Conceptual Density Functional Theory E-Book

Shubin Liu

• Herausgeber: Wiley-VCH GmbH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1260

Veröffentlichungsjahr: 2022

Beschreibung

Conceptual Density Functional Theory

A unique resource that combines experimental and theoretical qualitative computing methods for a new foundation of chemical reactivity

This two-volume reference book shows how conceptual density functional theory can reconcile empirical observations within silico calculations using density functional theory, molecular orbital theory, and valence bond theory. The ability to predict properties like electronegativity, acidity/basicity, strong covalent and weak intermolecular interactions as well as chemical reactivity makes DFT directly applicable to almost all problems in applied chemistry, from synthetic chemistry to catalyst design and materials characterization.

Edited by one of the most recognized experts in the field and contributed to by a panel of international experts, the work addresses topics such as:

• Qualitative methods that are capable of rationalizing chemical concepts derived from theory and computation
• Fundamental concepts like the computation of chemical bonding, weak interactions, and reactivity
• Computational approaches for chemical concepts in excited states, extended systems, and time-dependent processes

Theoretical chemists and physicists, as well as those applying theoretical calculations to empirical problems, will be able to use this book to gain unique insight into how theory intersects with experimental data in the field of qualitative computation.

Leseprobe

Conceptual Density Functional Theory

Towards a New Chemical Reactivity Theory

Volume 1 and Volume 2

Edited byShubin Liu

Editor

Prof. Shubin Liu

University of North Carolina

Research Computing Center

211 Manning Drive

CB# 3420

North Carolina

United States

Cover: © GarryKillian/Shutterstock; Courtesy of Dong–Xia Zhao

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Print ISBN: 978‐3‐527‐35119‐0

ePDF ISBN: 978‐3‐527‐82992‐7

ePub ISBN: 978‐3‐527‐82993‐4

oBook ISBN: 978‐3‐527‐82994‐1

This book is dedicated to the memory of the founding father of Conceptual Density Functional Theory,

Robert Ghormley Parr

Preface

Conceptual density functional theory (CDFT), pioneered by Robert G. Parr and coworkers, has been with us for over 30 years. As witnessed by the 32 chapters presented in this book, there have been tremendous interests and enormous developments on CDFT in the literature. Nevertheless, there has never been a book dedicated to this topic. With the sad passing of Bob Parr, our beloved teacher, mentor, collaborator, and friend, the goal for us to put collective efforts along that direction had become more urgent. That was the reason why we held CCTC2018 Symposium in Changsha, China, where a roundtable discussion led to a CDFT status report on Theoretical Chemical Accounts. When Aron Urbatsch of Wiley VCH approached me in January 2020 with the book idea and I subsequently consulted major players in CDFT about the possibility, the feedback that I obtained was overwhelmingly positive. That's how and when this book project got started in the first place.

Coincidently, that's also when the global pandemic of COVID‐19 got started. Only people who experienced it firsthand knew how hard and miserable life became during that period. Amazingly, we still could get this book done as proposed. I am deeply indebted and sincerely grateful to all authors who spent days and nights during this extremely tough time working on their contributions. I wish also to thank the publishers for their hard work and flexibility. Without the commitment and dedication of all of them, this book is simply impossible.

To wrap up, let us remember what Bob told us: “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.”

10 January 2022                      Shubin Liu

                             Chapel Hill, North Carolina, USA

Foreword

It was a great pleasure to read this book edited by Dr. Shubin Liu. The comprehensive collection of articles from leading scientist covers the field of conceptual density functional theory (CDFT) from historical perspectives, didactic presentations, chemical insights, critical analysis, and frontier ideas to computational software and applications in chemistry. This is the book to read for students and researchers.

Density functional theory (DFT) has become the most widely used method for computational chemistry and materials science because of the optimal balance between the accuracy of prediction on measurable physical quantities and the cost of computation. In view of the power of computational predictions from DFT, what is the role of CDFT? What is CDFT anyway?

Let us think over the meaning of conceptual. “The definition of conceptual is something having to do with the mind, or with mental concepts or philosophical or imaginary ideas”, according to a dictionary (https://www.yourdictionary.com/conceptual). “Something is conceptual when it deals primarily with abstract or original thoughts”, based on another one (https://www.vocabulary.com/dictionary/conceptual). This appears to fit our appreciation of CDFT.

Chemical concepts are indeed mental or philosophical ideas of chemists about molecules and their properties. Chemical concepts are therefore in general different from physical quantities, which are observables, such as the ionization potentials, electron affinities and the geometry of a molecule in its ground state. Chemical concepts are in the mind of a chemist and are apparently not directly related to observables. For examples, the concept of electronegativity describes the capability of a molecule to attract electrons, and the concept of chemical hardness is related to intrinsic chemical reactivities of a molecule. Chemical concepts are traditionally somewhat fuzzy, without a mathematical definition. They are a very useful part of the chemical language and chemical thinking framework.

In contrast, physical observables can often be directly computed from the electron density or the wavefunction of a molecule, or the density matrix or the Green's functions of the statistical ensemble of a condensed matter. While wave function methods can in principle rely on systematic improvement based on some natural orders or hierarchy in the theory, density functional methods had (and still have) to fight with a way to find good approximations. Thus, wave function methods had to find the best way to converge to the exact result, while the DFT framework stimulates one to find out what approximate methods really do.

When a method is created, it is intended for users ultimately. Of course, calculations should produce numbers that can be compared to experiments. Validation of computational predictions is critical for developing and accessing theoretical methods. It was the successful validation of DFT calculations that lead the broad application of DFT in chemistry and materials science and engineering. There is also a more ambitious objective: we would like our calculations to be predictive. Of course, “numbers don't lie”, but there is something more we would like to produce from our calculations. The users should get have some tools that they can apply on the results of the calculations that should allow them to think further, to connect the results with existing knowledge. In other words, to apply some concepts to the results obtained.

Let us exemplify the statement above. Ionization potentials and electron affinities are quantities that can be measured, and also be computed. The concept of reactivity is vague. However, it is widely used. There are several definitions for electronegativity, but this does not mean that a given definition is not based on quantities that cannot be derived from experimental measurements. For example, the electronegativity can be defined as half the sum of the ionization potential and the electron affinity.

Cleary different from the commonly used computational prediction aspect of DFT, as shown throughout the chapters in this book, CDFT is a framework of mathematical definitions of chemical concepts and application of the concepts to describing chemical systems, based mainly on DFT. Therefore, CDFT provides the quantitative connections to the concepts in the minds of chemists and the associated quantitative understanding of chemical reactivities, using electronic structure theory, mainly DFT. The 1978 identification by Parr and coworkers of electronegativity of atoms and molecules as the negative of chemical potentials averaged over the two limits of electron addition and removal is a great example, marking the birth of the field of CDFT.

What aspects of DFT are uniquely important for quantitative connections to chemical concepts? Using electron density, a reduced variable and much simpler object, leads to more direct connection to conceptual thinking. Furthermore, the definition on electronegativity highlights the feature of DFT in treating electron number as a continuous variable (fractionals). Yes, fractional number of electrons can occur as a grand canonical ensemble in quantum theory. However, there is no direct connection from a grand canonical ensemble description to an isolated molecule in its ground state. In other words, an isolated molecule in its ground state does not need an ensemble description. Similarly, in DFT, fractional numbers of electrons can occur in an ensemble. But there is an inherently more important role of fractional numbers of electrons in DFT; namely, fractional numbers of electrons are the manifestation in the electron density, a classical variable, of the quantum mechanical principle of state degeneracy. As a result of the linearity of the Schrodinger equation, any linear combination of degenerate eigenstate wavefunctions is also an eigenstate with the same energy. In DFT with the basic variable being the electron density, any convex linear combination of electron density for the degenerate states has the same total energy. This naturally leads to fractional charges and spins in the electron density and their corresponding exact conditions, which are critical for developing functional approximations.

As with electronegativity, a large part of CDFT has been developed for describing responses of the system to changes in electron numbers and/or external potentials. The corresponding total energy derivatives play a central role in CDFT. They have been applied in broad fields of chemistry and materials. On the other hand, many interesting mathematical definitions of chemical concepts are not expressed as such energy derivatives. Two outstanding examples are the electron localization function (ELF), capturing the electron localization features, and the non‐covalent interaction (NCI), revealing noncovalent interactions in molecules and bulk systems. Both ELF and NCI are not about measurable quantities, but they describe quantitatively what is in the minds of chemists. All these are featured in this book.

Many concepts used in interpreting results of density functional calculations could be also computed from experimental densities if they were accurate enough. Electronegativity, ELF and NCI are such examples. Even the fractional occupation numbers, a construct of the Kohn‐Sham method that shows up in situations when (near‐)degeneracy is important enter this category: once we have a density, we can construct the exact Kohn‐Sham potential, and check if fractional occupation numbers show up.

Looking forward, CDFT would benefit from developing sets of standards or benchmarks to evaluate concepts developed. The validation was critical for the development and success of computational DFT and should be expected to further promote the field of CDFT – it is important for CDFT to spreads out to users. This measures its success. However, challenges remain on how to develop such test sets.

The Editor of this book, Dr. Shubin Liu is a leader in CDFT, having been trained with the late Professor Robert G. Parr, the founding father of CDFT. Dr. Liu has assembled a team of distinguished scientists. Together, they provide a feast of CDFT for all to enjoy.

Durham, NC                   Weitao Yang

Paris,                      France Duke University

February 2022

                        Andreas Savin

                        CNRS and Sorbonne University

Part IFoundations

1Historic Overview

Paul Geerlings

Vrije Universiteit Brussel, Research Group of General Chemistry (ALGC), Faculty of Science and Bioengineering Science, Pleinlaan 2, Brussels 1050, Belgium

1.1 Introduction: From DFT to Conceptual DFT

Density Functional Theory goes back to the early days of Quantum Mechanics when, in 1926, Thomas and Fermi [1–3] presented a model to study the electronic structure of atoms on the basis of the electron density ρ (r) instead of the wave function. The simplification is spectacular: for an N‐electron system, one passes from an immensely complicated wave function Ψ(xN), a function of 4N variables (three spatial variables and one spin variable for each electron, gathered in a 4‐vector x) to just three variables in the density ρ(x, y, z). The results for atoms were encouraging, but the approach failed dramatically for (diatomic) molecules not being able to account for their stability. Was the loss of information when passing from a wave function to the density (in fact, an integration over 4N‐3 variables) too drastic? An important step was taken by Slater in the 1950s. In his Xα method [4], he presented a simplification of the Hartree–Fock method replacing the complicated nonlocal Fock operator with a local, single parameter, operator involving the density. The method turned out to be a quite efficient technique for electronic structure calculations on molecules and solids.

One has, however, to wait until 1964 when Hohenberg and Kohn [5] turned density‐based models into a full‐fledged theory through their two famous theorems. The first theorem is an existence theorem presenting the ground‐state energy of a system as functional of the density. The proof, based on a reductio ad absurdum, is, as quoted by Parr and Yang [6], “disarmingly simple.” The second theorem offers a variational principle, and so, at least in principle, a road to the “best” density: look for the one yielding the lowest energy, as known for decades in wave function quantum mechanics. The crux of the first theorem is that it is proven that for a given N‐electron system, its ground‐state density ρ(r) is compatible with a single external potential v(r), i.e. the potential felt by the electrons due to the nuclei, in the absence of external fields. This single external potential is equivalent to a unique constellation of nuclei: their number, position, and charge. To put it all succinctly: ρ determines v, and as it also determines N by integration, it also determines the Hamiltonian, and at least in principle, “everything.” Coming back to the second theorem, the variational procedure leads to the Euler equation of the problem

(1.1)

where FHK is the Hohenberg–Kohn functional and μ the Lagrangian Multiplier introduced during the variational procedure ensuring that the density remains properly normalized to N. Equation (1.1) is the analogue of the time‐independent Schrödinger equation Hψ = E ψ, which also can be obtained in a variational ansatz, where the Lagrangian Multiplier ensuring proper normalization of the wave function ψ is at the end identified with the system's energy E. The analogy is striking, but two aspects of this equation deserve further consideration. What is FHK, and what is the physical interpretation of μ? The Hohenberg–Kohn functional is a universal functional (i.e. v‐independent), which contains unknown parts governing electron correlation and exchange assembled in the exchange‐correlation functional Exc [ρ], which will be highlighted in other chapters in the “Fundamentals” part in this book. Quintessentially, it's the price to be paid for the simplification when passing from a wave function to the density, still retaining its essential information content. By introducing, in the context of a non‐interacting reference system, orbitals in the variational procedure, Kohn and Sham [7] were able to cast the variational equation into a series of pseudo‐one‐electron eigenvalue equations, similar to the Hartree–Fock equations, be it, again, that part of the concerned operator is unknown: the functional derivative of Exc with respect to ρ(r), δExc/δρ(r), termed the exchange‐correlation potential vxc (r).

The history of DFT is (among others) a quest for finding better and better approximations for this unknown vxc (r). The simplest approximation, of standard use, mainly by solid‐state physicists, in the 1970s and the 1980s was the local density approximation (LDA) [7], showing however substantial over‐binding in molecules [8]. Things became more interesting for chemists in the second half of the 1980s when the generalized gradient approximations (GGA) were launched [9, 10]. The great breakthrough, with the wide acceptance of DFT by the Quantum‐Chemical community, came in the early 1990s when hybrid functionals were introduced, in which a fraction of the GGA exchange was replaced with exact HF exchange, with as most prominent example the still ubiquitous B3LYP functional [8–10]. This approach yielded at that time unsurpassed quality/computing time ratios, the latter aspect being reinforced by its implementation in Pople's widely used GAUSSIAN package [11]. At that time, DFT was on its way to become the standard method for obtaining an optimal quality/cost ratio for studying properties and reactions of not too exotic systems of varying sizes. Afterward, its “popularity” grew at incredible pace. In his excellent 2012 Journal of Chemical Physics perspective, Burke [12] plots the number of papers retrieved from the Web of Science when searching for DFT as a function of time, reaching in 1996 about 1000 papers, 5000 in 2005, and 8000 in 2010. Nowadays, DFT is the workhorse “par excellence” used, not only by theoreticians but also by experimentalists, in combined experimental–computational papers, when exploring structure, stability, electronic properties, reactivity, and reactions of molecules, polymers, and solids in the most diverse subdomains of chemistry [8].

1.1.1 But, Where Is Conceptual DFT in This Story?

As highlighted at the very beginning of this chapter, the variational Eq. (1.1) stands central in DFT, just as the Schrödinger equation in wave function theory. Besides the quest for the exact Hohenberg–Kohn functional, we already mentioned that a second fundamental question in relation to this equation arises: what is the physical/chemical meaning of the Lagrangian multiplier μ? Its identification by Parr et al. [13] can be considered as the birth of Conceptual DFT.

This genesis, its early years, and evolution with a short reflection on its present status and its future will be described in the following paragraphs. Note that detailed explanations and derivations leading to the various concepts, formulas, and equations will mostly not be given in view of space limitations and because the reader will find them in Part I (Foundations) and, in case of more recent developments, in Part II (Extensions) of this book. This is also the reason why the number of references is kept to the most essential ones. For the most extensive reviews on Conceptual DFT, we can now already refer the reader to items [14–20] in the reference list.

1.2 The Birth of Conceptual DFT: The Identification of the Electronic Chemical Potential (1978)

In a landmark paper in 1978, Parr et al. [13] showed that the Lagrangian Multiplier in the DFT variational equation could be written as the partial derivative of the system's energy with respect to the number of electrons at fixed external potential.

(1.2)

The chemical importance of this demystification of the Lagrangian Multiplier shows up when going back to the early 1960s, when Iczkowski and Margrave [21] presented evidence, on the basis of experimental ionization energies and electron affinities, that the energy of an atom could reasonably well be written as a polynomial in n (the number of electrons N minus the nuclear charge) around n = 0 as

(1.3)

Assuming continuity and differentiability of E, the slope at n = 0 and at fixed nuclear charge Z, (∂E/∂n)n=0, could easily be seen as a measure of electronegativity χ of the neutral system

(1.4)

As it was recognized that the cubic and quartic terms were negligible, Mulliken's electronegativity definition [22]

(1.5)

where I and A are the first ionization and electron affinity, respectively, were regained as a special case, so that within this approximation

(1.6)

Generalizing the constant Z condition for atoms to a constant v condition for molecules, the Lagrangian Multiplier of the Euler Eq. (1.1) has now been identified with a cornerstone of (physical) chemistry: electronegativity. A bridge between Density Functional Theory and (concepts in) Chemistry has thereby been established.

The analogy between μ and (the expression for) the macroscopic chemical potential of component i in a system at given pressure p and temperature T is beautiful. Indeed, μi can be written as [23]

(1.7)

with G the Gibbs free energy and ni the number of moles of component i. The resemblance between Eqs (1.4) and (1.7) was at the origin of later, even up to the present moment, endeavor to scrutinize analogies between macroscopic thermodynamics and “microscopic” Conceptual DFT.

1.3 The Early Years (1978–1985): Completing the Launching of Conceptual DFT

Further exploring the E = E(N) function Parr and Pearson identified in 1983, quite soon after Parr's 1978 landmark paper, Pearson's hardness as the second derivative of E with respect to N at constant v, denoted as η [24]

(1.8)

Pearson had introduced the hardness concept in the early 1960s [25] in the context of the study of generalized acid–base reactions, where he proposed a classification of favorably interacting acids and bases, mainly built on the polarizability, terming low polarizable species as “hard” and highly polarizable species as “soft.” Combining this classification and the terminology then yields the famous hard‐soft acid‐base (HSAB) principle: hard acids preferentially interact with hard bases; soft acids preferentially interact with soft bases. But… no quantification of this hardness/softness concept was available, be it a way to calculate it. It was the identification of η as the second derivative Eq. (1.8), which paved the way to quantitative studies on the hardness of atoms and molecules and to use it as such or in the context of the HSAB principle: a second achievement where a chemical concept is linked to DFT, as indeed (∂2E/∂N2)vis nothing else than the N derivative of the Lagrangian Multiplier μ. Both μ and η are called global descriptors as they are associated to an overall characteristic of the system.

Soon after, in 1984, Parr and Yang [26] launched the first local, i.e. r‐dependent or varying from place to place descriptor, which further established the bridge between DFT and chemistry. They generalized and extended Fukui's frontier Molecular Orbital concept [27] by considering a mixed second‐order derivative f(r) = (∂2E/∂Nδv(r)). Its chemical significance is clear when realizing that it can be easily deduced from perturbation theory [6, 15] that

(1.9)

so that f(r) can also be written as (∂ρ(r)/∂N)v indicating how a system partitions the added or subtracted electrons in space. When the orbitals are kept unchanged (frozen) upon adding or subtracting electrons, it is easily seen that f(r) boils down to the highest occupied molecular orbital (HOMO) or lowest occupied molecular orbital (LUMO) density (for decreasing or increasing N, respectively) and can thereby directly be linked to the basic ingredients of Fukui's reactivity descriptors. In the early 1950s, Fukui emphasized the predominant role of the frontier orbitals in (certain types of) reactions. His Frontier MO theory was considered both a milestone and a guiding principle in studying chemical reactions and reactivity. One of the most prominent examples are the celebrated Woodward–Hoffmann rules [28], highlighting the conservation of orbital symmetry (with particular emphasis on the frontier MOs) in the course of concerted reactions. In honor of Fukui, this local descriptor f(r) was termed the Fukui function.

Note that when writing ρ(r) as (δE/δv(r))N, f(r) can also be written as (δμ /δv(r))N stressing again the link with the content of the variational Eq. (1.1).

Two points should be stressed. Again, a DFT routed quantity, the functional derivative of the Lagrangian Multiplier with respect to the external potential, has been connected to a chemical “cornerstone,” this time at stake when scrutinizing reactivity. On top of that, and almost unnoticed, the electron density itself also entered this series of descriptors, in fact, as Eq. (1.9) shows, as the “first” local one: the derivative of E with respect to v(r). It's useful to reconsider the fundamental equation of DFT, Eq. (1.1), again and to note that three of its main ingredients E, v(r), and μ