Approximate Kinetic Energy Functionals Research Articles

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.

Partially deorbitalized meta-GGA

Mejia-Rodriguez and Trickey recently proposed a procedure for removing the explicit dependence of meta-GGA exchange-correlation energy functionals Exc on the kinetic energy density τ. We present a simple modification to this approach in which the exact Kohn-Sham τ is used as input for Exc but the functional derivative of τ with respect to the density ρ, required to calculate the potential term ∫d3r′δExc∕δτ(r′)∣ρ⋅δτ(r′)∕δρ(r), is evaluated using an approximate kinetic energy density functional. This ‘half-way’ strategy ensures that the Kohn-Sham potential is a local multiplicative function (as opposed to the non-local potential of a generalized Kohn-Sham approach) while preserving the inherent non-locality of the functional itself. Electronic structure codes can be easily modified to use the new method. We validate it by quantifying the accuracy of the predicted lattice parameters, bulk moduli, magnetic moments and cohesive energies of a large set of periodic solids. An unanticipated benefit of this method is to gauge the quality of approximate kinetic energy functionals by checking if the self-consistent solution is indeed at the variational minimum.

Nuclear cusps and singularities in the nonadditive kinetic potential bifunctional from analytical inversion

The non-additive kinetic potential $v^{\text{NAD}}$ is a key quantity in density-functional theory (DFT) embedding methods, such as frozen density embedding theory and partition DFT. $v^{\text{NAD}}$ is a bi-functional of electron densities $\rho_{\rm B}$ and $\rho_{\rm tot} = \rho_{\rm A} + \rho_{\rm B}$. It can be evaluated using approximate kinetic-energy functionals, but accurate approximations are challenging. The behavior of $v^{\text{NAD}}$ in the vicinity of the nuclei has long been questioned, and singularities were seen in some approximate calculations. In this article, the existence of singularities in $v^{\text{NAD}}$ is analyzed analytically for various choices of $\rho_{\rm B}$ and $\rho_{\rm tot}$, using the nuclear cusp conditions for the density and Kohn-Sham potential. It is shown that no singularities arise from smoothly partitioned ground-state Kohn-Sham densities. We confirm this result by numerical calculations on diatomic test systems HeHe, HeLi$^+$, and H$_2$, using analytical inversion to obtain a numerically exact $v^{\rm NAD}$ for the local density approximation. We examine features of $v^{\rm NAD}$ which can be used for development and testing of approximations to $v^{\rm NAD}[\rho_{\rm B},\rho_{\rm tot}]$ and kinetic-energy functionals.

Adaptive Subsystem Density Functional Theory.

Subsystem density functional theory (DFT) is emerging as a powerful electronic structure method for large-scale simulations of molecular condensed phases and interfaces. Key to its computational efficiency is the use of approximate nonadditive noninteracting kinetic energy functionals. Unfortunately, currently available nonadditive functionals lead to inaccurate results when the subsystems interact strongly such as when they engage in chemical reactions. This work disrupts the status quo by devising a workflow that extends subsystem DFT's applicability also to strongly interacting subsystems. This is achieved by implementing a fully automated adaptive definition of subsystems which is realized during geometry optimizations or ab initio molecular dynamics simulations. The new method prescribes subsystem merging and splitting events redistributing the resources (both for work and data) in an efficient way making use of modern parallelization strategies and object-oriented programming. We showcase the method with examples probing from moderate-to-strong inter-subsystem interactions, opening the door to using subsystem DFT for modeling chemical reactions in molecular condensed phases with a black box computational tool.

Robust All-Electron Optimization in Orbital-Free Density-Functional Theory Using the Trust-Region Image Method.

We present a Gaussian-basis implementation of orbital-free density-functional theory (OF-DFT) in which the trust-region image method (TRIM) is used for optimization. This second-order optimization scheme has been constructed to provide benchmark all-electron results with very tight convergence of the particle-number constraint, associated chemical potential, and electron density. It is demonstrated that, by preserving the saddle-point nature of the optimization and simultaneously optimizing the density and chemical potential, an order of magnitude reduction in the number of iterations required for convergence is obtained. The approach is compared and contrasted with a new implementation of the nested optimization scheme put forward by Chan, Cohen, and Handy. Our implementation allows for semilocal kinetic-energy (and exchange-correlation) functionals to be handled self-consistently in all-electron calculations. The all-electron Gaussian-basis setting for these calculations will enable direct comparison with a wide range of standard high-accuracy quantum-chemical methods as well as with Kohn-Sham density-functional theory. We expect that the present implementation will provide a useful tool for analyzing the performance of approximate kinetic-energy functionals in finite systems.

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.

Accurate Reference Data for the Nonadditive, Noninteracting Kinetic Energy in Covalent Bonds.

The nonadditive, noninteracting kinetic energy (NAKE) is calculated numerically for fragments of H2, Li2, Be2, C2, N2, F2, and Na2 within partition density functional theory (PDFT). The resulting fragments are uniquely determined, and their sum reproduces the Kohn-Sham molecular density of the corresponding XC functional. We present the NAKE of these unique fragments as a function of internuclear separation and compare the use of fractional orbital occupation to the usual PDFT ensemble method for treating the fragment energies and densities. We also compare Thomas-Fermi and von Weizsäcker approximate kinetic energy functionals to the numerically exact solutions and find significant regions where the von Weizsäcker functional is nearly exact. In addition, we find that the von Weizsäcker approximation can provide accurate NAKE in stretched covalent bonds, especially in the cases of Li2 and Na2.

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.

Excitation energies from frozen-density embedding with accurate embedding potentials.

We present calculations of excitation energies within the time-dependent density functional theory (TDDFT) extension of frozen-density embedding (FDE) using reconstructed accurate embedding potentials. Previous applications of FDE showed significant deviations from supermolecular calculations; our current approach eliminates one potential error source and yields excitation energies of generally much better agreement with Kohn-Sham-TDDFT. Our results demonstrate that the embedding potentials represent the main error source in FDE-TDDFT calculations using standard approximate kinetic-energy functionals for excitations localized within one subsystem.

Accurate basis set truncation for wavefunction embedding

Density functional theory (DFT) provides a formally exact framework for performing embedded subsystem electronic structure calculations, including DFT-in-DFT and wavefunction theory-in-DFT descriptions. In the interest of efficiency, it is desirable to truncate the atomic orbital basis set in which the subsystem calculation is performed, thus avoiding high-order scaling with respect to the size of the MO virtual space. In this study, we extend a recently introduced projection-based embedding method [F. R. Manby, M. Stella, J. D. Goodpaster, and T. F. Miller III, J. Chem. Theory Comput. 8, 2564 (2012)] to allow for the systematic and accurate truncation of the embedded subsystem basis set. The approach is applied to both covalently and non-covalently bound test cases, including water clusters and polypeptide chains, and it is demonstrated that errors associated with basis set truncation are controllable to well within chemical accuracy. Furthermore, we show that this approach allows for switching between accurate projection-based embedding and DFT embedding with approximate kinetic energy (KE) functionals; in this sense, the approach provides a means of systematically improving upon the use of approximate KE functionals in DFT embedding.

Erratum: Properties of constraint-based single-point approximate kinetic energy functionals [Phys. Rev. B80, 245120 (2009)

Erratum: Properties of constraint-based single-point approximate kinetic energy functionals [Phys. Rev. B<b>80</b>, 245120 (2009)

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.

KINETIC ENERGY FUNCTIONALS: EXACT ONES FROM ANALYTIC MODEL WAVE FUNCTIONS AND APPROXIMATE ONES IN ORBITAL-FREE MOLECULAR DYNAMICS

We consider the problem of constructing kinetic energy functionals in density functional theory. We first discuss the functional generated through the application of local-scaling transformations to the exact analytic wavefunctions for the Hookean model of 4He (a finite mass three-particle system), and contrast this result with a previous one for ∞He (infinite mass system). It is shown that an exact non-Born-Oppenheimer treatment not only leads to mass-correction terms in the kinetic energy, but to a basically different functional expression. In addition, we report and comment on some recently advanced approximate kinetic energy functionals generated in the context of the constraint-based approach to orbital-free molecular dynamics. The positivity and non-singularity of a new family of kinetic energy functionals specifically designed for orbital-free molecular dynamics is discussed. Finally, we present some conclusions.

The weak covalent bond in NgAuF (Ng=Ar, Kr, Xe): A challenge for subsystem density functional theory

We have assessed the accuracy of a representative set of currently available approximate kinetic-energy functionals used within the frozen-density embedding scheme for the NgAuF (Ng=Ar, Kr, Xe) molecules, which we partitioned into a Ng and a AuF subsystem. Although it is weak, there is a covalent interaction between these subsystems which represents a challenge for this subsystem density functional theory approach. We analyzed the effective-embedding potentials and resulting electron density distributions and provide a quantitative analysis of the latter from dipole moment differences and root-mean-square errors in the density with respect to the supermolecular Kohn-Sham density functional theory reference calculation. Our results lead to the conclusion that none of the tested approximate kinetic-energy functionals performs well enough to describe the bond between the noble gas and gold adequately. This observation contributes to the growing evidence that the current procedure to obtain approximate kinetic-energy functionals by reparametrizing functionals obtained via the "conjointness" hypothesis of Lee, Lee, and Parr [Phys. Rev. A 44, 768 (1991)] is insufficient to treat metal-ligand interactions with covalent character.

Properties of constraint-based single-point approximate kinetic energy functionals

We present a substantial extension of our constraint-based approach for development of orbital-free (OF) kinetic-energy (KE) density functionals intended for the calculation of quantum-mechanical forces in multi-scale molecular dynamics simulations. Suitability for realistic system simulations requires that the OF-KE functional yield accurate forces on the nuclei yet be relatively simple. We therefore require that the functionals be based on DFT constraints, local, dependent upon a small number of parameters fitted to a training set of limited size, and applicable beyond the scope of the training set. Our previous "modified conjoint" generalized-gradient-type functionals were constrained to producing a positive-definite Pauli potential. Though distinctly better than several published GGA-type functionals in that they gave semi-quantitative agreement with Born-Oppenheimer forces from full Kohn-Sham results, those modified conjoint functionals suffer from unphysical singularities at the nuclei. Here we show how to remove such singularities by introducing higher-order density derivatives. We give a simple illustration of such a functional used for the dissociation energy as a function of bond length for selected molecules.

Condition on the Kohn–Sham kinetic energy and modern parametrization of the Thomas–Fermi density

We study the asymptotic expansion of the neutral-atom energy as the atomic number Z-->infinity, presenting a new method to extract the coefficients from oscillating numerical data. Recovery of the correct expansion yields a condition on the Kohn-Sham kinetic energy that is important for the accuracy of approximate kinetic energy functionals for atoms, molecules, and solids. For example, this determines the small gradient limit of any generalized gradient approximation and conflicts somewhat with the standard gradient expansion. Tests are performed on atoms, molecules, and jellium clusters using densities constructed from Kohn-Sham orbitals. We also give a modern, highly accurate parametrization of the Thomas-Fermi density of neutral atoms.

Generalized density functional theories using the k-electron densities: Development of kinetic energy functionals

Several explicit formulas for the kinetic energy of a many-electron system as a functional of the k-electron density are derived, with emphasis on the electron pair density. The emphasis is on general techniques for deriving approximate kinetic energy functionals and features generalized Weisacker bounds and methods using density-matrix reconstruction. Adapting results from statistical mechanics, a hierarchy of equations is derived that links electron pairs, triplets, quadruplets, etc.; this may be used to derive more accurate approximations. Several methods for defining the exact kinetic energy functional are presented, including the generalizations of the Levy and Lieb formulations of density-functional theory. Together with N-representability constraints on the k-density, this paper provides the basis for “generalized density functional theories” based on the electron pair density. There are also implications for conventional density-functional theory, notably regarding the development of more accurate density functionals for the kinetic energy.

Density-functional embedding using a plane-wave basis

The constrained electron density method of embedding a Kohn-Sham system in a substrate system (first described by P. Cortona, Phys. Rev. B {\bf 44}, 8454 (1991) and T.A. Wesolowski and A. Warshel, J. Phys. Chem {\bf 97}, 8050 (1993)) is applied with a plane-wave basis and both local and non-local pseudopotentials. This method divides the electron density of the system into substrate and embedded electron densities, the sum of which is the electron density of the system of interest. Coupling between the substrate and embedded systems is achieved via approximate kinetic energy functionals. Bulk aluminium is examined as a test case for which there is a strong interaction between the substrate and embedded systems. A number of approximations to the kinetic-energy functional, both semi-local and non-local, are investigated. It is found that Kohn-Sham results can be well reproduced using a non-local kinetic energy functional, with the total energy accurate to better than 0.1 eV per atom and good agreement between the electron densities.

Density functional theory with an approximate kinetic energy functional applied to study structure and stability of weak van der Waals complexes

In view of further application to the study of molecular and atomic sticking on dust particles, we investigated the capability of the “freeze-and-thaw” cycle of the Kohn–Sham equations with constrained electron density (KSCED) to describe potential energy surfaces of weak van der Waals complexes. We report the results obtained for C6H6⋯X (X=O2, N2, and CO) as test cases. In the KSCED formalism, the exchange-correlation functional is defined as in the Kohn–Sham approach whereas the kinetic energy of the molecular complex is expressed differently, using both the analytic expressions for the kinetic energy of individual fragments and the explicit functional of electron density to approximate nonadditive contributions. As the analytical form of the kinetic energy functional is not known, the approach relies on approximations. Therefore, the applied implementation of KSCED requires the use of an approximate kinetic energy functional in addition to the approximate exchange-correlation functional in calculations following the Kohn–Sham formalism. Several approximate kinetic energy functionals derived using a general form by Lee, Lee, and Parr [Lee et al., Phys. Rev. A. 44, 768 (1991)] were considered. The functionals of this type are related to the approximate exchange energy functionals and it is possible to derive a kinetic energy functional from an exchange energy functional without the use of any additional parameters. The KSCED interaction energies obtained using the PW91 [Perdew and Wang, in Electronic Structure of Solids ’91, edited by P. Ziesche and H. Eschrig (Academie Verlag, Berlin, 1991), p. 11] exchange-correlation functional and the kinetic energy functional derived from the PW91 exchange functional agree very well with the available experimental results. Other considered functionals lead to worse results. Compared to the supermolecule Kohn–Sham interaction energies, the ones derived from the KSCED calculations depend less on the choice of the approximate functionals used. The presented KSCED results together with the previous Kohn–Sham ones [Wesołowski et al., J. Phys. Chem. A 101, 7818 (1997)] support the use of the PW91 functional for studies of weakly bound systems of our interest.

Approximate kinetic-energy functionals

An approximate non-local independent fermion kinetic-energy density functional is constructed from a model for the density functional exchange hole. The functional is exact for one- and two-electron systems and gives a good approximation for a homogeneous electron gas. Reasonable results are also obtained for model systems.