Spontaneous symmetry breaking and the thermodynamic limit

 Spontaneous symmetry breaking is a fundamental concept in condensed matter and quantum field theory. Amongst philosophers of science the concept is receiving increasing attention, particularly in the context of discussions about emergence.

How do we understand the following two observations about a system at zero temperature?

At zero temperature for a finite-sized system there is no symmetry breaking. The ground state transforms as the trivial representation of the symmetry group of the Hamiltonian. It is non-degenerate.

In the thermodynamic limit, there is a family of degenerate ground states. They are related to one another by a transformation of the symmetry group. This concept is captured in picture below of the Mexican hat potential.

Motion around the trough is associated with the Goldstone mode. Motion perpendicular to the trough is associated with the "Higgs boson".

How does this picture connect with a finite system?

An intuitive picture is that the ball in the trough has a finite mass and so motion in the trough is like the quantum mechanics of a rotor with finite moment of inertia. Then there is a non-generate ground state with equal probability to be located at any angle. What might the moment of inertia be? For reasons described below it turns out to be related to the superfluid stiffness.

For the case of a Heisenberg antiferromagnet, the physics was worked out by Phil Anderson in 1952 where he introduced the concept of a "tower of states" that become degenerate in the thermodynamic limit. 

They are described by the following effective Hamiltonian

c is the speed of magnons (Goldstone bosons). vec(S) is the total spin, V is the volume of the system, and rho_s is the spin stiffness associated with the broken-symmetry. As the thermodynamic limit is approached the energy of these states scale with L^-d where d is the dimension of the system. In contrast, the magnon states scale with L^-1. Thus, for exact diagonalisation of sufficiently large systems, the "tower of states" should be clearly be below the magnons states.

 In 1992 all of the above was confirmed for the triangular lattice in numerical work by Bernu, Hluillier, and Pierre.  

In the figure below, the top panel shows the low-lying eigenstates. The lowest energy states do scale with S^2. The middle panel shows how these states do separate from the magnon states. 

The figure below shows how the moment of inertia [proportional to the denominator in the tower of states equation above] does scale with the system size.

 

More recently there has been some interesting work that explores how the tower of states appears in the entanglement entropy.

Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry

Max A. Metlitski, Tarun Grover

But for now, discussing that is above my pay grade 😀

I thank Gerard Milburn for asking me questions that led to me finally getting a better physical picture of the issues discussed here.

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.

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.

The quantum physics of life in red and green

Life is truly amazing!
Life is beautiful!
...and it involves quantum many-body physics...

There is a beautiful (short) review
Heme: From quantum spin crossover to oxygen manager of life 
Kasper Kepp

The article involves a plethora of topics that I have discussed before on this blog. I have included relevant links.

Kepp starts with the unique (chemically fine-tuned) properties of both iron and porphyrin that enable them to play a central role in two of the most important processes in life: respiration and photosynthesis. He has a beautiful paragraph (perhaps in the style of Roald Hoffmann):

Such ligand-field transitions of iron in porphyrin were familiar to our ancestors as the characteristic red color of blood that largely defines the human psychological and cultural connotations of the color representing courage, war, danger, and suffering. 

Incidentally, pi-pi* transitions within the porphyrin-derived chlorophylls are also responsible for the green color of plants, associated with nature, life and hope, so the reader may perhaps agree that porphyrin has had vast (but alas! rarely appreciated) cultural consequences.

The oxygen molecule is a spin triplet.
Iron(II) porphyrin is in a triplet spin state (S=1). The Fe(II) is a d6 configuration in a D_4h crystal field.
When they bind together the ground state is a spin-singlet.

There are two fundamental quantum chemistry questions that are discussed.

1. What is the electronic structure (many-body wave function) of the ground state for oxygen bound to heme?

2. What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

The first question has a long history. Like almost anything important and profound in quantum chemistry it goes back to Linus Pauling! In 1936 Pauling and Coryell argued that the ground state is

essentially a neutral O=O binding with two of its electrons to iron to produce a formally iron(II) if both the bonding electrons were confined to O2, corre- sponding to the non-bonding limit of neutral parts, but a formally iron(I) if the Fe–O bond were to be considered covalent. 

In 1960, McClure suggested a valence-bond formulation based on triplet–triplet coupling, which is appealing by the low promotion energies required to access these states, rather than the singlet states. In 1964, Weiss suggested, based on analogy to chemical reactions in aqueous solution, that the true ferrous hemeO2 adduct was mainly of the superoxo-iron(III) type caused by ‘‘electron transfer” from iron to O2. 

Goddard and Olafson suggested an ozone model of the adduct in 1975 which emphasized the four-electron three-center bond with maintained triplet state of dioxygen as in the McClure model with less electronic reorganization to explain the reversible binding. 

In 1977, Pauling maintained his original view again, the same year that Huynh, Case, and Karplus did a first attempt to bridge these views by performing early quantum chemical calculations that diplomatically emphasized the importance of both Weiss and Pauling resonance forms. 

However, interpretation depends on model language, orbital localization, and transformation between valence bond and orbital formalisms:  

In terms of molecular orbital theory, the wave function was a multi-configurational state dominated by the Pauling configuration; however, if one uses valence bond theory considerations, it can be interpreted as having large Weiss character. Thus, the multi-configurational state produced from CASPT2 is interpreted differently by different models. This partly explains why the trenches were so deeply dug during the exchange between Pauling, Goddard, McClure, and Weiss; all were right, and all were wrong. 

This is just another example of unnecessary conflicts about valence bond vs. molecular orbital (VB vs. MO). 

In terms of valence structures, the ground state was summarized by Shaik and Chen as having contributions from both Weiss, Pauling, and McClure forms, the first .. dominating. 

Ironically DFT ends up providing a useful language after all! 

The charge assignments to O2 are very dependent on calculation scheme, and both the orbitals, valence structures, and atomic charges that defined the Weiss-Pauling debate are non-observable. In contrast, the electron density is observable as are the geometries and spectroscopic data...

Molecular orbitals are not physical observables but calculational constructs. MO's don't exist.

In different words, one can take a many-body wave-function and make a linear unitary transformation of the molecular orbitals. The Slater determinants do not change. [The value of a determinant is invariant to a change of basis.]

Now. Question 2.
What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

Kepp talks about spin-orbit coupling and the fact that it is small for oxygen, motivating a discussion of a "broad crossing mechanism".  However, I am not sure this is relevant. I don't see the binding as necessarily spin forbidden. As the oxygen approaches the heme the two triplet states can mix to form a total spin singlet.
This is analogous to bringing two hydrogen atoms (each of which is spin 1/2) together to form a hydrogen molecule (which is spin zero). A multi-configurational wavefunction has no problem with this. But DFT-based approximations, which use a single determinant cannot describe this smooth crossover.

Other things of particular interest to me that are discussed in the review include the central role of back bonding and the success of the TPSSh functional in DFT calculations for organometallics.

Unfortunately, the review does not mention recent work by Weber et al, applying DMFT to the problem of oxygen binding to haemoglobin.

Superconducting order in organic charge transfer salts

A long-standing question for superconductivity in organic charge transfer salts concerns the symmetry of the superconducting order parameter. Is it unconventional (i.e. not s-wave) and if so are there nodes in the energy gap? Over the years there have been a wide range of claims, both theoretical and experimental.

Most recently a combined theory-STM experiment claimed the symmetry was d + s and that there were 8 nodes on the Fermi surface.

Two of my UQ colleagues recently posted a nice preprint that comes to a different conclusion.
Microwave Conductivity Distinguishes Between Different d-wave States: Umklapp Scattering in Unconventional Superconductors 
D. C. Cavanagh, B. J. Powell

Microwave conductivity experiments can directly measure the quasiparticle scattering rate in the superconducting state. We show that this, combined with knowledge of the Fermi surface geometry, allows one to distinguish between closely related superconducting order parameters, e.g., dx2−y2 and dxy superconductivity. We benchmark this method on YBa2Cu3O7−δ and, unsurprisingly, confirm that this is a dx2−y2 superconductor. We then apply our method to κ-(BEDT-TTF)2Cu[N(CN)2]Br, which we discover is a dxy superconductor.

In 2005 Ben Powell  (and others) showed that the simplest RVB theory gives such an order parameter with nodes required by symmetry.
[Aside: in our paper, this is denoted d_x2-y2, but that is because of how the x-y axes are defined].

Two results from quantum chemistry that physicists should/might worry about

Solid state physicists love model effective Hamiltonians such as Hubbard, Heisenberg, Anderson, and Holstein because they are simple but have rich properties, can explain diverse phenomena, and present a significant intellectual challenge.

However, it is worth considering key assumptions. The first is imbedded in the model and the second in approximate solutions.
Computational quantum chemistry does raise some questions that physicists rarely seem to think about. They are hard and scary.

1. Rigid orbitals.
Each lattice site is associated with some sort of atomic or molecular orbital. A beauty of the models is one does not have to know exactly what this orbital is.
Now consider different energy eigenstates of the model. These only differ in orbital occupations and the different coefficients in superposition of Slater determinants (or creation and annihilation operators). The localised orbitals do not change.
Similarly, when one changes lattice or vibrational co-ordinates in a Holstein or Su-Schrieffer-Heeger model, one assumes the orbitals associated with the lattice sites don't change.

2. Strong correlations can be captured with just a few Slater determinants.

Physicists like simple variational wave functions such as Gutzwiller, RVB, ...

Yet if one considers the treatment of simple molecules by high level quantum chemistry methods one finds two things need to be taken into account to obtain an accurate description of strong electron correlations.

A. Orbital relaxation can be significant.
If one considers the orbitals (whether localised or delocalised) that appear in many-body wave function for different eigenstates (e.g. singlet, triplet, low lying excited states) one sees these orbitals can be quite different. This is why one has methods such as the Breathing Orbital Valence Bond method, reviewed here. It discusses specific examples of how the quality of wave function can be improved by using different orbitals in covalent and ionic parts of a wave function and as bond lengths change.

If one wants to think about parametrising an effective Hamiltonian for the cuprates, such as the t-J model this paper is relevant.

Heisenberg exchange enhancement by orbital relaxation in cuprate compounds 
A.B. van Oosten, R. Broer, W.C. Nieuwpoort

B. Many Slater determinants are often required.

Chemists are very good at performing calculations that involve even hundreds of thousands of Slater determinants (configurations).
Consider the case of benzene. If one wants to get accurate energies for the ground and low-lying excited states. One finds that if one works with delocalised molecular orbitals one needs to include literally hundreds of thousands of orbitals.
However, if one works instead with localised orbitals and valence bond theory one finds one can use a handful of Slater determinants. For example, the ground state can be described in terms of the five structures below. Eighty per cent of the weight is in the first two. Their superposition forms the legendary resonating valence bond (RVB) structure. However, I want to stress that 20 per cent of the weight is in the rest, and this is not a trivial amount.

I doubt these issues matter if one just uses effective model Hamiltonians to obtain physical insights, describe qualitative behaviour and trends, and semi-quantitative comparisons with experiment. However, at some point one is not going to be able to get detailed quantitative description of real materials.

NF - mysteries of a small molecule

Nitrogen fluoride (NF) seems like a very simple molecule and you would think it would very well understood, particularly as it is small enough that it is accessible to high level quantum chemistry calculations. However, the molecule exhibits some subtle properties that present a theoretical challenge. There is limited experimental data because the molecule is only found as an intermediate in some chemical reactions.

Just like oxygen (O2), to which it is isoelectronic, the ground state is a triplet due to Hund's rule, as discussed for O2 here.

I just read a nice paper
A Valence Bond Study of the Low-Lying States of the NF Molecule 
Peifeng Su, Wei Wu, Sason Shaik, and Philippe C. Hiberty

Given that F is more electronegative than N one might expect the ground state to have a large electric dipole moment and this to increase as the molecule is stretched.
However, the ground state has only a small moment, it has the opposite direction to that expected from the electronegativity, and the direction changes sign when the bond is stretched.

Furthermore, unlike most molecules, the bond length is shorter and the dissociation energy larger in the low-lying excited states [which are singlets] than in the ground state.

The ground state has one sigma bond and six electrons in two pi orbitals.

The latter form three-electron bonds, which in the Valence Bond picture, involve exchange in position of an electron pair and an unpaired electron, where one goes from T1 above to the configurations shown below.

The authors consider 9 valence bond structures for the triplet ground state and 12 singlet VB structures for the two lowest singlet states. Their calculations lead to the following picture in terms of Lewis structures.

What insights are gained by the VB approach? Key is the idea of back donation or back bonding.

In a nutshell, the three lowest-lying states of NF can be considered as primarily bonded by a polar two-electron σ bond, complemented by π-bonding contributions of the three-electron bonding type for the ground state, and of the classical two-electron type for the excited states. In all three cases, the π-bonding contributions correspond to charge transfer from F to N, thus counterbalancing the σ polarization by back-donation. The tendencies of the bond lengths in the various electronic states comply with this simple model. Thus, all the computations and experimental measurements show that the bond length decreases consistently from the ³Σ⁻ state to ¹Δ, and from ¹Δ to ¹Σ⁺. This counterintuitive tendency, which implies that the more excited the molecule is, the more strongly it is bonded, is easily explained by the weights of the π-bonding Lewis structures, that sum up to 19 % in ³Σ⁻ state, to 28 % in the ¹Δ state, and to 37 % in the ¹Σ⁺ one (Figure 1 and Table 1 and Table 3). The same increase in the π-bonding contribution accounts again for the unusual fact that the bonding energy is far larger in the first excited state than in the ground state (96.4 vs 72.0 kcal mol⁻¹ at the VBCISD level). On the other hand, the bonding energy of the second excited state is now smaller than that of the first, as expected, since both excited states dissociate to the same products.

The tendencies in the dipole moment values of the various states are also readily rationalized by the simple VB model. The polar σ bond tends to tip the electron density towards the fluorine atom, and thus to favour negative values of the dipole moment (in the direction N⁺F⁻), while backdonation from the π systems has the opposite effect. The two effects compensate for each other in the ground state, where backdonation is moderate (19 % of π charge transfer). In the ¹Δ state, as the π charge transfer is increased relative to the ground state (28 % vs 19 %), backdonation wins over σ polarization, leading to a significant positive dipole moment, in the direction F⁺N⁻. The effect is further reinforced in the ¹Σ⁺ state, in which the π-charge transfer Lewis structure has such a large weight (37 %) that it completely overwhelms the σ polarization, ending up at a positive dipole moment of 0.728 D at the BOVB level.

I would like to see a basic description of these essential features in terms of a polarised two site Hubbard model with multiple orbitals and Hund's rule coupling, generalising the unpolarised two-orbital model here.

I got interested in this paper because of thinking about improper hydrogen (and halogen) bonds and wondering whether there is an "excited" diabatic state that has a shorter and stronger X-H bond than in the ground state. A general "ionic" (X^-H+) state will not have this property but if there is the option of back donation maybe something can happen....

A unified picture of weak chemical bonds: hydrogen, halogen, carbon...

Previously I posted about improper hydrogen bonds. These are weak hydrogen bonds that have the unusual property that in the X-H...Y system H-bonding leads to a shortening and hardening (blue shift) of the X-H bond. In contrast, for "proper" bonds, X-H lengthens and softens (red shift).

The past few years has seen a rapid increase in interest in an even broader class of weak bonds such as "halogen bonds",  denoted X-Z...Y where Z can now be not just H but a halogen (F, Cl, Br), chalcogen (O, S, Se, Te), or pnictogen (N, P, As, ..)....

There is an interesting paper that contains the helpful summary figure below
Negative hyperconjugation and red-, blue or zero-shift in X-Z---Y complexes
Jyothish Joy, Eluvathingal D. Jemmis and Kaipanchery Vidya

In trying to understand the paper I found reading the following older paper helpful
Electronic Basis of Improper Hydrogen Bonding:  A Subtle Balance of Hyperconjugation and Rehybridization
Igor V. Alabugin, Mariappan Manoharan, Scott Peabody, and Frank Weinhold

[Aside: note the senior author is Weinhold who has featured in some previous posts]

The basic idea is that there are two competing interactions. "Hyperconjugation" is Weinhold's view of proper H-bonds, via the Natural Bond Orbital donor-acceptor picture where the H-bond arises due to charge transfer from the lone pair orbital on Y to the σ* (anti-bonding) orbital associated with X-H. This lengthens and hardens X-H.
When this interaction is weak there is “X-H bond shortening” due to increase in the s-character (rehybridisation of the atomic orbital on X) and polarization of the X−H bond. This is associated with a shorter and harder X-H bond.

Bent's rule is central. It is one of the most general rules governing structure of organic molecules.

atoms tend to maximize the amount of s-character in hybrid orbitals aimed toward electropositive substituents and direct hybrid orbitals with the larger amount of p-character toward more electronegative substituents.

Increasing s-character generally leads to shorter bonds.
As the donor acceptor distance (X-Y) decreases the X-Z bond becomes more polarised and the s-character increases.
The authors note it should be possible to test predicted trends since the amount of s character in the X-Z bond can be measured from the relevant NMR coupling constant.

My question is whether this subtle competition can be captured by generalising my simple 2 diabatic state model for H-bonds to a 3 state model that includes the ionic character of the X-Z bond.

Chemistry is local. In praise of Wannier.

In several posts I have emphasised that Chemistry is local.
This is illustrated by the fact that specific bonds within a molecule have approximately the same length, energy, and vibrational frequency regardless of the details of the molecule, particularly the distant parts.
This locality leads to useful concepts and theoretical approaches such as Atoms in Molecules, Natural Bond Orbitals, Valence Bond theory.
However, this locality is at variance with Molecular Orbital theory and the Kohn-Sham orbitals in Density Functional Theory (DFT); the orbitals can be completely delocalised over the whole molecule.

What are the implications for solid state physics?
Band theory is the analogue of molecular orbital theory. Bloch electronic wave functions are completely delocalised throughout the crystal.
Wannier orbitals are the physics analogue of Boys orbitals in chemistry.

In 1984 Phil Anderson wrote:

The Wannier functions are still one of the most useful but underutilized methodologies of solid state physics, and in particular it is in the language of Wannier functions that I feel the chemical implications of band theory are most effectively expressed. 

Of course the reason is that most of the concepts of chemistry are local concepts, such as bonds, ions, complexes, etc., while band theory is a global structure, in which the wave functions permeate the entire system and the eigenenergies depend on the position of every atom everywhere. There is no a priori reason why band theory should lead to such a chemically intuitive result as that the carbon—carbon single bond should have roughly the same energy and bond length whenever it appears, the Oxygen anion should have a constant radius and negative electron affinity, etc. 

This same weakness is shared by band theory’s chemical equivalent, the molecular orbital theory of Hund and Mulliken. From the very first, there was a vain attempt to restore locality by the use of atomic states and the valence-bond idea, very much advocated by Pauling, but only Pauling’s great ingenuity in applying the vague concept of “resonance” and his enormous prestige kept this scheme afloat as long as it has been: it is just not a valid way of doing quantum mechanics, and fails completely in the case of metals and of the organic chemical equivalent of metals, namely aromatic compounds and graphite, and is not very useful elsewhere except in the hands of a master empiricist such as Pauling. 

Nonetheless many local basis: how is this compatible with quantum mechanics? This is the enigma to which Wannier functions give us a very precise and clear answer. Not only that, but with a bit of ingenuity it is possible to modify the local functions in such a way as to give one a simple, accurate and serviceable method for quantum chemical calculations.

P.W. Anderson, Chemical Pseudopotentials, Physics Reports 1984

The discussion of resonating valence bond (RVB) theory is a bit harsh ("it is not just a valid way of doing quantum mechanics") and ironic given that only three years later Anderson introduced his RVB theory of superconductivity!

Indeed, the past two decades has seen a resurgence of interest in and use of Wannier functions.
Here is a recent Reviews of Modern Physics on the subject.

Chemical bonding, blogs, and basic questions

Roald Hoffmann and Sason Shaik are two of my favourite theoretical chemists. They have featured in a number of my blog posts. I particularly appreciate their concern with using computations to elucidate chemical concepts.

In Angewandte Chemie there is a fascinating article, One Molecule, Two Atoms, Three Views, Four Bonds that is written as a three-way dialogue including Henry Rzepa.
The simple (but profound) scientific question they address concerns how to describe the chemical bonding in the molecule C2 [i.e. a diatomic molecule of carbon]. In particular, does it involve a quadruple bond?

The answer seems to be yes, based on a full CI [configuration interaction] calculation that is then projected down to a Valence Bond wave function.

The dialogue is very engaging and the banter back and forth includes interesting digressions such the role of Rzepa's chemistry blog, learning from undergraduates, the relative merits of molecular orbital theory and valence bond theory, the role of high level quantum chemical calculations, and why Hoffmann is not impressed by the Quantum Theory of Atoms in Molecules.