Philosophical perspectives on the emergence of molecular structure

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A new version of my review article on emergence

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

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

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

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

Quantum justification for classical discussions of potential energy surfaces in chemistry

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

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

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

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

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

What does the Born-Oppenheimer approximation mean for emergence?

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

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

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

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

The full Hamiltonian (given earlier) can be denoted by

 

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

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

 

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

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

In the BOA the nuclear probability distribution defined above is

 

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

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

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

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

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

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

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

The emergence of molecular structure from quantum theory

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

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

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

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

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

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

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

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

Full quantum solutions of the Hamiltonian

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

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

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

The ground state eigenfunction can be written as

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

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

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

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

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

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

A general quantum definition of molecular structure

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

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

Are chemical isomers emergent?

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

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

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

Structural isomers

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

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

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

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

Stereoisomers

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

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

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

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

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

Emergence?

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

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

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

Isomers illustrate the diversity of forms often associated with emergence.

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

Characteristics of static disorder can emerge from electron-phonon interactions

Electronic systems with large amounts of static disorder can exhibit distinct properties, including localisation of electronic states and sub-gap band tails in the density of states and electronic absorption. 

Eric Heller and collaborators have recently published a nice series of papers that show how these properties can also appear, at least on sufficiently long time scales, in the absence of disorder, due to the electron-phonon interaction. On a technical level, a coherent state representation for phonons is used. This provides a natural way of taking a classical limit, similar to what is done in quantum optics for photons. Details are set out in the following paper 

Coherent charge carrier dynamics in the presence of thermal lattice vibrations, Donghwan Kim, Alhun Aydin, Alvar Daza, Kobra N. Avanaki, Joonas Keski-Rahkonen, and Eric J. Heller

This work brought back memories from long ago when I was a postdoc with John Wilkins. I was puzzled by several related things about quasi-one-dimensional electronic systems, such as polyacetylene, that underwent a Peierls instability. First, the zero-point motion of the lattice was comparable to lattice dimerisation that produced an energy gap at the Fermi energy. Second, even in clean systems, there was a large subgap optical absorption. Third, there was no sign of the square-root singularity expected in the density of states, predicted by theories which treated the lattice classically, i.e., calculated electronic properties in the Born-Oppenheimer approximation.

I found that on the energy scales relevant to the sub-gap absorption, the phonons could be treated like static disorder and make use of known exact results for one-dimensional Dirac equations with random disorder. This explained the puzzles.

Effect of Lattice Zero-Point Motion on Electronic Properties of the Peierls-Fröhlich State

The disorder model can also be motivated by considering the Feynman diagrams for the electronic Green's function perturbation expansion in powers of the electron-phonon interaction. In the limit that the phonon frequency is small, all the diagrams become like those for a disordered system, where the strength of the static disorder is given by 

I then teamed up with another postdoc, Kihong Kim, who calculated the optical conductivity for this disorder model.

Universal subgap optical conductivity in quasi-one-dimensional Peierls systems

Two things were surprising about our results. First, the theory agreed well with experimental results for a range of materials, including the temperature dependence. Second,  the frequency dependence had a universal form. Wilkins was clever and persistent at extracting such forms, probably from his experience working on the Kondo problem.

The multi-faceted character of emergence (part 2)

In the previous post, I considered five different characteristics that are often associated with emergence and classified them as being associated with ontology (what is real and observable) rather than epistemology (what we believe to be true). 

Below I consider five more characteristics: self-organisation, unpredictability, irreducibility, contextuality and downward causation, and intra-stratum closure.

6. Self-organisation

Self-organisation is not a property of the system but a mechanism that a theorist says causes an emergent property to come into being. Self-organisation is also referred to as spontaneous order. 

In the social sciences self-organisation is sometimes referred to as an endogenous cause, in contrast to an exogenous cause. There is no external force or agent causing the order, in contrast to order that is imposed externally. For example, suppose that in a city there is no government policy about the price of a loaf of sliced wholemeal bread or on how many loaves that bakers should produce. It is observed that prices are almost always in the range of $4 to $5 per loaf, and that rarely are there bread shortages. This outcome is a result of the self-organisation of the free-market, and economists would say the price range and its stability has an endogenous cause. In contrast, if the government legislated the price range and the production levels that would be an exogenous cause. Friedrich Hayek emphasised the role of spontaneous order in economics. In biology, Stuart Kaufmann equates emergence with spontaneous order and self-organisation.

In physics, the periodicity of the arrangement of atoms in a crystal is a result of self-organisation and has an endogenous cause. In contrast, the periodicity of atoms in an optical lattice is determined by the laser physicist who creates the lattice and so has an exogenous cause.

Self-organisation shows how local interactions can produce global properties. In different words, short-range interactions can lead to long-range order. After decades of debate and study, the Ising model showed that this was possible. Other examples of self-organisation, include flocking of birds and teamwork in ant colonies. There is no director or leader but the system acts “as if” there is. 

7. Unpredictability

Ernst Mayr (This is Biology, p.19) defines emergence as “in a structured system, new properties emerge at higher levels of integration that could not have been predicted from a knowledge of the lower-level components.” Philip Ball also defines emergence in terms of unpredictability (Quanta, 2024).

More broadly, in discussions of emergence, “prediction” is used in three different senses: logical prediction, historical prediction, and dynamical prediction.

Logical prediction (deduction) concerns whether one can predict (calculate) the emergent (novel) property of the whole system solely from a knowledge of all the properties of the parts of the system and their interactions. Logical predictability is one of the most contested characteristics of emergence. Sometimes “predict” is replaced with “difficult to predict”, “extremely difficult to predict”, “impossible to predict”, “almost impossible to predict”, or “possible in principle, but impossible in practice, to predict.” 

As an aside, I note that philosophers distinguish between epistemological emergence and ontological emergence. They are associated with prediction that is "possible in principle, but difficult in practice" and "impossible in principle" respectively.

After an emergent property has been discovered experimentally sometimes it can be understood in terms of the properties of the system parts. In a sense “pre-diction” then becomes “post-diction.” An example is the BCS theory of superconductivity, which provided a posteriori, rather than a priori, understanding. In different words, development of the theory was guided by a knowledge of the phenomena that had already been observed and characterised experimentally. Thus, a keyword in the statement above about logical prediction is “solely”. 

Historical prediction. Most new states of matter discovered by experimentalists were not predicted even though theorists knew the laws that the microscopic components of the system obeyed. Examples include superconductivity (elemental metals, cuprates, iron pnictides, organic charge transfer salts, …), superfluidity in liquid 4He, antiferromagnetism, quasicrystals, and the integer and fractional quantum Hall states.

There are a few exceptions where theorists did predict new states of matter. These include are Bose-Einstein Condensates (BECs) in dilute atomic gases and topological insulators, the Anderson insulator in disordered metals, the Haldane phase in even-integer quantum antiferromagnetic spin chains, and the hexatic phase in two dimensions. It should be noted that prediction of BECs and topological insulators were significantly helped that theorists could predict them starting with Hamiltonians of non-interacting particles. Furthermore, all of these predictions involved working with effective Hamiltonians. None started with microscopic Hamiltonians for specific materials.

Dynamical unpredictability concerns what it means in chaotic dynamical systems, where it relates to sensitivity to initial conditions. I do not see this as an example of emergence as it can occur in systems with only a few degrees of freedom. However, some authors do associate dynamical unpredictability with complexity and emergence.

8. Irreducibility and singularities

An emergent property cannot be reduced to properties of the parts, because if emergence is defined in terms of novelty, the parts do not have the property. 

Emergence is also associated with the problem of theory reduction. Formally, this is the process where a more general theory reduces in a particular mathematical limit to a less general theory. For example, quantum mechanics reduces to classical mechanics in the limit where Planck’s constant goes to zero. Einstein’s theory of special relativity reduces to Newtonian mechanics in the limit where the speeds of massive objects become much less than the speed of light. Theory reduction is a subtle philosophical problem that is arguably poorly understood both by scientists [who oversimplify or trivialise it] and philosophers [who arguably overstate the problems it presents for science producing reliable knowledge]. Subtleties arise because the two different theories usually involve language and concepts that are "incommensurate" with one another. 

Irreducibility is also related to the discontinuities and singularities associated with emergent phenomena. As emphasised independently by Hans Primas and Michael Berry, singularities occur because the mathematics of theory reduction involves singular asymptotic expansions. Primas illustrates this by considering a light wave incident on an object and producing a shadow. The shadow is an emergent property, well described by geometrical optics, but not by the more fundamental theory of Maxwell’s electromagnetism. The two theories are related in the asymptotic limit that the wavelength of light in Maxwell’s theory tends to zero. This example illustrates that theory reduction is compatible with the emergence of novelty. Primas also considers how the Born-Oppenheimer approximation, which is central to solid state theory and quantum chemistry, is associated with a singular asymptotic expansion (in the ratio of the mass of an electron to the mass of an atomic nuclei in the system). 

Berry considers several other examples of theory reduction, including going from general to special relativity, from statistical mechanics to thermodynamics, and from viscous (Navier-Stokes) fluid dynamics to inviscid (Euler) fluid dynamics. He has discussed in detail how the caustics that occur in ray optics are an emergent phenomena and are associated with singular asymptotic expansions in the wave theory.

The philosopher of science Jeremy Butterfield showed rigorously that theory reduction occurred for four specific systems that exhibited emergence, defined by him as a novel and robust property. Thus, novelty is not sufficient for irreducibility.

9. Contextuality and downward causation

Any real system has a context. For example, it has boundary and an environment, both in time and space. In many cases the properties of the system are completely determined by the parts of the system and their interactions. Previous history and boundaries do not matter. However, in some cases the context may have a significant influence on the state of the system. An example is Rayleigh-Bernard convection cells and turbulent flow whose existence and nature are determined by the interaction of the fluid with the container boundaries. A biological example concerns what factors determine the structure, properties, and function that a particular protein (linear chain of amino acids) has. It is now known that the only factor is not just the DNA sequence that encodes for the amino acid sequence, in contradiction to some versions of the Central Dogma of molecular biology.  Other factors may be the type of cell that contains the protein and the network of other proteins in which the particular protein is embedded. Context sometimes matters.

Supervenience is the idea that once the micro level is fixed, macro levels are fixed too. The examples above might be interpreted as evidence against supervenience. Supervenience is used to argue against “the possibility for mental causation above and beyond physical causation.” 

Downward causation is sometimes equated with emergence, particularly in debates about the nature of consciousness. In the context of biology, Denis Noble defines downward causation as when higher level processes can cause changes in lower level properties and processes. He gives examples where physiological effects can switch on and off individual genes or signalling processes in cells, including maternal effects and epigenetics.

10. Intra-stratum closure: informational, causal, and computational

The ideas described below were recently developed by Rosas et al. from a computer science perspective. They defined emergence in terms of universality and discussed its relationship to informational closure, causal closure, and computational closure. Each of these are given a precise technical definition in their paper. Here I give the sense of their definitions. In considering a general system they do not pre-define the micro- and macro- levels of a system but consider how they might be defined so that universality holds, i.e., so that properties at the macro-level are independent of the details of the micro-level (i.e., are universal).

Informational closure means that to predict the dynamics of the system at the macroscale an observer does not need any additional information about the details of the system at the microscale. Equilibrium thermodynamics and fluid dynamics are examples. 

Causal closure means that the system can be controlled at the macroscale without any knowledge of lower-level information. For example, changing the software code that is running on a computer allows one to reliably control the microstate of the hardware of the computer regardless of what is happening with the trajectories of electrons in the computer.

Computational closure is a more technical concept, being defined in terms of “a conceptual device called the ε-(epsilon) machine. This device can exist in some finite set of states and can predict its own future state on the basis of its current one... for an emergent system that is computationally closed, the machines at each level can be constructed by coarse-graining the components on just the level below: They are, “strongly lumpable.” "

Rosas et al., show that informational closure and causal closure are equivalent and that they are more restrictive than computational closure. It is not clear to me how these closures relate to novelty as a definition of emergence.

In summary, emergence means different things to different people. I have listed ten different characteristics that have been associated with emergent properties. They are not all equivalent and so when discussing emergence it is important to be clear about which characteristic one is using to define emergence.

The multi-faceted character of emergence (part 1)

There is more to emergence than novel properties, i.e., where a whole system has a property that the individual components of the system do not have. Here I focus on emergent properties, but in most cases “property” might be replaced with state, phenomenon, or entity. I now discuss ten characteristics often associated with emergence, beyond novelty. Some people include one or more of these characteristics in their definitions of emergence. However, I do not include them in my definition because as I explain some of the characteristics are contentious. Some may not be necessary or sufficient for novel system properties.

The first five characteristics discussed below might be classified as objective (i.e., observable properties of the system) and the second five as subjective (i.e., associated with how an investigator thinks about the system). In different words, the first five are mostly concerned with ontology (what is real) and the second five with epistemology (what we know). The first five characteristics concern discontinuities, universality, diversity, mesoscales, and modification of parts. The second five concern self-organisation, unpredictability, irreducibility, downward causation, and closure. 

1. Discontinuities 

Quantitative changes in the system can become qualitative changes in the system. For example, in condensed matter physics spontaneous symmetry breaking only occurs in the thermodynamic limit (i.e., when the number of particles of the system becomes infinite). More is different. Thus, as a quantitative change in the system size occurs the order parameter becomes non-zero. In a system that undergoes a phase transition at a non-zero temperature, a small change in temperature can lead to the appearance of order and to a new state of matter. For a first-order phase transition, there is discontinuity in properties such as the entropy and density. These discontinuities define a phase boundary in the pressure-temperature diagram. For continuous phase transitions the order parameter is a continuous function of temperature, becoming non-zero at the critical temperature. However the derivative with respect to temperature may be discontinuous and/or thermodynamic properties such as the specific heat and susceptibility associated with the order parameter may approach infinite as the critical temperature is approached.

Two different states of a system are said to be adiabatically connected if one can smoothly deform one state into the other and all the properties of the system also change smoothly. The case of the liquid-gas transition illustrates subtle issues about defining emergence. A discontinuity does not imply a qualitative difference (novelty). On the one hand, there is a discontinuity in the density and entropy of the system as the liquid-gas phase boundary is crossed in the pressure-temperature diagram. On the other hand, there is no qualitative difference between a gas and a liquid. There is only a quantitative difference: the density of the gas is less than the liquid. Albeit sometimes the difference is orders of magnitude. The liquid and gas state can be adiabatically connected. There is a path in the pressure-temperature phase diagram that can be followed to connect the liquid and gas states without any discontinuities in properties.

The ferromagnetic state also raises questions, as illustrated by a debate between Rudolf Peierls and Phil Anderson about whether ferromagnetism exhibits spontaneous symmetry breaking. Anderson argued that it did not as, in contrast to the antiferromagnetic state, a non-zero magnetisation (order parameter) occurs for finite systems and the magnetic order does not change the excitation spectrum, i.e., produce a Goldstone boson. On the other hand, singularities in properties at the Curie temperature (critical temperature for ferromagnetism) only exist in the thermodynamic limit. Also, a small change in the temperature, from just above the Curie temperature to below, can produce a qualitative change, a non-zero magnetisation.

2. Universality

Properties often referred to as emergent are universal in the sense that it is independent of many of the details of the parts of the system. There may be many different systems that can have a particular emergent property. For example, superconductivity is present in metals with a diverse range of crystal structures and chemical compositions. 

Robustness is related to universality. If small changes are made to the composition of the system (for example replacing some of the atoms in the system with atoms of different chemical element) the novel property of the system is still present. In elementary superconductors, introducing non-magnetic impurity atoms has no effect on the superconductivity.

Universality is both a blessing and a curse for theory. Universality can make it easier to develop successful theories because it means that many details need not be included in a theory in order for it to successfully describe an emergent phenomenon. This is why effective theories and toy models can work even better than might be expected. Universality can make theories more powerful because they can describe a wider range of systems. For example, properties of elemental superconductors can be described by BCS theory and by Ginzburg-Landau theory, even though the materials are chemically and structurally diverse. The curse of universality for theory is that universality illustrates the problem of “under-determination of theory”, “over-fitting of data” and “sloppy theories” [Sethna et al.]. A theory can agree with the experiment even when the parameters used in the theory may be quite different from the actual ones. For example, the observed phase diagram of water can be reproduced, sometimes with impressive quantitative detail, by combining classical statistical mechanics with empirical force fields that assume water molecules can be treated purely being composed of point charges.

Suppose we start with a specific microscopic theory and calculate the macroscopic properties of the system, and they agree with experiment. It would then be tempting to think that we have the correct microscopic theory. However, universality suggests this may not be the case.

For example, consider the case of a gas of weakly interacting atoms or molecules. We can treat the gas particles as classical or quantum. Statistical mechanics gives exactly the same equation of state and specific heat capacity for both microscopic descriptions. The only difference may be the Gibbs paradox [the calculated entropy is not an extensive quantity] which is sensitive to whether or not the particles are treated as identical or not. Unlike the zeroth, first, and second law of thermodynamics, the third law does require that the microscopic theory be quantum. Laughlin discusses these issues in terms of “protectorates” that hide “ultimate causes” .  

In some physical systems, universality can be defined in a rigorous technical sense, making use of the concepts and techniques of the renormalisation group and scaling. These techniques provide a method to perform coarse graining, to derive effective theories and effective interactions, and to define universality classes of systems. There are also questions of how universality is related to the robustness of strata, and the independence of effective theories from the coarse-graining procedure.

3. Diversity

Even when a system is composed of a small number of different components and interactions, the large number of possible stable states with qualitatively different properties that the system can have is amazing. Every snowflake is different. Water is found in 18 distinct solid states. All proteins are composed of linear chains of 20 different amino acids. Yet in the human body there are more than 100,000 different proteins and all perform specific biochemical functions. We encounter an incredible diversity of human personalities, cultures, and languages. A stunning case of diversity is life on earth. Billions of different plant and animal species are all an expression of different linear combinations of the four base pairs of DNA: A, G, T, and C.

This diversity is related to the idea that "simple models can describe complex behaviour". One example is Conway’s Game of Life. Another example is how simple Ising models with a few competing interactions can describe a devil's staircase of ground states or the multitude of different atomic orderings found in binary alloys.

Goldenfeld and Kadanoff defined complexity [emergence] as “structure with variations”. Holland (VSI) discusses “perpetual novelty” giving the example of the game of chess, where are typical game may involve the order of 1050 move sequences. “Motifs” are recurring patterns (sequences of moves) in games. 

Condensed matter physics illustrates diversity with the many different states of matter that have been discovered. The underlying microscopics is “just” electrons and atomic nuclei interacting according to Coulomb’s law.

The significance of this diversity might be downplayed by saying that it is just a result of combinatorics. But such a claim overlooks the issue of the stability of the diverse states that are observed. In a system composed of many components each of which can take on a few states the number of possible states of the whole system grows exponentially with the number of components. For example, for a chain of ten amino acids there are 1013 different possible linear sequences. But this does not mean that all these sequences will produce a functional protein, i.e., a molecule that will fold rapidly (on the timescale of milliseconds) into a stable tertiary structure and perform a useful biochemical function such as catalysis of a specific chemical reaction or signal transduction.

4. Simple entities at the mesoscale 

A key idea in condensed matter physics is that of quasi-particles. A system of strongly interacting particles may have excitations, seen in experiments such as inelastic neutron scattering and Angle Resolved PhotoElectron Spectroscopy (ARPES), that can be described as weakly interacting quasi-particles. These entities are composite particles, and have properties that are quantitatively different, and sometimes qualitatively different, from the microscopic particles. Sometimes this means that the scale (size) associated with the quasi-particles is intermediate between the micro- and the macro-scales, i.e., it is a mesoscale. The existence of quasi-particles leads naturally to the technique of constructing an effective Hamiltonian [effective theory] for the system where effective interactions describe the interactions between the quasi-particles.

The economist Herbert Simon argued that a characteristic of a complex system is that the system can be understood in terms of nearly decomposable units. Rosas et al., argue that emergence is associated with there being a scale at which the system is “strongly lumpable”. Denis Noble has highlighted how biological systems are modular, i.e., composed of simple interchangeable components.

5. Modification of parts and their relationships

Emergent properties are often associated with the state of the system exhibiting patterns, order, or structure, terms that may be used interchangeably. This reflects that there is a particular relationship (correlation) between the parts which is different to the relationships in a state without the emergent property. This relationship may also be reflected in a generalised rigidity. For example, in a solid applying a force on one surface results in all the atoms in the solid experiencing a force and moving together. The rigidity of the solid defines a particular relationship between the parts of the system.

Properties of the individual parts may also be different. For example, in a crystal single-atom properties such as electronic energy levels change quantitatively compared to their values for isolated atoms. Properties of finite subsystems are also modified, reflecting a change in interactions between the parts. For example, in a molecular crystal the frequencies associated with intramolecular atomic vibrations are different to their values for isolated molecules. However, emergence is a sufficient but not a necessary condition for these modifications. In gas and liquid states, novelty is not present but there are still such changes in the properties of the individual parts.

As stated at the beginning of this section the five characteristics above might be associated with ontology (what is real) and objective properties of the system that an investigator observes and depend less on what an observer thinks about the system. The next five characteristics might be considered to be more subjective, being concerned with epistemology (how we determine what is true). In making this dichotomy I do not want to gloss over the fuzziness of the distinction or of two thousand years of philosophical debates about the relationship between ontology and epistemology, or between reality and theory.

In the next post, I will discuss the remaining five characteristics: self-organisation, unpredictability, irreducibility, contextuality and downward causation, and intra-stratum closure.

Thanks for reading this far!

Effective theories in classical and quantum mechanics

Working in quantum many-body theory, I slowly learned that many key concepts and techniques have predecessors and analogues in classical systems and one-body quantum systems. Examples include Green's functions, path integrals, cumulants, the linked cluster theorem, Hubbard-Stratonavich transformation (completing the square), mean-field theory, localisation due to disorder, and BBGKY hierarchy. Learning a full-blown quantum many-body version is easier if you first understand simpler analogues.

This post is about effective theories in classical systems and one-body quantum systems, following my earlier post about effective theories in quantum field theories of elementary particles.

Michèle Levi has a pedagogical article

Effective field theories of post-Newtonian gravity: a comprehensive review

This is motivated by the use of EFTs to describe gravitational waves produced by the inspiraling and merging of binary black holes and neutron stars. She discusses the different scales involved and how there are effective theories at each scale. She also puts these EFTs in the broader context of other fields.

Analogues in one-body quantum mechanics are also discussed  in

Effective Field Theories, Reductionism and Scientific Explanation, by Stephan Hartmann

"In his beautiful book Qualitative Methods in Quantum Theory, Migdal (1977) discusses an instructive example from quantum mechanics. Let S be a system which is composed of a fast subsystem Sf and a slow subsystem Ss, characterised by two frequencies of and os. It can be shown that the effects of Sf on Ss can be taken into account effectively by adding a potential energy term to the Hamiltonian operator of Ss. In this case, as well as in many other cases, one ends up with an effective Hamiltonian operator for the subsystem characterised by the smaller frequency (or energy)."

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

Emergence and the stratification of physics into sub-fields

The concept of emergence is central to understanding sub-fields of physics and how they are related, and not related, to other sub-fields.

The table below shows a stratum of sub-disciplines of physics. For each strata there are a range of length, time, and energy scales that are relevant. There are distinct entities that are composed of the entities from lower strata. These composite entities interact with one another via effective interactions that arise due to the interactions present at lower strata and can be described by an effective theory. Each sub-discipline of physics is semi-autonomous. Collective phenomena associated with a single strata can be studied, described, and understood without reference to lower strata.

Table entries are not meant to be exhaustive but to illustrate how emergence is central to understanding sub-fields of physics and how they are related to one another.

What do you think of the table? Is it helpful? Have you seen something like this before?

I welcome suggestions about entries that I could add.

Hollywood and a Physical Review paper

 I am not sure I have seen this before. If you watched the movie Oppenheimer, you may have noticed that a one point a student excitedly showed Oppenheimer the latest issue of Physical Review and the following image flashed across the movie screen.

On Continued Gravitational Contraction

J. R. Oppenheimer and H. Snyder

A beautiful blog post just appeared on 3 Quarks Daily,

 

September 1, 1939: A Tale Of Two Papers

by Ashutosh Jogalekar

The post describes the scientific and historical significance of the paper, including how it attracted no interest for twenty years, being eclipsed by a paper in the same issue of Physical Review.

The Mechanism of Nuclear Fission

Niels Bohr and John Archibald Wheeler

Have you ever seen a Hollywood movie that explicitly showed the page of a scientific journal article.

Seth Olsen (1975-2018): theoretical chemist

I was very sad to learn last week of the tragic death of Seth Olsen in an accident. He was a former collaborator and colleague at UQ.

Seth was an outstanding and energetic scientist who easily crossed discipline boundaries, especially between chemistry, physics, and molecular biology.

Much of what I know about computational quantum chemistry, fluorescent proteins, conical intersections, and diabatic states, I learnt from Seth. He played a significant role in this blog. A search revealed that his name is mentioned in more than 70 posts. Many posts were stimulated by his work, his questions, or his suggestions. He often wrote comments, covering a wide range of topics. I found his interest helpful and stimulating.

Seth grew up in the USA. He was a physics major at the College of William and Mary. In 2004 he completed a Ph.D in in Biophysics and Computational Biology at The University of Illinois at Urbana-Champaign. His thesis was entitled, ` Electronic Excited States of Green Fluorescent Protein Chromophore Models,'' and his advisor was Todd Martínez, now at Stanford.

I first met Seth in 2005 when he was a postdoc with Sean Smith at the Centre for Computational Molecular Science at University of Queensland. During that time he met Louise Kettle, a Ph.D student in chemistry, who he later married.

I was very happy when in 2008 I was able to persuade Seth to join my group as a Research Fellow. He helped my group expand from condensed matter into chemical physics.  In 2010 I was pleased when Seth was awarded a 5-year Australian Research Fellowship. We continued to collaborate, although in many ways I was the junior author.

A significant contribution of Seth was to use high-level quantum chemistry calculations to show that the low-lying excited electronic states of the chromophore molecule in the green fluorescent protein has a natural description in terms of the resonant colour theory of organic dyes developed in the middle of the twentieth century by Brooker, Platt, and Moffitt. In different words, he used quantum chemistry to justify and parametrise a simple effective Hamiltonian for a complex system. Furthermore, he provided a rigorous quantum chemical justification for the colour theory description of a very wide class of organic dyes based on the methine motif. These results provide chemical and physical insight, an understanding of trends, elucidate design principles, and make modeling in condensed environments such as proteins, solvents, and glasses much more feasible.

I had great respect for Seth's integrity, both personal and scientific. He carefully checked calculations and arguments, would not rush to publish, and would not indulge in hype. Much of my skepticism and caution about computational materials science I gained from Seth's critiques.

Seth had his priorities right, putting family first.
My kids thought Seth was pretty cool, particularly when he came to a group social at our house with a backpack that contained a home brew beer set up!

My sincere condolences to Louise and their three young children.

Don't know what else to say. This is the saddest blog post I have had to write.

Computational quantum chemistry in a nutshell

To the uninitiated (and particularly physicists) computational quantum chemistry can just seem to be a bewildering zoo of multiple letter acronyms (CCSD(T), MP4, aug-CC-pVZ, ...).

However, the basic ingredients and key assumptions can be simply explained.

First, one makes the Born-Oppenheimer approximation, i.e. one assumes that the positions of the N_n nuclei in a particular molecule are a classical variable [R is a 3N_n dimensional vector] and the electrons are quantum. One wants to find the eigenenergy of the N electrons. The corresponding Hamiltonian and Schrodinger equation is

The electronic energy eigenvalues E_n(R) define the potential energy surfaces associated with the ground and excited states. From the ground state surface one can understand most of chemistry! (e.g., molecular geometries, reaction mechanisms, transition states, heats of reaction, activation energies, ....)
As Laughlin and Pines say, the equation above is the Theory of Everything!
The problem is that one can't solve it exactly.

Second, one chooses whether one wants to calculate the complete wave function for the electrons or just the local charge density (one-particle density matrix). The latter is what one does in density functional theory (DFT). I will just discuss the former.

Now we want to solve this eigenvalue problem on a computer and the Hilbert space is huge, even for a simple molecule such as water. We want to reduce the problem to a discrete matrix problem. The Hilbert space for a single electron involves a wavefunction in real space and so we want a finite basis set of L spatial wave functions, "orbitals". Then there is the many-particle Hilbert space for N-electrons, which has dimensions of order L^N. We need a judicious way to truncate this and find the best possible orbitals.

The single particle orbitals can be introduced

where the a's are annihilation operators to give the Hamiltonian

These are known as Coulomb and exchange integrals. Sometimes they are denoted (ij|kl).
Computing them efficiently is a big deal.
In semi-empirical theories one neglects many of these integrals and treats the others as parameters that are determined from experiment.
For example, if one only keeps a single term (ii|ii) one is left with the Hubbard model!

Equivalently, the many-particle wave function can be written in this form.

Now one makes two important choices of approximations.

1. atomic basis set
One picks a small set of orbitals centered on each of the atoms in the molecule. Often these have the traditional s-p-d-f rotational symmetry and a Gaussian dependence on distance.

2. "level of theory"
This concerns how one solves the many-body problem or equivalently how one truncates the Hilbert space (electronic configurations) or equivalently uses an approximate variational wavefunction. Examples include Hartree-Fock (HF), second-order perturbation theory (MP2),  a Gutzwiller-type wavefunction (CC = Coupled Cluster), or Complete Active Space (CAS(K,L)) (one uses HF for higher and low energies and exact diagonalisation for a small subset of K electrons in L orbitals.
Full-CI (configuration interaction) is exact diagonalisation. This only possible for very small systems.

The many-body wavefunction contains many variational parameters, both the coefficients in from of the atomic orbitals that define the molecular orbitals and the coefficients in front of the Slater determinants that define the electronic configurations.

Obviously, one expects that the larger the atomic basis set and the "higher" the level of theory  (i.e. treatment of electron correlation) one hopes to move closer to reality (experiment). I think Pople first drew a diagram such as the one below (taken from this paper).

However, I stress some basic points.

1. Given how severe the truncation of Hilbert space from the original problem one would not necessarily to expect to get anywhere near reality. The pleasant surprise for the founders of the field was that even with 1950s computers one could get interesting results. Although the electrons are strongly correlated (in some sense), Hartree-Fock can sometimes be useful. It is far from obvious that one would expect such success.

2. The convergence to reality is not necessarily uniform.
This gives rise to Pauling points: "improving" the approximation may give worse answers.

3. The relative trade-off between the horizontal and vertical axes is not clear and may be context dependent.

4. Any computational study should have some "convergence" tests. i.e. use a range of approximations and compare the results to see how robust any conclusions are.

Emergence and singular asymptotic expansions

Seth Olsen kindly lent me his copy of Chemistry, Quantum Mechanics, and Reductionism by Hans Primas, published in 1981. I has a Foreword by Paul Feyerabend! 

[Primas died last October and there will be a symposium in his honour later this year]
This is a book I had wanted to read for a while since I had seen it referenced in various philosophical contexts. Besides some deep philosophy he has lots of polemical statements about theoretical chemistry.

Wanting to find an electronic version I could copy choice quotes from led me to a more dense, broader, and more recent (1998) article Emergence in exact natural science.

Here I mention a few highlights.

emergence and theory reduction are related.

Theory reduction is the process where a more general theory, such as quantum mechanics or special relativity, "reduces" in a particular mathematical limit to a less general theory such as classical mechanics. This is a subtle philosophical problem that is arguably poorly understood both by scientists [who oversimplify or trivialise it] and philosophers [who sometimes overstate the problem]. The subtleties arise because the two different theories usually involve concepts that are "incommensurate" with one another.

the distinction inside/outside is not covered by the most fundamental context-independent natural laws (first principles of physics). 

Here Primas is stresses the sometimes "arbitrary" value judgements that are made in distinguishing a "system" and its "environment". This involves distinguishing "patterns" and invoking "symmetry breaking".  He introduces notions of topology to try and make such distinctions more rigorous. I found this too technical to appreciate.

 Many inter-theoretical relations can be mathematically described by asymptotic expansions. Singular asymptotic expansions are never uniformly convergent in the intrinsic topology of the basic theory. This nonuniformity is not a disaster but an indication that the limiting case represents a caricature, suppressing irrelevant details and enhancing contextually relevant features. The discontinuous change in the limit leads to a discontinuous change in the semantics and therewith to a description in a new language in terms of emergent properties. In the same sense as a photograph can never replace a brilliant caricature, an asymptotic description can – for the intended purpose – be more adequate than the exact description.

Michael Berry also has a 1994 article that takes a similar point of view.

The assertions  

“something consists of elementary systems”,   

“something can be decomposed into “elementary systems”, 

“something can be described in terms of “elementary systems”, 

are not equivalent. 

When a light wave passes an object, a typical discontinuity – called the shadow – can be observed. However, in Maxwell’s electrodynamics – the fundamental theory for the propagation of light – shadows do not exist. Maxwell’s electrodynamics is governed by partial differential equations which have only continuous solutions. The discontinuities associated with shadows appear only in geometric optics, the limiting case of vanishing wavelength l , l → 0 .

He discusses at length how the notion of "molecular structure" in chemistry is an emergent concept. This relates to the issue of quantum entanglement between electrons and nuclei.

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. A more proper appreciation of the Born–Oppenheimer-description stresses its singular nature: it is an expansion about the singular point of infinite nuclear masses. An asymptotic expansion can be formulated in terms of the ratio e = (m/M)^1/4, where m is the mass of an electron and M is a mean nuclear mass of the molecular system. In the limiting case e = 0 the holistic correlations between nuclei and electrons are suppressed so the description of a molecule reduces to the description of the motion of electrons in the electric field of a classical nuclear framework. In this description the molecular structure is a property described by an emergent classical observable. The singular limiting case     e = 0 leads to a discontinuous change in the description and is the starting point for an asymptotic expansion in terms of the emergent property at higher levels of description. 

He then gives a another example that was new to me.

The transition from the more fundamental Lorentz-relativistic quantum mechanics to Galilei-relativistic quantum mechanics is governed by the contraction of the Lorentz group to the Galilei group – a highly singular limit. While the Lorentz group is semisimple, the Galilei group is not but has a more complicated mathematical structure. The emergent quantity associated with this contraction is the mass in the sense of a classical observable (which commutes with all other observables and can therefore be treated as a real parameter).

He also discusses how the concept of temperature is emergent, emphasising the centrality of the zeroth law of thermodynamics.