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.