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.

A model for light-induced spin-state trapping in spin-crossover materials

 An important challenge required to understand the physical properties of materials that are chemically and structurally complex is to ascertain which microscopic details are important. A related question is at what scale (length, number of atoms, energy) models should be developed.

A specific example is understanding the magnetic properties and state transitions of spin-crossover materials. This is difficult for equilibrium properties, let alone for non-equilibrium properties such as Light-Induced Excited Spin-State Trapping (LIESST). At low temperatures irradiation with light can induce a transition from the equilibrium low-spin state to a long-lived high-spin state, which is only an equilibrium state at higher temperatures. (LIESST gets a lot of attention because of the potential to make optical memories for information storage).

Some of my UQ colleagues recently published a nice paper that elucidates some of the key physics with the proposal and analysis of a (relatively simple) model that captures many details of the experimental data.

Toward High-Temperature Light-Induced Spin-State Trapping in Spin-Crossover Materials: The Interplay of Collective and Molecular Effects

M. Nadeem, Jace Cruddas, Gian Ruzzi, and Benjamin J. Powell

A molecular material and a model Hamiltonian with rich physics

Some of my UQ colleagues and Jaime Merino have written a series of nice papers inspired by an organometallic molecular material Mo3S7(dmit)3. They have considered possible model effective Hamiltonians to describe it and the different ground states that arise depending on the model parameters.
There is a rich interplay of strong correlations, Hund's rule coupling, spin frustration, spin-orbit coupling, flat bands, and Dirac cone physics.
Possible ground states include some sort of Mott insulator, a Haldane phase, semi-metal, ...

A good place to start is the following paper
Low-energy effective theories of the two-thirds filled Hubbard model on the triangular necklace lattice 
C. Janani, J. Merino, Ian P. McCulloch, and B. J. Powell

The figure below (taken from this paper) shows some of the molecular structure and some of the hopping integrals that are associated with an underlying decorated honeycomb lattice.

This model could be called kagomene, because it interpolates between the kagome lattice and the honeycomb lattice (graphene). The figure below is taken from this paper, which uses DFT and Wannier orbitals to estimate the tight-binding parameters and the spin-orbit coupling. Interaction driven topological insulator states are possible on this lattice.

There are a few things that are not "normal" about the physics, arising from the 4/3 band filling and the molecular orbitals that are delocalised over the triangles. Specifically, the orbital degeneracy does not arise from atomic orbital degeneracy (cf. d orbitals, or t2g and eg), but rather the E representation associated with C3 symmetry of the triangles.

Hund's rule coupling. 
This involves the E orbitals and arises purely from the Hubbard U on the non-degenerate orbital on a single lattice site.

Spin-orbital coupling.
This is Spin Molecular Orbital Coupling, where the electron spin couples to the angular momentum associated with motion around the triangle, not the angular momentum of degenerate atomic orbitals.

Haldane phase.
The associated spin-1's arise from the triplet ground state of four electrons on a triangle.
A DMRG study shows that this is the ground state of a three leg-ladder Hubbard model at 2/3 filling.

Many interesting and important open questions remain about the general phase diagram of the Hubbard model on the kagomene lattice. For example, the nature of the Mott insulator, different types of topological order, the possibility of superconductivity.....

Hopefully, these studies will stimulate new experimental studies and synthesis of new chemical compounds in this fascinating class of materials.

The challenge of multiferroism in organic Mott insulators

A theoretical picture of the Mott insulating phase of organic charge transfer salts [such as (BEDT-TTF)2X] is that they can be described by a single-band Hubbard model on an anisotropic triangular lattice at half filling. The spin excitations can then be described by the corresponding Heisenberg model. In these models, each lattice site corresponds to a single anti-bonding orbital on a pair (dimer) of BEDT-TTF molecules. Thus the internal structure of the dimer and the corresponding two-band Hubbard model at three-quarters filling is "integrated out" leaving a one-band picture.

However, there are some dielectric relaxation experiments that can be interpreted as inconsistent with the picture above.
The key question is whether there is charge order within the dimer, in particular, does it have a net dipole moment?
A 2010 theory paper by Hotta proposed this and an effective Hamiltonian for the Mott insulating phase where the spin on the dimer and the dipoles are coupled together. She suggested that a spin liquid phase could be driven by the dipoles, rather than spin frustration.
This picture also leads to the possibility of a multiferroic phase: coexisting ferromagnetic and ferroelectric phases.

There are two helpful recent reviews, presenting alternative views of the experiments.

Dielectric spectroscopy on organic charge-transfer salts
P Lunkenheimer and A Loidl

Ferroelectricity in molecular solids: a review of electrodynamic properties 
S Tomić and M Dressel

The figure below shows experimental measurements from this paper. (The authors of the first review above and Hotta are co-authors.) The figure shows the temperature dependence of the real part of the dielectric constant at different frequencies. Note how it becomes very large at low frequencies (almost static) near about 25 K, which coincidentally is the temperature at which this organic charge transfer salt becomes antiferromagnetic (with weak ferromagnetism due to spin canting).

The above dielectric behaviour is similar to what one sees at a ferroelectric transition.

However, one should be cautious about this interpretation for multiple reasons. These are tricky experiments.

Dielectric dispersion spectroscopy is a bulk probe, not a microscopic one. One is not measuring the electric dipole moment of a single unit cell but rather the electric polarisation of a bulk crystal that has surfaces and contains defects, and impurities. For example, charge accumulation on the sample surface can enhance the measured dielectric constant and lead to significant frequency dependence, even when the actual material has no intrinsic frequency dependence (This is known as Maxwell-Wagner polarisation or the space-charge effect).

There are reports of significant sample dependence; the dielectric constant can vary by up to two orders of magnitude!

The origin of the dependence of the results on the direction of the electric field is not clear (at least to me). One usually finds the largest effects when the electric field is parallel to the least conducting direction (i.e. perpendicular the layers) in the crystal.

The magnitude of the electric dipole moment that one deduces from the magnitude of the dielectric constant (by fitting the temperature dependence to a Curie form, as in the dashed line in the figure above) is an order of magnitude larger than the moment on single dimers that is deduced from infrared (IR) measurements. This last discrepancy is emphasized by the authors of the second review above.

(IR measures the vibrational frequencies of the BEDT-TTF molecules; spectral shifts are correlated with the charge density on the molecule. Splitting of spectral lines corresponds to the presence of charge order, as discussed here.)

If one does accept that charge order occurs, further questions that arise include:

How do we know that the charge order is occurring within the dimers not between dimers?

Are these dielectric properties necessary or relevant for the insulating, superconducting, and magnetic properties (antiferromagnetism or spin liquid) or is it just a second-order effect (causality or correlation)?

What is the relevant effective Hamiltonian in the Mott insulating phase?

How is this similar and different to multiferroic behaviour in inorganic materials?

What role does spin-orbit coupling [and specifically the Dzyaloshinskii-Moriya interaction] play?

What experimental signatures could be considered a "smoking gun" for the presence of electric dipoles on single dimers?

How does one understand the different experiments which probe the system on very different time scales?

Desperately seeking Weyl semi-metals

In 2011 it was proposed that pyrochlore iridates (such as Y2Ir2O7) could exhibit the properties of a Weyl semi-metal, the three-dimensional analog of the Dirac cone found in graphene.
Since the sociology of condensed matter research is driven by exotica this paper stimulated numerous theoretical and experimental studies.
However, as often is the case, things turn out to be more complicated and it seems unlikely that these materials  exhibit a Weyl semi-metal.

This past week I have read several nice papers that address the issue.

Variation of optical conductivity spectra in the course of bandwidth-controlled metal-insulator transitions in pyrochlore iridates
K. Ueda, J. Fujioka, and Y. Tokura

There is a very nice phase diagram which shows systematic trends as a function of the ionic radius of the rare earth element R=Y, Dy, Gd, ...
Most of the materials are antiferromagnetic insulators.

The colour shading describes the low energy spectral weight in the optical conductivity up to 0.3 eV.
Blue is an insulator and red actually means a very small low energy spectral weight.
N can be thought of as the number of charge carriers per unit cell. Specifically, if this was a simple weakly interacting Fermi liquid N=1. Thus, the value of 0.05 for Pr signifies strong electron correlations. [Unfortunately, the paper talks about this as "weak correlations"].

In fact, as shown below even in the metallic phase at T=50 K one cannot see the Drude peak down to 10 meV.
This presents a theoretical challenge to explain this massive redistribution of spectral weight.

Slater to Mott Crossover in the Metal to Insulator Transition of Nd2Ir2O7
M. Nakayama, Takeshi Kondo, Z. Tian, J. J. Ishikawa, M. Halim, C. Bareille, W. Malaeb, K. Kuroda, T. Tomita, S. Ideta, K. Tanaka, M. Matsunami, S. Kimura, N. Inami, K. Ono, H. Kumigashira, L. Balents, S. Nakatsuji, and S. Shin

This ARPES study does find band touching at the magnetic metal-insulator transition temperature but as the temperature is lowered the spectral weight is suppressed and there is no sign of Weyl points.

Phase Diagram of Pyrochlore Iridates: All-in–All-out Magnetic Ordering and Non-Fermi-Liquid Properties 
H Shinaoka, S Hoshino, M Troyer, P Werner

This LDA+DMFT study shows that a three-band description is important for the R=Y compound.
This sets the stage for describing the phase diagram above.

I thank Prachi Telang for discussions at IISER Pune about these materials and bad semi-metals that stimulated this post.

Novel spin-orbit coupling in the absence of local inversion symmetry

Normally we associate spin-orbit coupling with degenerate atomic orbitals (or energy bands) associated with d- or f-orbitals. However, in solid state physics a quite distinct type of spin-orbit coupling can occur and has attracted a lot of interest over the past decade.

In a seminal 2005 paper [which took 12 months for PRL to publish!] Kane and Mele proposed that in graphene a spin quantum Hall effect could occur due to spin-orbit coupling. Moreover, this paper proposed that this state was a topological insulator, starting a whole industry. I want to just focus on the spin-orbit coupling term in the Hamiltonian that is the first step in their argument.

This term arises because there are two carbon atoms per primitive unit cell in the crystal lattice. [A and B sub lattice]. It does not have local inversion symmetry.

How large is Delta_so ?
Kane and Mele estimated, based on a crude argument, that is was about 1.2 Kelvin. But, then they gave a renormalisation group argument, claiming that electron-electron interactions would increase the value to something like 7.5 K.
However, much more sophisticated analysis, such as this one, showed that Delta_so arose from subtle pi-sigma orbital mixing and was orders of magnitude smaller! Hence, the chance of seeing a quantum spin Hall effect in graphene are extremely unlikely.

Aside. This illustrates you can be wrong about something but still stimulate a whole new field. But, in fairness, everything is correct about the paper, except the parameter estimate for graphene. This is quite different to people who publish papers that are just plain wrong, but still stimulate positive outcomes.

What about other systems?
A nice example is monolayer MoS2, as discussed here.

A full three-dimensional crystal of MoS2 has inversion symmetry. However, a monolayer does not.
If you take a Mo atom as an inversion centre, a S atom is mapped onto an empty location.
Delta_so is estimated to be about 500 K.
It is orders of magnitude larger than graphene because the bare-spin orbit coupling is much larger due to the heavy Mo atoms.

A similar spin-orbit coupling has been proposed to occur in a quasi-one-dimensional metal, Li0.9Mo6O17.

Spin-orbit coupling and "triplet" superconductivity

My collaborators and I just finished a paper

Spin-orbit coupling and odd-parity superconductivity in the quasi-one-dimensional compound Li0.9Mo6O17
Christian Platt, Weejee Cho, Ross H. McKenzie, Ronny Thomale, and Sri Raghu

Here is the abstract.

We welcome any comments.

One thing I learnt and found interesting what the unusual spin-orbit coupling that arises due to lack of inversion symmetry in the four-atom unit cell. I will post separately about that next week as the story of the corresponding coupling in graphene is an interesting one.

Seeing the effects of relativity with the naked eye

Our natural tendency is to think that to see the effects of Einstein's special theory of relativity you have to be travelling at some significant fraction of the speed of light. However, this is not the case. In solid state physics I am aware of three concrete phenomena that are purely due to relativistic effects.

1. Gold metal is the colour "gold".
According to Wikipedia, "non-relativistic gold would be white. The relativistic effects are raising the 5d orbital and lowering the 6s orbital.^[11]"

2. Mercury is a liquid at room temperature.
This is nicely discussed in a recent blog post by Henry Rzepa concerning a recent paper that shows that relativistic effects shift the melting temperature by about 100 K.

3. Magnetic anisotropy and hysteresis in ferromagnets.
This results from spin-orbit coupling which is a consequence of relativity.

Deconstructing iridates and many-body time scales

Iridates such as Sr2IrO4 have attracted considerable attention because they are 5d systems that exhibit a strong interplay between spin-orbit coupling and strong electronic correlations.
[See this earlier post].

Sr2IrO4 is a focus because it has been argued that it is a J=1/2 Mott insulator, just like La2CuO4, the parent compound for cuprate superconductors.
A current holy grail is to dope this material in the hope of producing high-Tc superconductivity. Many are trying. No one is succeeding.

There is actually a whole series of layered compounds, the Ruddlesden-Popper perovskites that differ, not just in their stoichiometry, but also their crystal structure, and consequently how the Iridium ions are coupled together. Sr_n+1Ir_nO_3n+1, where n is the number of SrIrO3 perovskite layers sandwiched between extra SrO layers.

Resonant-Inelastic-X-ray-Scattering (RIXS) experiments show that spin excitations in the Mott insulating phase of Sr2IrO4 appear to be well described by a spin-1/2 Heisenberg model with a small amount of spin anisotropy due to crystal field effects. However, for Sr3Ir2O7, RIXS suggests a large spin gap and spin anisotropy.

However, a different experiment suggests a small anisotropy.
In contrast, SrIO3 is a metal.

A major theoretical challenge is describe this whole family of materials and the disparate results, starting just from the crystal structures.
This has been done in an impressive paper,

Effective J=1/2 insulating state in Ruddlesden-Popper iridates: An LDA+DMFT study
Hongbin Zhang, Kristjan Haule, and David Vanderbilt

The calculations are based on GGA+DMFT, and represent another landmark achievement for the combination of Dynamical Mean-Field Theory with Density Functional Theory methods.

A key ingredient to understanding the apparent inconsistency between the results of different experimental probes is that in the many-body treatment the matrix describing the hybridisation of the three d-orbitals is frequency dependent. This is in contrast to the static matrix associated with crystal field theory.

The X-ray and thermodynamic experiments probe the system on different time scales. Thus they are respectively, more sensitive to the high- and low-frequency part of the hybridisation matrix.

The Figure below shows the calculated optical conductivity for the first three compounds in the RP series.

They also show how 0.2 per cent epitaxial lattice strain can have a big effect. [Aside: These are the kind of calculations I would like to see to address thermal expansion in organic charge transfer salts.]

I thank Kristjan Haule for explaining this work to me.

Clarifying the "Rashba" effect

At the conference today Changyoung Kim gave a nice talk about how the "Rashba effect" [coupling of momentum and spin] on the surface of a semiconductor has a different physical origin to what is usually claimed. In particular, the energy scale for the relativistic Zeeman splitting proposed by Rashba gives a band splitting that is six orders of magnitude smaller than what is actually observed!

Here is the abstract of the associated PRL:

We propose that the existence of local orbital angular momentum (OAM) on the surfaces of high-Z materials plays a crucial role in the formation of Rashba-type surface band splitting. Local OAM state in a Bloch wave function produces an asymmetric charge distribution (electric dipole). The surface-normal electric field then aligns the electric dipole and results in chiral OAM states and the relevant Rashba-type splitting. Therefore, the band splitting originates from electric dipole interaction, not from the relativistic Zeeman splitting as proposed in the original Rashba picture. The characteristic spin chiral structure of Rashba states is formed through the spin-orbit coupling and thus is a secondary effect to the chiral OAM. 

Mott physics with spin-orbit coupling

There is a very nice and helpful review article
Correlated quantum phenomena in the strong spin-orbit regime
William Witczak-Krempa, Gang Chen, Yong Baek Kim, Leon Balents

Just a few things I learnt from quickly skimming it.

There are many outstanding and basic questions concerning the phase diagram of even the simplest possible two-orbital Hubbard model with spin-orbit coupling. There a many possible new phases waiting to be discovered [both experimentally and theoretically] or to be shown to not actually exist because their theoretical proposal is based on uncontrolled approximations. The figure below is a possible schematic phase diagram.

Much of the interesting physics requires spin-orbit coupling energies of the order of hundreds of meV, i.e. comparable to electronic band energy scales. Hence, this is irrelevant to many materials.
But the spin-orbit coupling can be quite strong in 5d transition metals. Iridates (iridium oxides) may be model compounds to realise this physics.

Table I provides a nice summary of the properties of the plethora of different new phases that have been proposed including axion insulator, Weyl semi-metal, fractional Chern insulator, ...

Na2IrO3 was originally proposed to be a realisation of the Heisenberg-Kitaev model and thus to have a possible spin liquid ground state. However, neutron scattering shows it has an unanticipated  magnetic ground state: "a zig-zag state with four-sublattice structure." This has led to new proposals as to the relevant effective spin Hamiltonian.

The review has only limited discussion of the role of Hunds rule.

It is repeatedly stated that spin-orbit coupling leads to entanglement of spin and orbital degrees of freedom. But the exact nature of this quantum entanglement is not clearly stated or calculated. For the case of entanglement arising via Hund's rule (not spin-orbit coupling) this is nicely discussed by Oles here.

The key thing that the spin-orbital coupling/entanglement does is remove (or at reduce) the coupling of the orbital degeneracies to the lattice which normally produces the Jahn-Teller effect and orbital ordering.

Sr2IrO4  is an approximate homolog of the parent material of the high-Tc cuprates, La2CuO4. It is a Jeff = 1/2 antiferromagnetic insulator and has an exchange constant J ∼1000 K comparable to that of the cuprates. This has led to a strong push to dope the material in the search of cuprate-type physics [high-Tc superconductivity, pseudogap, strange metal]. This has not occurred yet.

Both Sr2IrO4 and Sr3Ir2O7 have illustrated the power of the rapidly evolving technique of Resonant Inelastic X-ray Scattering (RIXS). It seems to be particularly suited for 5d compounds, and has been used to map out the full spin wave dispersion in both compounds.

I thank Tony Wright for bringing the review to my attention.

Trying to shed light on organic LED's?

The system below is just one example of an organometallic phosphorescent system which is the basis of some LED's.

Some of the questions that interest me are:

Suppose we ignore spin-orbit coupling and the environment (either a solvent or thin film)
what are the quantum numbers and energies of the different excited states?

If the complex has C3v symmetry excited states can be labelled by A1, A2, and E (the irreps of the group) . Spin rotational invariance means the states will be either singlets or triplets.

What is their oscillator strength? Which states can be identified with features in the optical absorption spectrum? and with the emission spectrum?

How much mixing between the ligand centred (LC) and metal-to-ligand-charge transfer (MLCT) states occurs due to hybridization of metal orbitals and ligand pi and pi* orbitals?

How does the large spin-0rbit coupling on the metal core change things?
This will mix singlet and triplet states, particularly states with significant metal character.
It will also allow intersystem crossing (i.e., transitions between singlet and triplet states and visa versa) and will give "triplet" states oscillator strength.

How does the environment change things?
It breaks the C3v symmetry and tends to localise excitations on the ligands.

What determines the quantum efficiency of phosphorescence? What processes compete with phosphorescence? What is the dynamics following photoexcitation?

It must be something like:

A1 (ground state) + photon ->
A1 or E (depending on polarisation of light) "singlet" ->
"MLCT" "singlet" via internal conversion (and a conical intersection), symmetry? ->
"MLCT" "triplet" via spin-orbit coupling (i.e., intersystem crossing), symmetry? ->
"LC" "triplet" with some "singlet" character from spin-orbit coupling-> A1 (ground state) + photon

There are well-defined selection rules from group theory for spin-orbit coupling and internal conversion. This may help identify the symmetry of the different states.

Can it be made any simpler?

Excited electronic states of organo-metallic complexes

When one sees spectra such as those above, is it possible to identify the electronic excited states associated with the different features?

This question is receiving considerable attention because complexes such as these are the basis for organic LED's and photovoltaic cells.



In March I gave a talk at the thursday COPE science meeting, based on the classic paper by Kober and Meyer, concerning complexes with three-fold symmetry.
A few take home points:

The group theory analysis helps define the quantum numbers of the different states, including the spin-orbit interaction which mixes singlet and triplet states. Furthermore, polarised light is sensitive to the symmetry of states (A, E). Note how the two spectra above are different.

It is possible to describe the spectra in terms of just a few parameter.

The magnitude of the exchange interaction (singlet-triplet splitting) is 1600 cm-1, comparable to other energy scalings, showing the importance of electronic correlations.

The spin-orbit interaction is comparable to the other energy scales as well.

The paper only treats metal-to-ligand charge transfer (MLCT) states. For understanding the emissive states of OLED's ligand-centred transitions may be just as important. A nice discussion of these issues is discussed in a recent review by Yersin and Finkenzeller .