Nonadditive Kinetic Potential Research Articles

Is the non-additive kinetic potential always equal to the difference of effective potentials from inverting the Kohn-Sham equation?

The relation used frequently in the literature according to which the non-additive kinetic potential which is a functional depending on a pair of electron densities is equal (up to a constant) to the difference of two potentials obtained from inverting two Kohn-Sham equations, is examined. The relation is based on a silent assumption that the two densities can be obtained from two independent Kohn-Sham equations, i.e., are vs-representable. It is shown that this assumption does not hold for pairs of densities: ρtot being the Kohn-Sham density in some system and ρB obtained from such partitioning of ρtot that the difference ρtot - ρB vanishes on a Lebesgue measurable volume element. The inversion procedure is still applicable for ρtot - ρB but cannot be interpreted as the inversion of the Kohn-Sham equation. It is rather the inversion of a Kohn-Sham-like equation. The effective potential in the latter equation comprises a "contaminant" that might even not be unique. It is shown that the construction of the non-additive kinetic potential based on the examined relation is not applicable for such pairs.

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.

Polarizable Frozen Density Embedding with External Orthogonalization.

We report a polarizable subsystem density functional theory to describe electronic properties of molecules embedded on a metal cluster. Interaction between the molecule and metal cluster is described using frozen density embedding (FDE). Substituting the nonadditive kinetic potential (NAKP) by approximate functionals is circumvented by enforcing external orthogonality (EO) through a projection operator. The computationally expensive freeze/thaw (FT) cycles are bypassed by including a polarization term in the embedding operator. Furthermore, the combination of polarization and EO permits supermolecular basis set calculations, which was not possible for strongly interacting systems with existing kinetic energy functionals. To test the method, we described the ground state density of pyridine, water, and benzene on a silver cluster. Performing FT on top of EO results in exact density embedding for this category of systems and is thus used for benchmarking the method. We find that the density is reproduced to within 0.15e, and the dipole and quadrupole moments are within 18% of the reference points for subsystem separations ranging from bonding to noninteracting distances. Additionally, our formalism allows the flexibility of incorporating different density functionals to the molecular and the metallic subsystems reducing the overall computational cost.

Nonadditive kinetic potentials from inverted Kohn–Sham problem

AbstractThe nonadditive kinetic potential is a key element in density‐dependent embedding methods. The correspondence between the ground‐state density and the total effective Kohn–Sham potential provides the basis for various methods to construct the nonadditive kinetic potential for any pair of electron densities. Several research groups used numerical or analytical inversion procedures to explore this strategy which overcomes the failures of known explicit density functional approximations. The numerical inversions, however, apply additional approximations/simplifications. The relations known for the exact quantities cannot be assumed to hold for quantities obtained in numerical inversions. The exact relations are discussed with special emphasis on such issues as: the admissibility of the densities for which the potential is constructed, the choice of densities to be used as independent variables, self‐consistency between the potentials and observables calculated using the embedded wavefunction, and so forth. The review focuses on how these issues are treated in practice. The review is supplemented with the analysis of the inverted potentials for weakly overlapping pairs of electron densities—the case not studied previously.

Communication: Density functional theory embedding with the orthogonality constrained basis set expansion procedure.

Density functional theory (DFT) embedding approaches have generated considerable interest in the field of computational chemistry because they enable calculations on larger systems by treating subsystems at different levels of theory. To circumvent the calculation of the non-additive kinetic potential, various projector methods have been developed to ensure the orthogonality of molecular orbitals between subsystems. Herein the orthogonality constrained basis set expansion (OCBSE) procedure is implemented to enforce this subsystem orbital orthogonality without requiring a level shifting parameter. This scheme is a simple alternative to existing parameter-free projector-based schemes, such as the Huzinaga equation. The main advantage of the OCBSE procedure is that excellent convergence behavior is attained for DFT-in-DFT embedding without freezing any of the subsystem densities. For the three chemical systems studied, the level of accuracy is comparable to or higher than that obtained with the Huzinaga scheme with frozen subsystem densities. Allowing both the high-level and low-level DFT densities to respond to each other during DFT-in-DFT embedding calculations provides more flexibility and renders this approach more generally applicable to chemical systems. It could also be useful for future extensions to embedding approaches combining wavefunction theories and DFT.

Accurate Dissociation of Chemical Bonds Using DFT-in-DFT Embedding Theory with External Orbital Orthogonality.

Our recent density functional theory (DFT)-in-DFT embedding protocol, which enforces intersubsystem (or external orbital) orthogonality, is used for the first time to investigate covalent bond dissociation and is shown to do so accurately. Full potential energy curves for the dissociation of a H-O bond in H2O and the C-C bond in H3C-CH3 have been constructed using the new embedding method, as have the challenging ionic bonds in LiH and LiF, and were found to match the reference Kohn-Sham (KS)-DFT curves to at least one part in 106. The added constraint of external orbital orthogonality allows for the formulation of an embedding protocol that does not rely on approximate kinetic energy functionals for the evaluation of the so-called nonadditive kinetic potential, does not introduce compensatory potentials, and does not require a total system calculation at any stage. The present work extends the demonstrated applicability of the external orthogonality variant of embedding theory by more than a factor of 2 to the interaction strength range of strong single bonds. In particular, it is demonstrated that homolytic cleavage of both covalent and ionic bonds into radicals can be accomplished.

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.

Self-consistency in frozen-density embedding theory based calculations

The bi-functional for the non-electrostatic part of the exact embedding potential of frozen-density embedding theory (FDET) depends on whether the embedded part is described by means of a real interacting many-electron system or the reference system of non-interacting electrons (see [Wesolowski, Phys. Rev. A. 77, 11444 (2008)]). The difference δΔF(MD)[ρ(A)]/δρ(A)(r<combining arrow>), where ΔF(MD)[ρ(A)] is the functional bound from below by the correlation functional E(c)[ρ(A)] and from above by zero. Taking into account ΔF(MD)[ρ(A)] in both the embedding potential and in energy is indispensable for assuring that all calculated quantities are self-consistent and that FDET leads to the exact energy and density in the limit of exact functionals. Since not much is known about good approximations for ΔF(MD)[ρ(A)], we examine numerically the adequacy of neglecting ΔF(MD)[ρ(A)] entirely. To this end, we analyze the significance of δΔF(MD)[ρ(A)]/δρ(A)(r<combining arrow>) in the case where the magnitude of ΔF(MD)[ρ(A)] is the largest, i.e., for Hartree-Fock wavefunction. In hydrogen bonded model systems, neglecting δΔF(MD)[ρ(A)]/δρ(A)(r<combining arrow>) in the embedding potential marginally affects the total energy (less than 5% change in the interaction energy) but results in qualitative changes in the calculated hydrogen-bonding induced shifts of the orbital energies. Based on this estimation, we conclude that neglecting δΔF(MD)[ρ(A)]/δρ(A)(r<combining arrow>) may represent a good approximation for multi-reference variational methods using adequate choice for the active space. Doing the same for single-reference perturbative methods is not recommended. Not only it leads to violation of self-consistency but might result in large effect on orbital energies. It is shown also that the errors in total energy due to neglecting δΔF(MD)[ρ(A)]/δρ(A)(r<combining arrow>) do not cancel but rather add up to the errors due to approximation for the bi-functional of the non-additive kinetic potential.

Comment on “Accurate frozen-density embedding potentials as a first step towards a subsystem description of covalent bonds” [J. Chem. Phys. 132, 164101 (2010)

Several assertions which are incorrect or might be misleadingly interpreted as well as omissions of issues concerning the non-additive kinetic energy potential made by Fux et al. are analyzed. They concern issues of great importance for any computational method based on the orbital-free embedding theory: evaluation of the total energy, approximating the non-additive kinetic potential, exact properties of non-additive kinetic energy potential. In a nutshell, the authors do not distinguish between two different quantities: the functional, i.e., the correspondence assigning the non-additive kinetic potential to a pair of electron densities and the function (the potential itself).

Embedded density functional theory for covalently bonded and strongly interacting subsystems

Embedded density functional theory (e-DFT) is used to describe the electronic structure of strongly interacting molecular subsystems. We present a general implementation of the Exact Embedding (EE) method [J. Chem. Phys. 133, 084103 (2010)] to calculate the large contributions of the nonadditive kinetic potential (NAKP) in such applications. Potential energy curves are computed for the dissociation of Li(+)-Be, CH(3)-CF(3), and hydrogen-bonded water clusters, and e-DFT results obtained using the EE method are compared with those obtained using approximate kinetic energy functionals. In all cases, the EE method preserves excellent agreement with reference Kohn-Sham calculations, whereas the approximate functionals lead to qualitative failures in the calculated energies and equilibrium structures. We also demonstrate an accurate pairwise approximation to the NAKP that allows for efficient parallelization of the EE method in large systems; benchmark calculations on molecular crystals reveal ideal, size-independent scaling of wall-clock time with increasing system size.

Exact nonadditive kinetic potentials for embedded density functional theory

We describe an embedded density functional theory (DFT) protocol in which the nonadditive kinetic energy component of the embedding potential is treated exactly. At each iteration of the Kohn-Sham equations for constrained electron density, the Zhao-Morrison-Parr constrained search method for constructing Kohn-Sham orbitals is combined with the King-Handy expression for the exact kinetic potential. We use this formally exact embedding protocol to calculate ionization energies for a series of three- and four-electron atomic systems, and the results are compared to embedded DFT calculations that utilize the Thomas-Fermi (TF) and the Thomas-Fermi-von Weisacker approximations to the kinetic energy functional. These calculations illustrate the expected breakdown due to the TF approximation for the nonadditive kinetic potential, with errors of 30%-80% in the calculated ionization energies; by contrast, the exact protocol is found to be accurate and stable. To significantly improve the convergence of the new protocol, we introduce a density-based switching function to map between the exact nonadditive kinetic potential and the TF approximation in the region of the nuclear cusp, and we demonstrate that this approximation has little effect on the accuracy of the calculated ionization energies. Finally, we describe possible extensions of the exact protocol to perform accurate embedded DFT calculations in large systems with strongly overlapping subsystem densities.