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."

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.

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.

Emergence in Chemistry

It is important to be clear what the system is. Most of chemistry is not really about isolated molecules. A significant amount of chemistry occurs in an environment, often within a solvent. Then the system is the chemicals of interest and the solvent. For example, when it is stated that HCl is an acid, this is not a reference to isolated HCl molecules but a solution of HCl in water, and then the HCl dissociates into H+ and Cl- ions. Chemical properties such as reactivity can change significantly depending on whether a compound is in the solid, liquid, or gas state, or on the properties of the solvent in which it is dissolved.

Scales

The time scales for processes, which range from molecular vibrations to chemical reactions, can vary from femtoseconds to days. Relevant energy scales, corresponding to different effective interactions, can vary from tens of eV (strong covalent bonds) to microwave energies of 0.1 meV (quantum tunnelling in an ammonia maser).

Other scales are the total number of atoms in a compound, which can range from two to millions, the total number of electrons, and the number of different chemical elements in the compound. As the number of atoms and electrons increases, so does the dimensionality of the Hilbert space of the corresponding quantum system.

Novelty

All chemical compounds are composed of a discrete number of atoms, usually of different type. For example, acetic acid, denoted CH3COOH, is composed of carbon, oxygen, and hydrogen atoms. The compound usually has chemical and physical properties that the individual atoms do not have.

Chemistry is all about transformation. Reactants combine to produce products, e.g. A + B -> C. C may have chemical or physical properties that A and B did not have.

Chemistry involves concepts that do not appear in physics. Roald Hoffmann argued that concepts such as acidity and basicity, aromaticity, functional groups, and substituent effects have great utility and are lost in a reductionist perspective that tries to define them precisely and mathematicise them.

Diversity

Chemistry is a wonderland of diversity, as it arranges chemical elements in a multitude of different ways that produce a plethora of phenomena. Much of organic chemistry just involves three different atoms: carbon, oxygen, and hydrogen.

Molecular structure

Simple molecules (such as water, ammonia, carbon dioxide, methane, benzene) have a unique structure defined by fixed bond lengths and angles. In other words, there is a well-defined geometric structure that gives the locations of the centres of atomic nuclei. This is a classical entity. This emerges from the interactions between the electrons and nuclei of the constituent atoms.

In philosophical discussions of emergence in chemistry, molecular structure has received significant attention. Some claim it provides evidence of strong emergence. The arguments centre around the fact that the molecular structure is a classical entity and concept that is imposed, whereas a logically self-consistent approach would treat both electrons and nuclei quantum mechanically.

The molecular structure of ammonia (NH3) illustrates the issue. It has an umbrella structure which can be inverted. Classically, there are two possible degenerate structures. For an isolated molecule, quantum tunnelling back and forth between the two structures can occur. The ground state is a quantum superposition of two molecular structures. This tunnelling does occur in a dilute gas of ammonia at low temperature, and an associated quantum transition (at a wavelength of 1.2 cm) is the basis of the maser, the forerunner of the laser. This example of ammonia was discussed by Anderson at the beginning of his seminal More is Different article to illustrate how symmetry breaking leads to well-defined molecular structures in large molecules. 

Figure is taken from here.

Born-Oppenheimer approximation 

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

Singularity. The Born-Oppenheimer approximation is justified by an asymptotic expansion in powers of (m/M)^1/4, where m is the mass of an electron and M the mass of an atomic nucleus in the molecule. This has been discussed by Primas and Bishop.

The rotational and vibrational degrees of freedom of molecules also involve a separation of time and energy scales. Consequently, one can derive separate effective Hamiltonians for the vibrational and rotational degrees of freedom.

Qualitative difference with increase in molecular size

Consider the following series with varying chemical properties: formic acid (CH2O2), acetic acid (C2H4O2), propionic acid (C3H6O2), butyric acid (C4H8O2), and valerianic acid (C5H10O2), whose members involve the successive addition of a CH2 radical. The Marxist Friedrich Engels used these examples as evidence for Hegel’s law: “The law of transformation of quantity into quality and vice versa”.

In 1961, Platt discussed properties of large molecules that “might not have been anticipated” from properties of their chemical subgroups. Table 1 in Platt’s paper lists “Properties of molecules in the 5- to 50-range that have no counterpart in diatomics and many triatomics.” Table 2 lists “Properties of molecules in the 50- to 500-atom range and up that go beyond the properties of their chemical sub-groups.” The properties listed included internal conversion (i.e., non-radiative decay of excited electronic states), formation of micelles for hydrocarbon chains with more than ten carbons, the helix-coil transition in polymers, chromatographic or molecular sorting properties of polyelectrolytes such as those in ion-exchange resins, and the contractility of long chains.

Platt also discussed the problem of molecular self-replication. Until 1951, it was assumed that a machine could not reproduce itself,f and this was the fundamental difference between machines and living systems. However, von Neumann showed that a machine with a sufficient number of parts and a sufficiently long list of instructions can reproduce itself. Platt pointed out that this suggested there is a threshold for autocatalysis: “this threshold marks an essentially discontinuous change in properties, and that fully-complex molecules larger than this size differ from all smaller ones in a property of central importance for biology.” Thus, self-replication is an emergent property. A modification of this idea has been pursued by Stuart Kauffman with regard to the origin of life, that when a network of chemical reactions is sufficiently large, it becomes self-replicating.

The green comet and quantum chemistry

The comet C/2022 E3 (ZTF) getting a lot of attention, pointed out to me by my friend Alexey. Why is it green? This basic question turns out to be scientifically rich and has only recently been answered.

The green glow comes from a triplet excited state of diatomic carbon, C2. This got my interest because a decade ago I blogged on debates by quantum chemists about whether C2 involves a quadruple bond. Back in 1995, Roald Hoffmann wrote an interesting column in The American Scientist (and reproduced in his beautiful book Same and Not the Same) about the molecule and how it is present in various organometallic compounds and inorganic crystals.

Recent advances in understanding the photophysics of C2 were reported in 2021 in this paper.

Photodissociation of dicarbon: How nature breaks an unusual multiple bond

Jasmin Borsovszky, Klaas Nauta, Jun Jiang, Christopher S. Hansen, Laura K. McKemmish, Robert W. Field, John F. Stanton, Scott H. Kable, and Timothy W. Schmidt 

Here is a summary of the significance and content of the paper from Chemistry World.

..as dicarbon streams out of the comet core, it is destroyed by sunlight – this is why the comet tail, unlike the coma, is colourless. However, the precise mechanism of this supposed photodissociation had remained unclear.

Researchers in Australia and the US have now for the first time observed diatomic carbon’s photodissociation in the lab. The team produced dicarbon by photolysing tetrachloroethylene, and then breaking it apart with laser pulses. This allowed them to determine its bond dissociation energy with the same precision as for oxygen and nitrogen. Previous measurements for dicarbon had uncertainties an order of magnitude higher than for other diatomic molecules.

To break its quadruple bond, the molecule must absorb two photons and undergo two ‘forbidden’ transitions, those that break spectroscopic rules. Cometary dicarbon, the researchers calculated, has a lifetime of around two days until sunlight breaks it apart – the reason why its colour is visible in the coma but not in the tail.

The wonders of gallium

 A friend recently showed me that solid gallium can melt in your hand.

I did not know this. I was quite familiar with liquid mercury, but not gallium. 

The existence of elemental gallium was predicted by Mendeleev in 1869 after he constructed the periodic table. It was discovered within six years. He was able to predict that it would have a low-melting temperature, based on extrapolations from the known melting temperatures of elements close to it in the periodic table.

Solid gallium is soft enough to be cut with a knife.

Three different stable crystal structures for solid gallium are shown below.

The phase diagram of pure gallium is shown below.

Note the negative slope of the phase boundary between the liquid and the solid alpha-Ga. This is like water. It follows from the Clausius-Clapeyron equation that the solid state has lower density than the liquid state. Gallium is the only elemental metal with this property. (The semi-metals antinomy and bismuth also do).

Gallium remains liquid over a wider range of temperatures (2373 K) than any other known substance.

The figures above are taken from the following paper from 2020.

Ab initio phase diagram and nucleation of gallium  Haiyang Niu, Luigi Bonati, Pablo M. Piaggi, and Michele Parrinello

Unfortunately, that paper does not provide much insight into the low melting temperature. The key is that the solid state contains dimers of Ga, that are weakly bonded to each other. A helpful discussion is the introduction to the following paper.

On the bonding of Ga2, structures of Gan clusters and the relation to the bulk structure of gallium 

N. Gaston and A.J. Parker

The image above is from the entry on Gallium in the beautiful book The Elements by Theodore Gray.

I thank my young friend Alexey for introducing me to the wonders of gallium.

A guide through hype about computational chemistry on quantum computer

One of the many problems with hype in science is that it glosses over problems that means they do not get addressed which ultimately hinders real scientific progress. 

There is a lot of hype about how quantum computers will be able to solve problems in materials science that are of industrial significance and thus "herald a new era of chemical research". Such claims are carefully examined and deconstructed in the following preprint. Most of the authors are at Schrodinger, Inc.

How will quantum computers provide an industrially relevant computational advantage in quantum chemistry?

V.E. Elfving, B.W. Broer, M. Webber, J. Gavartin, M.D. Halls, K. P. Lorton, A. Bochevarov

The article is also a useful guide to current state-of-the-art computational chemistry on classical computers.

I reproduce most of the paper abstract below as it is helpful summary.

Numerous reports claim that quantum advantage, which should emerge as a direct consequence of the advent of quantum computers, will herald a new era of chemical research because it will enable scientists to perform the kinds of quantum chemical simulations that have not been possible before. Such simulations on quantum computers, promising a significantly greater accuracy and speed, are projected to exert a great impact on the way we can probe reality, predict the outcomes of chemical experiments, and even drive design of drugs, catalysts, and materials. 

In this work we review the current status of quantum hardware and algorithm theory and examine whether such popular claims about quantum advantage are really going to be transformative. We go over subtle complications of quantum chemical research that tend to be overlooked in discussions involving quantum computers. 

We estimate quantum computer resources that will be required for performing calculations on quantum computers with chemical accuracy for several types of molecules. In particular, we directly compare the resources and timings associated with classical and quantum computers for the molecules H2 for increasing basis set sizes, and Cr2 for a variety of complete active spaces (CAS) within the scope of the CASCI and CASSCF methods. The results obtained for the chromium dimer enable us to estimate the size of the active space at which computations of non-dynamic correlation on a quantum computer should take less time than analogous computations on a classical computer. Using this result, we speculate on the types of chemical applications for which the use of quantum computers would be both beneficial and relevant to industrial applications in the short term.

The authors present a useful typology of claims of quantum advantage that are irrelevant.

1. Irrelevance due to availability of accurate experimental results. 

2. Irrelevance due to availability of conventional computational results. 

3. Irrelevance due to real world complexity:

When simulated chemical processes are very complicated and involve potentially hundreds of intermediates, conformations, or reaction paths, as in catalytic and metabolic pathways, the real research bottleneck lies in a combinatorial explosion of possibilities to probe with simulation.

4. Irrelevance to industrial applications

Many of the issues discussed in the preprint are not unrelated to those associated with hype about using machine learning in computational materials science, and are beautifully critiqued by Roald Hoffmann and Jean-Paul Malrieu.

Tutorial on modelling quantum dynamics in biomolecules

This week I am giving two (virtual) talks at a meeting

Light-matter Interactions from scratch: Theory and Experiments at the Border with Biology 

supported by the ICTP (International Center for Theoretical Physics) in Trieste.

In the ICTP tradition, one talk is a tutorial and the second talk is about my research.

Here are the slides for the tutorial on Effective Model Hamiltonians for Quantum Dynamics in Complex Molecular Materials. Feedback is welcome.

The research talk is about hydrogen bonding. I will post slides for that later.

Sage wisdom on computational materials science

Roald Hoffmann and Jean-Paul Malrieu are two of my favourite living theoretical chemists. Both greatly value the role of concepts and intellectual clarity in theory. Hoffmann has featured in 22 posts on this blog.

They recently published a wonderful trilogy in  Angewandte Chemie.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. 

Part A. Stage Setting

Part B. The March of Simulation, for Better or Worse

Part C. Toward Consilience

I add this trilogy to my list of 5 papers every computational chemistry student should read, suggested by me a decade ago. [Malrieu is author of one of those and Hoffmann co-author of another.]

Although the trilogy addresses and uses specific examples from computational quantum chemistry it is just as relevant to anyone interested in computational materials science. Actually, I hope that anyone interested in materials science would read and digest it as it gives a sober and balanced perspective about the relationship between theory, simulation, and understanding.

Articles are timely as they address hype about how AI techniques will "revolutionise" materials theory. 

The articles are beautifully written and engage with broader themes such as philosophy of science, culture, art, and politics.

Finally, I just love this photo of the two authors, both in their eighties. the photo reflects some of the joy they find in science, so beautifully expressed in these articles.

I thank Ben Powell for bringing the papers to my attention.

Implicit versus explicit beliefs

 How can we design a room-temperature superconductor? How can a government stimulate economic growth? How can an NGO help reduce domestic violence? Why do communities become segregated on racial lines? How can I improve my mental health?

These important questions may seem unrelated. However, I propose that often there is a common issue about the strategies that people (whether individuals, professions, NGOs, funding agencies, governments, ...) propose to find answers or when definite answers are proposed.

Many strategies and answers involve a heavy dose of implicit beliefs. These are assumptions that are never stated. They may be elements of a worldview, which according to one definition, is

a commitment, a fundamental orientation of the heart, that can be expressed as a story or in a set of presuppositions (assumptions which may be true, partially true, or entirely false) which we hold (consciously or subconsciously, consistently or inconsistently) about the basic construction of reality, and that provides the foundation on which we live and move and have our being.

 James W. Sire, The Universe Next Door: A basic worldview catalog

These implicit beliefs may relate to values and morality. But I want to focus more on implicit beliefs that are related to academic disciplines such as philosophy of science, psychology, political science, theology, economics, anthropology,  sociology, ...  Most of us have never studied these disciplines and some of us may be skeptical about some of them. But, my point is that everyone has implicit ideas about what is true with regard to the objects these disciplines study. Everyone has a philosophy of science. Everyone has ideas about how minds work and how to change societies. It is just that these beliefs are rarely stated. 

Why does this matter? If implicit beliefs are never stated, they can never be tested, evaluated, critiqued, refined, or rejected. I believe that implicit beliefs are too often based on intuition, prejudice, common sense, or culture (social pressure to conform to accepted wisdom). This is not necessarily bad. Sometimes intuition, common sense, and culture are helpful and correct. We could not survive in life if we did not have them. We simply don't have the time, energy, and resources to constantly question and validate everything. On the other hand, if there is a vacuum, it will get filled with something. A major lesson from scientific history is that intuition, prejudice, common sense are sometimes wrong.

I now give three concrete examples of implicit beliefs. They cover computational materials science, public policy, and social activism.

Understanding materials using computers

Amongst others, there are two things, we would like more computational power to be able to do. One is to do reliable ab initio calculations of the properties of complex molecules and solids, from proteins to crystals with unit cells containing large numbers of atoms.  Another is to do exact diagonalisation (or some alternative reliable method) of many-body Hamiltonians, such as the Hubbard model, on large enough lattices that finite-size effects are minimal or can be reliably accounted for.

Over the past decade, there has been a lot of hype about how quantum computers and/or machine learning techniques will solve these problems and thus initiate a new era of materials understanding, discovery, and design with significant technological and economic benefits. My problem is that these claims usually seem to have the implicit belief that the only obstacle to progress is one of computational power. This fallacy has recently been deconstructed and critiqued in detail in three beautiful essays by Roald Hoffmann and Jean-Paul Malrieu, Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond.

Public policy

National economies around the world have been battered by the covid-19 pandemic. In response, governments of prosperous countries are spending big on stimulus packages. This involves taking on massive amounts of debt and significant government intervention in "free" market economies. Will these initiatives achieved their goals, particularly in the long term? Could they actually make things worse? Responses from pundits, both for and against, are laden with implicit beliefs. Unfortunately, economists cannot agree on the answer to the basic question, "Does government stimulus spending actually produce economic growth?" This issue is nicely discussed in a pre-pandemic podcast at Econtalk. 

NGOs and social activism

Many NGOs are about change. They aim to build a better world, addressing problems such as domestic violence, poverty, climate change, corruption, racism,... They aim to promote education, human rights, good governance, democracy, health, transparency, .. I love NGOs. I support many: philosophically, financially, and practically. To survive most NGOs have to raise funds, whether from many small donors or large philanthropies. This requires a well-honed pitch that aims to inspire potential donors to give. Furthermore, the whole operation of most NGOs is laden with implicit beliefs, whether those of the founders, staff or donors.

Consider a hypothetical NGO whose goal is to reduce the number of murders in a country. I chose this example because it may at first appear less controversial and contentious than some. Almost everyone thinks murder is wrong (always) and societies should stop reduce it. But why do murders occur? Revenge, passion, drugs, alcohol, money, politics, racism, ... Will making the purchase of guns difficult reduce the murder rate? Gun lobbyists will claim "Guns don't kill people. Criminals do! Law-abiding citizens need guns for self-defense." (cringe). There are many other alternative strategies: increasing penalties (longer jail terms or even the death penalty), the number of police, weapons for police, community policing, drug rehabilitation, breaking up gangs, ... Wow! It's complicated. My main point is that the hypothetical NGO will probably have an implicit belief that one particular strategy is the best one. Furthermore, if you identify and question this belief reasonable debate may not follow, but it may even be claimed that you don't care about stopping murder.

Some NGOs and philanthropies have become mindful of these issues. In response to a grant application that I helped an NGO write we were asked what our "theory of change" was? I discovered that there is a whole associated "industry. According to theoryofchange.org (!)

One organisation which began to focus on these issues was the US-based Aspen Institute and its Roundtable on Community Change. ... [leading] to the publication in 1995 of New Approaches to Evaluating Comprehensive Community Initiatives. In that book, Carol Weiss, ... hypothesized that a key reason complex programs are so difficult to evaluate is that the assumptions that inspire them are poorly articulated. She argued that stakeholders of complex community initiatives typically are unclear about how the change process will unfold and therefore give little attention to the early and mid-term changes that need to happen in order for a longer term goal to be reached.  

This led to software designed to help organisations plan initiatives, with a particular emphasis on teasing out assumptions embedded in plans. A related method is construction of a logframe matrix [logical framework].

All models are wrong but some are useful. I first learnt this aphorism from Scott Page, in his wonderful course Model Thinking at Coursera.  Models help us think more clearly. Simple quantitative models, such as agent-based models, in the social sciences, have the value that their assumptions can be clearly stated, and then the consequences of these assumptions can be investigated in a rigorous manner.

What do you think? Are there examples that you think involve implicit beliefs that need to be stated explicitly?

Mean-field theories: helpful or misleading? From Hubbard to COVID-19 models

Mean-field theory (self-consistent field theory) is incredibly valuable. It gives significant insights into what is possible with a particular model.
What kind of phases and broken symmetries may be possible?
How does the phase diagram depend on different parameters in a model?
Indeed, mean-field theory is the basis of the whole Landau paradigm for spontaneous symmetry breaking and phase transitions.
Implementations of Density Functional Theory (DFT) in computational materials science are basically mean-field theories. Most of computational quantum chemistry involves some sort of mean-field theory.

Mean-field theories do not take into account fluctuations, dynamic or spatial.
Basically, a many-body problem is reduced to a one-body problem.

A good mean-field theory can win you a Nobel Prize. That's what Anderson, BCS, Ginsberg, Abrikosov, and Leggett all did!
Can you think of others?

However, mean-field theory does have its limitations.
It is usually quantitatively wrong. It often gives unreliable values for transition temperatures. In spatial dimensions less than four, mean-field theory gives the wrong values for the critical exponents near a phase transition.

An even bigger problem is that mean-field can be qualitatively wrong.
For many models (e.g. the Ising model or Heisenberg model) mean-field theory always gives a transition from a disordered to an ordered phase at a non-zero temperature.
However, in one dimension the Ising model has no phase transition in one dimension. For a Heisenberg ferromagnet or antiferromagnet, there is no transition at finite temperature in two dimensions.
The Mermin-Wagner theorem states that in two dimensions a superconductor or superfluid never has long-range order at finite temperature. Instead, there is a Kosterlitz-Thouless transition, to a distinct state of matter, with power-law correlations.

Mean-field theory can also fail to predict the existence of states of matter. For example, for Hubbard models, mean-field theory can produce several states: a Fermi liquid metal, a ferromagnetic metal, an antiferromagnetic metal, and a spin-density-wave insulator. But it is quite possible the model also can have non-magnetic Mott insulating phases, superconductivity, non-Fermi liquid metals, and pseudogap states.

In the next post, I will discuss some issues that arise in mean-field theories used in modeling the COVID-19 epidemic.

Estimating the Ising interaction in spin-crossover compounds

I previously discussed how one of the simplest model effective Hamiltonians that can describe many physical properties of spin-crossover compounds is an Ising model in an external "field". The s_i=+/-1 is a pseudo-spin denoting the low-spin (LS) and high-spin (HS) states of a transition metal molecular complex at site i.

The ``external field" is one half of the Gibbs free energy difference between the LS and HS states. The physical origin of the J interaction is ``believed to be'' elastic, not magnetic interactions. A short and helpful review of the literature is by Pavlik and Boca.

Important questions are:

1. What is a realistic model that can explain how J arises due to elastic interactions?
2. How does one calculate J from quantum chemistry calculations?
3. How does one estimate J for a specific material from experimental data?
4. What are typical values of J?

I will focus on the last two questions.
One can do a mean-field treatment of the Ising model, leading to a model free energy for the whole system that has the same form as that of an ideal binary mixture of two fluids where
x = (1 + av(s_i))/2, is the relative fraction of low spins. 
This model free energy was proposed in 1972 by Slichter and Drickmamer.
The free energy of interaction between the two "fluids" is of the form -Gamma x^2.
Gamma is often referred to as the ``co-operativity" parameter.
Minimising the free energy versus x gives a self-consistent equation for x(T).
This can be compared to experimental data for x vs T, e.g. from the magnetic susceptibility, and a Gamma value extracted for a specific material.

Values for Gamma obtained in this way for a wide-range of quasi-one-dimensional materials [with covalent bonding (i.e. strong elastic interactions) between spin centres] are given in Tables 1 and 2 of Roubeau et al. The values of Gamma are in the range 2-10 kJ/mol. In temperature units this corresponds to 240-1200 K.

My calculations [which may be wrong] give that Gamma = 4 J z, where z is the number of nearest neighbours in the Ising model. This means that (for a 1d chain with z=2) that J is in the range of 0.3-1.5 kJ/mol, or 30-150 K.

In many spin-crossover materials, the elastic interactions are via van der Waals, hydrogen bonding, or pi-stacking interactions. In that case, we would expect smaller values of J.
This is consistent with the following.
An analysis of a family of alloys by Jakobi et al. leads to a value of Gamma of 2 kJ/mol.
[See equation 9b. Note B=Gamma=150 cm^-1.  Also in this paper x is actually denoted gamma and x denotes the fraction of Zn in the material.].

I thank members of the UQ SCO group for all they are teaching me and the questions they keep asking.

What is the mass of a molecular vibration?

This is a basic question that I have been puzzling about. I welcome solutions.

Consider a diatomic molecule containing atoms with mass m1 and m2. It has a stretch vibration that can be described by a harmonic oscillator with a reduced mass mu given by

.

Now consider a polyatomic molecule containing N atoms.
It will have 3N-6 normal modes of vibration.
[The 6 is due to the fact that there are 6 zero-frequency modes: 3 rigid translations and 3 rotations of the whole molecule].
In the harmonic limit, the normal mode problem is solved below.
[I follow the classic text Wilson et al., Molecular Vibrations].
The problem is also solved in matrix form in Chapter 6 of Goldstein, Classical Mechanics].

One now has a collection of non-interacting harmonic oscillators. All have mass = 1. This is because the normal mode co-ordinates have units of length * sqrt(mass).

The quantum chemistry package Gaussian does more. It calculates a reduced mass mu_i for each normal mode i using the formula below.

This is discussed in these notes on the Gaussian web site. From mu_i and the normal mode frequency_i it then calculates the spring constant for each normal mode.

I have searched endlessly, and tried myself, but have not been able to answer the following basic questions:

1. How do you derive this expression for the reduced mass?
2. Is this reduced mass physical, i.e. a measurable quantity?

Similar issues must also arise with phonons in crystals.

Any recommendations?

Fun facts about phonons

Today we just take it for granted that crystals are composed of periodic arrays of interacting atoms. However, that was only established definitively one hundred years ago.
I have been brushing up on phonons with Marder's nice textbook, Condensed Matter Physics.
There are two historical perspectives that I found particularly fascinating. Both involve Max Born.

In a solid the elastic constants completely define the speeds of sound (and the associated linear dispersion relationship). In a solid of cubic symmetry, there are only three independent elastic constants, C_11, C_44, and C_12.
Cauchy and Saint Venant showed that if all the atoms in a crystal interact through pair-wise central forces then C_44=C_12. However, in a wide range of elemental crystals, one finds that C_12 is 1-3 times larger than C_44. This discrepancy caused significant debate in the 19th century but was resolved in 1914 by Born who showed that angular forces between atoms could explain the violation of this identity. From a quantum chemical perspective, these angular forces arise because it costs energy to bend chemical bonds.

The first paper on the dynamics of a crystal lattice was by Born and von Karman in 1912. This preceded the famous x-ray diffraction experiment of von Laue that established the underlying crystal lattice. In 1965, Born reflected

The first paper by Karman and myself was published before Laue's discovery. We regarded the existence of lattices as evident not only because we knew the group theory of lattices as given by Schoenflies and Fedorov which explained the geometrical features of crystals, but also because a short time before Erwin Madelung in Göttingen had derived the first dynamical inference from lattice theory, a relation between the infra-red vibration frequency of a crystal and its elastic properties.... 

Von Laue's paper on X-ray diffraction which gave direct evidence of the lattice structure appeared between our first and second paper. Now it is remarkable that in our second paper there is also no reference to von Laue. I can explain this only by assuming that the concept of the lattice seemed to us so well established that we regarded von Laue's work as a welcome confirmation but not as a new and exciting discovery which it really was.

This raises interesting questions in the philosophy of science. How much direct evidence do you need before you believe something? I can think of two similar examples from more recent history: the observation of the Higgs boson and gravitational waves. Both were exciting, and rightly earned Nobel Prizes.
However, many of us were not particularly surprised.
The existence of the Higgs boson made sense because it was a necessary feature of the standard model, which can explain so much.
Gravitational waves were a logical consequence of Einstein's theory of general relativity, which had been confirmed in many different ways. Furthermore, gravitational waves were observed indirectly through the decay of the orbital period of binary pulsars.

A critique of DFT calculations for spin crossover materials

A basic question concerning spin crossover compounds is what are the energy difference and entropy difference between the low spin (LS) and high spin (HS) states.

The relative magnitude of these two quantities determines the crossover temperature from the LS to HS state.

From experiment typical values of the energy difference Delta H are of the order of 1-5 kcal/mol (4-20 kJ/mol). Entropy differences are typically about 30-60 kJ/mol/K. (See table 1 in the Kepp paper below).
This relatively small difference in energy presents a challenge for computational quantum chemistry,
such as calculations based on density functional theory, because of the strong electron correlations associated with the transition metal ions,

Over the past few years some authors have done nice systematic studies of a wide range of compounds with a wide range of DFT exchange-correlation (XC) functionals. Here I will focus on two papers.

Benchmarking Density Functional Methods for Calculation of State Energies of First Row Spin-Crossover Molecules 
Jordi Cirera, Mireia Via-Nadal, and Eliseo Ruiz

Theoretical Study of Spin Crossover in 30 Iron Complexes 
Kasper P. Kepp

First, these studies are refreshing and important. Too many computational chemistry calculations are dubious because they do not do systematics. 
Here I will just discuss the first paper.

Cirera et al. use 8 different XC functionals to study 20 different compounds. They find that only one (!) functional (TPSSh) correctly gives a low spin ground state for all the compounds, i.e. Delta H is positive.

The figure below nicely summarises the results.

Before one gets too excited that one has now found the "right" functional, one should note that when one uses TPSSh to calculate the crossover temperature there is little correlation with the experimental values.

To put all this in a broader context consider the hierarchal figure below which is in the spirit of the metaphor of Jacob's ladder proposed by John Perdew. [The figure is from here]. However, I do not think Jacob's ladder is the best Biblical metaphor.

This highlights the ad hoc nature of DFT based calculations and that one is a long way from anything that should seriously be considered to be a true ab initio calculation.

It should also be noted that all these calculations are for a single molecule in vacuum. However, the experiments are in the solid state (or solution) and so the energetics can be shifted by electrostatic screening and/or solvation. The crossover temperature (which can become a first-order phase transition) may also be shifted by intermolecular elastic interactions.

Strategies for minimal effective Hamiltonians

An important step in understanding any class of complex materials is to find/discover the simplest possible effective Hamiltonian that can be used to describe the main properties of interest (e.g. a phase diagram).

Doing this well is a non-trivial and subjective process. I am thinking about this because I am currently trying to figure out the appropriate Hamiltonian for spin-crossover compounds.

Here are some key elements of the process. 

"Simplest possible" means having the fewest possible degrees of freedom and parameters.

1. What are the key degrees of freedom (molecular orbitals, vibrations, spins, ...)?

2. What are the key interactions and the associated Hamiltonian?

3. What approximation scheme can be used to calculate properties of the many-body Hamiltonian (ground state, thermodynamics, electronic, magnetic, ...)?

4. How do the calculated properties compare to experiment?

5. Can we estimate the values of the Hamiltonian parameters from the comparison of the calculated properties with experiments? 

6. Can we estimate the values of the Hamiltonian parameters from ab initio electronic structure methods, such as those based on density functional theory (DFT)?

Inevitably, things do not work out perfectly, sometimes qualitatively and always somewhat quantitatively. Then one has to face the difficult task of deciding what the problem is and what the next step is. There are several options.

A. There are some missing degrees of freedom in the original Hamiltonian.

B. There are some missing interactions.

C. The approximation scheme used to calculate properties was not reliable enough.

D. There is a problem with the experiments.

E. This is really the best one can hope to do and you should move on to other problems. i.e, know when to quit and face the law of diminishing returns.

This plethora of options is why falsifiability is so hard in the theory of strongly correlated electron materials. But, it does not mean we should give up on it.

The flow diagram below is one way of looking at the process. Some people like the picture. Others do not. As usual, real science is not quite so algorithmic.

Origin of the strong spin-phonon coupling in transition metal complexes

Spin crossover (SCO) materials are fascinating and raise many interesting questions.
Here I want to address the underlying physics of why there is a large change in bond length (typically 10-20 per cent) when the spin state of the complex changes. Basically, it is because the ligand field splitting Delta changes significantly with bond length. The change in spin state is associated with electrons moving between from the upper d -levels on the metal ion  (e_g in an octahedral complex) to the upper levels (t_g2).

What is the physical origin of this splitting?

How does Delta vary with the distance between the metal (M) and the ligand L?

One can answer the second question theoretically with quantum chemistry computations and experimentally by changing the ligand L, which leads to changes in bond length. The figure below is taken from the book, Ligand Field Theory and its Applications.

The variation of Delta and the bond length with ligand reflects the spectrochemical series.

Quantum chemistry computations show a similar variation for a given complex by varying the M-L distance. For example, see Figure S5 in the Supplementary Material for this paper.

What is the physical origin of this splitting?

A first guess is from "crystal field theory" that associates the energy level splitting with classic electrostatic effects. This gives a value that falls of as the sixth power of the M-L distance, and makes the concrete (and roughly correct ) prediction that for an octahedral complex the e_g levels move up by 3/2 times the amount that the t_2g levels move down. For a tetrahedral complex the opposite happens. However, there are two significant problems with this prediction. First, the predicted splitting is an order of magnitude too small. Second, this model predicts the opposite trend to the spectrochemical series.

A better description is obtained from "ligand field theory" where the splitting arises from covalent bonding between the d-orbitals on the metal and the p-orbitals on the ligand. For an octahedral complex, the t_2g (e_g) orbitals have positive (zero or negative) overlap with the ligand orbitals.

Aside.

It is interesting (and disturbing?) that the authors of the figure above compare the data for Delta vs. R to power laws, 1/R^6 and 1/R^5.  For the data R varies by about 10 per cent. To distinguish between power laws one should be comparing data over several orders of magnitude!

In reality, the data is just as consistent with a linear decay.

What is of interest to me is the magnitude of the decay, G= 1 eV/Angstrom. The next step is to argue why this is "large". The change in bond length with spin crossover will be approximately G/B where B is the elastic constant for the bond.