Measuring the social, ethical, and political values of different AI models

I continue to enjoy reading my hard copy of The Economist every week. Occasionally, I post examples of insightful graphics presented in articles.

Here are some graphics that struck me recently. They are taken from a fascinating article. AI models’ values are very different from most people’s

The first figure shows a comparison of the answers given to the World Values Survey by different Large Language Models and people from different regions (cultures) of the world. The vertical axis goes from traditional values at the bottom to secular values at the top. The horizontal axis goes from survival (or communal) values on the left to self-expression on the right.

Note how the different cultures do not overlap. Furthermore, the LLMs are distant from almost everyone. Just a few are close to the English speaking world. 

The second graph compares answers to the VOTER survey of political questions. The horizontal and vertical axes correspond to Social and Economic issues respectively. Positive and negative values correspond to conservative and liberal respectively. Answers from LLMs are compared to a sample of Trump and Biden voters from the 2020 USA Presidential election. Note the clear political polarisation. Furthermore, the AI models are all liberal.

I offer no comments on whether any of this is good or bad. I do suggest two implications. First, extensive use of AI will tend to shift peoples values in a particular direction, just like media (music, TV, movies, art) does. Second, given the conflict of values with most people these biases will offend or concern many communities and increase the backlash against AI companies and its widespread adoption. Consequently, some will modify the training of their models so they align more with the values of vocal communities. Whether or not that is seen as a good thing, will depend on whether you think some of those values should be affirmed.

What do you think?

Macroscopic quantum effects in superconductors and superfluids

Quantisation of magnetic flux in a superconductor

Magnets and electrical currents produce magnetic fields, regions of space where other magnets and electrical wires experience a mechanical force. For a circle of wire in the presence of a magnetic field the magnetic flux is defined as the strength of the magnetic field passing through the circle multiplied by the area of the circle. A law of electromagnetism states that if the field varies with time, then a voltage is produced in the wire with a magnitude that is proportional to the rate at which the magnetic flux through the circle changes. This is the physics behind all electrical motors and electrical generators. In the everyday world magnetic flux can have any value and can be varied continuously by changing the strength of magnetic field. In the quantum world that is not the case. Magnetic flux is quantised.

In 1961, two experimental groups independently reported the first observation of a macroscopic quantum effect, the quantisation of the magnetic flux passing through a superconducting cylinder (Figure 30). One team was Bascom Deaver and William Fairbank and the other Robert Doll and Martin Nabauer. A tall thin cylinder made of tin was placed in a magnetic field and cooled down to a low enough temperature that it entered the superconducting state. The magnetic flux passing through the cylinder was then measured as the magnetic field was varied. The resulting graph has four noteworthy features. First, there are clear steps, showing that the magnetic flux has discrete values. In contrast, in the normal metallic state the graph was a straight line. Secondly, the magnitude of the steps was the same, to within about one per cent, suggesting quantisation of a single unit of magnetic flux. Thirdly, the value of this quantum of magnetic flux was equal to the value of h/2e. Thus, it was completely determined by the two fundamental constants, h and e, Planck’s constant and the charge on an electron, respectively. And fourthly, graphs with the same three features noted above were later observed in other superconducting materials and cylinders. This showed that flux quantisation is independent of details such as the chemical composition and dimensions of the cylinder. This flux quantisation is a macroscopic quantum effect. It is macroscopic because the system is macroscopic, and the magnetic flux is a macroscopic property. It is quantum as and the magnitude of the quantisation is determined by Planck’s constant.

                                                                       (b)

Figure 30. Quantisation of magnetic flux in a superconducting cylinder. (a) A tall thin cylinder of tin was placed in a magnetic field. (b) The graph shows the value of the magnetic flux passing through the cylinder as the magnetic field was varied. Note the step like structure, showing quantisation of the flux.

The quantum of magnetic flux is denoted Φ0 (= h/2e) and has the value 2.067833848...×10−15   tesla (metre)2. This number also determines the scale of quantum interference effects between two superconductors, as we will see shortly. The flux quantum is also relevant to vortices that form when some superconductors are placed in a magnetic field (Figure 20). A persistent electrical current flows around the vortex and the magnetic field penetrates the core of the vortex. It can also be shown, both theoretically and experimentally, that the magnetic flux associated with each vortex is exactly equal to one quantum of flux. Something similar happens in superfluids.

Macroscopic quantum effects in superfluids

When a cylinder containing a fluid is rotated about an axis passing down the centre of the cylinder the fluid will also rotate. The faster the cylinder is rotated the faster the fluid rotates. A physical quantity known as the circulation is proportional to the speed of rotation and the diameter of the cylinder. With a variable speed motor, the rotation speed can be continuously varied and in normal fluids the circulation has continuous values. But not in a superfluid, as shown in a beautiful experiment done by W.F. Vinen in 1961 using liquid 4He. He observed that when the liquid was cooled below the superfluid transition temperature that the circulation could only take on discrete values. Furthermore, these discrete values are multiples of h/M where h is Planck’s constant and M is the mass of one atom of helium. This value was predicted by Lars Onsager in 1949 who identified h/M with the circulation of a single vortex in the superfluid. This is another macroscopic quantum effect.

The quantisation of magnetic flux in superconductors and of circulation in superfluids showed that both superconductors and superfluids can be classified as quantum states of matter. The close similarity of these quantum phenomena, even though superconductivity occurs in solids and superfluidity in liquids. This indicates a deep underlying unity, demonstrated through the study of condensed matter physics. 

This is an extract from Chapter 7, Quantum Matter, in Condensed Matter Physics: A Very Short Introduction.

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.

What is a quasiparticle?

 An example of emergent entities in condensed matter physics are quasiparticles. The concept can be described with the following analogue. When a horse gallops through the desert it stirs up a dust cloud that travels with it. The motion of the horse cannot be separated from the accompanying dust cloud. They act as one entity. Similarly, in a system consisting of many interacting particles, when one particle moves it carries with it a “cloud” of other particles. This composite entity is referred to as a quasiparticle. It turns out to be easiest to understand the whole system of particles in terms of the quasiparticles rather than in terms of the individual particles.

Quasiparticles are composite objects. Like the constituent particles in the system, quasiparticles each have properties such as charge, mass, and spin. However, these properties of a single quasiparticle may be different from those of the individual particles of which it is constituted. An example is holes in semiconductors; the many electrons in a crystal act collectively to produce a hole (the absence of a single electron), a quasiparticle with the opposite charge to that of a single electron. A more striking example is for the fractional quantum Hall states; the charge of the quasiparticles can be a fraction of the charge on a single electron.

Different musical instruments produce distinct sounds because they are made of different materials, and they vibrate in different ways in response to different stimuli. In general, the vibrations of a medium reflect something about the medium itself. Chapter 3 discussed how in a crystal the number of distinct ways that sound can travel through a crystal reflects the symmetry and ordering of the atoms in the crystal.

When the skin on a drum is hit by a drumstick the skin vibrates at particular frequencies. Similarly, a state of matter responds to external stimuli such as light, sound or heat, by oscillating at particular frequencies. These vibrations travel through the matter as waves. The properties of these waves reflect the particular order present in the state of matter. Here is a specific example. When a neutron with a particular energy and momentum is absorbed by a ferromagnetic crystal the interaction of the magnetism of the neutron with that of the atoms in the crystal produces a collective oscillation of the magnetic state of the crystal in time and space. Known as a spin wave, this oscillation has a particular frequency and wavelength. In quantum theory, waves and particles are equivalent to one another. The energy and momentum of a particle are related to the waves’ frequency and wavelength, respectively. Particles equivalent to light waves are known as photons; particulate equivalents of sound waves are known as phonons. And similarly, the particle equivalent of a spin wave is known as a magnon. These collective excitations are quasiparticles. Whereas the particles in a system may interact strongly with one another, the quasiparticles may interact weakly with one another. This makes analysis and understanding of the relevant theories more tractable.

The quasiparticle concept is a powerful theoretical tool in condensed matter physics. It is the basis for the construction of models that enable emergent phenomena to be understood in terms of the effective interactions between components such as quasiparticles, rather than in terms of the actual constituent particles and their interactions. This approach requires profound physical insight in order to discern what the truly essential components of a system are. Lev Landau was one of the first theoretical physicists to take this approach, introducing the idea of quasiparticles in his theories of superfluidity in 4He and of liquid 3He. This approach was also central to the BCS theory of superconductivity. Phil Anderson was also a master of the approach, using intuition to propose models that were simple enough for analysis and yet complex enough to capture the essential physics associated with a particular state of matter. In 1977 he was awarded the Nobel Prize for work using this approach to understand two specific systems: magnetic atoms in metals and the motion of electrons in materials that are not crystals and are dirty in the sense of containing many impurities.

An extract from Chapter 9, "Emergence: More is Different", in Condensed Matter Physics, A Very Short Introduction

A more detailed and technical discussion is in Section 8.2 of my review article on emergence.

Biological evolution and emergence

 The theory of evolution explains the origin of biological diversity and levels of similarity between species. A characteristic of emergence is that many iterations of a simple law (natural selection of the fittest to reproduce) can produce novel, diverse and rich structures. In biological evolution many generations in a population can produce new traits and species. 

Many of the most debated issues about evolution relate to the different characteristics of emergence and are briefly discussed below.

Scales

Central to emergence are the ideas of “many” and of scales. The former can take two forms: a system composed of many interacting components, or a system that undergoes many iterations according to a rule that is repeated many times. For evolution, both forms of “many” are relevant and have several dimensions. Evolution occurs in a population, i.e., a community of many members of a species living in a specific environment. Each member of the population has a specific genotype (many genes), which largely determines biological characteristics, from proteins to organs, defined as the phenotype. The environment also consists of many interacting species. Natural selection can act at multiple levels: on genes, cells, organisms, species, and groups of species.

Microscopic and macroscopic scales can also manifest in different ways. In terms of length, the micro- and macro- scales can be defined in terms of genotypes and phenotypes, respectively. In terms of time, microevolution and macroevolution roughly correspond to directly observable timescales and geological timescales, respectively. They are associated with the emergence of new traits within a species and new species, respectively.

Novelty

Development of new traits and species occurs over many generations, due to the repetition of the rule of natural selection.

Evolution theory uses concepts such as natural selection, survival of the fittest, niches, and hierarchical trees, that are not present in chemistry and physics. 

Connecting micro- and macro- properties

As for other systems, this is one of the great challenges of emergence. Genotypes and phenotypes are extremely well characterised. Genotype-phenotype maps seek to connect these micro- and macro- levels. A detailed understanding of how microevolution leads to macroevolution is a challenge.

Discontinuities

In microevolution, new traits occur within a species due to (continuous) adaptation to the environment. In contrast, in macroevolution, new organs and species can occur suddenly (at least on geological timescales). An example is the Cambrian explosion of new life forms. Extinctions can also represent discontinuities.

Evolution of a population occurs in response to changes in an environment. New traits, new species, and extinctions can be viewed as qualitative changes due to quantitative changes. For example, small changes in the oxygen concentration in the atmosphere is one (among many) hypotheses for the cause of the Cambrian explosion.

Using techniques from statistical physics, the transition of a species from survival to extinction can be viewed as a non-equilibrium phase transition to an absorbing state. The order parameter is the population and a toy model is directed population.1

Diversity with limitations

All species are based on the same biochemistry of DNA and proteins. Yet from these same building blocks there is an incredible diversity: more than 8 million distinct species, including more than 10,000 species of birds and more than 15,000 species of ants. Darwin said nature produces “endless forms most beautiful.”

But there are limitations. For example, the number of species with more than one head, brain, heart, or liver is limited. There are many more genotypes than phenotypes. 

The dominant view is that evolution is driven by random genetic mutations. Debates have arisen about how much evolution is limited (constrained) by morphology and environment.

Ball stated: (p. 332)

“convergent evolution is often regarded as a sign that certain shapes or structures are ideal adaptations to particular environments for physical reasons: wings consisting of flat, thin membranes are best for flying, torpedo-shaped bodies a streamlined for efficient swimming, and so on… There is a tendency in evolutionary biology to regard natural selection as a process with an infinite palette: anything is possible so long as it doesn't break the laws of physics. But the laws of physics might impose more constraint than that, precisely because biology uses rather than merely suffers them.”

Universality

Not all mutations produce a change in phenotype. There are neutral mutations. There are many more genotypes than phenotypes. In other words, genotype-phenotype maps are many-to-one.

Species that are unrelated or distantly related (in the tree of life) sometimes have traits or behaviours that are similar. Convergent evolution is the hypothesis that natural selection produced the same outcome in a different context. 

Modularity at the mesoscale

The economist Simon pointed out that evolution can occur on much faster time scales than might be expected because of modularity. According to Clune et al.

“A long-standing, open question in biology is how populations are capable of rapidly adapting to novel environments, a trait called evolvability [1]. A major contributor to evolvability is the fact that many biological entities are modular, especially the many biological processes and structures that can be modelled as networks, such as metabolic pathways, gene regulation, protein interactions and animal brains [1–7].”

Ball highlighted how domains in proteins provide functional modules that evolution uses: (pp. 174-5)

“the evolution of metazoan proteins is not so much a slow affair of letting random genetic mutations change one amino acid for another and seeing what effect it produces. Rather, it constitutes a reshuffling of already functional modules to produce multidomain molecules with new potential - a strategy much more likely to yield successful results…  the “unit” of molecular evolution here is not really the base pair of DNA or the amino acid or protein, or the gene itself, by the peers at a scale intermediate between the two: the module of a domain. It seems that this shuffling, rather than the slow mutation of primary base sequences, is what has driven the evolution of animals.”

Johnston et al. considered an algorithmic picture of evolution that 

“suggests that symmetric structures preferentially arise not just due to natural selection but also because they require less specific information to encode and are therefore much more likely to appear as phenotypic variation through random mutations… many genotype–phenotype maps are exponentially biased toward phenotypes with low descriptional complexity. A preference for symmetry is a special case of this bias… Lower descriptional complexity also correlates with higher mutational robustness, which may aid the evolution of complex modular assemblies of multiple components.”

Self-organisation

Complex biological structures, from proteins to organisms, have formed spontaneously due to evolution over millions of years. Their intricacy and functionality have led to claims of purpose and design. However, this is argued to be an “apparent” design, just like an economy whose self-organisation appears “as if” it is guided by an “invisible hand.”

Kauffman claimed that self-organisation is as important as natural selection in driving evolution.

Unpredictability

A contested question about evolution is the role of contingency (historical accidents) and whether the evolution of complex life forms, particularly humans, was an accident of history or inevitable.

Irreducibility

Until recently, evolutionary biology has been dominated by a reductionist gene-centric view, popularised by Dawkins. However, recent discussions about systems biology, evo-devo, and epigenetics have questioned this view. Some characterise these alternative views as a form of structuralism.

Complexity

An algorithmic picture of evolution suggests that simplicity spontaneously emerges as many genotype-phenotype maps may be biased towards phenotypes with low descriptional complexity. 

Toy models

An earlier post discussed the key role that toy models, such as “bean bag” genetics, have played in evolutionary theory.

Cross-fertilisation of fields

Ideas from evolution have stimulated the development of genetic algorithms in computer science.

Drossel has reviewed connections between evolution and statistical physics, including a wide range of toy models. Examples include spin glass models that give rise to rugged landscapes for fitness and can describe hierarchical structures, comparable to Darwin’s tree of life. Goldenfeld and Woese argued that evolution can be viewed as a collective phenomenon far from equilibrium. The toy model central to their discussion is directed percolation.

I welcome comments. My knowledge of biology is limited, and scientifically some the ideas above can be contentious. (Never mind philosophy, politics, or theology!)

Critical points in condensed matter illuminate universality

Every person is unique. No two people are identical. We differ in physical appearance, personality, fingerprints, heartbeat, gait, and DNA. Such differences are used to identify criminals and in video surveillance of citizens by nation states. Yet in other ways all humans are the same. We all have brains, hearts, and lungs. All our bodies use the same biochemistry to stay alive: whether to breathe oxygen, digest food, or fight infections. On some level we have common aspirations: to survive, to be loved, to be happy, and to find meaning and purpose. Yet these aspirations find many expressions. Humans have certain universal qualities and properties, yet at a finer level of detail there is a particularity of each of these properties. They are at one level the same but are not the same at another level. 

All academic disciplines search for universals; they develop categories, concepts, and theories that overarch particularities. Biologists classify species of plants and animals and types of cells and viruses. All biological systems use the same molecules (DNA, RNA, and proteins) and chemical reactions. The same genetic code uses the information encoded in a piece of DNA to make proteins with specific functions. Anthropologists study the immense diversity of human cultures and societies. This diversity can be described in terms of universal concepts such as kinship, family, ritual, community, economics, law, and morality. Linguists study the common structures and grammars of the thousands of different human languages.  Although the world we live in is diverse, disciplines have each discovered some universals.

Condensed matter physicists study diverse states of matter and the transitions between them. A surprising discovery is that there is much more universality than might be expected, particularly given the chemical and structural diversity of materials. In this chapter, I will discuss the nature of this universality, how it emerges, and the length scales associated with transitions between different states of matter. Landau’s great insight was that many of the chemical and structural details of materials are irrelevant to understanding phase transitions. Furthermore, a precise classification of different types of phase transitions, into what are called universality classes, can be made. For example, superconducting, superfluid, and a subset of magnetic transitions are in the same class. The determinants of the universality classes are the symmetry of the state and the spatial dimensionality of the system. None of the other details matter.

Many phase diagrams (such as the Figure above) include a critical point, located at the end of a boundary between two different states of matter. A common example is the critical point that occurs at a specific temperature and pressure for a transition between a liquid and a gas. Understanding the physical properties of a material close to its critical point was a great challenge for theoretical physics, lasting a hundred years, and was only solved in the 1970s. The powerful theoretical ideas and techniques that were developed provide a quantitative way to relate the properties of a system at one length scale to properties at a different length scale. These techniques also have application to a wide range of other problems and fields including elementary particle physics, chaos theory, fractals, polymers, and machine learning. New insights were gained into universality and emergent phenomena.

An extract from "The Critical Point," chapter 6, Condensed Matter Physics: A Very Short Introduction

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.

Condensed matter physics in flatland

Adventures in Flatland

In everyday life we think of most objects as having three dimensions. But what would life be like in a two-dimensional world? For one thing, it would be harder to move around. We could no longer step over things but would have to move around them. In 1884 Edwin Abbott published Flatland: A Romance of Many Dimensions, under the pseudonym, A. Square, a satirical novella about social life in Victorian England. People are represented by geometrical objects. Men are represented by shapes such as triangles and hexagons. Women are represented by lines. The social status of men increases with the number sides that their shape has and how many of the sides are of the same length. Abbott’s book created limited interest and was largely forgotten by the 1920s. Interest revived when theoretical physicists started to think about worlds in different dimensions. This interest was stimulated by Albert Einstein’s theories of relativity, that proposed that we live in a four-dimensional world, not a three-dimensional one. Time is the fourth dimension, and there is an intimate and concrete connection between time and space. Attempts to unify gravity with other fundamental forces has led to physicists proposing and studying theories with more than four dimensions.

Changing the number of spatial dimensions leads to different physics because it changes what is mathematically possible. In three dimensions, there were only five highly symmetrical shapes known as Platonic solids (tetrahedron, cube, octahedron, icosahedron, and dodecahedron). In contrast, in two dimensions it is possible to make an infinite number of symmetrical shapes, known as regular polygons, shapes made of straight lines of equal length such as squares or hexagons. Similarly, the number of Bravais lattices differ in two and three dimensions. Changing the number of spatial dimensions changes both what is mathematically possible and what is physically possible.

What would condensed matter physics be like in Flatland? This question received limited attention before the 1970s. Occasionally, theoretical physicists would investigate mathematical models of crystals or magnets in one or two dimensions just because the mathematics was simpler and more tractable than in three dimensions. The goal was to obtain insight into physics in three dimensions. We will consider a famous example, the Ising model. 

In the 1970s, several surprising developments led to significant interest in condensed matter physics in spatial dimensions different from the usual three. First, it became possible to make a wide range of material systems that were two-dimensional. Secondly, theoretical work showed that states of matter, and phase transitions between them, can be qualitatively different in one, two, and three spatial dimensions. And thirdly, considering different numbers of spatial dimensions turned out to be very fruitful for theory, particularly for understanding phase transitions near critical points. 

An extract from "Adventures in Flatland," chapter 5 in Condensed Matter Physics: A Very Short Introduction

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.

Condensed matter physics is about how order emerges from disorder

 The order of things

Life and the world around us sometimes appears chaotic and random. We may feel this way about traffic, weather, economics, social change, politics, or our personal relationships. Perhaps that is why many yearn for regularity, predictability, order, and stability. Science is a search for patterns and order in the natural world. Condensed matter physics is about how order emerges from disorder.

This chapter explores how different states of matter are associated with different types of ordering of the atoms in the material. The symmetry of the state reflects the type of ordering, i.e., the patterns associated with the state. There is also a rigidity associated with the ordering and the rigidity determines the nature of the deviations from perfect ordering and results in entities such as vortices that are central to the physical properties of the state of matter.

The association of a state of matter with a specific type of ordering is illustrated in Figure 15 by an analogue with the dodgem bumper cars at an amusement park. A quiet day at the park is not much fun as collisions between cars are rare. In other words, there is little correlation between the relative locations and speeds of the cars. In comparison, on a busy day at the park the spatial separation of the cars is small, and their positions and speeds are more correlated with one another than on a quiet day. But, in both cases, there is no ordered arrangement of the cars. In contrast, after the park closes the cars are parked and arranged in an orderly manner. There is a rigidity associated with their spatial arrangement. One car cannot be moved without moving others. These three states of the dodgem cars are an analogue of three states of matter: gas, liquid, and crystal. 

Figure 15. A dodgem car analogue for the three states of matter: crystal, liquid, and gas. The only ordered arrangement is for the crystal (car park after hours) and this is associated with a specific symmetry and rigidity. The liquid and gas (busy and quiet day) only differ in density and the amount of correlation between the positions of the different atoms (dodgem cars).

In the dodgem car analogue, there are other possible types of ordering. In some amusement parks there is a track, and the cars are meant to all go in the same direction. The symmetry between clockwise and anti-clockwise of the track is then broken.  In the car park, Figure 15 shows cars that are symmetrical with respect to front and back. However, real cars have a front and back, and so can be parked either front first or back first. Hence, several types of ordering are possible: all cars park back first, all cars park front first, cars are front first or back first at random, alternating patterns of front first and back first as one goes along a row, alternating rows of front first and back first, and so on. These different types of ordering in the car park all have analogues in different solid states of matter.

Liquid crystals involve unique types of ordering. These materials are composed of elongated organic molecules, such as those shown in Figure 16. At high temperatures the material is in a liquid state and the orientations and positions of the molecules are random. The liquid has both continuous translational and rotational symmetry. At low temperatures the molecules form a solid crystal without the continuous translational and rotational symmetry of the liquid state. As the crystal is heated the temperature increases and there is a phase transition to the liquid crystal state, in which all the molecules point in the same direction, but their positions are random. Hence, the liquid crystal state has the continuous translational symmetry of the liquid, but not its continuous rotational symmetry, like the crystal. As the temperature increases further there is a transition to the liquid state (Figure 16). In terms of the dodgem car analogue the liquid crystal state is similar to when cars park in a field all pointing in the same direction but there are no grid lines, and their positions are then random.

The existence of a state in between a liquid and crystal was first proposed in 1888 by botanist and chemist Friedrich Reinitzer who was doing research on cholesterol at the Institute for Plant Physiology in Prague. He performed a heating experiment similar to that described in Figure 4. Instead of one melting transition he observed transitions at two distinct temperatures. 

Figure 16. Liquid crystals. (a) An example of the type of elongate organic molecule found in these materials. Each molecule can be represented by an oval shape. (b) In the nematic liquid crystal state, the molecules tend to point in the same direction, but their positions are random. 

There are multiple alternative orderings for liquid crystals with names such as nematic, smectic, chiral nematic, discotic, and chlorestic. In the smectic phase molecules form layers of oriented molecules. The character of the liquid crystal state can be detected by shining polarised light on the material. Liquid crystal displays (LCDs) in electronic devices use the property that an electric field can orient the molecules, and this changes the interaction of the material with polarised light.

For solid crystals the nature of the ordering and the symmetry associated with a specific crystal structure is clear once the spatial arrangements of the atoms in the crystal are determined, such as by X-ray diffraction. For other states of matter, such as superconductors, superfluids, and antiferromagnets, the nature of the ordering and the symmetry is often not apparent and has only been determined with significant scientific insight. 

An extract from "The order of things," chapter 4 in Condensed Matter Physics: A Very Short Introduction.

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.

Symmetry matters in condensed matter physics

 Snowflakes form incredibly diverse structures, seen when they condense onto a plate of glass. Every snowflake is different. On the other hand, every snowflake is the same. They are all composed of ice, a solid state of water. Every snowflake is composed of units that have a six-fold symmetry (Figure 8). Every snowflake is composed solely of water molecules. This paradox of the particular and the universal is at the heart of condensed matter physics. Although diversity prevails anything is not possible. No snowflake has five-fold symmetry. Snowflakes have enchanted scientists for a long time. The astronomer Johannes Kepler studied them and in 1611 wrote a small book about them as a gift for his patron. Kepler suggested snowflakes provided clues to deeper questions about the composition of matter. Today, Kenneth Libbrecht, a physicist at Caltech, has spent most of his career studying snowflakes and has produced beautiful volumes of photographs of them.

Figure 8. A snowflake shows a six-fold symmetry, just like a hexagon. The snowflake appears identical when it is rotated by an angle of sixty degrees about an axis passing through its centre and perpendicular to the page.

Condensed matter physicists ask several questions about snowflakes. What is the reason for the six-fold symmetry of the snowflake? What is the connection between the macroscopic properties of snowflakes and the properties of the underlying microscopic constituents, molecules of H2O? How is the diversity of snowflake shapes possible? Is there a phase diagram that defines the external conditions under which the different shapes form?

There is a long history in art, architecture, philosophy, and science, of associating symmetry with beauty and perfection. The ancient Greek philosopher Plato was a proponent of this view. He studied a particular class of solid shapes: cube, tetrahedron, octahedron, icosahedron, and dodecahedron. Plato identified the first four shapes with the four “elements”: earth, wind, fire, and water, respectively, and the fifth with the heavens. Each of these solid shapes is highly symmetric. Every face of a Platonic solid is the same shape (square, triangle, pentagon,...) and each of those shapes has edges of equal length. 

Like Plato, Kepler believed that “God is a geometer” and that God’s creation should reflect the perfection of God. These convictions led Kepler to propose in 1597 that the orbits of the planets around the Sun were circular and that the Platonic solids determined the relative size of the orbits. Later this model for the solar system was shown not to be true. In fact, Kepler himself became famous because he showed that the planets moved in elliptical, not circular orbits. Nevertheless, Kepler’s model was the beginning of a long history of successfully relating physical laws to symmetry and geometry.

A key discovery in physics from the past century is that symmetry is central to understanding a wide range of physical phenomena, whether colliding billiard balls, the allowed energies of an atom, the fundamental forces of nature, or different states of matter. Symmetries determine what is physically possible. For example, that energy cannot be created or destroyed is a consequence of the fact that physical laws do not change with time.

In this Chapter I explore three key ideas. First, transitions between different states of matter are associated with changes in symmetry. Thus, symmetry provides a criterion for specifying the qualitative difference between distinct states of matter. Second, for a specific state of matter the relevant symmetry constrains what is physically possible. Third, symmetry is central to making connections between the macroscopic and microscopic properties of a state of matter. The next chapter will explore how symmetry is associated with the type of ordering that occurs in a state of matter.

An extract from Chapter 3, Condensed Matter Physics: A Very Short Introduction

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.