the excited state calculation step without compromising the accuracy. This approximate scheme makes it possible to use a first principles approach to investigate environment effects with specific interactions at coupled-cluster level of theory at a cost comparable to that of calculations of the individual subsystems in vacuum.

Quantum-Chemical Electron Densities of Proteins and of Selected Protein Sites from Subsystem Density Functional Theory.

The ability to calculate accurate electron densities of full proteins or of selected sites in proteins is a prerequisite for a fully quantum-mechanical calculation of protein-protein and protein-ligand interaction energies. Quantum-chemical subsystem methods capable of treating proteins and other biomolecular systems provide a route to calculate the electron densities of proteins efficiently and further make it possible to focus on specific parts. Here, we evaluate and extend the 3-partition frozen-density embedding (3-FDE) scheme [Jacob, C. R.; Visscher, L. J. Chem. Phys.2008, 128, 155102] for this purpose. In particular, we have extended this scheme to allow for the treatment of disulfide bridges and charged amino acid residues and have introduced the possibility to employ more general partitioning schemes. These extensions are tested both for the prediction of full protein electron densities and for focusing on the electron densities of a selected protein site. Our results demonstrate that 3-FDE is a promising tool for the fully quantum-chemical treatment of proteins.

Semilocal and hybrid density embedding calculations of ground-state charge-transfer complexes

We apply the frozen density embedding method, using a full relaxation of embedded densities through a freeze-and-thaw procedure, to study the electronic structure of several benchmark ground-state charge-transfer complexes, in order to assess the merits and limitations of the approach for this class of systems. The calculations are performed using both semilocal and hybrid exchange-correlation (XC) functionals. The results show that embedding calculations using semilocal XC functionals yield rather large deviations with respect to the corresponding supermolecular calculations. Due to a large error cancellation effect, however, they can often provide a relatively good description of the electronic structure of charge-transfer complexes, in contrast to supermolecular calculations performed at the same level of theory. On the contrary, when hybrid XC functionals are employed, both embedding and supermolecular calculations agree very well with each other and with the reference benchmark results. In conclusion, for the study of ground-state charge-transfer complexes via embedding calculations hybrid XC functionals are the method of choice due to their higher reliability and superior performance.

Bond energy decomposition analysis for subsystem density functional theory

We employed an explicit expression for the dispersion (D) energy in conjunction with Kohn-Sham (KS) density functional theory and frozen-density embedding (FDE) to calculate interaction energies between DNA base pairs and a selected set of amino acid pairs in the hydrophobic core of a small protein Rubredoxin. We use this data to assess the accuracy of an FDE-D approach for the calculation of intermolecular interactions. To better analyze the calculated interaction energies we furthermore propose a new energy decomposition scheme that is similar to the well-known KS bond formation analysis [F. M. Bickelhaupt and E. J. Baerends, Rev. Comput. Chem. 15, 1 (2000)], but differs in the electron densities used to define the bond energy. The individual subsystem electron densities of the FDE approach sum to the total electron density which makes it possible to define bond energies in terms of promotion energies and an explicit interaction energy. We show that for the systems considered only a few freeze-and-thaw cycles suffice to reach convergence in these individual bond energy components, illustrating the potential of FDE-D as an efficient method to calculate intermolecular interactions.

Towards systematically improvable models for actinides in condensed phase: the electronic spectrum of uranyl in Cs2UO2Cl4 as a test case

In this work we explore the use of frozen density embedding [Gomes et al., Phys. Chem. Chem. Phys., 2008, 10, 5353] as a way to construct models of increasing sophistication for describing the low-lying electronic absorption spectra of UO2(2+) in the Cs2UO2Cl4 crystal. We find that a relatively simple embedding model, in which all but the UO2(2+) unit are represented by an embedding potential, can already describe the main spectral features and the main environment effects can be attributed to the four chloride ions situated at the UO2(2+) equatorial plane. Contributions from species further away, albeit small, are found to be important for reaching a close agreement with experimentally observed quantities such as the excited states' relative positions. These findings suggest that such an embedding approach is a viable alternative to supermolecular calculations employing larger models of actinyl species in condensed phase. Nevertheless, we observe a slight red shift of the excitation energies calculated with our models compared to experimental results, and attribute this discrepancy to inaccuracies in the underlying structural parameters.

Calculation of electronic excitations using wave-function in wave-function frozen-density embedding

Recently, a general framework suitable for general frozen-density embedding (FDE) methods was published [S. Höfener, A. S. P. Gomes, and L. Visscher, J. Chem. Phys. 136, 044104 (2012)]. In the present article, we report the fragmentation of a supermolecule while treating all subsystems with coupled-cluster theory and the interaction of the subsystems with density-functional theory. This variant is denoted wave-function theory in wave-function theory FDE, or coupled-cluster theory in coupled-cluster theory FDE. Main target of this approach is not the embedding of a single molecule in large solvation shells, but rather the possibility to divide a complex system consisting of several molecules when all subsystems are to be treated with, e.g., coupled-cluster methods to provide a balanced and unbiased description. We present numerical results for hydrogen-bonded complexes which exhibit rather strong interactions. Cases with weakly interacting subsystems are expected to exhibit even higher accuracy. This facilitates the study of properties of larger complexes such as DNA base pairs with coupled-cluster methods.

Exact non-additive kinetic potentials in realistic chemical systems

In methods based on frozen-density embedding theory or subsystem formulation of density functional theory, the non-additive kinetic potential (v(t)(nad)(r)) needs to be approximated. Since v(t)(nad)(r) is defined as a bifunctional, the common strategies rely on approximating v(t)(nad)[ρ(A),ρ(B)](r). In this work, the exact potentials (not bifunctionals) are constructed for chemically relevant pairs of electron densities (ρ(A) and ρ(B)) representing: dissociating molecules, two parts of a molecule linked by a covalent bond, or valence and core electrons. The method used is applicable only for particular case, where ρ(A) is a one-electron or spin-compensated two-electron density, for which the analytic relation between the density and potential exists. The sum ρ(A) + ρ(B) is, however, not limited to such restrictions. Kohn-Sham molecular densities are used for this purpose. The constructed potentials are analyzed to identify the properties which must be taken into account when constructing approximations to the corresponding bifunctional. It is comprehensively shown that the full von Weizsäcker component is indispensable in order to approximate adequately the non-additive kinetic potential for such pairs of densities.

The electronic spectrum of CUONg4 (Ng = Ne, Ar, Kr, Xe): New insights in the interaction of the CUO molecule with noble gas matrices

The electronic spectrum of the CUO molecule was investigated with the IHFSCC-SD (intermediate Hamiltonian Fock-space coupled cluster with singles and doubles) method and with TD-DFT (time-dependent density functional theory) employing the PBE and PBE0 exchange-correlation functionals. The importance of both spin-orbit coupling and correlation effects on the low-lying excited-states of this molecule are analyzed and discussed. Noble gas matrix effects on the energy ordering and vibrational frequencies of the lowest electronic states of the CUO molecule were investigated with density functional theory (DFT) and TD-DFT in a supermolecular as well as a frozen density embedding (FDE) subsystem approach. This data is used to test the suitability of the FDE approach to model the influence of different matrices on the vertical electronic transitions of this molecule. The most suitable potential was chosen to perform relativistic wave function theory in density functional theory calculations to study the vertical electronic spectra of the CUO and CUONg(4) with the IHFSCC-SD method.

On the accuracy of frozen density embedding calculations with hybrid and orbital-dependent functionals for non-bonded interaction energies

We analyze the accuracy of the frozen density embedding (FDE) method, with hybrid and orbital-dependent exchange-correlation functionals, for the calculation of the total interaction energies of weakly interacting systems. Our investigation is motivated by the fact that these approaches require, in addition to the non-additive kinetic energy approximation, also approximate non-additive exact-exchange energies. Despite this further approximation, we find that the hybrid/orbital-dependent FDE approaches can reproduce the total energies with the same accuracy (about 1 mHa) as the one of conventional semi-local functionals. In many cases, thanks to error cancellation effects, hybrid/orbital-dependent approaches yield even the smallest error. A detailed energy-decomposition investigation is presented. Finally, the Becke-exchange functional is found to reproduce accurately the non-additive exact-exchange energies also for non-equilibrium geometries. These performances are rationalized in terms of a reduced-gradient decomposition of the non-additive exchange energy.

Quantum-chemical embedding methods for treating local electronic excitations in complex chemical systems

Quantum chemistry has become an invaluable tool for studying the electronic excitation phenomena underlying many important chemical, biological, and technological processes. Here, we review quantum-chemical approaches for modeling such phenomena. In particular, embedding methods can be particularly useful for treating localized excitations in complex chemical systems. These split the total system into a number of interacting subsystems. The electronic excitations processes occurring in the subsystem of interest are then treated with high accuracy, while its environment is taken into account in a more approximate way. In this review, we use a formulation based on the formally exact frozen-density embedding theory as our starting point. This provides a common framework for discussing the different embedding approaches that are currently available. Moreover, it also forms the basis of emerging methods that allow for a seamless coupling of density-functional theory and wavefunction based approaches, both for ground and excited states. These provide new possibilities for studying electronic excitations in large systems with predictive quantum-chemical methods.

Modelling charge transfer reactions with the frozen density embedding formalism

The frozen density embedding (FDE) subsystem formulation of density-functional theory is a useful tool for studying charge transfer reactions. In this work charge-localized, diabatic states are generated directly with FDE and used to calculate electronic couplings of hole transfer reactions in two π-stacked nucleobase dimers of B-DNA: 5'-GG-3' and 5'-GT-3'. The calculations rely on two assumptions: the two-state model, and a small differential overlap between donor and acceptor subsystem densities. The resulting electronic couplings agree well with benchmark values for those exchange-correlation functionals that contain a high percentage of exact exchange. Instead, when semilocal GGA functionals are used the electronic couplings are grossly overestimated.