Formalism

Orbital-dependent density functionals

Optimally-tuned range-separated hybrid functionals

Ensemble-generalized functionals

 

 

Orbital-dependent density functionals

First principles electronic structure calculations, based only on the periodic table and the laws of quantum mechanics, have made large strides in recent decades and have become the foundation for the understanding of a huge variety of physical and chemical systems.

Much of this progress has been due to density functional theory (DFT), which has emerged as the work-horse approach for real-world materials (as opposed to model systems). DFT is an approach to the many-electron problem in which the electron density, rather than the many-electron wave function, plays a central role. It has become the method of choice for electronic structure calculations across an unusually wide variety of fields, from organic chemistry to condensed matter physics. There are two main reasons for the spectacular success of DFT:

  • First and foremost, DFT offers the only currently known practical way for first principles calculations of systems with many thousands of electrons.
  • Second, it enhances our understanding by relying on relatively simple, accessible quantities that are easily visualized even for very large systems.

DFT has progressed from a formal approach to a practical one by virtue of the Kohn-Sham equations. These constitute a mapping of the original N-electron Schrödinger equation into an effective set of N one-electron Schrödinger-like equations, where all non-classical electron interactions (i.e., exchange and correlation) are subsumed into an additive one-electron potential, known as the exchange-correlation potential. The latter is the functional derivative of the exchange-correlation energy, which is a functional of the density. This mapping is exact in principle, but always approximate in practice. Progress therefore hinges critically on our ability to obtain more accurate approximations for exchange-correlation functionals that are applicable across a wide range of systems.

We believe that research into orbital-dependent density functionals is one of the most promising arenas in modern density functional theory. In such functionals, the exchange-correlation energy is expressed explicitly in terms of Kohn-Sham orbitals and is only an implicit functional of the density. This allows maximal freedom in functional construction and offers a real hope for alleviating some of the most serious difficulties associated with present day treatments of exchange and correlation within DFT. Furthermore, orbital-dependent functionals can be employed fully within the original Kohn-Sham framework, in which case the exchange-correlation potential is derived using the optimized effective potential equation. However, they can be - and usually are - employed using the generalized Kohn-Sham framework, in which case one obtains a non-local potential that corresponds to mapping the original many-electron problem into one of partially interacting electrons. A leading example of those, although not always recognized as such, is the so-called hybrid functionals, where a fraction of exact exchange is “mixed in” with a fraction of explicitly density-dependent exchange. While the Kohn-Sham mapping is unique, there are many generalized Kohn-Sham maps. This additional flexibility allows one to choose the best mapping for a given task.

Our group is actively engaged in constructing, testing, benchmarking, and applying to complex systems several important classes of orbital-dependent functionals.

  • For a comprehensive review article on the topic, see:
  • For work extending the scope and examining the properties of generalized Kohn-Sham theory, see:

Optimally-tuned range-separated hybrid functionals

In recent years we have been developing and employing functionals based on the concept of optimal tuning of a range-separated hybrid functional. In this approach, one separates the electron-repulsion into short- and long-range components, treating the short-range so as to achieve a good balance between exchange and correlation, using semi-local approximations (possibly with short-range exact-exchange), but emphasizing exact-exchange in the long-range so as to obtain the correct asymptotic potential. Optimal tuning means that the range-separation parameter (roughly, the cross-over point from short to long range) is an adjustable, system-dependent parameter (rather than a universal one). This parameter is obtained non-empirically based on the satisfaction of physical constraints, typically the ionization potential theorem and related properties. This allowed us to solve several related problems that plagued density functional theory, including the infamous gap problem and the charge-transfer excitation problem, for both molecules and solid-state systems.

Ensemble-generalized functionals

Usually, the reference fictitious electron gas of DFT, into which the original system is mapped, is a pure quantum mechanical state. However, there are important cases where use of a reference ensemble state, i.e., a statistical mixture of pure quantum states, is either desirable or outright necessary. Four scenarios where ensemble reference states arise are: (i) systems with degenerate ground states; (ii) systems possessing a fractional number of electrons; (iii) systems in an excited stationary state; (iv) systems at a finite temperature. Collectively, approaches dealing with any of the four scenarios (and related ones) are known as ensemble DFT (EDFT). Our work in this area focuses on developing a unified framework for the extension of standard approximate density functionals into novel ensemble density functionals, a process which we call “ensemblization”, and to use that to solve problems with which traditional DFT struggles.