Calculation of nuclear magnetic resonance shieldings using frozen-density embedding

We have extended the frozen-density embedding (FDE) scheme within density-functional theory [T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, 8050 (1993)] to include external magnetic fields and applied this extension to the nonrelativistic calculation of nuclear magnetic resonance (NMR) shieldings. This leads to a formulation in which the electron density and the induced current are calculated separately for the individual subsystems. If the current dependence of the exchange-correlation functional and of the nonadditive kinetic-energy functional are neglected, the induced currents in the subsystems are not coupled and each of them can be determined without knowledge of the induced current in the other subsystem. This allows the calculation of the NMR shielding as a sum of contributions of the individual subsystems. As a test application, we have calculated the solvent shifts of the nitrogen shielding of acetonitrile for different solvents using small geometry-optimized clusters consisting of acetonitrile and one solvent molecule. By comparing to the solvent shifts obtained from supermolecular calculations we assess the accuracy of the solvent shifts obtained from FDE calculations. We find a good agreement between supermolecular and FDE calculations for different solvents. In most cases it is possible to neglect the contribution of the induced current in the solvent subsystem to the NMR shielding, but it has to be considered for aromatic solvents. We demonstrate that FDE can describe the effect of induced currents in the environment accurately.     

Exploring the ability of frozen-density embedding to model induced circular dichroism

In this study, we present calculations of the circular dichroism (CD) spectra of complexes between achiral and chiral molecules. Nonzero rotational strengths for transitions of the nonchiral molecule are induced by interactions between the two molecules, which cause electronic and/or structural perturbations of the achiral molecule. We investigate if the chiral molecule (environment) can be represented only in terms of its frozen electron density, which is used to generate an effective embedding potential. The accuracy of these calculations is assessed in comparison to full supermolecular calculations. We can show that electronic effects arising from specific interactions between the two subsystems can reliably be modeled by the frozen-density representation of the chiral molecule. This is demonstrated for complexes of 2-benzoylbenzoic acid with (-)-(R)-amphetamine and for a nonchiral, artificial amino acid receptor system consisting of ferrocenecarboxylic acid bound to a crown ether, for which a complex with l-leucine is studied. Especially in the latter case, where multiple binding sites and interactions between receptor and target molecule exist, the frozen-density results compare very well with the full supermolecular calculation. We also study systems in which a cyclodextrin cavity serves as a chiral host system for a small, achiral molecule. Problems arise in that case because of the importance of excitonic couplings with excitations in the host system. The frozen-density embedding cannot describe such couplings but can only capture the direct effect of the host electron density on the electronic structure of the guest. If couplings play a role, frozen-density embedding can at best only partially describe the induced circular dichroism. To illustrate this problem, we finally construct a case in which excitonic coupling effects are much stronger than direct interactions of the subsystem densities. The frozen density embedding is then completely unsuitable.     

NMR solvent shifts of acetonitrile from frozen density embedding calculations

We present a density functional theory (DFT) study of solvent effects on nuclear magnetic shielding parameters. As a test example we have focused on the sensitive nitrogen shift of acetonitrile immersed in a selected set of solvents, namely water, chloroform, and cyclohexane. To include the effect of the solvent environment in an accurate and efficient manner, we employed the frozen-density embedding (FDE) scheme. We have included up to 500 solvent molecules in the NMR computations and obtained the cluster geometries from a large set of conformations generated with molecular dynamics. For small solute-solvent clusters comparison of the FDE results with conventional supermolecular DFT calculations shows close agreement. For the large solute-solvent clusters the solvent shift values are compared with experimental data and with values obtained using continuum solvent models. For the water --> cyclohexane shift the obtained value is in very good agreement with experiments. For the water --> chloroform NMR solvent shift the classical force field used in the molecular dynamics simulations is found to introduce an error. This error can be largely avoided by using geometries taken from Car-Parrinello molecular dynamics simulations.     

Frozen-density embedding-based many-body expansions

Fragmentation methods allow for the accurate quantum chemical (QC) treatment of large molecular clusters and materials. Here we explore the combination of two complementary approaches to the development of such fragmentation methods: the many-body expansion (MBE) on the one hand, and subsystem density-functional theory (DFT) or frozen-density embedding (FDE) theory on the other hand. First, we assess potential benefits of using FDE to account for the environment in the subsystem calculations performed within the MBE. Second, we use subsystem DFT to derive a density-based MBE, in which a many-body expansion of the electron density is used to calculate the system's total energy. This provides a correction to the energies calculated with a conventional energy-based MBE that depends only on the subsystem's electron densities. For the test case of clusters of water and of aspirin, we show that such a density-based MBE converges faster than the conventional energy-based MBE. For our test cases, truncation errors in the interaction energies are below chemical accuracy already with a two-body expansion. The density-based MBE thus provides a promising avenue for accurate QC calculation of molecular clusters and materials.     

Orbital-free embedding applied to the calculation of induced dipole moments in CO2...X (X = He, Ne, Ar, Kr, Xe, Hg) van der Waals complexes

The orbital-free frozen-density embedding scheme within density-functional theory [T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, 8050 (1993)] is applied to the calculation of induced dipole moments of the van der Waals complexes CO2...X (X = He, Ne, Ar, Kr, Xe, Hg). The accuracy of the embedding scheme is investigated by comparing to the results of supermolecule Kohn-Sham density-functional theory calculations. The influence of the basis set and the consequences of using orbital-dependent approximations to the exchange-correlation potential in embedding calculations are examined. It is found that in supermolecular Kohn-Sham density-functional calculations, different common approximations to the exchange-correlation potential are not able to describe the induced dipole moments correctly and the reasons for this failure are analyzed. It is shown that the orbital-free embedding scheme is a useful tool for applying different approximations to the exchange-correlation potential in different subsystems and that a physically guided choice of approximations for the different subsystems improves the calculated dipole moments significantly.     

Potential-energy surfaces of local excited states from subsystem- and selective Kohn-Sham-TDDFT

Calculating excited-state potential-energy surfaces for systems with a large number of close-lying excited states requires the identification of the relevant electronic transitions for several geometric structures. Time-dependent density functional theory (TDDFT) is very efficient in such calculations, but the assignment of local excited states of the active molecule can be difficult. We compare the results of the frozen-density embedding (FDE) method with those of standard Kohn-Sham density-functional theory (KS-DFT) and simpler QM/MM-type methods. The FDE results are found to be more accurate for the geometry dependence of excitation energies than classical models. We also discuss how selective iterative diagonalization schemes can be exploited to directly target specific excitations for different structures. Problems due to strongly interacting orbital transitions and possible solutions are discussed. Finally, we apply FDE and the selective KS-TDDFT to investigate the potential energy surface of a high-lying π → π^∗ excitation in a pyridine molecule approaching a silver cluster.     

The electronic spectrum of CUONg(4) (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.     

Legendre-transform functionals for spin-density-functional theory

We provide a rigorous proof that the Hohenberg-Kohn theorem holds for spin densities by extending Lieb's Legendre-transform formulation to spin densities. The resulting spin-density-functional theory resolves several troublesome issues. Most importantly, the present paper provides an explicit construction for the spin potentials at any point along the adiabatic connection curve, thus providing a formal basis for the use of exchange-correlation functionals of the spin density in the Kohn-Sham density-functional theory (DFT). The practical implications of this result for unrestricted Kohn-Sham DFT calculations is considered, and the existence of holes below the Fermi level is discussed. We argue that an orbital's energy tends to increase as its occupation number increases, which provides the basis for a computational algorithm for determining the occupation numbers in Kohn-Sham DFT and helps explain the origin of Hund's rules and holes below the Fermi level.     

A flexible implementation of frozen-density embedding for use in multilevel simulations

A new implementation of frozen-density embedding (FDE) in the Amsterdam Density Functional (ADF) program package is presented. FDE is based on a subsystem formulation of density-functional theory (DFT), in which a large system is assembled from an arbitrary number of subsystems, which are coupled by an effective embedding potential. The new implementation allows both an optimization of all subsystems as a linear-scaling alternative to a conventional DFT treatment, the calculation of one active fragment in the presence of a frozen environment, and intermediate setups, in which individual subsystems are fully optimized, partially optimized, or completely frozen. It is shown how this flexible setup can facilitate the application of FDE in multilevel simulations.     

Is it worthwhile to go beyond the local-density approximation in subsystem density functional theory?

Frozen density embedding (FDE) theory is one of the major techniques aiming to bring modeling of extended chemical systems into the realm of high accuracy calculations. To improve its accuracy it is of interest to develop kinetic energy density functional approximations specifically for FDE applications. In the study reported here we focused on optimizing parameters of a generalized gradient approximation-like kinetic energy functional with the purpose of better describing electron excitation energies. We found that our optimized parametrizations, named excPBE and excPBE-3 (as these are derived from a Perdew-Burke-Ernzerhof-like parametrization), could not yield improvements over available functionals when applied on a test set of systems designed to probe solvatochromic shifts. Moreover, as several different functionals yielded very similar errors to the simple local-density approximation (LDA), it is questionable whether it is worthwhile to go beyond the LDA in this context.     

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.     

Unambiguous optimization of effective potentials in finite basis sets

The optimization of effective potentials is of interest in density-functional theory (DFT) in two closely related contexts. First, the evaluation of the functional derivative of orbital-dependent exchange-correlation functionals requires the application of optimized effective potential methods. Second, the optimization of the effective local potential that yields a given electron density is important both for the development of improved approximate functionals and for the practical application of embedding schemes based on DFT. However, in all cases this optimization turns into an ill-posed problem if a finite basis set is introduced for the Kohn-Sham orbitals. So far, this problem has not been solved satisfactorily. Here, a new approach to overcome the ill-posed nature of such finite-basis set methods is presented for the optimization of the effective local potential that yields a given electron density. This new scheme can be applied with orbital basis sets of reasonable size and makes it possible to vary the basis sets for the orbitals and for the potential independently, while providing an unambiguous potential that systematically approaches the numerical reference.     

A subsystem density-functional theory approach for the quantum chemical treatment of proteins

We present an extension of the frozen-density embedding (FDE) scheme within density-functional theory [T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, 8050 (1993)] that can be applied to subsystems connected by covalent bonds, as well as a practical implementation of such an extended FDE scheme. We show how the proposed scheme can be employed for quantum chemical calculations of proteins by treating each constituting amino acid as a separate subsystem. To assess the accuracy of the extended FDE scheme, we present calculations for several dipeptides and for the protein ubiquitin.     

Wave‐function frozen‐density embedding: Approximate analytical nuclear ground‐state gradients

We report the derivation of approximate analytical nuclear ground‐state uncoupled frozen density embedding (FDEu) gradients for the resolution of identity (RI) variant of the second‐order approximate coupled cluster singles and doubles (RICC2) as well as density functional theory (DFT), and an efficient implementation thereof in the KOALA program. In order to guarantee a computationally efficient treatment, those gradient terms are neglected which would require the exchange of orbital information. This approach allows for geometry optimizations of single molecules surrounded by numerous molecules with fixed nuclei at RICC2‐in‐RICC2, RICC2‐in‐DFT, and DFT‐in‐DFT FDE level of theory using a dispersion correction, required due to the DFT‐based treatment of the interaction in FDE theory. Accuracy and applicability are assessed by the example of two case studies: (a) the Watson‐Crick pair adenine‐thymine, for which the optimized structures exhibit a maximum error of about 0.08 Å for our best scheme compared to supermolecular reference calculations, (b) carbon monoxide on a magnesium oxide surface model, for which the error amount up to 0.1 Å for our best scheme. Efficiency is demonstrated by successively including environment molecules and comparing to an optimized conventional supermolecular implementation, showing that the method is able to outperform conventional RICC2 schemes already with a rather small number of environment molecules, gaining significant speed up in computation time. © 2016 Wiley Periodicals, Inc.     

Parameters for excess electron transfer in DNA. Estimation using unoccupied Kohn-Sham orbitals and TD DFT

We show that the energetics and electronic couplings for excess electron transfer (EET) can be accurately estimated by using unoccupied Kohn-Sham orbitals (UKSO) calculated for neutral pi stacks. To assess the performance of different DFT functionals, we use MS-PT2 results for seven pi stacks of nucleobases as reference data. The DFT calculations are carried out by using the local spin density approximation SVWN, two generalized gradient approximation functionals BP86 and BLYP, and two hybrid functionals B3LYP and BH&HLYP. Best estimations within the UKSO approach are obtained by the B3LYP and SVWN methods. TD DFT calculations provide less accurate values of the EET parameters as compared with the UKSO data. Also, the excess charge distribution in the radical anions is well described by the LUMOs of neutral systems. In contrast, spin-unrestricted DFT calculations of radical anions considerably overestimate delocalization of the excess electron. The excellent results obtained for the ground and excited states of the radical anions (excitation energy, transition dipole moment, electronic coupling, and excess electron distribution) by using UKSO of neutral dimers suggest an efficient strategy to calculate the EET parameters for DNA pi stacks.     

Comparing ab initio density-functional and wave function theories: the impact of correlation on the electronic density and the role of the correlation potential

The framework of ab initio density-functional theory (DFT) has been introduced as a way to provide a seamless connection between the Kohn-Sham (KS) formulation of DFT and wave-function based ab initio approaches [R. J. Bartlett, I. Grabowski, S. Hirata, and S. Ivanov, J. Chem. Phys. 122, 034104 (2005)]. Recently, an analysis of the impact of dynamical correlation effects on the density of the neon atom was presented [K. Jankowski, K. Nowakowski, I. Grabowski, and J. Wasilewski, J. Chem. Phys. 130, 164102 (2009)], contrasting the behaviour for a variety of standard density functionals with that of ab initio approaches based on second-order M?ller-Plesset (MP2) and coupled cluster theories at the singles-doubles (CCSD) and singles-doubles perturbative triples [CCSD(T)] levels. In the present work, we consider ab initio density functionals based on second-order many-body perturbation theory and coupled cluster perturbation theory in a similar manner, for a range of small atomic and molecular systems. For comparison, we also consider results obtained from MP2, CCSD, and CCSD(T) calculations. In addition to this density based analysis, we determine the KS correlation potentials corresponding to these densities and compare them with those obtained for a range of ab initio density functionals via the optimized effective potential method. The correlation energies, densities, and potentials calculated using ab initio DFT display a similar systematic behaviour to those derived from electronic densities calculated using ab initio wave function theories. In contrast, typical explicit density functionals for the correlation energy, such as VWN5 and LYP, do not show behaviour consistent with this picture of dynamical correlation, although they may provide some degree of correction for already erroneous explicitly density-dependent exchange-only functionals. The results presented here using orbital dependent ab initio density functionals show that they provide a treatment of exchange and correlation contributions within the KS framework that is more consistent with traditional ab initio wave function based methods.     

The merits of the frozen-density embedding scheme to model solvatochromic shifts

We investigate the usefulness of a frozen-density embedding scheme within density-functional theory [J. Phys. Chem. 97, 8050 (1993)] for the calculation of solvatochromic shifts. The frozen-density calculations, particularly of excitation energies have two clear advantages over the standard supermolecule calculations: (i) calculations for much larger systems are feasible, since the time-consuming time-dependent density functional theory (TDDFT) part is carried out in a limited molecular orbital space, while the effect of the surroundings is still included at a quantum mechanical level. This allows a large number of solvent molecules to be included and thus affords both specific and nonspecific solvent effects to be modeled. (ii) Only excitations of the system of interest, i.e., the selected embedded system, are calculated. This allows an easy analysis and interpretation of the results. In TDDFT calculations, it avoids unphysical results introduced by spurious mixings with the artificially too low charge-transfer excitations which are an artifact of the adiabatic local-density approximation or generalized gradient approximation exchange-correlation kernels currently used. The performance of the frozen-density embedding method is tested for the well-studied solvatochromic properties of the n-->pi(*) excitation of acetone. Further enhancement of the efficiency is studied by constructing approximate solvent densities, e.g., from a superposition of densities of individual solvent molecules. This is demonstrated for systems with up to 802 atoms. To obtain a realistic modeling of the absorption spectra of solvated molecules, including the effect of the solvent motions, we combine the embedding scheme with classical molecular dynamics (MD) and Car-Parrinello MD simulations to obtain snapshots of the solute and its solvent environment, for which then excitation energies are calculated. The frozen-density embedding yields estimated solvent shifts in the range of 0.20-0.26 eV, in good agreement with experimental values of between 0.19 and 0.21 eV.