Chemical Concepts from Density Functional Theory

LIU Shubin , ZHANG Xiaojuan

1 Research Computing Center, University of North Carolina, Chapel Hill, NC 27599-3420, USA.

2 Editorial Office of Acta Physico-Chimica Sinica, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, P. R. China.

Emails: shubin@email.unc.edu (L.S.); xjzhang@pku.edu.cn (Z.X.)

1 Introduction

Chemical concepts such as stability, bonding, acidity,basicity, redox potential, hardness, softness, aromaticity,electrophilicity, nucleophilicity, regioselectivity,stereoselectivity, hydrophobicity, hydrophilicity, steric effect,anomeric effect, allosteric effect, synergetic effect, etc., are widely used in chemistry to appreciate structure, bonding, and reactivity properties of electronic systems including molecules,condensed phases and surface processes. Even though modern computational chemistry is well established from the perspective of accuracy and complexity, how to quantify these chemical concepts is a still unresolved task1.

Pinnacled by half of the Nobel Prize in Chemistry in 1998,density functional theory (DFT)1–4has been generally accepted as the most successful development in theoretical and computational chemistry in the past few decades. Nevertheless,for most people, it is merely a computational approach. It falls well behind in providing chemical understanding and conceptual insights. According to the basic theorems of DFT,the electron density of a system should contain adequate information to determine everything in the ground state,including all properties related to stability, bonding, and chemical reactivity. One early example of such efforts was Bader’s atoms-in-molecules (AIM) theory5. Other relevant but more recent works along the line include Becke’s electron localization function (ELF)6,7and the noncovalent interaction(NCI) index by Yang and coworkers8. It is, however, Robert G.Parr of University of North Carolina, who pioneered the field with the establishment of the framework called conceptual density functional theory (CDFT)1–4.

In CDFT, when changes in the total number of electrons and external potential take place during a chemical process,subsequent variations in the total energy of the system can be expressed by a Taylor perturbation expansion, where the first and second order derivatives of the total energy with respect to the total electron number and the external potential have been employed to quantify the chemical concepts such as electronegativity, hardness/softness, Fukui function, response function, etc. Recent progresses in formulating electrophilicity,dual descriptor, etc., together with many applications in various chemical phenomena and transformations have been observed in the literature1,4,9.

In this Special Issue, to promote the efforts to quantify chemical understanding in terms of DFT, we invite experts from across the world to present their recent results on this topic. This Special issue is not simply just another occasion to bring together people with the same interest. Rather, it serves as a reminder to our readers-especially students-that newcomers are welcome to contribute and opportunities are abundant. Our aim is simple and clear—we cannot just compute; we need understand.

2 Extension and generalization of conceptual density functional theory

The original theoretical framework of CDFT was for the electronic system in the pure state in the canonical ensemble,where the natural variables are the total number of electrons and the external potential. Its extension to grand canonical and other ensembles via Legendre transformations is possible. Also,the Taylor expansion first considered was only up to the second order. Higher order terms are likely. In this Special Issue, four articles touched these points. Geerlings et al.10examined the response function as the second-order derivative of the Taylor expansion and considered its representation in all four ensembles. The work by von Szentpály11extends the original idea of obtaining electron affinity using the traditional parabola curve of the total energy with respect to the total electron number. González et al.12proposed a machine-learning approach to quantify electrophilicity so that it best fits the experimental values, whereas Franco-Pérez and coworkers13generalize the dual descriptor, which is third-order quantity from the aforementioned Taylor expansion, to be temperature dependent in the grand canonical ensemble.

3 New approaches to appreciate chemical concepts

A few recent developments to appreciate chemical concepts involving structure, bonding and reactivity have emerged. Eight articles are included in this Special Issue as illustrative demonstrations of some new trends in the literature. In the perspective, Contreras-Carcía and Yang14review the recent status on how to decode the chemical information from the electron density and other ingredients used in semilocal functionals. Heidar-Zadeh et al.15proposed a generalized Hirshfeld scheme to partition atoms in molecules. Lu et al.16highlight how the valence electron density, instead of the total electron density, can be employed to valuate molecular electronic structure, whereas Jiang and coworkers17employ blocked localized orbitals to appreciate weak interactions such as hydrogen bonding. A recent endeavor attracting more attentions is the information-theoretic approach. In this Special Issue, four articles are related to this relatively new approach.Nalewajski18outlines chemical reactivity descriptions using the information-theoretic approach, Zhong et al.19, Alipour20,and Yu et al.21demonstrate the usefulness of this approach in appreciating alkane isomer stability, water nanocluster, and aromaticity propensity, respectively.

4 Recent theoretical developments

DFT is a ground state theory. To further our understanding to appreciate chemical reactivity in excited states and time-dependent processes, more theoretical developments are in need. In this Special Issue, we collect five articles in this regard. Nagy22presents a phase space view of ensembles for the excited state. Ayers and Levy23show that it is possible to perform the Levy constrained search in Fock space with a noninteger electron number. Polkosnik and Lou24proposed a kernel energy method for Kohn-Sham density matrix. And Finzel and Bultinck25discussed their new findings of the relationship between kinetic and exchange-correlation functionals in the orbital-free DFT framework. More importantly, using a simple model of two same-spin non-interacting fermions in a one-dimensional box with an opaque wall, Savin26elucidates the scope of usefulness and limitation of chemical bonding and Lewis electron pairs in time-dependent processes.

5 Applications of conceptual density functional theory

Applying conceptual DFT ideas to real chemical systems to provide physiochemical insights for molecular phenomena and chemical processes has been widely pursued ever since the birth of Conceptual DFT, and the effort is still ongoing. Ten contributions27–36are collected in this Special Issue to showcase the most recent focus of this research direction. They include treatments of biological systems27, nanotubes28,reactions29,30, toxicity31, catalysis32,33, resonance states34, new chemical species35, and extreme conditions36.

6 Conclusions and outlook

As the first attempt to appreciate chemical concepts in DFT,Conceptual DFT is a powerful and general framework to appreciate and predict molecular property and chemical reactivity. It has been used, for instance, to understand aromaticity, acidity, basicity, catalysis, mechanism, reactivity,stability, toxicity, among many others. The theory was built through first and second-order response functions as the derivatives of the energy and other quantities with respect to the total number of electrons and other variables stemmed from Taylor expansions under the framework of the four classical thermodynamics-like ensembles. Insightful analyses have provided the derivatives relevant chemical meanings in chemical language like electronegativity, hardness/softness,Fukui function, electrophilicity, etc. The development of the Conceptual DFT framework is not over. It is still evolving. For example, are there other third-order descriptors which may play significant roles in explaining chemical processes? Can they be applicable to systems with weak interactions, where electron transfer is minimal? How about extensions of Conceptual DFT to excited states and time-dependent domains?

Another recent trend is the attempt to develop quantitative DFT-based descriptors without the perturbative approach.Rather, it employs simple density functionals instead. The steric effect was recently formulated in this manner using the Weizsäcker kinetic energy density functional. Quantifying electrophilicity, nucleophilicity, and regioselectivity in terms of the Kullback-Leibler divergence is another example. The information-theoretic approach is being developed in the same spirit as well. It remains to be seen if the quantification in this way is generally applicable in understanding real chemistry problems and whether there are other quantities in chemistry that can be quantified in a similar manner. Keeping in mind that according to the basic theorems of DFT, the electron density alone is adequate to determine all properties of an electronic system in the ground state, we are confident that such an endeavor is not only completely justified but also well feasible.

Finally, we conclude by quoting a narration by Robert G.Parr, the inventor of Conceptual DFT, who recently passed away. “There is another whole side of DFT which has concerned and still concerns many of us, the ‘conceptual’ side.This side is rich in potential, and it is not without accomplishment. The concepts of DFT neatly tie into older chemical reasoning, and they are useful for discussing molecules in course of reaction as well as for molecules in isolation. Where solid-state physics has Fermi energy, chemical potential, band gap, density of states, and local density of states, quantum chemistry has ionization potential, electron affinity, hardness, softness, and local softness. Much more too”37.

Acknowledgments:We are immensely indebted to the Editor-in-Chief as well as the entire crew of the Editorial Office of Acta Physico-Chimica Sinica for making the Special Issue possible. I am in particular grateful to Dr. Xiaojuan Zhang, the Managing Editor, and Dr. Ying Xiong, the Executive Editor of this Special Issue, for their dedication, hard-working, and professionalism. We are also grateful to all authors of this Special Issue for their willingness to contribute in a timely fashion. Without their original work, this Special Issue would have never been possible.

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