Rudolph Marcus (1923-2026): theoretical chemical physicist

Rudolph Marcus died last week. He was 102. There is a nice obituary in The New York Times. He was best known for his theory of electron transfer, for which he was the sole recipient of the Nobel Prize in Chemistry in 1992.

Although the theory was proposed for electron transfer in a polar solvent it applies to a wide range of other systems, where two quantum states are coupled to one another and to an environment. One example is for Forster transfer of excitons between molecules, which is central to photosynthesis.

I will give a physics perspective, based on a talk I gave in Slovenia back in 2013. Slides are here.

Basically, Marcus proposed an effective Hamiltonian for two diabatic states and calculated a transition rate in a limit that is relevant to most chemical contexts.

The Hamiltonian can be viewed as the spin-boson model, in which a two-level system is coupled to a bath of harmonic oscillators. 

[The dense book by Weiss on Quantum Dissipative Systems makes the connection explicit in detail. A more accessible treatment may be chapter 16 in the book Chemical Dynamics in Condensed Phases by Nitzan.]

The Hamiltonian is 

This defines a spectral density, which is important for quantum decoherence, but not so much here, except it defines a timescale that determines the classical limit, which Marcus assumed.

This can be used to define a quantity central to Marcus' theory, the reorganisation energy.

The transition rate between the two quantum states is given by

I consider this to be one of the most important equations in chemical physics, particularly for the understanding and design of functional materials.

Aside. Much of this is equivalent to Holstein's 1959 treatment of incoherent polaron transport (see Mahan, Many-body physics).

A key experiment by John R. Miller, Lidia T. Calcaterra and Gerhard L. Closs in 1984 showed how good the theory was and how its predictions were counter-intuitive. The authors considered a family of molecules that allowed them to tune epsilon, the energy difference between the two electronic states. [On the graph below epsilon = - Delta G].

The vertical scale is the reaction rate on a logarithmic scale. It varies by four orders of magnitude.

What is surprising? As stated at the Nobel prize award ceremony:

"The quadratic equation predicts that electron transfer reactions will occur more slowly the larger the driving force of the reaction is. This phenomenon received its own name, “the inverted region.” To a chemist, the phenomenon is just as unexpected as when a skier finds himself gliding more slowly down a slope the steeper it is."

The theory implies an important design principle for functional materials: if optimising functionality means maximising the reaction rate, then tune the energy difference epsilon to equal the reorganisation energy E_R.

The theory illustrates two important aspects of emergence: effective theories and universality. Many different systems can be described by the same theory. The environment may involve many degrees of freedom and its coupling to the system is characterised by many parameters (the M_alpha above). However, only one parameter matters, the reorganisation energy.

For Australians, there is some ambivalence about the way Marcus' name is often solely associated with electron transfer theory. We often refer to it as Marcus-Hush or Hush-Marcus theory because Noel Hush did similar work around the same time. Some of the history is recounted here by Ian Rae and Jeff Reimers. There are also subtle debates about whether the electron transfer is adiabatic or non-adiabatic.

My only personal interaction with Marcus was in 2011 when I visited the chemistry department at Caltech. Marcus kindly took me to lunch at the faculty club, along with his research group. Then he was 84 years old. He kept publishing papers until he died.

Philosophical perspectives on the emergence of molecular structure

 In philosophical discussions of emergence and reductionism in chemistry, molecular structure has received significant attention and debate. Sometimes molecular structure is used to justify strong emergence, i.e., that molecular cannot be predicted, even in principle, solely from quantum theory.

Primas was one of the first to claim that molecular structure could not be reduced to quantum physics. Consider the following statements.

“From a physical point of view the crucial point of a Born–Oppenheimer description is not a simplification of the mathematical problem, but the replacement of the basic theory by a related but qualitatively new one…

[the structure of benzene] does not exist in a full quantum-theoretical description since electrons and nuclei are entangled by Einstein-Podolsky–Rosen correlations. The concept of molecular structure does not follow from first principles - all molecules with the same empirical formula have the same Schrödinger equation, so that, at this level, the shape of a molecule as the main feature of molecular chemistry is simply not in evidence. In a quantum theoretical description the molecular shape emerges by abstracting from the actually existing Einstein-Podolsky–Rosen correlations between the electrons and the nuclei. Historically, the structure concept has been introduced into quantum chemistry by the so called Born-Oppenheimer approximation. But this terminology is misleading since the main issue is not an approximation, but the breaking of a holistic symmetry.”

I have italicised claims that are contestable and that I will discuss further below.

Cartwright has given philosophical arguments as to why chemistry cannot be reduced to physics. In this context, she claims (pp. 106-7)

“The typical method of quantum-mechanical treatment of molecules begins with the Born–Oppenheimer approximation…

This approximation treats the atomic nucleus as a classical particle. But this fundamentally violates quantum mechanics which, following the Heisenberg uncertainty principle, maintains that we cannot have a simultaneous assignment of fixed positions and fixed momenta. The approximations that provide the reduction violate the very theory that the chemistry is being reduced to… the success of quantum chemistry relies fundamentally on assumptions that belong to classical chemistry.” 

This claim that the BOA violates the Heisenberg uncertainty principle was rebutted in an earlier post and discussed in more detail by Huggett et al. Nevertheless, Lombardi et al. are not satisfied by the rebuttal.

Hendry claimed molecular structure is evidence of strong emergence and/or downward causation. In a similar spirit, Ellis and Drossel argued that crystal structures in solid state physics are evidence of strong emergence. The arguments of Hendry have been criticised by Seifert. The arguments centre around the fact that the molecular structure is a classical entity and a concept that is imposed, whereas a logically self-consistent approach would treat both electrons and nuclei quantum mechanically. It is claimed that the existence of molecular structures is assumed and not derived in quantum chemistry calculations as they assume the Born-Oppenheimer approximation (BOA). 

Scerri responded to these arguments claiming that chemistry (particularly the concept of molecular structure) is irreducible to quantum physics. He claimed these arguments are not valid because they misunderstand the role of the BOA. It does not violate the uncertainty principle and in practice chemists can and do perform non-BOA approximations. 

Fortin et al. rejected the view that decoherence can explain isomerism, as decoherence does not resolve issues associated with the quantum measurement problem. 

Franklin and Seifert claim “the problem of molecular structure just is the quantum measurement problem.” This is debatable. Most molecular structures can be understood in terms of the nuclear probability density having a unique maximum and decoherence then is not relevant. Decoherence and the collapse of the nuclear wavefunction are only relevant in systems such as ammonia and stereoisomers in which there are molecular structures with equal energy and separated by high energy barriers.

I now respond to some of the contestable claims of Primas.

“electrons and nuclei are entangled by Einstein-Podolsky–Rosen correlations”

It is possible to quantify and calculate the amount of entanglement between electrons and nuclei in a specific quantum state. In an EPR state the entanglement is maximal. In the BOA wavefunction entanglement is present, but is absent in the crude BOA. The entanglement has been estimated in benzene and is very small. The only molecules where the entanglement may be significant are those, such as isomers, where there are two degenerate molecular structures and the overlap of the associated nuclear wavefunctions is small (i.e., the tunnel splitting is small). But then, in most chemical situations decoherence will wash out this entanglement.

“all molecules with the same empirical formula have the same Schrödinger equation, so that, at this level, the shape of a molecule as the main feature of molecular chemistry is simply not in evidence.”

This is the problem of isomers. It is resolved because isomers are present in the solution to the Schrödinger equation, as I argued earlier.

“the crucial point of a Born–Oppenheimer description is not a simplification of the mathematical problem, but the replacement of the basic theory by a related but qualitatively new one… the main issue is not an approximation, but the breaking of a holistic symmetry.”

This seems subjective to me. I see the BOA as just a well-justified approximation. The electrons and nuclei are treated differently because they are. They have vastly different masses and this affects how they can be treated in any solution to the full Hamiltonian. Nevertheless, the BOA is a quantum theory and the nuclei are described by a wavefunction.

As discussed earlier, for small molecules in practise it is no longer necessary to use the BOA and the electrons and nuclei can be treated on an equal footing. Molecular structure is present in solutions to the full Schrödinger equation.

I wonder if the objection to use of the BOA is any different to the use of approximations in other theories? For example, consider theoretical treatments of the motion of planets in the solar system. The effects of all the planets are not treated on an equal footing. The effect of other planets on a planet of interest are treated perturbatively.

In conclusion, the arguments that molecular structure is evidence of strong emergence are weak. Some confusion may partly arise from misinterpreting the significance of the following valid observations.

i. Molecular structures were first conjectured before quantum theory was proposed.

ii. The BOA was proposed to explain molecular structure from quantum theory.

iii. Today, almost all calculations of molecular structure assume BOA.

iv. Chemists talk about molecular structures largely in classical not quantum terms.

However, the scientific reality is that for small molecules their structure, can be understood, described, and calculated in purely quantum terms. Yet, that understanding provides a strong justification for the validity of the BOA and for the convenience of using classical language to describe molecular structure.

I stress that the weakness of the arguments for the strong emergence of molecular structure, does not mean that an emergent perspective is not relevant to chemistry. Molecular structure is emergent, when defined in terms of novelty. This leads to effective theories defined in terms of potential energy surfaces. Furthermore, as the next section argues chemistry exhibits novel properties, concepts, and theories that are not present in physics.

This post is extracted from Section 15, of my review article "Emergence: from physics to biology, sociology, and computer science."

A new version of my review article on emergence

On the arXiv, I have posted a new version of my review article, Emergence: from physics to biology, sociology, and computer science.

I have added expanded sections on molecular structure, quantitative measures of causal emergence, and biological evolution.

There are also many minor additions and corrections. I hope the hyperlinked Table of Contents is helpful.

I welcome feedback and suggestions. I am sure there is much more to do.

Quantum justification for classical discussions of potential energy surfaces in chemistry

 In computational quantum chemistry, the Born-Oppenheimer approximation (BOA) is used to determine potential energy surfaces (PES) for electronic states of molecules. It is standard practise to identify local minima on a PES with molecular structures. Molecular binding energies are identified with the difference in energy between these minima and the energies of the isolated atoms of which the molecule is composed.

 A chemical reaction between two molecules A and B to produce C can be understood in terms of the PES for the composite system consisting of all the atoms in A and B. The dynamics of the chemical reaction can be described in terms of a path on the PES that goes from the local minimum associated with A and B infinitely far apart to the minimum associated with the structure C. The path will pass through a saddle point on the PES and this is identified with a transition state in the chemical reaction and its energy determines the activation energy for the chemical reaction. The local curvature of the PES near a minima can be used to determine force constants for harmonic motion and the associated vibrational frequencies of a molecule.

This picture is a completely classical one and so may motivate a claim that chemists mix classical and quantum concepts and calculations in an ad-hoc manner. (This kind of argument is often used by philosophers to claim that chemistry cannot be reduced to physics). This is unfair because a quantum description of the nuclear dynamics can be given in terms of quantum wave-packets, consistent with the Heisenberg uncertainty principle, and whose dynamics is defined by the quantum equation for nuclear motion that is given by the BOA. Furthermore, the dynamics of the centre of a wave-packet is given by classical equations of motion (Ehrenfest’s theorem). Thus, the classical language used by chemists can be viewed as a justified and compact version of a quantum description.

Structural isomers. Isomers are associated with different local minima on the electronic ground state potential energy surface for a given combination of atoms. To understand a quantum description of isomers, consider a reaction coordinate associated with an isomerisation reaction (i.e. conversion of one isomer to the other). There are three energy scales of relevance: the energy difference between the ground state energy of the two isomers, the magnitude of the barrier height (activation energy), and the quantum zero-point energy associated with vibrations in the direction of the reaction coordinate. Denote these energies as dE, Eb , and Ezp, respectively. If dE ~ Ezp << Eb then the nuclear probability density rho(R) for the vibrational ground state will have two local maxima, corresponding to the geometries of the two isomers. If dE >> Ezp then the nuclear probability density rho(R) for the vibrational ground state will have only one local maxima, corresponding to the lower energy isomer geometry. However, the geometry of the higher energy isomer can be found as a local maximum in the nuclear probability density rho(R) for one of the excited vibrational ground states. 

 Figure. Potential energy surface associated with the two structural isomers of HOCO. TS_n denote different transition states associated with the chemical reaction OH + CO -> H + CO2. Taken from Bui et al.

What does the Born-Oppenheimer approximation mean for emergence?

Most philosophical debates about the emergence of molecular structure centre around the issue of irreducibility. Specifically, can the existence of structures be predicted from quantum theory without assuming their existence or invoking classical concepts? I will argue that the answer is yes, contrary to much of the philosophical literature, which relies heavily on the widespread use of the Born-Oppenheimer approximation (BOA) in quantum chemistry calculations. However, the fact that these arguments for irreducibility are weak does not mean that emergence (defined in terms of novelty) is not central to chemistry.

In a previous post, I discussed recent work showing how the BOA is not necessary for quantum chemistry and that molecular structure can be defined independently of it.

However, since the BOA plays a central role in the philosophical arguments, it is worth reviewing what it is and what it does and does not assume or mean.

In 1927, Born and Oppenheimer introduced an approximation to allow the solution of the full quantum equations for electrons interacting with charged nuclei. Without the BOA, much of theoretical chemistry and solid-state physics would be incredibly difficult in practice. The approximation is based on the separation of time and energy scales associated with electronic and nuclear motion. It leads to the concept of potential energy surfaces for electronic states. They define an effective theory for the dynamics of the atomic nuclei in a molecule or solid.

The full Hamiltonian (given earlier) can be denoted by

 

where the first term is the kinetic energy operator for the nuclei. In the Born-Oppenheimer approximation (BOA) the full wavefunction is written as a product of a nuclear wavefunction and an electronic wavefunction.

Substituting this in the eigenvalue equation for the full Hamiltonian leads to separate eigenvalue equations for the electronic and nuclear wavefunctions, assuming terms depending on gradients with respect to R of the electronic terms can be neglected.

 

In the first equation, the nuclear co-ordinates appear as parameters not as operators. This is central to the philosophical debates.

The second equation can be viewed as an effective Hamiltonian for the nuclear degrees of freedom. The function E_e(R) defines the potential energy surface of the molecule. 

In the BOA the nuclear probability distribution defined above is

 

As discussed in the earlier post, the structure of many molecules can be defined in terms of the value of R at which the probability is maximum. 

I make four points about the BOA that are relevant to philosophical debates about whether molecular structure is predictable in a logically consistent manner from quantum theory.

1. The BOA does treat the nuclear degrees of freedom quantum mechanically. They are described by the nuclear wavefunction Phi(R), which is determined by the second eigenvalue equation. Consequently, the BOA does not violate Heisenberg’s uncertainty principle, contrary to some claims in the philosophy literature.

2. The BOA is not ad hoc. Corrections to it can be calculated and have been for many molecules. These corrections are typically small, being of order (me/Mi)^1/2. Exceptions, such as near conical intersections (where the potential energy surfaces for two electronic states touch) are well-known and well-studied.

3. For most small molecules, the results of BOA calculations compare favourably with wave-functions obtained from solutions of the full quantum Hamiltonian. When there are differences, they are largely small quantitative differences. When the differences are qualitative, they have largely been anticipated from knowledge of the limitations of the BOA.

4. The Born-Oppenheimer approximation is an example of a general approach to quantum mechanics problems, discussed by Migdal. Consider a system composed of two subsystems that have dynamics on two vastly different time scales, termed fast and slow. The effects of the fast system on the slow system can be treated by adding a potential energy term to the Hamiltonian operator of the slow system. 

In forthcoming posts, I will discuss quantum justifications for classical descriptions of nuclear dynamics on potential energy surfaces and then discuss philosophers' views about the BOA and molecular structure.

The emergence of molecular structure from quantum theory

Most debates about the emergence of molecular structure centre around the issue of irreducibility. Specifically, can the existence of molecular structures be predicted from quantum theory without assuming their existence or invoking classical concepts?

Consider a molecule that contains Ne electrons and Nn atomic nuclei (ions). The full quantum-mechanical Hamiltonian for the system is 

where e is the electronic charge, rj is the position of the j-th electron, Zi  and Mi are the charge and mass, respectively, of the i’th ion with position co-ordinate Rj. This is the Hamiltonian that Laughlin and Pines dubbed “The Theory of Everything” because if the solution (i.e., eigenstates and eigenvalues of the Hamiltonian operator) could be found it would describe almost all of chemistry and materials science.

This Hamiltonian treats the electrons and nuclei on an equal footing. 

For isomers, the Hamiltonian is identical. However, as will be discussed in a later post, that does not preclude solutions to the Hamiltonian that can describe isomers. 

The Hamiltonian has global translational and rotational symmetry, where all the particles undergo the same rotation or translation. In contrast, molecular structures may have discrete rotational symmetries. However, this is not necessarily a problem, as an eigenstate of a quantum problem can transform according to a non-trivial irreducible representation of the symmetry. For example, except the s-orbitals all the orbitals of the hydrogen atom are spatially anisotropic.

The electrons are identical particles and so have permutation symmetry. They are fermions with spin-1/2 and so any eigenstate must be antisymmetric under the exchange of two electrons. The energies associated with this exchange are crucial to the formation of chemical bonds and the stability of molecular structures.

If two or more atoms in the molecule are identical, then any exact eigenstate must be consistent with permutation symmetry. If a nucleus is composed of an even (odd) number of nucleons, then it is a boson (fermion) with integer (half-integer) spin, and eigenstates must be symmetric (antisymmetric) under exchange of identical nuclei. However, the corresponding exchange energies are relatively small (because the quantum delocalisation of the nuclei is small) and consequently most practical calculations of the eigenstates do not make this requirement of the eigenstates. Nevertheless, if the electrons and nuclei are treated on equal footing, this should be done. Although this is challenging, it has been done recently, as discussed below. 

Full quantum solutions of the Hamiltonian

In most computational quantum chemistry, the Hamiltonian is solved in the Born-Oppenheimer approximation, which will be introduced and discussed later. This is a source of some confusion and contention in philosophical discussions about the emergence of molecular structure.

Due to advances in methodology and computational power over the past few decades, it has become possible in practise to solve the full quantum Hamiltonian for small molecules. There are three levels of complication associated with this: quantum nuclear motion, rotational symmetry, and some nuclei being identical particles. There are also two challenges: first, finding the eigenstates and second, deducing the molecular structure from the eigenstates.

To begin, I consider the simplest case and ignore the complications associated with rotational symmetry or identical nuclei. This provides some insight and undermines some objections in the philosophical literature.

The ground state eigenfunction can be written as

where r and R are 3Ne and 3Na -dimensional vectors, respectively. Note that this function will have a complicated structure as it will depend on the spin states of all the electrons, denoted by s.

A probability distribution (reduced density matrix) for the positions of the nuclei is given by

where the sum is over all the electron spin degrees of freedom.

For many molecules, but not all, this probability distribution will have a unique global maximum at the coordinates R_0. This set of coordinates defines the geometry of the molecular structure. The physics underlying the existence of well-defined maxima is that the mass of the nuclei is much larger than the mass of the electrons, and as a result, the zero-point motions of the nuclei are much smaller than the separation of the nuclei in the molecular structure.

Note that the nuclear probability distribution is regularly measured in scattering experiments (using X-rays, neutrons, or electrons), and its maxima are used to determine the structures of molecules and crystals. The Debye-Waller factor is a measure of the width of the probability distribution. At low temperatures, it is determined by quantum zero-point motion. In other words, it is well established experimentally that classical molecular structures are an approximation to a fluctuating quantum structure.

Not every molecule will have a probability distribution with a unique maximum. An example is ammonia. As discussed further below, it has two maxima; each represents an umbrella geometry, and they are related by an inversion symmetry. The ground state wavefunction of the whole system is a superposition of two quantum states, each being associated with one of the two umbrella geometries, and the electronic and nuclear degrees of freedom are entangled with one another.

A general quantum definition of molecular structure

Lang et al. have recently overcome the challenges mentioned above to determine molecular structure in a manner that treats the electrons and nuclei on an equal footing with regard to quantum theory. They have considered both rotational symmetry and nuclear permutation symmetry and given a general definition of molecular structure involving nuclear probability densities calculated from the full wavefunction. They have explicitly performed these calculations for D3+, (where D is deuterium). The result is that the molecule has the same triangular structure that is observed experimentally and calculated using the Born-Oppenheimer approximation. This work is significant because it explicitly shows that molecular structure can be predicted in practice, not just in principle, from quantum theory.

In a forthcoming post, I will discuss the Born-Oppenheimer approximation and some of the confusion associated with it.

Are chemical isomers emergent?

In discussions of emergence, particularly in chemistry, isomers are often given as an example of an emergent phenomenon. In Anderson's original "More is Different" article, he discussed the chirality of sugar molecules as an example of symmetry breaking. More recently, isomers (and the associated concept of molecular structure) are invoked to justify contentious claims about strong emergence and downward causality.

Here, I explain what isomers are and consider whether they are emergent in the sense of novelty, i.e., they have properties that are qualitatively different from their constituents.

In a later post, I hope to address the more general and knotty problems of molecular structure and the Born-Oppenheimer approximation.

Structural isomers

These occur when a specific collection of atoms (chemical formula) can have more than one molecular structure. An example, shown below, is C3H4.

Each structure has different chemical and physical properties. Aggregates of each molecule can have different properties such as boiling and melting points.

Some isomers are more stable than others. They may be able to interconvert, but sometimes not on laboratory time scales.

From the point of view of a ground state potential energy surface, the different isomer structures correspond to different local minima on the surface.

Stereoisomers

The simplest example is HFClBr. There are two stable structures shown below. They are related by a chiral (mirror) symmetry. They differ physically in that they rotate the plane of polarisation of incident light in opposite directions. 

The isomers, known as enantiomers, have the same ground state energy. In terms of a potential energy surface, they correspond to two different minima and are separated by a high-energy barrier. In principle, the two forms can quantum-tunnel between each other.

Chemically, the two isomers differ in how they react with other chiral molecules.

Chirality is central to molecular biology. Proteins are made of amino acids, and in nature they all have the L-form. Most forms of DNA involve double helices with right-handed chirality. 

The chirality of drug molecules matters, as tragically found with thalidomide in the 1950s. 

Emergence?

The constituent components of these molecules can be viewed as electrons and atomic nuclei. Alternatively, the components could be viewed as the atoms they are made of. In both cases, the parts of the system do not have the structure and properties that the system does. The atoms, nuclei, and electrons all have spherical symmetry, whereas the molecules do not. Another argument is that since the isomers are qualitatively different from one another, at least one of them must be qualitatively different from the components. Hence, these molecular structures can be viewed as emergent.

However, this goes against the view that we generally associate emergence with systems with many interacting parts. If we take two massive particles interacting by gravity, they can form a stable orbit. Neither particle has this property, but we don't generally claim that such orbits are emergent.

[I am grateful to a commenter on an old post who pointed this out].

Isomers illustrate the diversity of forms often associated with emergence.

There are subtleties associated with the stability of enantiomers and the associated breaking of chiral symmetry. This is similar to the issue of ammonia having a stable pyramidal structure. (Also discussed by Anderson in "More is Different"). An isolated molecule in a vacuum will have no chirality. The ground state is a quantum superposition of both enantiomers. However, in the laboratory, the interaction of each molecule with its environment, such as other molecules, leads to decoherence that prevents quantum tunnelling. In that case, there are an infinite number of degrees of freedom associated with the environment, and they are crucial for the emergence of enantiomers.

Responding to scientific uncertainty

Science provides an impressive path to certainty in some areas, particularly in physics. However, as scientists seek to describe increasingly complex entities, moving from chemistry to biology, and then to humans and societies, the level of uncertainty increases.

One observes a wide range of responses to scientific knowledge being uncertain. Here are a few.

Denial. Science is about facts and absolute truth. There really isn’t a problem. We should just trust the scientists.

Minimisation. There is some uncertainty, but it isn’t anything to be concerned about. Some scientists will also minimise any uncertainty about their own research. This may occur because of career ambition. Others will minimise public discussion of uncertainty to try and avoid promoting the science scepticism discussed below.

Optimistic perseverance. The uncertainty is openly acknowledged. Some of the uncertainty does not matter for what we need to know. Other uncertainties can be reduced by further scientific work, such by more precise measurements with new instruments or by developing more sophisticated theories.

Total scepticism. There is a suspicion about the validity of most scientific knowledge, particularly that which is perceived to have philosophical, religious, or political implications.

Suspicion about science

In spite of the success of science at describing the material world and leading to powerful and useful technologies, there is much public suspicion of science. On the one hand, this is understandable given that science has led to technologies with undesirable health, environmental and social consequences. Some scientists, governments and companies have lied about these consequences and hidden them from the public. Human subjects have been abused in medical experiments. Drugs that were claimed to be effective and safe turned out to be ineffective or have undesirable side effects. Science has been used for ideological purposes. Sometimes scientists have faked results to advance their own careers. However, these failures should not undermine our trust in reliable scientific knowledge. Distinctions should be made between the bodies of knowledge, the applications of that knowledge, and the actions of institutions. I now discuss several common claims in public discussion that are used to justify scepticism of scientific knowledge.

Science is always changing. 

One day, scientists tell you that chocolate is good for your health, and the next year they say it is bad for you. And that is just the start. Then there are eggs, wine, running marathons, and cheese. They just can’t make up their mind. So why should we trust them? At one time, they believed in phlogiston and the aether. Now they say they don’t exist. Aristotle was replaced by Newton, who was replaced by Einstein. So why believe in human-induced climate change, biological evolution, vaccines, the Big Bang theory, or Einstein’s theories?

It is true that scientific knowledge does develop and change over time. However, today we have incredibly detailed observations and theories in physics, astronomy, chemistry, biology, and geology. Any future changes will be relatively minor because they will have to be consistent with all the knowledge we have now. Furthermore, when theories change, such as when Einstein superseded Newton, they don’t show that the old theory was completely wrong, but rather that it applied in a limited domain. For example, Newton’s theories of motion and gravity are extremely reliable when it comes to objects that are much larger than atoms, less dense than a black hole, and are moving at speeds less than about 10,000 kilometres per second. This is why engineers spend years learning Newton’s theories, not Einstein’s. If you want to build a good bridge or a rocket, Newton is good enough. He is not wrong.

Update. (Jan. 19). I just discovered that the NY Times had a recent op-ed Science Keeps Changing. So Why Should We Trust It?

“Well, that’s just a theory.” 

In popular debate, such a refrain may be applied to the theory of biological evolution, the Big Bang theory in cosmology, or human-induced climate change. The claimant usually wants to dismiss a particular theory as just idle speculation. Here, the term “theory” is used in the same sense as everyday speculations, such as “I have a theory as to why the president resigned,” or “I have a theory about why my computer is running so slowly.” These are just stories that sound somewhat plausible. In contrast, scientific theories in physics, such as quantum theory and Einstein’s theories of relativity, have precisely defined mathematical formulations that have been checked for logical consistency, made specific predictions, and tested to great precision in experiments. They are not “just theories.” For example, for the Big Bang theory about the beginning of the universe and Darwin’s theory of biological evolution and diversity, there are many independent lines of evidence that are consistent with each theory.  

Scientists cannot be trusted. 

They are not committed to the truth, but rather to their own interests and agendas, related to their careers, politics, and religion. They close ranks and support the status quo of current scientific “dogma”, rather than being open to original thinkers who critique it and propose alternative theories. They don’t want to lose their well-paid jobs and lucrative grants. 

On the one hand, scientists can be conservative and resistant to new ideas. On the other hand, there are significant career incentives to overturn existing knowledge and have your radical new theory accepted. That is how some scientists become famous and win Nobel Prizes. The reasons it does not happen very often are not necessarily for social or ideological reasons. Many of the theories we have today can explain an awful lot. It requires a lot of evidence, carefully acquired and checked, to convince people that those theories need to be modified, let alone abandoned. This may take decades. But it does happen. An example is the Big Bang theory of the universe, whose acceptance was initially resisted because it went against the prevailing view that the universe did not have a beginning. In biology, the discovery in 1970 of the enzyme reverse transcriptase went against a popular version of the “Central dogma” of molecular biology that DNA was always converted to RNA and not the reverse. That discovery led to a Nobel Prize.

I don’t trust scientists. I will do my own research. There is lots of good material from unbiased sources on the internet.

The internet provides a range of information and perspectives on practically any issue imaginable, including science. The material is particularly vast and controversial on biological evolution, the beginning of the universe, fundamental physics, the age of the earth, climate change, and medicine. Since the covid-19 pandemic, scepticism of the effectiveness and safety of vaccines has increased. 

Ivermectin is a drug that was developed as a treatment for parasite worms. Its incredible success was recognised by the award of the 2015 Nobel Prize in Physiology or Medicine to William Campbell and Satoshi Omura, who discovered the drug. During the pandemic, high-profile politicians and social media influencers promoted ivermectin as a treatment for covid-19, even after systematic medical studies showed it was ineffective. Recently, it has gained a reputation as a “miracle” drug that can even cure cancer, but this is being suppressed by the medical establishment. All clinical trials have shown the drug is ineffective for human ailments, beyond deworming. Nevertheless, there are groups on social media with hundreds of thousands of members that discuss the conspiracy, how to get the drug, and the experiences of participants using it to treat a wide range of ailments. Danny Lemoi, a founder of one of the largest groups, died in 2023 after taking massive daily doses of the drug for several years to treat a heart condition. Afterwards, one member of the group wrote “No one can convince me that he died because of ivermectin. He ultimately died because of our failed western medicine which only cares about profits and not the cure.”

Fans of ivermectin claim that they are escaping the biases and vested interests of the medical establishment and Big Pharma as they pursue the truth. However, they are not escaping bias and vested interests. Successful social influencers build their reputations and million-dollar incomes from promoting scepticism. If there is no conspiracy, just scientific uncertainty and occasional incompetence and malpractice, their following collapses. Populist politicians build their careers on criticism of and stoking resentment towards elites, such as the medical establishment. The authority of the medical establishment is replaced with the authority of the popular opinion of a group of people whose views are shaped by social media algorithms, intuition, and anecdotal experience.

My purpose in giving the example of Ivermectin is not to start a detailed critique of science scepticism. Rather, it is to illustrate the role that the interplay of trust, authority, and tradition plays in how we determine what is true and what to act on. There are two competing traditions here: the populism of alternative medicine and the elitism of professional medicine. Each has its own sources of authority. In the end, it boils down to who we trust. We do not have the time, energy, resources or inclination to check the veracity of every single piece of information we have access to. We take shortcuts. This is what tradition does for us, for better and worse. Thus, we cannot escape tradition. We are all swimming in traditions, many of which are in conflict with one another. The question is whether we are aware of it and what we do with that awareness.

Elastic interactions and complex patterns in binary systems

One of the many beauties of condensed matter physics is that it can reveal and illuminate how two systems or phenomena that at first appear to be quite different actually involve similar physics. This is an example of universality: for emergent phenomena, many details don't really matter. One example is the similarities between superconductivity and superfluidity. A consequence of universality is that the same concepts, techniques, toy models, and effective theories can be used to describe a wide range of systems.

The complex organometallic molecules, known by the misnomer "spin crossover" compounds, exhibit a rich range of phase transitions and types of spatial order. Key aspects of the physics are the following.

• Each transition metal ion can be in one of two possible states: low-spin or high-spin. 
• The size of each molecular complex depends on the spin state.
• Consequently, the molecules interact with their neighbours via elastic interactions.

A toy model that can describe this is expanding balls connected by springs. Various versions of this type of model are reviewed here. The simplest version is the chain model below.

It turns out there are other classes of systems described by similar models. As far as I am aware, this was first pointed out in Consequences of Lattice Mismatch for Phase Equilibrium in Heterostructured Solids Layne B. Frechette, Christoph Dellago, Phillip L. Geissler

That paper is motivated by experiments on the growth of semiconductor quantum dots, by ion exchange, such as when CdSe is bathed in an Ag-rich solution and Ag2Se is produced with heterostructures (i.e., patterns of Ag and Se ions) that are different from the bulk crystal.

They consider the balls and springs model above on a triangular lattice.

They also point out how similar physics is relevant to binary metal alloys, e.g, AgCu, citing 

Ising model for phase separation in alloys with anisotropic elastic interaction—I. Theory, P. Fratzl and O. Penrose

Those authors consider a square lattice with elastic interactions associated with bond stretching along the edges and diagonals of the squares and bending of the square angles.

Frechette et al. also mention experiments on thin films of  DNA modified metallic nanoparticles. Compared to atomic systems these can tolerate larger lattice-mismatch before the formation of defects due to lattice strain.

Other systems (not mentioned) described by similar Ising models are metal-hydrogen systems, where the Ising pseudospin signifies whether a hydrogen atom is present at a particular site in the metallic crystal.

Frechette et al. start with the ball and springs model and "integrate out" the springs to obtain an effective Hamiltonian, which is an Ising model.

The spatial range of the interaction between Ising spins is shown in the colour-shaded plot below.

The interaction has two components.

One is an infinite range "ferromagnetic" part, seen as the light blue below.

The second is a short-range interaction which is mostly "antiferromagnetic" (i.e., red), but extends over several lattice sites. (Note, this interaction will be frustrated on the triangular lattice).

Using this toy model, Frechette et al. can obtain complex patterns (heterostructures) similar to those seen in quantum dots grown by ion exchange.

There is some subtle (and confusing) physics associated with deriving the Ising model from the ball and springs model. 

Due to the long-range nature of elastic interactions, the boundary conditions matter. 

The infinite range part of the Ising interaction arises from dealing with the lattice constant for the crystal, depending on the net "magnetisation" of the "spins". But that is a story for another day.

Confusing bottom-up and top-down approaches to emergence

Due to emergence, reality is stratified. This is reflected in the existence of semi-autonomous scientific disciplines and subdisciplines. A major goal is to understand the relationship between different strata. For example, how is chemistry related to physics? How is genetics related to cell biology?

Before describing two alternative approaches —top-down and bottom-up —I need to point out that in different fields, these terms are used in opposite senses. That can be confusing!

In the latest version of my review article on emergence, I employ the same terminology traditionally used in condensed matter physics, chemistry, and biology. It is also consistent with the use of the term “downward causation” in philosophy. 

Top-down means going from long-distance scales to short-distance scales, i.e., going down in the diagrams shown in the figure above. In contrast, in the quantum field theory of elementary particles and fields (high-energy physics), “top-down” means the opposite, i.e., going from short to long distance length scales. This is because practitioners in that field tend to draw diagrams with high energies at the top and low energies at the bottom.

Bottom-up approaches aim to answer the question: how do properties observed at the macroscale emerge from the microscopic properties of the system? 

History suggests that this question may often be best addressed by identifying the relevant mesoscale at which modularity is observed and connecting the micro- to the meso- and connecting the meso- to the macro. For example, high-energy degrees of freedom can be "integrated out" to give an effective theory for the low-energy degrees of freedom.

Top-down approaches try to surmise something about the microscopic from the macroscopic. This has a long and fruitful history, albeit probably with many false starts that we may not hear about, unless we live through them or read history books. Kepler's snowflakes are an early example. Before people were completely convinced of the existence of atoms, the study of crystal facets and of Brownian motion provided hints of the atomic structure of matter. Planck deduced the existence of the quantum from the thermodynamics of black-body radiation, i.e. from macroscopic properties. Arguably, the first definitive determination of Avogadro's number was from Perrin's experiments on Brownian motion, which involved mesoscopic measurements. Comparing classical statistical mechanics to bulk thermodynamic properties gave hints of an underlying quantum structure to reality. The Sackur-Tetrode equation for the entropy of an ideal gas hinted at the quantisation of phase space. The Gibbs paradox hinted that fundamental particles are indistinguishable. The third law of thermodynamics hints at quantum degeneracy. Pauling’s proposal for the structure of ice was based on macroscopic measurements of its residual entropy. Pasteur deduced the chirality of molecules from observations of the facets in crystals of tartaric acid. Sometimes a “top-down” approach means one that focuses on the meso-scale and ignores microscopic details.

The top-down and bottom-up approaches should not be seen as exclusive or competitive, but rather complementary. Their relative priority or feasibility depends on the system of interest and the amount of information and techniques available to an investigator. Coleman has discussed the interplay of emergence and reductionism in condensed matter. In biology, Mayr advocated a “dual level of analysis” for organisms. In social science, Schelling discussed the interplay of the behaviour of individuals and the properties of social aggregates. In a classic study of complex organisations in business, understanding this interplay was termed differentiation and integration.

I thank Jeremy Schmit for requesting clarification of this terminology.

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. 

------

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.

Science job openings in sunny Brisbane, Australia

Bribie Island, just north of Brisbane.

The University of Queensland has just advertised several jobs that may be of interest to readers of this blog, particularly those seeking to flee the USA.

There is a junior faculty position for a theorist working at the interface of condensed matter, quantum chemistry, and quantum computing.

There is also a postdoc to work on the theory of strongly correlated electron systems with my colleagues Ben Powell and Carla Verdi.

There is a postdoc in experimental condensed matter, to work on scanning probe methods, such as STM, with my colleague Peter Jacobson.

Glasshouse Mountains. Just north of Brisbane.

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 an associated quantum transition (at a wavelength of 1.2 cm) 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.

Pattern formation and emergence

Patterns in space and/or time form in fluid dynamics (Rayleigh-Bénard convection and Taylor-Couette flow), laser physics, materials science (dendrites in the formation of solids from liquid melts), biology (morphogenesis), and chemistry (Belousov-Zhabotinsky reactions). External constraints, such as temperature gradients, drive most of these systems out of equilibrium. 

Novelty. 

The parts of the system can be viewed as the molecular constituents or small uniform parts of the system. In either case, the whole system has a property (a pattern) that the parts do not have.

Discontinuity. 

When some parameter becomes larger than a critical value, the system transitions from a uniform state to a non-uniform state. 

Universality. 

Similar patterns, such as convection rolls in fluids, can be observed in diverse systems regardless of the microscopic details of the fluid. Often, there is a single parameter, such as the Reynolds number, which involves a combination of fluid properties, that determines the type of patterns that form. Cross and Hohenberg highlighted how the models and mechanisms of pattern formation across physics, chemistry, and biology have similarities. Turing’s model for pattern formation in biology associated it with concentration gradients of reacting and diffusing molecules. However, Gierer and Meinhardt showed that it is sufficient to have a network with competition between short-range positive feedback and long-range negative feedback. This could occur in a circuit of cellular signals.

Self-organisation. 

The formation of a particular pattern occurs spontaneously, resulting from the interaction of the many components of the system.

Effective theories. 

A crystal growing from a liquid melt can form shapes such as dendrites. This process involves instabilities of the shape of the crystal-liquid interface. The interface dynamics are completely described by a few partial differential equations that can be derived from macroscopic laws of thermodynamics and heat conduction. A helpful review is by Langer. 

Diversity. 

Diverse patterns are observed, particularly in biological systems. In toy models, such as the Turing model, with just a few parameters, a diverse range of patterns, both in time and space, can be produced by varying the parameters. Many repeated iterations can lead to a diversity of structures. This may result from a sensitive dependence on initial conditions and history. For example, every snowflake is different because, as it falls, it passes through a slightly different environment, with small variations in temperature and humidity, compared to others.

Toy models. 

Turing proposed a model for morphogenesis in 1952 that involved two coupled reaction-diffusion equations. Homogeneous concentrations of the two chemicals become unstable when the difference between the two diffusion constants becomes sufficiently large. A two-dimensional version of the model can produce diverse patterns, many resembling those found in animals. However, after more than seventy years of extensive study, many developmental biologists remain sceptical of the relevance of the model, partly because it is not clear whether it has a microscopic basis. Kicheva et al., argue that “pattern formation is an emergent behaviour that results from the coordination of events occurring across molecular, cellular, and tissue scales.” 

Other toy models include Diffusion Limited Aggregation, due to Witten and Sander, and Barnsley’s iterated function system for fractals that produces a pattern like a fern.

Here is a beautiful lecture on Pattern Formation in Biology by Vijaykumar Krishnamurthy