Adam Wasserman

Density Functional Partition Theory with Fractional Occupations

Partition theory (PT) is a formally exact methodology for calculating the density of any molecule or solid via separate calculations on individual fragments. Just as Kohn-Sham density functional theory (DFT) introduces noninteracting fermions in an effective potential that is defined to yield the exact density of the interacting problem, in PT a global effective potential is found that ensures that the sum of the fragment densities is that of the full system. By combining the two, density functional partition theory (DFPT) produces a DFT scheme that yields the (in principle) exact molecular density and energy via Kohn-Sham calculations on fragments. We give the full formalism and illustrate DFPT in the general case of noninteger fragment occupations.

Molecular binding energies from partition density functional theory.

Approximate molecular calculations via standard Kohn-Sham density functional theory are exactly reproduced by performing self-consistent calculations on isolated fragments via partition density functional theory [P. Elliott, K. Burke, M. H. Cohen, and A. Wasserman, Phys. Rev. A 82, 024501 (2010)]. We illustrate this with the binding curves of small diatomic molecules. We find that partition energies are in all cases qualitatively similar and numerically close to actual binding energies. We discuss qualitative features of the associated partition potentials.

Rydberg Transition Frequencies from the Local Density Approximation

A method is given that extracts accurate Rydberg excitations from density functional calculations in the local density approximation, despite the short-ranged potential. For the case of He and Ne, the asymptotic quantum defects predicted by the calculations are in less than 5% error, yielding transition frequency errors of less than 0.1 eV.

Density-based partitioning methods for ground-state molecular calculations.

With the growing complexity of systems that can be treated with modern electronic-structure methods, it is critical to develop accurate and efficient strategies to partition the systems into smaller, more tractable fragments. We review some of the various recent formalisms that have been proposed to achieve this goal using fragment (ground-state) electron densities as the main variables, with an emphasis on partition density-functional theory (PDFT), which the authors have been developing. To expose the subtle but important differences between alternative approaches and to highlight the challenges involved with density partitioning, we focus on the simplest possible systems where the various methods can be transparently compared. We provide benchmark PDFT calculations on homonuclear diatomic molecules and analyze the associated partition potentials. We derive a new exact condition determining the strength of the singularities of the partition potentials at the nuclei, establish the connection between charge-transfer and electronegativity equalization between fragments, test different ways of dealing with fractional fragment charges and spins, and finally outline a general strategy for overcoming delocalization and static-correlation errors in density-functional calculations.

Fragment-based time-dependent density functional theory.

Using the Runge-Gross theorem that establishes the foundation of time-dependent density functional theory, we prove that for a given electronic Hamiltonian, choice of initial state, and choice of fragmentation, there is a unique single-particle potential (dubbed time-dependent partition potential) which, when added to each of the preselected fragment potentials, forces the fragment densities to evolve in such a way that their sum equals the exact molecular density at all times. This uniqueness theorem suggests new ways of computing the time-dependent properties of electronic systems via fragment-time-dependent density functional theory calculations. We derive a formally exact relationship between the partition potential and the total density, and illustrate our approach on a simple model system for binary fragmentation in a laser field.

The Importance of Being Inconsistent

We review the role of self-consistency in density functional theory (DFT). We apply a recent analysis to both Kohn-Sham and orbital-free DFT, as well as to partition DFT, which generalizes all aspects of standard DFT. In each case, the analysis distinguishes between errors in approximate functionals versus errors in the self-consistent density. This yields insights into the origins of many errors in DFT calculations, especially those often attributed to self-interaction or delocalization error. In many classes of problems, errors can be substantially reduced by using better densities. We review the history of these approaches, discuss many of their applications, and give simple pedagogical examples.

Resonance Lifetimes from Complex Densities

The ab initio calculation of resonance lifetimes of metastable anions challenges modern quantum chemical methods. The exact lifetime of the lowest-energy resonance is encoded into a complex “density” that can be obtained via complex-coordinate scaling. We illustrate this with one-electron examples and show how the lifetime can be extracted from the complex density in much the same way as the ground-state energy of bound systems is extracted from its ground-state density.

Time-Dependent Density Functional Calculation of e-H Scattering

Phase shifts for single-channel elastic electron-atom scattering are derived from time-dependent density functional theory. The H- ion is placed in a spherical box, its discrete spectrum found, and phase shifts deduced. Exact exchange yields an excellent approximation to the ground-state Kohn-Sham potential, while the adiabatic local density approximation yields good singlet and triplet phase shifts.

Continuum states from time-dependent density functional theory

Linear response time-dependent density functional theory is used to study low-lying electronic continuum states of targets that can bind an extra electron. Exact formulas to extract scattering amplitudes from the susceptibility are derived in one dimension. A single-pole approximation for scattering phase shifts in three dimensions is shown to be more accurate than static exchange for singlet electron-He(+) scattering.

Transferability of Atomic Properties in Molecular Partitioning: A Comparison

For a given choice of fragmentation of a molecule, Partition Density Functional Theory (PDFT) provides fragment densities that add up to the correct molecular density, and produce the-in principle exact-molecular energy. Using a simple model system of a heteronuclear diatomic molecule, we investigate the transferability of the resulting PDFT fragment densities by examining how their shapes and dipoles are preserved as the environment changes, and compare with other partitioning schemes. Our results show that (1) the transferability of PDFT densities is about an order of magnitude higher than that of real-space partitioning schemes, and (2) the PDFT dipoles are about an order of magnitude more transferable than Hirshfeld dipoles in regions of chemical relevance.