Frozen density embedding with non-integer subsystems' particle numbers

Split electrons in partition density functional theory

Partition density functional theory is a density embedding method that partitions a molecule into fragments by minimizing the sum of fragment energies subject to a local density constraint and a global electron-number constraint. To perform this minimization, we study a two-stage procedure in which the sum of fragment energies is lowered when electrons flow from fragments of lower electronegativity to fragments of higher electronegativity. The global minimum is reached when all electronegativities are equal. The non-integer fragment populations are dealt with in two different ways: (1) An ensemble approach (ENS) that involves averaging over calculations with different numbers of electrons (always integers) and (2) a simpler approach that involves fractionally occupying orbitals (FOO). We compare and contrast these two approaches and examine their performance in some of the simplest systems where one can transparently apply both, including simple models of heteronuclear diatomic molecules and actual diatomic molecules with two and four electrons. We find that, although both ENS and FOO methods lead to the same total energy and density, the ENS fragment densities are less distorted than those of FOO when compared to their isolated counterparts, and they tend to retain integer numbers of electrons. We establish the conditions under which the ENS populations can become fractional and observe that, even in those cases, the total charge transferred is always lower in ENS than in FOO. Similarly, the FOO fragment dipole moments provide an upper bound to the ENS dipoles. We explain why and discuss the implications.

Description of intermolecular charge transfer with subsystem density-functional theory

Efficient quantum-chemical methods that are able to describe intermolecular charge transfer are crucial for modeling organic semiconductors. However, the correct description of intermolecular charge transfer with density-functional theory (DFT) is hampered by the fractional charge error of approximate exchange-correlation (xc) functionals. Here, we investigate the charge transfer induced by an external electric field in a tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) complex as a test case. For this seemingly simple model system, a supermolecular DFT treatment fails with most conventional xc functionals. Here, we present an extension of subsystem DFT to subsystems with a fractional number of electrons. We show that within such a framework, it becomes possible to overcome the fractional charge error by enforcing the correct dependence of each subsystem's total energy on the subsystem's fractional charge. Such a subsystem DFT approach allows for a correct description of the intermolecular charge transfer in the TTF-TCNQ model complex. The approach presented here can be generalized to larger molecular aggregates and will thus allow for modeling organic semiconductor materials accurately and efficiently.

Hartree and Exchange in Ensemble Density Functional Theory: Avoiding the Nonuniqueness Disaster

Ensemble density functional theory is a promising method for the efficient and accurate calculation of excitations of quantum systems, at least if useful functionals can be developed to broaden its domain of practical applicability. Here, we introduce a guaranteed single-valued "Hartree-exchange" ensemble density functional, E_{Hx}[n], in terms of the right derivative of the universal ensemble density functional with respect to the coupling constant at vanishing interaction. We show that E_{Hx}[n] is straightforwardly expressible using block eigenvalues of a simple matrix [Eq. (14)]. Specialized expressions for E_{Hx}[n] from the literature, including those involving superpositions of Slater determinants, can now be regarded as originating from the unifying picture presented here. We thus establish a clear and practical description for Hartree and exchange in ensemble systems.