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.

How many states of matter are there?

Diamond and graphite are distinct solid states of carbon. They have qualitatively different physical properties, at both the microscopic and the macroscopic scale. Condensed matter physics is all about states of matter. In science classes at school, you were probably taught that there are only three states of matter: solid, liquid, and gas. Like other things you were told in school, this is incorrect. There are endless, unlimited, distinct states of matter. 

Consider the “liquid crystals” that are the basis of LCDs (Liquid Crystal Displays) in the screens of televisions, computers, and smartphones. How can something be both a liquid and a crystal? A liquid crystal is a distinct state of matter. Solids can be found in many different states. We have already seen that there are two different solid states of carbon: graphite and diamond. In everyday life ice means simply solid water. But there are in fact eighteen different solid states of water, depending on the temperature of the water and the pressure that is applied to the ice. In each of these eighteen states there is a unique spatial arrangement of the water molecules and there are qualitative differences in the physical properties of the different solid states. Welcome to the world of condensed matter...

Extract from Chapter 1, Condensed Matter Physics: A Very Short Introduction

Classifying objects, people, and societies requires making qualitative distinctions. One book is easy to understand, and another is hard. One person is kind, and another is mean. One society is egalitarian, and another is not. Justifying such qualitative distinctions is hard. Not everyone will agree. Are there definitive criteria to justify a particular quality? Some claim they can quantify qualities such as these but that is contentious. In contrast, in condensed matter physics it is possible to give objective criteria that distinguish different states of matter. A state can only exist under specific external conditions, including defined ranges of parameters such as temperature and pressure. This chapter describes the clear signatures of transitions between different states that are observed as these parameters are varied. Some of the many known states of matter will be introduced including superconductors, superfluids, and magnets. On the way we will learn about “dry ice”, how to convert graphite into diamond, and how freeze-dried food is made.

Abrupt changes in properties

If you put some ice cubes in one empty glass and water in another, the ice does not change its shape, whereas water takes the shape of the glass. Solids are rigid and liquids are not. The distinct change from one state to another can be detected by observing an abrupt change or discontinuity in physical properties. For example, ice (solid water) has a different density to liquid water. This is evident because ice floats. The solid state of water has a lower density than the liquid state. To put it another way, water expands when it freezes. That’s why water pipes can burst if they freeze in cold weather.

A transition between two distinct states of matter is an example of a tipping point: a small change in a system variable can produce large changes in the system. For example, changing the temperature of water from +1 °C to -1 °C can produce a qualitative change in the system's properties. The water changes from liquid to solid. Tipping points occur in a wide range of physical, biological, and social systems. Examples include a stock market crash, the outbreak of an epidemic, and the operation of a room thermostat. Tipping points show that quantitative differences can become qualitative differences.

Extract from Chapter 2, Condensed Matter Physics: A Very Short Introduction

What is condensed matter physics?

 Every day we encounter a diversity of materials: liquids, glass, ceramics, metals, crystals, magnets, plastics, semiconductors, foams, … These materials look and feel different from one another. Their physical properties vary significantly: are they soft and squishy or hard and rigid? Shiny, black, or colourful? Do they absorb heat easily? Do they conduct electricity? The distinct physical properties of different materials are central to their use in technologies around us: smartphones, alloys, semiconductor chips, computer memories, cooking pots, magnets in MRI machines, LEDs in solid state lighting, and fibre optic cables. Consequently, the science of materials attracts researchers in a wide range of disciplines: physics, chemistry, biology, mathematics, and the varieties of engineering (electrical, chemical, mechanical, material…). But why do different materials have different physical properties? 

There are more than one hundred different types of atoms, or chemical elements, in the universe. Any material is composed of a specific collection of different atoms, and they are arranged in a particular spatial pattern within the material. A central question is: 

How are the physical properties of a material related to the properties of the atoms from which the material is made?

Extract from Chapter 1, Condensed Matter Physics: A Very Short Introduction

A mystery about science is that humans can do it

We are surrounded by scientific knowledge and have become so used to it that we often take science for granted. We may rarely reflect on the amazing revelations of science—and so miss the opportunity to recognize the awesome nature of the universe. Things that we know, learn, and do today in science would have been inconceivable decades, let alone centuries, ago. 

Einstein said, “The most incomprehensible thing about the universe is that it is comprehensible.”  For Einstein, the success of science was a wonderful mystery. As he wrote to his friend Maurice Solovine: 

. . . I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way . . . the kind of order created by Newton’s theory of gravitation, for example, is wholly different.  

There are several dimensions to the comprehensibility of the universe being mysterious. Einstein highlighted the first mystery, which is that there is order in the world, as reflected in scientific laws, such as Newton’s theory of gravity, and that this order can be succinctly stated in the language of mathematics. To the best of our knowledge, these laws hold for all time and everywhere in the universe. The existence of the orderly behaviour encoded in scientific laws is necessary for science to work, which leads to the second mystery. Why have we been able to discover these laws?

A second dimension that makes science possible is the intellectual abilities of humans. Humans not only have the rational ability to do science—to reason, to understand, to communicate—but also the ability to design instruments, such as telescopes and microscopes. There seems to be a connection between the rationality of the universe and human rationality. The idea that there may be harmony between the structures of the universe and those of the human mind has a long history.  In the Renaissance, it was encapsulated in the metaphor of the “music of the spheres”. In his book, Harmonies of the World (1619), Johannes Kepler connected music and his explanations of planetary orbits. Einstein said that “Mozart’s music is so pure and beautiful that I see it as a reflection of the inner beauty of the universe.” 

Humans might have been different. Suppose that the average human intelligence was lower than it is today, and the variation of human intelligence was smaller. Then, there might have been no Galileo, Isaac Newton, Robert Boyle, Charles Darwin, Albert Einstein, Richard Feynman, Phil Anderson, or Linus Pauling. Without these brilliant figures in scientific history, scientific progress would have been slow. 

The third dimension is that human language enables scientists to formulate, represent, and communicate ideas, theories, and the results of scientific experiments. This language sometimes involves mathematics, graphs, or tables of data. Scientists can understand one another. Even though there can be misunderstandings, these can be resolved. There is a scientific culture that transcends the diversity of cultures associated with different countries, linguistic groups, and ethnicities.

The fourth dimension is the physical dexterity of humans. I am a theoretical physicist not an experimental physicist. I am “all thumbs” and not particularly good in the lab. Consequently, I have done no laboratory work since I was a Ph.D. student. In contrast, some gifted scientists have an ability to do things in a laboratory that most people cannot. Their manual dexterity allows them to fabricate precision instruments, grow pure crystals, blow exquisite glassware, see faint images, and fine-tune electronic instruments in extraordinary ways. If some humans did not have such amazing abilities, scientific progress would have been much slower—or possibly non-existent.

A fifth dimension that makes science possible is the availability and processability of materials that have been central to scientific progress. Making instruments requires specific materials, such as metals, glass, rubber, insulators, plastics, and semiconductors. If we lived in a world where some of these materials were very rare or could not be processed to the purity or malleability required for scientific instruments, we would not have supercomputers, electron microscopes, or the James Webb Space Telescope today. We might be struggling to make even the simple telescopes used by Galileo.

These five dimensions are all required for humans to be able to do science. There are several additional mysteries of science.  These can be divided into two classes: what science can do and what we can learn about the universe from science. Science allows us to know certain things about reality (epistemology) and also to understand the nature of that reality (ontology). In other words, science helps us make maps of physical reality. The terrain represented by those maps is amazing. And the fact that we can make the maps is amazing.

Scandals in Australian universities

In Australia, scandals about the management of public universities continue to be covered in the media. A recent one is the use of billions of dollars to pay consulting firms to tell management which staff to sack and courses to cut because they are not making a profit.

Below is a recent episode of an ABC (Australian equivalent of BBC or PBS in USA) show on the topic, Chaos on Campus.

I tend to avoid engaging too much with media lamenting the state of unversities as I find it too disturbing. However, I was asked to reference the show in something I was asked to write and so felt I should watch it. It was painful.

This is definitely a scandal. However, it got me reflecting on something that I think gets virtually no media coverage and when I talk to people outside the university, they are pretty surprised and shocked. Anecdotal evidence from my colleagues is that attendance at lectures is now typically around 10-30 per cent of enrolment. Even before COVID-19, lectures at UQ were all recorded. Faculty have no choice. But only a few per cent of students watch the videos. This is quite demoralising for faculty.

What does this low level of student engagement mean for learning outcomes?

What is happening elsewhere? 

Here are some links from Google AI. 

I did not find them that insightful.

One link is an article from The Guardian in Australia from last year. It highlights how moving things to online and lowering standards is driven by financial incentives. This ties in with the scandals in the video. The values of Australian universities are money, marketing, management, and metrics.

What is your own experience with the level of disengagement? How do you think this is affecting student learning? How are you and your colleagues adapting? Are academic standards being lowered? Any suggestions on ways forward?