Tackling the Long-Standing Energy Gap Problem in Kohn-Sham Density Functional Theory
Over the past two decades, Kohn-Sham density functional theory (KS-DFT) has been one of the most popular theoretical methods for the study of the ground-state properties of large electronic systems. Although the essential ingredient of KS-DFT, the exact exchange-correlation (XC) energy functional Exc[r], has not been known, functionals based on the local density approximation (LDA) and generalized gradient approximations (GGAs), are reasonably accurate for properties governed by short-range XC effects, and are computationally favorable for large systems. Besides, more advanced and accurate functionals have been successively developed to extend the applicability of KS-DFT to a wide variety of systems. Consequently, Prof. Walter Kohn (UC Santa Barbara), one of the main developers of KS-DFT, was awarded the Nobel Prize in Chemistry in 1998. However, as existing density functional methods are approximate, they can exhibit qualitative failures in certain situations. For nanoscale applications, it is tremendously important to resolve these qualitative failures by developing accurate density functionals in KS-DFT and/or new computational methods going beyond KS-DFT, at as low computational cost as possible.
The prediction of the fundamental gap has been an important and challenging topic in KS-DFT. For an atomic or molecular system, the fundamental gap is the difference between the vertical ionization potential and electron affinity. By contrast, the KS gap is defined as the energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the system. The KS gap has been frequently used as a measure of the fundamental gap. However, as there is a considerable difference between the fundamental gap and the KS gap due to the derivative discontinuity (DD) of Exc[r], the KS gap (even with the exact functional) severely underestimates the fundamental gap, leading to the well-known energy gap problem in KS-DFT.
Aiming to resolve this long-standing fundamental problem, a research team led by Dr. Jeng-Da Chai, Associate Professor of the Department of Physics, has recently published a paper entitled “Restoration of the Derivative Discontinuity in Kohn-Sham Density Functional Theory: An Efficient Scheme for Energy Gap Correction” in the prestigious physics journal Physical Review Letters. In the article, the authors proposed a systematic approach from the perspective of perturbation theory to evaluate the DD using the KS orbitals and their energies, wherein the exact DD can be obtained by the infinite-order perturbation scheme. Truncation of the perturbation series at low order yields an efficient scheme for obtaining the approximate DD. While the zeroth-order scheme yields a vanishing DD, the first-order estimate of the DD can be expressed as an explicit universal (i.e., system-independent) functional of the ground-state density and the KS LUMO density, allowing very efficient and accurate calculations of the DD. The fundamental gap can be predicted by adding the estimated DD to the KS gap.