Nav: Home

Localized orbital scaling correction functional ushering DFT to a new level of accuracy

September 13, 2017

Kohn-Sham density functional theory has been one of the most successful theories applied to chemistry. It is formally rigorous; its relatively low computational cost plus its competitive accuracy in small and medium-sized systems make it one of the most popular methods in electronic structure calculations, and perhaps the only choice for modeling quantum effects of electrons in large chemical and biological systems. Nevertheless, the exact functional is not explicitly known and the state-of-the-art density functional approximations (DFAs) suffer from systematic errors. One of the dominant errors in DFAs is the delocalization error, which is ubiquitous and manifested in various ways. This has been a longstanding and challenging open problem for decades. Recently, Chen Li, Neil Qiang Su and Weitao Yang from Duke University and Xiao Zheng from the University of Science and Technology of China made exciting progress and developed a novel localized orbital scaling correction (LOSC) framework, which demonstrates systematic elimination of the delocalization error.

The major consequences of delocalization error in DFAs can be categorized in two aspects: (1) Kohn-Sham (KS) orbital energy error and (2) total energy error. The delocalization error leads to important failures in practical applications such as the unphysical underestimation of band gaps, the excessively delocalized electron distributions and wrong charge transfers. Regarding these two aspects of errors, both are manifested in a size-dependent manner. In particular, among the KS orbital energies, the highest occupied molecular orbital (HOMO) energy, εHOMO, is supposed to agree with the (negative) vertical ionization potential (-Ive), as required by the exact functional. Furthermore, the Ive as calculated by DFAs are supposed to agree with the experimental value, Iexp.

For commonly used DFAs, the above mentioned two conditions are not satisfied. As shown by the helium cluster calculations in the subfigure (a), the HOMO energy by the local density approximation (LDA) shows a positive error compared to -Ive while Ive exhibits a negative error, and the two errors are displayed in a size-dependent manner, suggesting that the delocalization error of DFAs has to show up in one way or the other, or both, while the sum of the magnitudes remains unchanged. Moreover, this is prevalent in all kinds of DFAs, including the most popular B3LYP functional, as shown in green.

To understand these errors, one can map them into the errors of a fractionally charged He atom, as illustrated in the subfigure (b). The fractional system energies are supposed to scale linearly with the fractional electron number n (0?n<1) for the exact functional. Shown in subfigure (b) is the energy deviation from the linearity condition for a single He atom, where the fractional system energies are underestimated- this defines the delocalization error. Moreover, the fractional charge error in (b) has 1-1 correspondence with the total energy error ΔI = Ive - Iexp. In the limit of M goes to infinity, one can deduce that ΔI agrees with the slope error of the E(N) curve at the integer, which is exactly the HOMO energy error of a single He atom. Therefore, all the errors are inter-connected. It is thus possible to eliminate delocalization error systematically through (1) removing the errors in (b) and (2) guaranteeing size-consistent removal of the error in all systems.

In the present work of LOSC, the authors invented novel local variables, called orbitallets, which are localized orbitals (LOs) that achieve locality in both spatial and energy spaces. Through these orbitallets, they were able to express the electron density and KS density matrix in the local representation, where the local occupation matrix composed of fractional numbers arises naturally. Therefore, these local variables are capable of capturing local fractional information precisely and completely in the orbital space, and serve as the perfect building blocks for the LOSC functional. Furthermore, by rewriting the terms in the GSC and LSC functionals in terms of curvature correction formulas and making analogy to the LOSC, they introduced a local curvature matrix, whose elements are designed as a functional of the orbitallets, to be in one-to-one correspondence with the local occupation matrix. Finally, the LOSC formula is written as an explicit and elegant expression over the parent functional in terms of these two matrices.

The LOSC achieves all the desirable features. In the above figure, the LOSC-LDA errors are essentially negligibly small, indicating the validity and size-consistency of the correction. These have also been validated by the much improved dissociation curves of diatomic molecular cations, ranging from H2+, He2+, water dimer cation and benzene dimer cation. The HOMO, LUMO energy errors have been much reduced and the good performance maintains as system size grows. This has been demonstrated by tests on polyacene and trans-polyacetylene oligomers. Furthermore, the LOSC is capable of retrieving the right electron density when the parent DFAs makes it qualitatively wrong, as shown by the example of a solvated chlorine anion when deprived of an electron.

The performance of LOSC thus shows promise of eliminating delocalization error systematically within the KS density functionals. It is worth noticing that unlike traditional paradigm of designing functionals using the density, the density gradients, the kinetic energy density, etc., the LOSC framework involves completely new ingredients, the orbitallets, which themselves are implicit functionals of the KS density matrix, and demonstrated their unique capability of addressing longstanding problems associated with traditional DFAs. This reflects a paradigm change in designing functionals, and greatly broadens the way in the exploration for the exact functional in its own living space. In this sense, LOSC has ushered the trend in developing new generation of density functional approximations, promoting density functional theory to a new level of accuracy.
-end-
See the article: Localized Orbital Scaling Correction for Systematic Elimination of Delocalization Error in Density Functional Approximations
Chen Li, Xiao Zheng, Neil Qiang Su, Weitao Yang
Natl Sci Rev, 2017, doi: 10.1093/nsr/nwx111
https://doi.org/10.1093/nsr/nwx111

Science China Press

Related Energy Articles:

Wave energy researchers dive deep to advance clean energy source
One of the biggest untapped clean energy sources on the planet -- wave energy -- could one day power millions of homes across the US.
A new energy source within the cells
Scientists at the Centre for Genomic Regulation in Barcelona, Spain, find evidence of a new energy source within cell nucleus.
MIT Energy Initiative welcomes Exelon as member for clean energy research
MIT Energy Initiative announces that national energy provider Exelon joins MITEI as a member to focus research support through MITEI's Low-Carbon Energy Centers.
Clean energy from water
Fuel cells generate electrical energy through a chemical reaction of hydrogen and oxygen.
Determinant factors for energy consumption and perception of energy conservation clarified
Change in lifestyle is a key component to realizing a low-carbon society.
Lactate for brain energy
Nerve cells cover their high energy demand with glucose and lactate.
Evidence shows low energy sweeteners help reduce energy intake and body weight
Use of low energy sweeteners (LES) in place of sugar, in children and adults, leads to reduced calorie intake and body weight - and possibly also when comparing LES beverages to water -- according to a review led by researchers at the University of Bristol published in the International Journal of Obesity today.
ASU professor honored for work on energy and social aspects of energy policy
Martin 'Mike' Pasqualetti, an Arizona State University professor and an expert on energy and social components of energy development, will be awarded 2015 Alexander and Ilse Melamid Memorial Medal by the American Geographical Society.
Stanford's Global Climate and Energy Project awards $9.3 million for energy research
GCEP has awarded scientists at Stanford and four other universities funding to develop a suite of promising energy technologies.
Energy efficiency upgrades ease strain of high energy bills in low-income families
Low-income families bear the brunt of high-energy costs and poor thermal comfort from poorly maintained apartment buildings.

Related Energy Reading:

Energy: Heat, Light, and Fuel (Amazing Science)
by Darlene R. Stille (Author), Sheree Boyd (Illustrator)

The forms of energy are discovered, including chemical energy, kinetic energy, and solar energy. View Details


The Energy Bus: 10 Rules to Fuel Your Life, Work, and Team with Positive Energy
by Jon Gordon (Author), Ken Blanchard (Foreword)

The Energy Bus, an international best seller by Jon Gordon, takes readers on an enlightening and inspiring ride that reveals 10 secrets for approaching life and work with the kind of positive, forward thinking that leads to true accomplishment - at work and at home. Jon infuses this engaging story with keen insights as he provides a powerful roadmap to overcome adversity and bring out the best in yourself and your team. When you get on The Energy Bus you’ll enjoy the ride of your life! View Details


Energy Medicine: Balancing Your Body's Energies for Optimal Health, Joy, and Vitality
by Donna Eden (Author), David Feinstein (Author), Caroline Myss (Foreword)

In this updated and expanded edition of her alternative-health classic, Eden shows readers how they can understand their body's energy systems to promote healing. View Details


Energy: A Beginner's Guide (Beginner's Guides)
by Vaclav Smil (Author)

With his famous equation, E=mc2, Einstein showed us that all matter can be described as energy. It is everywhere; it is everything. In this engaging introduction, renowned author and scientist Vaclav Smil covers a wide range of topics, from the inner workings of the human body to the race for more efficient and eco-friendly fuels. Shedding light on the science behind global warming and efforts to prevent it, and how our daily decisions affect energy consumption, this book highlights the importance of energy in both past and present societies. Whether you're after insight or dinner table... View Details


The Path of Energy: Awaken Your Personal Power and Expand Your Consciousness
by Synthia Andrews (Author), Barbara Marciniak (Foreword)

Learning to consciously interact in the domain of subtle energy is the next step in personal and planetary transformation. The Path of Energy is a unique handbook of principles, practices and exercises to help you access your energy awareness and live a more empowered life. In the book are 13 meditations that activate patterns of light within your energy body to awaken specific abilities and levels of awareness. Each meditation includes benefits and purpose, step-by-step written directions augmented with line drawings, and an interpretive illustration of the completed energy pattern.... View Details


Energy and Civilization: A History (MIT Press)
by Vaclav Smil (Author)

A comprehensive account of how energy has shaped society throughout history, from pre-agricultural foraging societies through today's fossil fuel--driven civilization.

Energy is the only universal currency; it is necessary for getting anything done. The conversion of energy on Earth ranges from terra-forming forces of plate tectonics to cumulative erosive effects of raindrops. Life on Earth depends on the photosynthetic conversion of solar energy into plant biomass. Humans have come to rely on many more energy flows -- ranging from fossil fuels to photovoltaic generation of... View Details


Time, Talent, Energy: Overcome Organizational Drag and Unleash Your Team’s Productive Power
by Michael C. Mankins (Author), Eric Garton (Author)

Managing Your Scarcest Resources

Business leaders know that the key to competitive success is smart management of scarce resources. That?s why companies allocate their financial capital so carefully. But capital today is cheap and abundant, no longer a source of advantage. The truly scarce resources now are the time, the talent, and the energy of the people in your organization?resources that are too often squandered. There?s plenty of advice about how to manage them, but most of it focuses on individual actions. What?s really needed are organizational solutions that... View Details


The All-Day Energy Diet: Double Your Energy in 7 Days
by Yuri Elkaim (Author)

      With The All-Day Energy Diet you will reclaim your health and vitality, feel alive and youthful again, and accomplish all of your fitness goals. Follow it to the letter, or choose your favorite energy pick-me-ups—either way, you’ll look and feel great in just seven days’ time. Yuri knows, because he has experienced it in his own life and seen it happen with thousands of clients. It doesn’t matter whether you’re the unhealthiest, most lethargic person in the world or a super-healthy fitness buff. If you’re a human from planet Earth, this can work for you, too.View Details


Energy: What Everyone Needs to Know®
by Jose Goldemberg (Author)

Without a doubt, the topic of energy--from coal, oil, and nuclear to geothermal, solar and wind--is one of the most pressing across the globe. It is of paramount importance to policy makers, economists, environmentalists, and industry as they consider which technologies to invest in, how to promote use of renewable energy sources, and how to plan for dwindling reserves of non-renewable energy.

In Energy: What Everyone Needs to Know®, José Goldemberg, a nuclear physicist who has been hailed by Time magazine as one of the world's top "leaders and visionaries on the... View Details


The Way of Energy: Mastering the Chinese Art of Internal Strength with Chi Kung Exercise (A Gaia Original)
by Master Lam Kam-Chuen (Author)

For centuries, the Chinese exercise system called Chi Kung has been shrouded in secrecy. The Way of Energy is the first comprehensive guide to the most powerful form of Chi Kung ever developed—“standing like a tree”.

A gentle, yet profoundly beneficial form of exercise, which requires very little movement, this form of Chi Kung unlocks deep reserves of energy builds internal stamina, strengthens immunity, relieves chronic illness, and promotes the natural regeneration of the nervous system.

Using step-by-step instructions and more than 100 drawings and full-color... View Details

Best Science Podcasts 2017

We have hand picked the best science podcasts for 2017. Sit back and enjoy new science podcasts updated daily from your favorite science news services and scientists.
Now Playing: TED Radio Hour

Simple Solutions
Sometimes, the best solutions to complex problems are simple. But simple doesn't always mean easy. This hour, TED speakers describe the innovation and hard work that goes into achieving simplicity. Guests include designer Mileha Soneji, chef Sam Kass, sleep researcher Wendy Troxel, public health advocate Myriam Sidibe, and engineer Amos Winter.
Now Playing: Science for the People

#448 Pavlov (Rebroadcast)
This week, we're learning about the life and work of a groundbreaking physiologist whose work on learning and instinct is familiar worldwide, and almost universally misunderstood. We'll spend the hour with Daniel Todes, Ph.D, Professor of History of Medicine at The Johns Hopkins University, discussing his book "Ivan Pavlov: A Russian Life in Science."