My review article on emergence

I just posted on the arXiv a long review article on emergence

Emergence: from physics to biology, sociology, and computer science

The abstract is below.

I welcome feedback. 

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Many systems of interest to scientists involve a large number of interacting parts and the whole system can have properties that the individual parts do not. The system is qualitatively different to its parts. More is different. I take this novelty as the defining characteristic of an emergent property. Many other characteristics have been associated with emergence are reviewed, including universality, order, complexity, unpredictability, irreducibility, diversity, self-organisation, discontinuities, and singularities. However, it has not been established whether these characteristics are necessary or sufficient for novelty. A wide range of examples are given to show how emergent phenomena are ubiquitous across most sub-fields of physics and many areas of biology and social sciences. Emergence is central to many of the biggest scientific and societal challenges today. Emergence can be understood in terms of scales (energy, time, length, complexity) and the associated stratification of reality. At each stratum (level) there is a distinct ontology (properties, phenomena, processes, entities, and effective interactions) and epistemology (theories, concepts, models, and methods). This stratification of reality leads to semi-autonomous scientific disciplines and sub-disciplines. A common challenge is understanding the relationship between emergent properties observed at the macroscopic scale (the whole system) and what is known about the microscopic scale: the components and their interactions. A key and profound insight is to identify a relevant emergent mesoscopic scale (i.e., a scale intermediate between the macro- and micro- scales) at which new entities emerge and interact with one another weakly. In different words, modular structures may emerge at the mesoscale. Key theoretical methods are the development and study of effective theories and toy models. Effective theories describe phenomena at a particular scale and sometimes can be derived from more microscopic descriptions. Toy models involve minimal degrees of freedom, interactions, and parameters. Toy models are amenable to analytical and computational analysis and may reveal the minimal requirements for an emergent property to occur. The Ising model is an emblematic toy model that elucidates not just critical phenomena but also key characteristics of emergence. Many examples are given from condensed matter physics to illustrate the characteristics of emergence. A wide range of areas of physics are discussed, including chaotic dynamical systems, fluid dynamics, nuclear physics, and quantum gravity. The ubiquity of emergence in other fields is illustrated by neural networks, protein folding, and social segregation. An emergent perspective matters for scientific strategy, as it shapes questions, choice of research methodologies, priorities, and allocation of resources. Finally, the elusive goal of the design and control of emergent properties is considered.

Emergence in Chemistry

It is important to be clear what the system is. Most of chemistry is not really about isolated molecules. A significant amount of chemistry occurs in an environment, often within a solvent. Then the system is the chemicals of interest and the solvent. For example, when it is stated that HCl is an acid, this is not a reference to isolated HCl molecules but a solution of HCl in water, and then the HCl dissociates into H+ and Cl- ions. Chemical properties such as reactivity can change significantly depending on whether a compound is in the solid, liquid, or gas state, or on the properties of the solvent in which it is dissolved.

Scales

The time scales for processes, which range from molecular vibrations to chemical reactions, can vary from femtoseconds to days. Relevant energy scales, corresponding to different effective interactions, can vary from tens of eV (strong covalent bonds) to microwave energies of 0.1 meV (quantum tunnelling in an ammonia maser).

Other scales are the total number of atoms in a compound, which can range from two to millions, the total number of electrons, and the number of different chemical elements in the compound. As the number of atoms and electrons increases, so does the dimensionality of the Hilbert space of the corresponding quantum system.

Novelty

All chemical compounds are composed of a discrete number of atoms, usually of different type. For example, acetic acid, denoted CH3COOH, is composed of carbon, oxygen, and hydrogen atoms. The compound usually has chemical and physical properties that the individual atoms do not have.

Chemistry is all about transformation. Reactants combine to produce products, e.g. A + B -> C. C may have chemical or physical properties that A and B did not have.

Chemistry involves concepts that do not appear in physics. Roald Hoffmann argued that concepts such as acidity and basicity, aromaticity, functional groups, and substituent effects have great utility and are lost in a reductionist perspective that tries to define them precisely and mathematicise them.

Diversity

Chemistry is a wonderland of diversity, as it arranges chemical elements in a multitude of different ways that produce a plethora of phenomena. Much of organic chemistry just involves three different atoms: carbon, oxygen, and hydrogen.

Molecular structure

Simple molecules (such as water, ammonia, carbon dioxide, methane, benzene) have a unique structure defined by fixed bond lengths and angles. In other words, there is a well-defined geometric structure that gives the locations of the centres of atomic nuclei. This is a classical entity. This emerges from the interactions between the electrons and nuclei of the constituent atoms.

In philosophical discussions of emergence in chemistry, molecular structure has received significant attention. Some claim it provides evidence of strong emergence. The arguments centre around the fact that the molecular structure is a classical entity and concept that is imposed, whereas a logically self-consistent approach would treat both electrons and nuclei quantum mechanically.

The molecular structure of ammonia (NH3) illustrates the issue. It has an umbrella structure which can be inverted. Classically, there are two possible degenerate structures. For an isolated molecule, quantum tunnelling back and forth between the two structures can occur. The ground state is a quantum superposition of two molecular structures. This tunnelling does occur in a dilute gas of ammonia at low temperature, and the associated quantum transition is the basis of the maser, the forerunner of the laser. This example of ammonia was discussed by Anderson at the beginning of his seminal More is Different article to illustrate how symmetry breaking leads to well-defined molecular structures in large molecules. 

Figure is taken from here.

Born-Oppenheimer approximation 

Without this concept, much of theoretical chemistry and condensed matter would be incredibly difficult. It is based on the separation of time and energy scales associated with electronic and nuclear motion.  It is used to describe and understand the dynamics of nuclei and electronic transitions in solids and molecules. The potential energy surfaces for different electronic states define effective theory for the nuclei. Without this concept, much of theoretical chemistry and condensed matter would be incredibly difficult.

Singularity. The Born-Oppenheimer approximation is justified by an asymptotic expansion in powers of (m/M)^1/4, where m is the mass of an electron and M the mass of an atomic nucleus in the molecule. This has been discussed by Primas and Bishop.

The rotational and vibrational degrees of freedom of molecules also involve a separation of time and energy scales. Consequently, one can derive separate effective Hamiltonians for the vibrational and rotational degrees of freedom.

Qualitative difference with increase in molecular size

Consider the following series with varying chemical properties: formic acid (CH2O2), acetic acid (C2H4O2), propionic acid (C3H6O2), butyric acid (C4H8O2), and valerianic acid (C5H10O2), whose members involve the successive addition of a CH2 radical. The Marxist Friedrich Engels used these examples as evidence for Hegel’s law: “The law of transformation of quantity into quality and vice versa”.

In 1961, Platt discussed properties of large molecules that “might not have been anticipated” from properties of their chemical subgroups. Table 1 in Platt’s paper lists “Properties of molecules in the 5- to 50-range that have no counterpart in diatomics and many triatomics.” Table 2 lists “Properties of molecules in the 50- to 500-atom range and up that go beyond the properties of their chemical sub-groups.” The properties listed included internal conversion (i.e., non-radiative decay of excited electronic states), formation of micelles for hydrocarbon chains with more than ten carbons, the helix-coil transition in polymers, chromatographic or molecular sorting properties of polyelectrolytes such as those in ion-exchange resins, and the contractility of long chains.

Platt also discussed the problem of molecular self-replication. Until 1951, it was assumed that a machine could not reproduce itself,f and this was the fundamental difference between machines and living systems. However, von Neumann showed that a machine with a sufficient number of parts and a sufficiently long list of instructions can reproduce itself. Platt pointed out that this suggested there is a threshold for autocatalysis: “this threshold marks an essentially discontinuous change in properties, and that fully-complex molecules larger than this size differ from all smaller ones in a property of central importance for biology.” Thus, self-replication is an emergent property. A modification of this idea has been pursued by Stuart Kauffman with regard to the origin of life, that when a network of chemical reactions is sufficiently large, it becomes self-replicating.

The green comet and quantum chemistry

The comet C/2022 E3 (ZTF) getting a lot of attention, pointed out to me by my friend Alexey. Why is it green? This basic question turns out to be scientifically rich and has only recently been answered.

The green glow comes from a triplet excited state of diatomic carbon, C2. This got my interest because a decade ago I blogged on debates by quantum chemists about whether C2 involves a quadruple bond. Back in 1995, Roald Hoffmann wrote an interesting column in The American Scientist (and reproduced in his beautiful book Same and Not the Same) about the molecule and how it is present in various organometallic compounds and inorganic crystals.

Recent advances in understanding the photophysics of C2 were reported in 2021 in this paper.

Photodissociation of dicarbon: How nature breaks an unusual multiple bond

Jasmin Borsovszky, Klaas Nauta, Jun Jiang, Christopher S. Hansen, Laura K. McKemmish, Robert W. Field, John F. Stanton, Scott H. Kable, and Timothy W. Schmidt 

Here is a summary of the significance and content of the paper from Chemistry World.

..as dicarbon streams out of the comet core, it is destroyed by sunlight – this is why the comet tail, unlike the coma, is colourless. However, the precise mechanism of this supposed photodissociation had remained unclear.

Researchers in Australia and the US have now for the first time observed diatomic carbon’s photodissociation in the lab. The team produced dicarbon by photolysing tetrachloroethylene, and then breaking it apart with laser pulses. This allowed them to determine its bond dissociation energy with the same precision as for oxygen and nitrogen. Previous measurements for dicarbon had uncertainties an order of magnitude higher than for other diatomic molecules.

To break its quadruple bond, the molecule must absorb two photons and undergo two ‘forbidden’ transitions, those that break spectroscopic rules. Cometary dicarbon, the researchers calculated, has a lifetime of around two days until sunlight breaks it apart – the reason why its colour is visible in the coma but not in the tail.

A few things I have learnt from professional editors

 Until a few years ago I had never engaged with or received feedback from my writing from a professional editor. This is because the only genre I wrote that involved an editor was papers for scientific journals. But the editors of journals are not really editors in the literary sense. They are more like gatekeepers. Colleagues and collaborators may provide feedback on written work, but again they are amateurs.

In the past few years, I have been writing some popular articles and a popular book and have been part of a writing group. In the process, I have engaged with several professional editors. They were getting paid to make my writing better. I have learnt a lot. Here are a few of the things. On the one hand, some of this may not seem that relevant to scientific articles and grant applications. On the other hand, think of the joy of reading a beautiful scientific article, such as those by Roald Hoffmann. Think of how many papers you try to read and you cannot figure out what they are actually about. Also, I think this is particularly relevant to writing review articles, somewhat of a lost art.

Can it be shorter? Most of the writing I have worked with editors on had a strict word limit. I struggled to stay within it. However, the editors forced/helped me in two ways. First, the fixed word limit helped me structure the work and be realistic about the volume of content. For example, for my Very Short Introduction, I broke down the 35,000-word limit to ten chapters, each of about 3500 words. This made the writing quite manageable. Second, editors helped by cutting out content that was not essential, even when I loved it. Third, editors rewrote some of my sentences making them both shorter and clearer. Seeing their improvements I became aware of some of my bad habits.

Find your voice and tell a story. We are all unique and each piece of writing is unique and is making a unique point. Don't try and be someone else. A grant application needs to make the case that your proposed project is unique and that you are uniquely qualified to do it. Your writing will be more engaging and compelling if it expresses your unique perspective and there is a natural narrative.

A few of these suggestions overlap with some of Stephen King's writing tips. 

Sage wisdom on computational materials science

Roald Hoffmann and Jean-Paul Malrieu are two of my favourite living theoretical chemists. Both greatly value the role of concepts and intellectual clarity in theory. Hoffmann has featured in 22 posts on this blog.

They recently published a wonderful trilogy in  Angewandte Chemie.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. 

Part A. Stage Setting

Part B. The March of Simulation, for Better or Worse

Part C. Toward Consilience

I add this trilogy to my list of 5 papers every computational chemistry student should read, suggested by me a decade ago. [Malrieu is author of one of those and Hoffmann co-author of another.]

Although the trilogy addresses and uses specific examples from computational quantum chemistry it is just as relevant to anyone interested in computational materials science. Actually, I hope that anyone interested in materials science would read and digest it as it gives a sober and balanced perspective about the relationship between theory, simulation, and understanding.

Articles are timely as they address hype about how AI techniques will "revolutionise" materials theory. 

The articles are beautifully written and engage with broader themes such as philosophy of science, culture, art, and politics.

Finally, I just love this photo of the two authors, both in their eighties. the photo reflects some of the joy they find in science, so beautifully expressed in these articles.

I thank Ben Powell for bringing the papers to my attention.

Implicit versus explicit beliefs

 How can we design a room-temperature superconductor? How can a government stimulate economic growth? How can an NGO help reduce domestic violence? Why do communities become segregated on racial lines? How can I improve my mental health?

These important questions may seem unrelated. However, I propose that often there is a common issue about the strategies that people (whether individuals, professions, NGOs, funding agencies, governments, ...) propose to find answers or when definite answers are proposed.

Many strategies and answers involve a heavy dose of implicit beliefs. These are assumptions that are never stated. They may be elements of a worldview, which according to one definition, is

a commitment, a fundamental orientation of the heart, that can be expressed as a story or in a set of presuppositions (assumptions which may be true, partially true, or entirely false) which we hold (consciously or subconsciously, consistently or inconsistently) about the basic construction of reality, and that provides the foundation on which we live and move and have our being.

 James W. Sire, The Universe Next Door: A basic worldview catalog

These implicit beliefs may relate to values and morality. But I want to focus more on implicit beliefs that are related to academic disciplines such as philosophy of science, psychology, political science, theology, economics, anthropology,  sociology, ...  Most of us have never studied these disciplines and some of us may be skeptical about some of them. But, my point is that everyone has implicit ideas about what is true with regard to the objects these disciplines study. Everyone has a philosophy of science. Everyone has ideas about how minds work and how to change societies. It is just that these beliefs are rarely stated. 

Why does this matter? If implicit beliefs are never stated, they can never be tested, evaluated, critiqued, refined, or rejected. I believe that implicit beliefs are too often based on intuition, prejudice, common sense, or culture (social pressure to conform to accepted wisdom). This is not necessarily bad. Sometimes intuition, common sense, and culture are helpful and correct. We could not survive in life if we did not have them. We simply don't have the time, energy, and resources to constantly question and validate everything. On the other hand, if there is a vacuum, it will get filled with something. A major lesson from scientific history is that intuition, prejudice, common sense are sometimes wrong.

I now give three concrete examples of implicit beliefs. They cover computational materials science, public policy, and social activism.

Understanding materials using computers

Amongst others, there are two things, we would like more computational power to be able to do. One is to do reliable ab initio calculations of the properties of complex molecules and solids, from proteins to crystals with unit cells containing large numbers of atoms.  Another is to do exact diagonalisation (or some alternative reliable method) of many-body Hamiltonians, such as the Hubbard model, on large enough lattices that finite-size effects are minimal or can be reliably accounted for.

Over the past decade, there has been a lot of hype about how quantum computers and/or machine learning techniques will solve these problems and thus initiate a new era of materials understanding, discovery, and design with significant technological and economic benefits. My problem is that these claims usually seem to have the implicit belief that the only obstacle to progress is one of computational power. This fallacy has recently been deconstructed and critiqued in detail in three beautiful essays by Roald Hoffmann and Jean-Paul Malrieu, Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond.

Public policy

National economies around the world have been battered by the covid-19 pandemic. In response, governments of prosperous countries are spending big on stimulus packages. This involves taking on massive amounts of debt and significant government intervention in "free" market economies. Will these initiatives achieved their goals, particularly in the long term? Could they actually make things worse? Responses from pundits, both for and against, are laden with implicit beliefs. Unfortunately, economists cannot agree on the answer to the basic question, "Does government stimulus spending actually produce economic growth?" This issue is nicely discussed in a pre-pandemic podcast at Econtalk. 

NGOs and social activism

Many NGOs are about change. They aim to build a better world, addressing problems such as domestic violence, poverty, climate change, corruption, racism,... They aim to promote education, human rights, good governance, democracy, health, transparency, .. I love NGOs. I support many: philosophically, financially, and practically. To survive most NGOs have to raise funds, whether from many small donors or large philanthropies. This requires a well-honed pitch that aims to inspire potential donors to give. Furthermore, the whole operation of most NGOs is laden with implicit beliefs, whether those of the founders, staff or donors.

Consider a hypothetical NGO whose goal is to reduce the number of murders in a country. I chose this example because it may at first appear less controversial and contentious than some. Almost everyone thinks murder is wrong (always) and societies should stop reduce it. But why do murders occur? Revenge, passion, drugs, alcohol, money, politics, racism, ... Will making the purchase of guns difficult reduce the murder rate? Gun lobbyists will claim "Guns don't kill people. Criminals do! Law-abiding citizens need guns for self-defense." (cringe). There are many other alternative strategies: increasing penalties (longer jail terms or even the death penalty), the number of police, weapons for police, community policing, drug rehabilitation, breaking up gangs, ... Wow! It's complicated. My main point is that the hypothetical NGO will probably have an implicit belief that one particular strategy is the best one. Furthermore, if you identify and question this belief reasonable debate may not follow, but it may even be claimed that you don't care about stopping murder.

Some NGOs and philanthropies have become mindful of these issues. In response to a grant application that I helped an NGO write we were asked what our "theory of change" was? I discovered that there is a whole associated "industry. According to theoryofchange.org (!)

One organisation which began to focus on these issues was the US-based Aspen Institute and its Roundtable on Community Change. ... [leading] to the publication in 1995 of New Approaches to Evaluating Comprehensive Community Initiatives. In that book, Carol Weiss, ... hypothesized that a key reason complex programs are so difficult to evaluate is that the assumptions that inspire them are poorly articulated. She argued that stakeholders of complex community initiatives typically are unclear about how the change process will unfold and therefore give little attention to the early and mid-term changes that need to happen in order for a longer term goal to be reached.  

This led to software designed to help organisations plan initiatives, with a particular emphasis on teasing out assumptions embedded in plans. A related method is construction of a logframe matrix [logical framework].

All models are wrong but some are useful. I first learnt this aphorism from Scott Page, in his wonderful course Model Thinking at Coursera.  Models help us think more clearly. Simple quantitative models, such as agent-based models, in the social sciences, have the value that their assumptions can be clearly stated, and then the consequences of these assumptions can be investigated in a rigorous manner.

What do you think? Are there examples that you think involve implicit beliefs that need to be stated explicitly?

The quantum physics of life in red and green

Life is truly amazing!
Life is beautiful!
...and it involves quantum many-body physics...

There is a beautiful (short) review
Heme: From quantum spin crossover to oxygen manager of life 
Kasper Kepp

The article involves a plethora of topics that I have discussed before on this blog. I have included relevant links.

Kepp starts with the unique (chemically fine-tuned) properties of both iron and porphyrin that enable them to play a central role in two of the most important processes in life: respiration and photosynthesis. He has a beautiful paragraph (perhaps in the style of Roald Hoffmann):

Such ligand-field transitions of iron in porphyrin were familiar to our ancestors as the characteristic red color of blood that largely defines the human psychological and cultural connotations of the color representing courage, war, danger, and suffering. 

Incidentally, pi-pi* transitions within the porphyrin-derived chlorophylls are also responsible for the green color of plants, associated with nature, life and hope, so the reader may perhaps agree that porphyrin has had vast (but alas! rarely appreciated) cultural consequences.

The oxygen molecule is a spin triplet.
Iron(II) porphyrin is in a triplet spin state (S=1). The Fe(II) is a d6 configuration in a D_4h crystal field.
When they bind together the ground state is a spin-singlet.

There are two fundamental quantum chemistry questions that are discussed.

1. What is the electronic structure (many-body wave function) of the ground state for oxygen bound to heme?

2. What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

The first question has a long history. Like almost anything important and profound in quantum chemistry it goes back to Linus Pauling! In 1936 Pauling and Coryell argued that the ground state is

essentially a neutral O=O binding with two of its electrons to iron to produce a formally iron(II) if both the bonding electrons were confined to O2, corre- sponding to the non-bonding limit of neutral parts, but a formally iron(I) if the Fe–O bond were to be considered covalent. 

In 1960, McClure suggested a valence-bond formulation based on triplet–triplet coupling, which is appealing by the low promotion energies required to access these states, rather than the singlet states. In 1964, Weiss suggested, based on analogy to chemical reactions in aqueous solution, that the true ferrous hemeO2 adduct was mainly of the superoxo-iron(III) type caused by ‘‘electron transfer” from iron to O2. 

Goddard and Olafson suggested an ozone model of the adduct in 1975 which emphasized the four-electron three-center bond with maintained triplet state of dioxygen as in the McClure model with less electronic reorganization to explain the reversible binding. 

In 1977, Pauling maintained his original view again, the same year that Huynh, Case, and Karplus did a first attempt to bridge these views by performing early quantum chemical calculations that diplomatically emphasized the importance of both Weiss and Pauling resonance forms. 

However, interpretation depends on model language, orbital localization, and transformation between valence bond and orbital formalisms:  

In terms of molecular orbital theory, the wave function was a multi-configurational state dominated by the Pauling configuration; however, if one uses valence bond theory considerations, it can be interpreted as having large Weiss character. Thus, the multi-configurational state produced from CASPT2 is interpreted differently by different models. This partly explains why the trenches were so deeply dug during the exchange between Pauling, Goddard, McClure, and Weiss; all were right, and all were wrong. 

This is just another example of unnecessary conflicts about valence bond vs. molecular orbital (VB vs. MO). 

In terms of valence structures, the ground state was summarized by Shaik and Chen as having contributions from both Weiss, Pauling, and McClure forms, the first .. dominating. 

Ironically DFT ends up providing a useful language after all! 

The charge assignments to O2 are very dependent on calculation scheme, and both the orbitals, valence structures, and atomic charges that defined the Weiss-Pauling debate are non-observable. In contrast, the electron density is observable as are the geometries and spectroscopic data...

Molecular orbitals are not physical observables but calculational constructs. MO's don't exist.

In different words, one can take a many-body wave-function and make a linear unitary transformation of the molecular orbitals. The Slater determinants do not change. [The value of a determinant is invariant to a change of basis.]

Now. Question 2.
What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

Kepp talks about spin-orbit coupling and the fact that it is small for oxygen, motivating a discussion of a "broad crossing mechanism".  However, I am not sure this is relevant. I don't see the binding as necessarily spin forbidden. As the oxygen approaches the heme the two triplet states can mix to form a total spin singlet.
This is analogous to bringing two hydrogen atoms (each of which is spin 1/2) together to form a hydrogen molecule (which is spin zero). A multi-configurational wavefunction has no problem with this. But DFT-based approximations, which use a single determinant cannot describe this smooth crossover.

Other things of particular interest to me that are discussed in the review include the central role of back bonding and the success of the TPSSh functional in DFT calculations for organometallics.

Unfortunately, the review does not mention recent work by Weber et al, applying DMFT to the problem of oxygen binding to haemoglobin.

Don't be written off!

One of the most basic skills needed to succeed, or even survive, in professional life is to be able to write well. This is true whether you work in science, industry, business, or an NGO.
Of course, there are exceptions where an individual is incredibly gifted at the technical side of a job and can't even write a coherent paragraph. But, sorry, that individual is probably not you! Furthermore, even they need a collaborator or manager who is good at writing.

Most young scientists struggle to write a paper or a grant application, particularly when English is not their first language.

Here are a few suggestions on how to improve your writing skills over time.

First, accept that writing is hard work. Even John Grisham says that!

Accept that developing your writing skills is a project of a lifetime. This means starting early.

If you are an undergrad, take some humanities courses that require writing essays. Take writing lab reports seriously.
If you have to write a thesis, start writing it now.
Take a writing course. Take another.

Practise.
Write papers yourself. If you are the first author you really should write the first draft, including the introduction yourself. Don't let your boss (or someone more experienced) do it or expect them to. Your draft may be poor and get heavily edited or even discarded completely. But you will learn from the process and with time confidence and competence will follow.
Write a blog, even if no one reads it.

Learn by osmosis.
Read scientific authors known for the clarity and beauty of their writing. eg. David Mermin and Roald Hoffmann.
Read a lot and read broadly publications (newspapers and magazines) that are known for their excellent writing: The New York Times, The Economist, The New Yorker,...
Read famous novels and non-fiction books.

Read slowly and thoughtfully. Don't just skim everything.
I also suspect you may be better off reading hard copies.
Try to notice whether a piece of writing: makes sense, is hard to understand, is enjoyable to read?

Any other suggestions?

Searching for conical intersections for singlet fission

Previously I have posted about the fascinating challenge of understanding singlet fission [and the inverse process of triplet-triplet annihilation] in large organic molecules.  A key feature to understand is how fission can occur in less than 100 femtoseconds, suggestive of a conical intersection between excited state potential energy surfaces.

In Telluride Nandini Ananth gave a nice talk about work described in the paper

The Low-Lying Electronic States of Pentacene and Their Roles in Singlet Fission 
Tao Zeng,  Roald Hoffmann , and Nandini Ananth

Diabatic states provide a natural and powerful approach to understanding what is going on.
The authors perform high level quantum chemistry calculations to describe the relevant electronic excited states. They claim that for a pair of pentacene molecules one needs to include at least six diabatic states. Their dominant electronic configuration is shown in the schematic below.

We find that only one of the two charge-transfer states, ac, is engaged in the SF [singlet fission] in pentacene; it is the low-lying charge-transfer state that gets closer to the multi- and single-exciton states. Moreover, the ac diabat can move into degeneracy with the single-exciton states, more effectively mediating the mixing of the bright single- to and dark multiexciton diabats. This finding is different from the basic assumption of high-lying charge-transfer states in the superexchange model, emphasizing the need to adapt the general SF model to specific cases.

Aside: I wonder if this is one the few papers that Hoffmann has co-authored where strong electron correlations are central.

In more recent work, the authors have tried to pin down what is the relevant nuclear co-ordinate [vibrational mode] associated with a conical intersection. It is not the intermolecular separation but may be instead the relative orientation [twisting] of the two penatacene molecules. This has included some constructive interaction with the experimental group of Luis Campos.

Teaching enhances research enhances teaching

This is the main point of a nice article by Roald Hoffmann, that I posted about in the early days of this blog.

I experienced this a few weeks ago. I have been working on a paper with my postdoc Nandan Pakhira about the viscosity of strongly correlated fermion fluids, focussing on the Hubbard model. A basic issue I got quite confused about is the relation between the momentum, Bloch wave vector, and velocity of an electron in a Bloch state. Yet, I when I taught this to my solid state physics class I was reminded of the correct result.

Chemical bonding, blogs, and basic questions

Roald Hoffmann and Sason Shaik are two of my favourite theoretical chemists. They have featured in a number of my blog posts. I particularly appreciate their concern with using computations to elucidate chemical concepts.

In Angewandte Chemie there is a fascinating article, One Molecule, Two Atoms, Three Views, Four Bonds that is written as a three-way dialogue including Henry Rzepa.
The simple (but profound) scientific question they address concerns how to describe the chemical bonding in the molecule C2 [i.e. a diatomic molecule of carbon]. In particular, does it involve a quadruple bond?

The answer seems to be yes, based on a full CI [configuration interaction] calculation that is then projected down to a Valence Bond wave function.

The dialogue is very engaging and the banter back and forth includes interesting digressions such the role of Rzepa's chemistry blog, learning from undergraduates, the relative merits of molecular orbital theory and valence bond theory, the role of high level quantum chemical calculations, and why Hoffmann is not impressed by the Quantum Theory of Atoms in Molecules.

Computational chemistry versus chemical concepts

Robert Mulliken was one of the founders of quantum chemistry. In 1965 he gave a conference talk
Molecular Scientists and Molecular Science: Some Reminiscences. In it he made a commonly quoted statement highlighted below. I reproduce it in context.

....I would have liked first to say something about Molecular Quantum Mechanics (MQM) problems.  .... The general idea [of Lowdin's conferences] was that with old-fashioned chemical concepts, which at first seemed to have their counterparts in MQM, the more accurate the calculations became the more the concepts tended to vanish into thin air. So we have to ask, should we try to keep these concepts-do they still have a place-or should they be relegated to chemical history. Among such concepts are electronegativity....., hybridization, population analysis, charges on atoms, even the idea of orbitals, ....

Roald Hoffmann has argued these concepts do have a role. I would certainly agree. Computations should support, elucidate, and clarify concepts, not eliminate them. The issues are nicely discussed in 5 papers every computational chemistry student should read.

I thank Anna Painelli for bringing this quote to my attention.

Writing effective papers

Weston Borden's article 40 years of fruitful chemical collaborations has an significant observation concerning writing effective papers: focus on the physical explanation of the results rather than on the details of the methodology.
He recounts how he he learnt this, while starting out as an Assistant Professor at Harvard, in a collaboration with Lionel Salem. Borden had performed some calculations using the Pariser-Parr-Pople (PPP) model for the electronic structure of conjugated organic molecules [for physicists an extended Hubbard model with long-range Coulomb interactions].

Lionel read my draft, and he promptly rewrote it. Lionel’s revised version, which was the one that we published, focused much more than my draft had on the explanation of the PPP results, rather than on the details of the calculations. This experience taught me a valuable lesson. Although describing the details of calculations and the results obtained from them is certainly important, it is even more important to write a clear, physical explanation of the results. 

This was also the lesson that I learned from the papers that Roald Hoffmann published in the late 1960s and early 1970s. Although it was well-known that the Extended Hückel (EH) method that Roald used was quantitatively unreliable, Roald provided such convincing qualitative explanations of his EH results that it always seemed to me Roald’s EH results must be correct.

I think these observations are just as relevant and important for physicists.

Aside: an earlier post sung the praises of Hoffmann's paper titles.

Borden then makes the important and worrying observation:

Perhaps the tremendous increase in the accuracy of electronic structure calculations during the past 40 years has had the undesirable consequence that computational chemists feel less obliged to provide the kind of detailed physical explanations of their results than Roald routinely furnished 40 years ago.

The basics of electronegativity

Electronegativity is a simple and profound concept for understanding and giving a semi-quantitative description of broad classes of chemical bonds. It reflects the brilliant intuition of Linus Pauling who first introduced it in 1932. The bold idea is to assign a single number to each element of the periodic table; the relative value of the number between two elements then determines the magnitude of the polarity (charge distribution) in a chemical bond between two different elements.

I have been reviewing the concept because it is a key element to understanding hydrogen bonds, the recent IUPAC definition stating:

the hydrogen bond is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment X-H in which X is more electronegative than H, ...

Unlike earlier definitions it does not require that the acceptor be more electronegative than H, only that for X-H...Y-Z

the acceptor is an electron-rich region such as, but not limited to, a lone pair in Y or a pi-bonded pair in Y-Z. 

The Wikipedia page on electronegativity is a helpful introduction but I found section 6.4 in the classic Coulson's Valence extremely helpful.

There are several alternative definitions of electronegativity (Pauling, Mulliken, Alfred-Ronnow, Allen, ...). This highlights a few things:

   -like most intuitive chemical concepts they are not something that can be defined rigorously, without ambiguity, or in a reductionist manner (a point highlighted by Roald Hoffmann)

  -the concepts are useful for understanding semi-quantitative trends

  -these different definitions actually highlight the power of the concept because they show how a wide range of chemical and physical properties (bonding energies, dipole moments, charge distributions, ...) are correlated.
The graph below shows Pauling vs. Mulliken electronegativities

The "clearest" and most "precise" definition is that of Mulliken, where the electronegativity is the average of the ionisation energy and the electron affinity of the atom. This equals half of the ground state energy difference between the cation and the anion.
This means that if A and B have the same electronegativity that the ionic valence bond (VB) structures A+B- and A-B+ will have the same energy and so contribute equally to the full VB wave function, leading to no charge polarity.

To me the success of the concept also highlights the fact that predominantly chemical bonding is local.

Like most chemical concepts there are exceptions to their naive application. For example, carbon is less electronegative than oxygen, and so one might expect that in carbon monoxide (CO) there would be a net negative charge on the oxygen atom. However, the opposite is true.

What is the ground state of solid hydrogen?

The Journal of Chemical Physics website has a fascinating podcast with Roald Hoffmann, Neil Ashcroft, and Vanessa Labet talking about a series of 4 papers they have just published about "molecular" hydrogen under pressure. They illustrate some very rich and subtle physics and chemistry.

It highlights the importance of both physical and chemical insight, simple models, and how there are still these old problems waiting to be solved.

Will a chemist ever win the Nobel Prize in Physics?

There is an interesting editorial by Roald Hoffmann, What, Another Nobel Prize in Chemistry to a Nonchemist? in the latest issue in Angewandte Chemie International. 

Hoffmann, thoughtfully argues that chemists should not be upset [some are] that the chemistry prize seems to be increasingly awarded to people from outside chemistry departments [esp. biochemistry and molecular biology, but also physics and materials science].

He also asks the interesting question: will a chemist ever win the Nobel Prize in physics? He argues that Bednorz and Muller who discovered superconductivity in cuprate compounds might be considered chemists. I don't buy that. Their education, employment, and publications were clearly in the physics.

I welcome possible answers to Hoffmann's question.
My answer might be: in principle, yes; but in practice no. I think this may be partly because of the arrogant reductionism of influential parts of the physics community.
Possible areas impacting physics and to which chemists make important contributions include synthesis of new materials with exotic ground states, single molecule electronics, single molecule spectroscopy, glasses, ....

I thank Seth Olsen for bringing the article to my attention.

The role of theory in chemistry

I just read a nice article Theory in Chemistry by Roald Hoffmann. Although, written in 1974, I consider it just as poignant (or perhaps even more so) today. It should be read in full. But here are a few highlights to motivate you to read the 3 pages.

Is not the design and analysis of a beautiful experiment theory?

Given the human character, rationalisation poses no problem. Prediction is another matter. By and large, theory has not predicted much chemistry.

some exceptions.... I would call a credibility nexus - in which a group of experimentalists, otherwise skeptical of theory, suddenly found itself faced with the success of a simple theory. That set of specialists quickly became concerts, often zealots... 

the most important role of theory in chemistry is to provide a framework in which to think, to organize experimental knowledge

Hoffmann identifies six examples of "credibility nexus"

• Huckel's rules for the stability of different conjugated organic molecules
• Walsh's rules (1953) which showed how the geometry changes in excited states of polyatomic molecules could be derived from simple symmetry and bonding diagrams
• Huckel theory descriptions of the spin distribution in anion radicals
• crystal field theory for transition metal complexes which led to a renaissance of inorganic chemistry
• critical phenomena
• orbital symmetry conservation [for which Hoffmann later received the Nobel Prize]

I came across the article via a review article, "A Different Story of pi-Delocalization" by Shaik, Shurty, Davidovich, and Hiberty who claim that valence bond theory provides a  "credibility nexus".

5 Papers every computational chemistry student should read

I have a dream. That every advisor (supervisor) who gets a student to perform a computational chemistry calculation will have them read the following five papers. The papers are from a range of eras and with different emphasis. But, a common theme is the importance of calculations aiding concept development and being aware of the limitations these calculations.
Reading these papers should be like reading the road rules before you get your drivers license.
I list the papers in chronological order.

Present state of molecular structure calculations
C.A. Coulson (1960)
Quantum chemistry and its unachieved missions
Jean-Paul Malrieu (1998)
Is my chemical universe localized or delocalized? is there a future for chemical concepts?
Sason Shaik (2007)
Predicting Molecules - More realism , please!
Roald Hoffmann, Paul Schleyer, and Fritz Schaefer (2008)
Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexed
John P. Perdew, Adrienn Ruzsinszky, Lucian Constantin, Jianwei Sun, and Gabor Csonka (2009)

I welcome alternative suggestions. Later I may write more about the individual papers and why I think they are important.

Breaking down the physics-chemistry divide

Chemists and physicists tend to talk different languages, including when discussing the same thing. One important parallel is the common concept of the molecular orbitals in a molecule and the energy bands in a crystal. Specifically, the Huckel method to describe electronic properties of conjugated organic molecules is identical to the tight-binding method in solid state physics.

Yet these important parallels seem to rarely be pointed out in textbooks. [One exception is a brief mention in Walter Harrison's Electronic Structure and the Properties of Solids].
Recently when I have taught solid state I have pointed out the connection and sometimes worked through the nice treatment of Huckel theory in chapter 8 of the classic book Coulson's Valence by Roy McWeeny.

Besides showing the molecule-solid connection this can illustrate a few useful things including:

• How Bloch's theorem works in a finite system.
• How energy bands emerge in the thermodynamic limit (see above).
• The correspondence between bonding (anti-bonding) orbitals in a molecule and valence (conduction) bands in a crystal.
• The potential importance of electron-electron interactions, which are completely neglected in both Huckel and tight-binding approximations. Valence bond theory takes these interactions into account.

Any other ideas?

Some of the above parallels are explored in more detail in a beautiful article How Chemistry and Physics meet in the Solid State by Roald Hoffmann, and in a forthcoming book chapter by Ben Powell.

Entitled to a reading

Here are a few reasons why you should work hard at picking the title of your papers.

* They are one of your only chances to get people interested in actually reading your paper.

* When people are reviewing your CV many will just look at the title of your papers, as well as the journal they are published in. Interesting, diverse, informative, and understandable titles create a good impression. Boring, repetitive, and highly technical titles create a bad impression. Make sure all your papers don't have essentially the same title!

* They are fun.

What I generally do is to write down as many as five possible titles for the paper and then consider their relative merits and discuss them with co-authors and colleagues. This helps sharpen the title.

If you want to see some good examples look over the publication list of Roald Hoffmann. Here are just a few from the past decade:

A Molecular Perspective on Lithium-Ammonia Solutions

A Little Bit of Lithium Does a Lot for Hydrogen

A Bonding Quandary—A Demonstration of the Fact That Scientists Are Not Born With Logic

The Contributions of Through-Bond Interactions to the Singlet-Triplet Energy Difference in 1,3-Dehydrobenzene

“Half Bonds” in an Unusual Coordinated (S4)2- Rectangle

Emergent reduction of electronic state dimensionality in dense ordered Be-Li alloys

A Quantum Mechanically Guided View of Mg44Rh7

The Many Ways to Have a Quintuple Bond

A Pnictogen of Peculiar Posture

On the other hand you could go the Ph.D comics route: