How important are electron-electron interactions in organic molecules?

There is a really nice old (1975) paper Why is azulene blue and anthracene white? by Michl and Thulstrup, which highlights the relevant issues.

They point out that these two molecules 1 and 3 have similar ionisation energies (IP) and electron afinities (EA) but their lowest lying singlet (S) and triplet (T) excited states have quite different energies (see the Table below). If a simple molecular orbital (Huckel) picture which ignores electron-electron interactions was valid than the energy difference between the LUMO and HOMO would equal IP-EA = T = S.

However, the Table clearly shows this is not the case.

At the Hartree-Fock level the discrepancies provide a measure of

J = Coulomb integral ~ Hubbard U ~ 5 eV

2K= Exchange integral = Singlet-Triplet splitting ~ 1 eV.

Michl and Thulstrup give a nice simple explanation of why the exchange integral K is smaller in azulene than in anthracene. (This is what leads to the different colours). Azulene is a non-alternating hydrocarbon (5 ring + 7 ring) and so the spatial overlap of the HUMO and LUMO is much smaller than for anthracene.

The fact that Hartree-Fock captures some of the above does not mean that electron correlations are not important. I think the fact that J is comparable to the energy splitting of the HOMO and LUMO estimated from IP and EA suggests configuration interaction in the singlet excited states may be significant.

p.s. There is an old post Am I LUMO-phobic? plus comments which discusses the issues. A recent paper by Paul Schwenn, Paul, Burn, and Ben Powell showed that some of the claimed "success" at DFT based methods calculating IP and EA is due to a fortuitous cancellation of errors.

## Monday, April 30, 2012

## Saturday, April 28, 2012

### Is the economy heating up?

I enjoyed reading the article An economic analogy to thermodynamics by Wayne Saslow. It proposes mapping different economic variables to thermodynamic ones. It learnt a little about economics which was nice. He suggests the following analogies between thermodynamic and economic variables. Wealth, utility, surplus, price, and number of goods, correspond to free energy, internal energy, TS, chemical potential, and particle number respectively. The Maxwell relations correspond to what in economics are known as Slutsky conditions!

The economic temperature T is identified with the level of economic development.

Economic entropy is expected to be related to "economic variety, which in turn may be a measure of the economic value of leisure."

Here is a small extract to give you the flavour. It considers the application of the analogue of the Gibbs-Duhem relation Ndmu = -SdT + VdP

I did not think the papers review of thermodynamic concepts was particularly insightful. But, that is what I think about most textbooks. I think the following two points may be particularly relevant for economic analogues.

First, the zeroth law tells us that there is

Second,

The economic temperature T is identified with the level of economic development.

Economic entropy is expected to be related to "economic variety, which in turn may be a measure of the economic value of leisure."

Here is a small extract to give you the flavour. It considers the application of the analogue of the Gibbs-Duhem relation Ndmu = -SdT + VdP

I did not think the papers review of thermodynamic concepts was particularly insightful. But, that is what I think about most textbooks. I think the following two points may be particularly relevant for economic analogues.

First, the zeroth law tells us that there is

**a single variable that tells us whether or not two systems will change when they are brought into thermal contact.**This allows us to define temperature. So does there exist a single economic variable that will tell us whether two isolated economic systems will change when they are allowed to interact (e.g., two countries start trading with one another)?Second,

**entropy is all about irreversibility.**It tells us when the system is isolated whether or not we can get from one equilibrium state to another. Economic systems do undergo irreversible changes. Is there a single variable that can describe the ordering of economic equilibrium states?## Friday, April 27, 2012

### Should you be suspicious of papers written by a rugby team?

I wonder if over the past decade the number of co-authors of condensed matter papers in high profile journals has increased significantly. It seems quite common now to have more than ten authors.

Is this really justified?

Have all of these people really made a substantial contribution to the paper?

Are they all willing to stake their scientific reputations on all of the results in the paper?

Many of these journals require a statement of the contributions of the different authors. But, most of the statements I read are quite generic.

I realise that some of these papers involve theory and experimental collaborations. Some report measurements using complementary probes (e.g., x-rays + neutrons + optical spectroscopy). However, for many of these papers I would have thought that the numbers would be:

1-2 people make the sample

2-3 make the measurements

1 is a friendly theorist who helps in the interpretation.

This adds up to 4-6 not 15!

Why should we care? Here are some possible concerns.

Is this really justified?

Have all of these people really made a substantial contribution to the paper?

Are they all willing to stake their scientific reputations on all of the results in the paper?

Many of these journals require a statement of the contributions of the different authors. But, most of the statements I read are quite generic.

I realise that some of these papers involve theory and experimental collaborations. Some report measurements using complementary probes (e.g., x-rays + neutrons + optical spectroscopy). However, for many of these papers I would have thought that the numbers would be:

1-2 people make the sample

2-3 make the measurements

1 is a friendly theorist who helps in the interpretation.

This adds up to 4-6 not 15!

Why should we care? Here are some possible concerns.

- Some junior people who do the bulk of the work are not getting due credit.
- There is a dilution of responsibility for the content of the paper.
- The long author list may be hoodwinking us into thinking that the paper reports a more substantial body of work than it does.
- There is a problem with joint theory-experiment papers.

Is this something we should care about?

## Thursday, April 26, 2012

### Student misconceptions about entropy

I am slowly learning that many students think that because of the second law of thermodynamics that the entropy of a system must increase in

Nice counter examples at fixed temperature and pressure are

These are possible because there is a net decrease in the Gibbs free energy.

The entropy of the system plus surroundings increases.

I am trying to address this by continually testing understanding of this point with online quizzes and in class "clicker" quizzes.

**any process.**They forget or ignore that this is only true for an**isolated**system.Nice counter examples at fixed temperature and pressure are

- freezing of a liquid
- condensation of vapour
- slow compression of a gas
- many chemical reactions: e.g., combination of hydrogen gas and oxygen gas to form liquid water in a fuel cell

These are possible because there is a net decrease in the Gibbs free energy.

The entropy of the system plus surroundings increases.

I am trying to address this by continually testing understanding of this point with online quizzes and in class "clicker" quizzes.

### Strange metal. Strange insulator. Strange material.

Jaime Merino and I just finished a paper Effective Hamiltonian for the electronic properties of the quasi-one-dimensional material Li0.9Mo6O17

This is a really strange and interesting material. It has featured in earlier posts. We discuss how the observed properties of both the "metallic" phase and the "insulating" phase are quite unusual and don't seem to fit into any "standard" picture [Fermi liquid, Luttinger liquid, quantum critical, ...]. We then propose the simplest possible lattice model Hamiltonian that might capture its properties. This is worthy of further study.

We thank Nigel Hussey for getting us interested in this fascinating material. He has a forthcoming PRL about the unconventional (possibly triplet) superconductivity.

This is a really strange and interesting material. It has featured in earlier posts. We discuss how the observed properties of both the "metallic" phase and the "insulating" phase are quite unusual and don't seem to fit into any "standard" picture [Fermi liquid, Luttinger liquid, quantum critical, ...]. We then propose the simplest possible lattice model Hamiltonian that might capture its properties. This is worthy of further study.

We thank Nigel Hussey for getting us interested in this fascinating material. He has a forthcoming PRL about the unconventional (possibly triplet) superconductivity.

Comments welcome.

## Tuesday, April 24, 2012

### Thanks to Mac and Time Machine

8 days ago the hard drive on my MacBook Pro died. It was 2 and half years old. Fortunately, I had Time Machine backups and had purchased the Apple Care Protection plan which lasts 3 years. I had not got around to registering it and so had to. This all went pretty smoothly. An Apple technician came and picked up the computer, repaired it, and returned. This took 8 days, but there was a 2 day delay because I had to get the registration done and approved.

The Time Machine backup worked beautifully.

This is a lot better experience than I ever had with PCs that died. Repair always took much much longer and restoration of backup was much messier.

The Time Machine backup worked beautifully.

This is a lot better experience than I ever had with PCs that died. Repair always took much much longer and restoration of backup was much messier.

## Monday, April 23, 2012

### What actually is the Born-Oppenheimer approximation?

It can actually be different things to different people.

The same applies to the "adiabatic" approximation.

There is a nice old article from 1977, What does the term "vibronic coupling" mean?.

It clearly distinguishes different approximations that often get called "Born-Oppenheimer" such as crude adiabatic, Condon, Born-Huang,...

Then there are "corrections" to "Born-Oppenheimer" such as Herzberg-Teller.

The paper is useful because it clearly defines everything and has two nice Tables. One gives the relevant equations for the different approximations. Another compares the actual terminology used by different authors (pre-1975).

Does it matter?

Besides Born-Oppenheimer breaking down near conical intersections it also matters for

The same applies to the "adiabatic" approximation.

There is a nice old article from 1977, What does the term "vibronic coupling" mean?.

It clearly distinguishes different approximations that often get called "Born-Oppenheimer" such as crude adiabatic, Condon, Born-Huang,...

Then there are "corrections" to "Born-Oppenheimer" such as Herzberg-Teller.

The paper is useful because it clearly defines everything and has two nice Tables. One gives the relevant equations for the different approximations. Another compares the actual terminology used by different authors (pre-1975).

Does it matter?

Besides Born-Oppenheimer breaking down near conical intersections it also matters for

- "intensity borrowing" where "forbidden" electronic transitions gain oscillator strength by coupling to the vibrational degrees
- electronic isotope effects: isotope exchange reactions can occur [at the level of the BO approx. the isotopes have the same potential energy surface].
- Resonant Raman scattering.

I thank Jeff Reimers for bringing the article to my attention.

## Friday, April 20, 2012

### The squeaky wheel gets the most oil

If you want someone else to do something then the reality is the more you are in their presence the more likely they are to act on it. Everyone is busy and has many people trying to get their attention. I think this is particularly true of technical staff and many faculty, particularly those with a lot of admin responsibility. Hence, depending on how urgent your request stopping by someones office every day (every couple of hours?) is a good idea. Of course, you have to be careful not to be too pushy that they get turned off and will drag their feet. But this is also why

So, don't assume that just because you emailed a request a week ago they are working on it. They are probably working on the request of the last person who walked into their office.

I believe this can be a difficult issue for people from non-Western countries. It can be quite hard for international students to be so demanding. However, the painful reality is that if they are not making polite regular requests for attention they may just get neglected.

**physical presence**is better than email because you can sense their mood.So, don't assume that just because you emailed a request a week ago they are working on it. They are probably working on the request of the last person who walked into their office.

I believe this can be a difficult issue for people from non-Western countries. It can be quite hard for international students to be so demanding. However, the painful reality is that if they are not making polite regular requests for attention they may just get neglected.

## Thursday, April 19, 2012

### Students should write their own formula sheets

With in class exams there are several options on what access to background material that students should have:

1. Students can bring any material (texts, notes, assignment solutions).

2. Students can just bring a copy of the text without annotations.

3. Students can bring a one or two page sheet of formulas that they write themselves.

4. The lecturer provides a formula sheet.

Which do you favour?

1. Provides the most realistic "real world" type of assessment, but it can be hard to write new and suitable questions. Given the choice very few students will choose this option.

I have used 2. before. I found it surprising (and disappointing) that I could set questions whose answer could be found in the text but there were still a significant fraction of students who could not do them.

I have been using 3. lately for my solid state physics class (4th year undergraduates). The students hand in their formula sheets with their exam answers. I suspect I started doing this partly to reduce my workload and I thought it would be good revision exercise for the students. However, I am discovering that

The student sheets give an idea of how hard it is for students to identify key pieces of knowledge. Some equations are of tangential relevance and key ones are omitted. For example, in a recent exam only 1 out of 7 included Bloch's theorem in their formula sheet. Surely, this is the most important equation in the whole course! This underscores to me that we have to not just tell students what the key facts and concepts are but also (somehow)

1. Students can bring any material (texts, notes, assignment solutions).

2. Students can just bring a copy of the text without annotations.

3. Students can bring a one or two page sheet of formulas that they write themselves.

4. The lecturer provides a formula sheet.

Which do you favour?

1. Provides the most realistic "real world" type of assessment, but it can be hard to write new and suitable questions. Given the choice very few students will choose this option.

I have used 2. before. I found it surprising (and disappointing) that I could set questions whose answer could be found in the text but there were still a significant fraction of students who could not do them.

I have been using 3. lately for my solid state physics class (4th year undergraduates). The students hand in their formula sheets with their exam answers. I suspect I started doing this partly to reduce my workload and I thought it would be good revision exercise for the students. However, I am discovering that

**it is quite revealing to look over the student sheets to see what students do and don't include.**Some are very well prepared and others are not. Some don't include fundamental constants they will need and then try to use this as an excuse for not completing a question.The student sheets give an idea of how hard it is for students to identify key pieces of knowledge. Some equations are of tangential relevance and key ones are omitted. For example, in a recent exam only 1 out of 7 included Bloch's theorem in their formula sheet. Surely, this is the most important equation in the whole course! This underscores to me that we have to not just tell students what the key facts and concepts are but also (somehow)

**train them to be able to recognise what is really important**.## Tuesday, April 17, 2012

### Seeing the positive charge of holes

When students first learn solid state physics the concept of holes and its utility is not easy to grasp. I find it really helpful to use the Solid state simulations program

One can consider the motion of electrons in Bloch states for a band structure with the energy contours shown in green in the figure below. One can vary the external magnetic and electric field.

If one goes to preset 6 (which has zero electric field) one can start an electron on an electron or a hole Fermi surface. One sees that the motion in a magnetic field has the opposite circulation for electrons and holes, in both real and Bloch wave vector space.

Hence, the holes really do act like positively charged particles.

*ziman*very helpful to illustrate the difference between electrons and holes.One can consider the motion of electrons in Bloch states for a band structure with the energy contours shown in green in the figure below. One can vary the external magnetic and electric field.

If one goes to preset 6 (which has zero electric field) one can start an electron on an electron or a hole Fermi surface. One sees that the motion in a magnetic field has the opposite circulation for electrons and holes, in both real and Bloch wave vector space.

Hence, the holes really do act like positively charged particles.

### Why temperature and pressure?

Before introducing the Gibbs free energy in my thermodynamics class I asked the students to say which variables they thought were generally the "easiest" to control in chemistry and physics

Many students thought volume and temperature. (Maybe because what they mostly learn about is gases!).

I think the "correct" answer is pressure and temperature because

* these are environmental not system variables

* it is very hard to control the volume of a solid.

Am I right?

I now realise this is a rather subtle point and worth getting students to think about.

Appreciating it makes students think about experiments rather than mathematics and helps motivate why the Gibbs free energy is actually the most useful thermodynamic function.

**experiments:**volume and temperature, volume and energy, pressure and temperature, ...., or all?Many students thought volume and temperature. (Maybe because what they mostly learn about is gases!).

I think the "correct" answer is pressure and temperature because

* these are environmental not system variables

* it is very hard to control the volume of a solid.

Am I right?

I now realise this is a rather subtle point and worth getting students to think about.

Appreciating it makes students think about experiments rather than mathematics and helps motivate why the Gibbs free energy is actually the most useful thermodynamic function.

## Monday, April 16, 2012

### Delocalised molecular orbitals are not superior

A recent issue of the Journal of Chemical Education has several excellent articles responding to a recent controversial article by Alexander Grushow suggesting that hybrid atomic orbitals have no real physical and chemical basis and so should not be included in the undergraduate chemistry curriculum.

There are good articles rejecting this argument. Two of the articles are written by pairs of my favourite quantum chemists Landis and Weinhold, and Hiberty and Shaik.

They are worth reading because they highlight some key issues in quantum chemistry and chemical bonding.

At the heart of the matter is whether one favours a

Neither picture is incorrect. They are complementary ways of looking at the same chemical reality. [See for example this post discussing the old MO-VB rivalry].

Both sets of authors emphasize this by stressing that a single Slater determinant is invariant under a unitary transformation. Such a transformation takes one from one molecular orbital representation to antoher. e.g., from canonical molecular orbitals to hybrid atomic orbitals. Both give the same total electron density.

This debate also highlights that chemical bonding is a quantum many-body effect and must be viewed as such. For example, the one electron orbitals are not the key quantity but the many-body wavefunction. Also, photoelectron spectroscopy does not measure orbital energies but the difference in energy between the ground state and a cation.

I thank Seth Olsen for bringing these articles to my attention.

There are good articles rejecting this argument. Two of the articles are written by pairs of my favourite quantum chemists Landis and Weinhold, and Hiberty and Shaik.

They are worth reading because they highlight some key issues in quantum chemistry and chemical bonding.

At the heart of the matter is whether one favours a

**delocalised**picture (with canonical molecular orbitals) or a**localised**picture (**hybrid atomic orbitals**and valence bonds).Neither picture is incorrect. They are complementary ways of looking at the same chemical reality. [See for example this post discussing the old MO-VB rivalry].

Both sets of authors emphasize this by stressing that a single Slater determinant is invariant under a unitary transformation. Such a transformation takes one from one molecular orbital representation to antoher. e.g., from canonical molecular orbitals to hybrid atomic orbitals. Both give the same total electron density.

This debate also highlights that chemical bonding is a quantum many-body effect and must be viewed as such. For example, the one electron orbitals are not the key quantity but the many-body wavefunction. Also, photoelectron spectroscopy does not measure orbital energies but the difference in energy between the ground state and a cation.

I thank Seth Olsen for bringing these articles to my attention.

## Friday, April 13, 2012

### Is it mathematics or physics?

I think that Mathematical Physics as a research area struggles with its identity at times. Is it mathematics or physics or neither?

I don't think Chemical Physics has a comparable identity crisis, with its value being appreciated by both communities.

Many? theoretical physicists (particularly great ones like Feynman and Anderson) will claim that most big breakthroughs in theory occur using old mathematics and with little regard to mathematical rigour. Sometimes a focus on mathematical formalism is an impediment rather than a aid to real progress. (See for example, this post about Anderson's views).

Whether string theory is actually theoretical physics or just beautiful mathematics is debatable (Anderson had a nice review of Peter Woit's book a while back).

However, there are times where Mathematical Physics does really produce some nice new mathematics. A recent case is the work by Stanislav Smirnov on conformal invariance and was honoured by award of a 2010 Fields Medal.

I don't think Chemical Physics has a comparable identity crisis, with its value being appreciated by both communities.

Many? theoretical physicists (particularly great ones like Feynman and Anderson) will claim that most big breakthroughs in theory occur using old mathematics and with little regard to mathematical rigour. Sometimes a focus on mathematical formalism is an impediment rather than a aid to real progress. (See for example, this post about Anderson's views).

Whether string theory is actually theoretical physics or just beautiful mathematics is debatable (Anderson had a nice review of Peter Woit's book a while back).

However, there are times where Mathematical Physics does really produce some nice new mathematics. A recent case is the work by Stanislav Smirnov on conformal invariance and was honoured by award of a 2010 Fields Medal.

## Wednesday, April 11, 2012

### Sometimes it is best to hire nobody

This may seem a strange claim.

You have a grant or your department has a position. You fail to attract the quality of candidates you hoped, particularly after some attrition as the better candidates accepted positions elsewhere. But, you should hire someone. Otherwise the grant will go to waste, you will lose the position, or at least everything will be delayed for a year. Voices (both internal and external) will say you should just hire someone.

No! This can be a mistake.

Sometimes there will be people (whether graduate students or faculty members) who will turn out to be a

Fortunately, I have never experienced this first hand. However, I have seen disasters happen. Sometimes for this reason I have postponed hiring someone. Once I never filled a funded Ph.D position since I just could not find anyone suitable.

No candidate is perfect. Be willing to compromise your hopes and expectations. But, never hire anyone out of desperation.

Cartoon is by Kerry Soper

You have a grant or your department has a position. You fail to attract the quality of candidates you hoped, particularly after some attrition as the better candidates accepted positions elsewhere. But, you should hire someone. Otherwise the grant will go to waste, you will lose the position, or at least everything will be delayed for a year. Voices (both internal and external) will say you should just hire someone.

No! This can be a mistake.

Sometimes there will be people (whether graduate students or faculty members) who will turn out to be a

**net drain**on your resources (not just money but time, energy, and lab consumables) and relationships (harmonious research group or departmental). Be particularly wary of people who have a history of not getting along with others. You may wish you never hired them.Fortunately, I have never experienced this first hand. However, I have seen disasters happen. Sometimes for this reason I have postponed hiring someone. Once I never filled a funded Ph.D position since I just could not find anyone suitable.

No candidate is perfect. Be willing to compromise your hopes and expectations. But, never hire anyone out of desperation.

Cartoon is by Kerry Soper

## Tuesday, April 10, 2012

### Are you impressed or depressed?

Previously I posted about just how hard it is to predict new phases of matter, particularly in a specific material. I more or less claimed this has never be done. I was incorrect. Ben Powell pointed out to me two significant counter examples: Bose Einstein Condensates (BECs) and Topological Insulators. Both represent monumental and profound achievements. But, how impressed (or smug) should we be?

After all, both these examples involve

After all, both these examples involve

**non-interacting particles**, or at least particles just interacting at the mean-field level. Hence, this just further underscores to me just how hard it is to actually predict truly emergent phenomena, involving "non-trivial" quantum many-body physics.### Is science a noun or a verb?

I just encountered this simple and helpful question in the context of how and what we teach students.

If we teach science as a static body of knowledge (particularly facts, theories, and techniques) we are acting as if science is a

By contrast, if we focus on teaching students to think scientifically and critically, to solve problems, and to ask questions, then we act as if science is a

If we teach science as a static body of knowledge (particularly facts, theories, and techniques) we are acting as if science is a

**noun.**By contrast, if we focus on teaching students to think scientifically and critically, to solve problems, and to ask questions, then we act as if science is a

**verb.**

## Thursday, April 5, 2012

### Measuring a free energy change

It is just the voltage from a battery (electrochemical cell)!

Soon I will give a lecture introducing the free energy in a thermodynamics course. It is an incredibly important concept. We try and get the students to learn the course mantra, "the Gibbs free energy must be minimised (at constant temperature and pressure)".

Two significant points for students to learn about the Gibbs free energy

-it is directly measurable

-where many of the tabulated values for delta G for chemical reactions come from

The voltage of an electrochemical cell V (in Volts) is related to delta G (in Joules per mole) by

V = -delta G/n F

where F = Faraday constant = 96485 J/mol/V and n=number of electrons transferred at the electrode.

In my lecture I will show two really nice videos from Chemistry Comes Alive on the electrolysis of water. I particularly like how the second video (below) shows how the ratio of the volume of the gas produced at the cathode (hydrogen) is twice that produced at the anode (oxygen).

Soon I will give a lecture introducing the free energy in a thermodynamics course. It is an incredibly important concept. We try and get the students to learn the course mantra, "the Gibbs free energy must be minimised (at constant temperature and pressure)".

Two significant points for students to learn about the Gibbs free energy

-it is directly measurable

-where many of the tabulated values for delta G for chemical reactions come from

The voltage of an electrochemical cell V (in Volts) is related to delta G (in Joules per mole) by

V = -delta G/n F

where F = Faraday constant = 96485 J/mol/V and n=number of electrons transferred at the electrode.

In my lecture I will show two really nice videos from Chemistry Comes Alive on the electrolysis of water. I particularly like how the second video (below) shows how the ratio of the volume of the gas produced at the cathode (hydrogen) is twice that produced at the anode (oxygen).

## Wednesday, April 4, 2012

### Effective Hamiltonians for a family of organic dyes

Seth Olsen and I have just finished a paper An Effective Hamiltonian for Symmetric Diarylmethanes from a Series of Analogous Quantum Chemical Models.

Finding simple effective Hamiltonians for classes of complex chemical systems is not easy. Justifying them from quantum chemistry is even harder. We consider a family of organic dye molecules related to Michler's hydrol blue, including auramine-O and malachite green. These molecules are increasingly being used as sensors of the local environment in biomolecules.

The three lowest singlet states can be described by a 3x3 matrix Hamiltonian whose parameters vary across the family of the dyes. This variation appears to be correlated with an empirical parameter [Brown-Okamoto] used to characterise the effect of substituents. There is also a subtle and interesting variation in the character of the diabatic states as one traverses the family of dyes.

Finding simple effective Hamiltonians for classes of complex chemical systems is not easy. Justifying them from quantum chemistry is even harder. We consider a family of organic dye molecules related to Michler's hydrol blue, including auramine-O and malachite green. These molecules are increasingly being used as sensors of the local environment in biomolecules.

The three lowest singlet states can be described by a 3x3 matrix Hamiltonian whose parameters vary across the family of the dyes. This variation appears to be correlated with an empirical parameter [Brown-Okamoto] used to characterise the effect of substituents. There is also a subtle and interesting variation in the character of the diabatic states as one traverses the family of dyes.

## Monday, April 2, 2012

### A spin liquid in my favourite frustrated spin model

At last weeks cake meeting (condensed matter group meeting/journal club) I gave a talk about an interesting recent paper Incommensurate correlations in the anisotropic triangular Heisenberg lattice by Andreas Weichselbaum and Steve White.

The model is the spin-1/2 Heisenberg model on the anisotropic triangular lattice with antiferromagnetic interactions. By varying the relative strength (or spatial anisotropy) of the interactions the model can interpolate between the square lattice, triangular lattice, and weakly coupled chains (with a frustrated interchain interaction J'). Back in 1998 I argued that this is the minimal model for the spin excitations in the Mott insulating phase of a family of organic superconductors. I have since written 7 papers on the model. A recent review article looks at the model in light of theoretical and experimental studies, which reveal its richness including the possibility of spin liquid ground states.

Weichselbaum and White perform extensive DMRG (density matrix renormalisation group) studies considering lattices as large as 64 x 8. This is arguably the highest power numerical study of the model to date. They mostly focus on the very specific question of whether in the coupled chain limit (J' much less than J, the intrachain coupling) the spin correlations ever become commensurate. [An Sp(N), large N study (not referenced) suggested this was the case]. This limit is particularly relevant to the material Cs2CuCl4, which has J'/J ~0.3, and is a candidate material to have deconfined spinon excitations, a connection that is clearly mentioned in the paper.

Here are a few of the interesting results in the paper. First, the results are quite sensitive to the boundary conditions and to the system size, even for these relatively large systems.

For

For this same parameter regime J'/J ~ 1-1.2 a ground state which breaks translation symmetry is seen. The figure below shows the magnitude of the spin correlations on different links in the lattice. A similar ground state was found in a recent DMRG study of a four leg ladder.

The results vary significantly depending on whether the system size in the vertical direction is 4n or 4n+2. Periodic boundary conditions are applied in this direction. The authors state:

Huckel's rule states that rings with 4n+2 pi electrons (e.g. benzene) are stable and do not dimerise. They are

In contrast, rings with 4n electrons (e.g. cyclobutadiene) are unstable to dimerisation (i.e., they have alternating single and double bonds). They are

This 4n/4n+2 dichotomy can also be formulated in valence bond theory, as emphasized passionately by Shaik and Hiberty.

The model is the spin-1/2 Heisenberg model on the anisotropic triangular lattice with antiferromagnetic interactions. By varying the relative strength (or spatial anisotropy) of the interactions the model can interpolate between the square lattice, triangular lattice, and weakly coupled chains (with a frustrated interchain interaction J'). Back in 1998 I argued that this is the minimal model for the spin excitations in the Mott insulating phase of a family of organic superconductors. I have since written 7 papers on the model. A recent review article looks at the model in light of theoretical and experimental studies, which reveal its richness including the possibility of spin liquid ground states.

Weichselbaum and White perform extensive DMRG (density matrix renormalisation group) studies considering lattices as large as 64 x 8. This is arguably the highest power numerical study of the model to date. They mostly focus on the very specific question of whether in the coupled chain limit (J' much less than J, the intrachain coupling) the spin correlations ever become commensurate. [An Sp(N), large N study (not referenced) suggested this was the case]. This limit is particularly relevant to the material Cs2CuCl4, which has J'/J ~0.3, and is a candidate material to have deconfined spinon excitations, a connection that is clearly mentioned in the paper.

Here are a few of the interesting results in the paper. First, the results are quite sensitive to the boundary conditions and to the system size, even for these relatively large systems.

For

**all system sizes**the authors see**a spin gap**in the parameter range of J'/J ~ 1-1.2. [see the yellow curve in the figure above which is for 64 x 6.] This range and the magnitude of the gap do vary significantly with the size of the system. The authors claim this gap will vanish in the thermodynamic limit. However, I am not convinced, partly because series expansions do produce a gap in this parameter range (see Figure 10 in this PRB).For this same parameter regime J'/J ~ 1-1.2 a ground state which breaks translation symmetry is seen. The figure below shows the magnitude of the spin correlations on different links in the lattice. A similar ground state was found in a recent DMRG study of a four leg ladder.

The results vary significantly depending on whether the system size in the vertical direction is 4n or 4n+2. Periodic boundary conditions are applied in this direction. The authors state:

Overall, the dimerization seen here suggests a qualitative difference of the systems of width 4n+2 (symmetry-broken systems), with n an integer, to systems of width 4n (uniform systems), while nevertheless, a two-chain periodicity perpendicular to the chains is maintained in either case. Equivalently, this translates into an even-odd effect in the number of laterally coupled zigzag chains.No mention is made of a very famous phenomena in organic chemistry, which must be related. But the precise connection is not completely clear to me, because in that case 4n+2 tends to be uniform and 4n tends to break symmetry.

Huckel's rule states that rings with 4n+2 pi electrons (e.g. benzene) are stable and do not dimerise. They are

**aromatic.**The electrons are delocalised.In contrast, rings with 4n electrons (e.g. cyclobutadiene) are unstable to dimerisation (i.e., they have alternating single and double bonds). They are

**anti-aromatic.**This 4n/4n+2 dichotomy can also be formulated in valence bond theory, as emphasized passionately by Shaik and Hiberty.

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