Friday, September 30, 2011

Why you might worry about classical models of water

A nagging question for the whole field of (classical) molecular dynamics simulations of biomolecules is whether they can have an adequate description of water. This is particularly important because almost all biomolecular processes involve subtle interactions of the biomolecule of interest with its aqueous environment.

I learnt a lot from reading the article On the origin of the redshift of the OH stretch in Ice Ih: evidence from the momentum distribution of the protons and the infrared spectral density, by C. J. Burnham, G. F. Reiter, J. Mayers, T. Abdul-Redah, H. Reichert and H. Dosch.

They highlight several problems and state:
Clearly there is something missing from water models. All of the above difficulties have been consistently tackled by experimentalists, but they remain either unrecognized or unacknowledged by much of the simulation community.
Here are the 4 main difficulties they list concerning the differences between the properties of a water monomer and the water molecule in ice Ih [most common phase of ice]:

1. the magnitude of the measured anharmonicity parameter of the OH stretch X_OH (=difference between 1/2 of the second overtone frequency and the fundamental) of the OH stretch in the condensed phase is increased from 87 cm-1 in the gas-phase to 134 cm-1 in ice Ih. This behavior cannot be reproduced by a simple anharmonic oscillator (such as a Morse oscillator), for which elongation of the OH stretch by the ice H-bond results in a decrease in |XOH|.

2. the enormous observed increase (25 fold) of the integrated IR intensity in the OH stretch mode in ice compared to that of the gas-phase monomer. Most water models (even including polarizable ones) predict almost no increase at all.

3. The gas-phase molecular dipole moment derivative with respect to the OH stretch is observed to be in a direction some 25 degrees outside of the OH stretch vector. In contrast, it is observed that this derivative becomes nearly parallel to the OH vector in ice.

4. The HOH angle increases from the gas-phase value of 104.5 to 107 degrees in ice. This is in contrast to virtually all empirical water models, which predict a lowering of the HOH angle from the gas-phase value.

The authors propose their own (classical) solution to these problems.
I am not in a position to judge the validity or reasonableness of the solution.
However, I have a prejudice that ultimately the origins of these problems is that the H-bond has a significant covalent and quantum character.

Wednesday, September 28, 2011

Scepticism and caution is usually the best response

All the media attention to recent anomalies in the speeds of neutrinos raises some interesting scientific and sociological questions. Should you go to the media before you have results published in a peer reviewed journal?

A few things to bear in mind.

These deviations from the speed of light are one part in 100 thousand. They require measuring the distance travelled to the same precision, i.e. an accuracy of 20 cm!

Relative measurements rather than absolute measurements are usually more reliable. As the Nature News (n.b. not paper) article said
Most troubling for OPERA is a separate analysis of a pulse of neutrinos from a nearby supernova known as 1987a. If the speeds seen by OPERA were achievable by all neutrinos, then the pulse from the supernova would have shown up years earlier than the exploding star's flash of light; instead, they arrived within hours of each other.
An earlier anomaly in neutrino physics was of the 17 keV neutrino that was "discovered" in 1985. It is worth reading the historical reviews in Reviews of Modern Physics and Nature which discuss what went wrong. These articles should sober theorists who jump on  a bandwagon once some apparent experimental anomaly is observed.

On the other hand, the solar neutrino problem is a case where experimental anomalies did lead to interesting new physics. But note, there the anomalies were by a factor of three in the observed neutrino flux.

Tuesday, September 27, 2011

A quantum chemist tries to solve cuprate superconductivity

Journal of Physical Chemistry Letters has a paper Origin of the Pseudogap in High-Temperature Cuprate Superconductors  by Jamil Tahir-Kheli and William A. Goddard, III

I can't say I really follow the details here. The doping dependence of the pseudogap they calculate seems to be a percolation effect. A key issue is whether they can produce the anisotropy in the pseudogap in momentum space. It is not clear they do since everything seems to be done in real space.

There is some interesting history here, going back to the early days of high-Tc.
Sparks fly over conflicting theories from Chemical and Engineering News in 1988, discusses conflict between Phil Anderson and Goddard.
A clash erupts between scientific subculture is a New York Times article from 1989 which quotes a Dr. Anderson who is presumably Phil Anderson.

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

Friday, September 23, 2011

Quantum computational matter

Stephen Bartlett gave a colloquium on this topic today.
The ground states of quantum antiferromagnets contain substantial amounts of entanglement (although how much is hard to quantify). Hence, one might hope they are a resource that might be used in quantum computation. Stephen described some recent work [see this PRL and this PRL ] which considers how this might be done. The focus seems to be on gapped systems which have hidden symmetries and can be described as tensor network states. The prime example is the Haldane spin-1 antiferromagnetic Heisenberg chain which is adiabatically connected to the AKLT model.

I was reminded of some work I was involved in a few years ago (described in this PRL) which showed how one could take any spin singlet state [not just the ground state!] from a spin-1/2 chain and perform projective Bell measurements (on pairs of spins) along the chain and teleport quantum states with perfect fidelity along the chain. I was wondering what the relationship (similarities and differences) of our work was with this more recent work.

Where is theoretical chemistry going?

There is an Editorial Theoretical Chemistry - Quo Vadis? by Walter Thiel in Angewandte Chemie International Edition.
["Quo Vadis" is latin for "where are you going?"]

It is worth reading. It is particularly good that he offers some concrete precautions for experimentalists running computational codes.
I thought the following characterisation of theoretical chemistry was disappointingly narrow:
Theoreticians are primarily interested in testing the performance and limitations of newly developed computational methods, for example by systematic validation on established benchmarks or through proof-of-principle calculations. 
Overall I prefer the articles I highlighted in 5 papers every computational chemistry student should read, together with Hoffmann's 1974 article on theory in chemistry and Zewail's article on the future of chemical physics.
Those articles place a much greater emphasis on theory providing unifying concepts.

I thank Seth Olsen for bringing the article to my attention.

Thursday, September 22, 2011

Writing effective personal statements

Most applications for scholarships or admission to graduate school require a personal statement. I found a very useful site at Penn State, Writing Personal Statements online.

I read the two sample statements from applicants for a Marshall Scholarship. Both were very impressive. They illustrate several key ingredients
  • Distinctly personal. It is about you. Only you could write this. Avoid generics "I am really interested in subject X and want to study at University Y because it is a world class university."
  • Personally engaging. A natural outcome is that the reader should want to meet you.
  • Specific connection between applicant and the target institution/program. Mentioning specific courses, faculty, and research projects is key.
  • Polished and well written. This means writing and rewriting and getting feedback on drafts.
For those of us reading and reviewing applications I see two important consequences of this material being freely available.

First, the bar is higher. Any student with a little "get up and go" can Google "personal statements scholarship applications" and find material such as this. They should then aim to produce something of comparable quality.

Second, we need to be wary of plagiarism and so running applications through Turnitin or Googling suspect sentences may be a necessary precaution.

As an aside, I mention the movie Spanglish has an amusing opening scene [which I could not find on YouTube] where an admissions officer at Princeton is reading personal statements from undergraduate applicants.

Wednesday, September 21, 2011

The case for effective Hamiltonians

When trying to understand complex molecular materials, the dominant approach in chemistry to is to do DFT-based calculations for the system of interest. However, a case needs to be made for an alternative "physics" approach. Recent Anthony Jacko, Ben Powell and I wrote an article Models of organometallic complexes which makes the case below for effective Hamiltonians.

Another approach to modeling the optoelectronic properties of organometallic complexes is to construct a model with fewer states but an accurate treatment of the electronic correlations. This contrasts with first principles calculations, which include several basis states for each atom but neglect some electronic correlations. The small number of degrees of freedom in such semi-empirical models allows one to make fewer approximations on the interactions and correlations in the model. It also allows one to identify key trends that describe broad classes of materials. This approach has proven itself incredibly powerful in wide areas of materials science. For example, the Anderson single impurity model can describe a wide range of systems including magnetic impurities in metals, quantum dots in semiconductor heterostructures, carbon nanotubes , and single molecule transistors.47,48
In principal an effective model Hamiltonian is found by starting with the exact Hamiltonian and ‘integrating out’ high energy states.49 This procedure is computationally expensive,50 so often one simply chooses a reduced basis set, motivated by the physical processes one wishes to capture.49 DFT can be used to estimate the values (or trends in values) of some of the parameters of these effective models. The model Hamiltonian can then be solved, retaining correlations that the approximate DFT functional does not include.51–55
Identifying the frontier orbitals which dominate the photophysics is one of the most significant steps of the effective model approach. In this reduced basis set one can define an effective Hamiltonian with just a few parameters. Conjugated polyenes have been investigated in this way via the Hückel, Hubbard, Heisenberg and Pariser-Parr-Pople models.49 This approach has been applied to organometallic complexes, for example mixed valence binuclear systems including magnetic atoms in proteins (Hubbard and double exchange models; Ref. 56), molecular magnets (Ref. 57), Anderson impurity models for cobalt based valence tautomers (Ref. 58), and a series of metal-coredbipyridine complexes (Ref. 22). It has also been shown recently that this approach naturally explains the sensitivity of the photophysical properties of organometallic complexes to small chemical changes.18
To correctly describe the character of the excited states the model must capture the key interactions. There are many important features of the system that might be included in such a model, for example electronic ‘hopping’ terms between the frontier molecular orbitals, direct Coloumb interactions between electrons in those orbitals, spin interactions, and relativistic effects such as spin–orbit coupling. The relative energy scales of these various interactions will define the composition of the excited states and therefore their properties. 

I welcome comments and suggestions on any review articles that persuasively make the case for effective Hamiltonians as an important tool in materials modelling.

Monday, September 19, 2011

Seeking a new phase of matter in organic charge transfer salts

An important finding of recent cluster DMFT studies concerns the nature of the metallic state in the Hubbard model at half filling. Near the Mott transition it is found that the scattering rate is  anisotropic over the Fermi surface (a Fermi liquid with momentum space differentiation) just as it is in the doped Hubbard model.
Emanuel Gull showed a general phase diagram (U/t vs. doping) in his talk at the Ringberg meeting.
(I could not find it in any of his papers, such as this PRB, but he kindly provided a copy). This "momentum space differentiated" phase is intermediate between the pseudogap state and the isotropic Fermi liquid, both as a function of doping and U/t.

This (temperature dependent) anisotropy should be observable in angle dependent magnetoresistance (ADMR) measurements on organic charge transfer salts in the kappa-(BEDT-TTF)2X family. These materials are all at half filling. A candidate material is X= Cu[N(CN)2]Br which lies close to the Mott transition. [By deuteration it can be tuned into the Mott phase]. Previous papers [e.g. see Section 3.4  in a recent review article I wrote with Ben Powell] have considered evidence for a pseudogap state in the organics. However, I am unaware of any significant attention being paid to this distinct idea of an anisotropic scattering rate in the organics.

In a 2007 PRL Singleton et al. found that that experimental data for X=Cu(SCN)2 can be adequately modelled by an isotropic scattering rate with a Fermi liquid temperature
dependence. Is the co-efficient for the quadratic temperature dependence consistent with what Dressel finds for the quadratic frequency dependence of the scattering rate  from
the optical conductivity? [This earlier post discusses the general issue of the relation between the temperature and frequency dependence of the scattering rate in Fermi liquid theory].
Thus, in order to see the variation in the scattering rate one may have to go even closer to the Mott insulating phase by considering X= Cu[N(CN)2]Br, (as in this Nature paper on the Nernst effect).

Topological insulators in prime time TV

The popular sit-com The Big Bang Theory continues to break new ground by introducing prime time TV viewers to the latest developments in condensed matter physics. This clip from an episode The Thespian Catalyst begins with Sheldon attempting to teach a class of Caltech graduate students about topological insulators. Note that the equations on the board are authentic, as usual.

Sheldon finds, as many of us do, there is a weak correlation between our perceptions of our teaching performance and our students perceptions of the quality of our performance. The show also discusses important issues about what initiatives we should take to address this issue and to improve the quality of our teaching.

Friday, September 16, 2011

Thursday, September 15, 2011

What is the "mechanism" of superconductivity in the cuprates?

Thomas Maier gave a nice talk at the conference. He considered this question [or at least what is the mechanism of superconductivity in the Hubbard model?] via a Bethe-Salpeter equation (some of the results are described in this PRB and this PRB).
where Gamma^pp is the irreducible particle-particle vertex and chi_0 is

Their calculations are for a cluster DMFT treatment of the doped Hubbard model. They find that the irreducible particle-particle vertex peaks at a wavevector of (pi,pi) as does the spin susceptibility chi(K-K'). Indeed they find that this vertex is to a good approximation related to the spin susceptibility via an RPA type relation.
where U(T) is a temperature dependent renormalised Hubbard U.

But why does the superconductivity go away as one approaches the Mott insulator? After all, the spin fluctuations are increasing! This is because chi_0 ~ GG [Thomas had a name for this that I did not catch] is decreasing because of the suppresion of quasi-particle weight as the Mott insulator is approached.

I would be interested to see this approach applied to an extended Hubbard model on the square lattice at one quarter filling, near the charge ordering transition [see this PRL and PRB for the context]. Two questions the approach could answer are
  • Is there superconductivity? If so, is it d_xy symmetry?
  • Is it mediated by spin or charge fluctuations?
The latter question is relevant because there is spin ordering associated with the charge ordering and when the charge order melts so will the spin order, potentially leading to significant spin  fluctuations.

Wednesday, September 14, 2011

The long road from the Hubbard model to experiment

An outstanding theoretical challenge remains (reliably) calculating thermodynamic and transport properties of the one band Hubbard model. This is necessary to answer questions such as: is it the relevant effective Hamiltonian for cuprate superconductors?

Emanuel Gull gave a nice talk at the conference about work on cluster DMFT treatment of the doped Hubbard model. This PRB paper contains a nice comparison of how the results vary with cluster size.  They consider cluster geometries up to 16 sites. At least 8 sites are necessary to cover the important region around wavevector (pi/2,pi/2) where one hopes to find nodes in a pseudogap, superconducting gap, and cold spots on the Fermi surface.
The calculations produce four distinct phases:
  • Mott insulator
  • An isotropic Fermi liquid (FL)
  • A FL with momentum space differentiation
  • A momentum selective Mott insulator (a pseudogap state)
Of particular interest to me is the existence of the third state which may correspond to the anisotropic Marginal Fermi liquid state seen in overdoped cuprates. It is desirable to compare the doping, temperature, and frequency dependences of the self energies they calculate [Figures 8-11] to the model form that Jure Kokalj and I have considered.

Tuesday, September 13, 2011

Mapping out Fermi surface anisotropies in overdoped cuprates

Here are the slides for the talk, Angle-dependent magnetoresistance as a probe of Fermi surface anisotropies I am giving this afternoon at the conference.
Much of the talk is based on this preprint.

Key elements of Iron pnictide superconductors

David Singh gave a nice talk today about the iron pnictide superconductors. Here are just a few of the points that he emphasized.

This is a big family of materials - high Tc is a very robust phenonema. Common features
  • Fe atoms are in a square lattice
  • near divalent Fe
  • tetrahedral coordination (2 Fe per unit cell)
The connection of the pnictides to the cuprates is weak, contrary to what some people asserted in the early days [because of the proximity of superconductivity and antiferromagnetism]
-doping not essential to superconductivity
-Mott physics is not relevant (no Mott insulating state)
-magnetic order & superconductivity do co-exist?
-multiple orbitals are present [they may be important for avoiding Mott insulator]
-many materials are electronically "three-dimensional" rather than two-dimensional
[ARPES and dHvA see significant corrugation of the Fermi surface].

Proximity of superconductivity to a magnetically ordered phase should not be viewed as necessarily implying a magnetic mechanism for Cooper pairing. He gave a summary [in the form of a phase diagram] of the 1966 paper by Berk and Schrieffer who argued that Pd was actually not superconducting due to proximity to a ferromagnetic state.
An important case is fullerene superconductors. They are due to electron-phonon pairing, but can be close to an antiferromagnetic Mott insulating phase.

How long will it take us to solve high-Tc superconductivity?

How long after the experimental discovery of a quantum many-body phenomena does it take to develop a theory that will be eventually accepted as the correct one?

Superconductivity was first observed in 1911 and BCS theory arrived in 1957.

At the conference in the castle today, Sasha Lichtenstein pointed out that the Kondo effect was first observed in 1933, Kondo described some of the relevant physics in 1964, but it wasnt until the 1980s that the problem was really solved. Thats 50 years. He pointed out that it is 25 years since the discovery of high-Tc superconductivity in the cuprates and suggested maybe we have 25 more years to get the theory right!

On the other hand, Laughlin found the correct theory of the fractional quantum Hall effect within a year of its experimental discovery!

Monday, September 12, 2011

How bad can a metal get?

A striking feature of strongly correlated electron materials is the existence of bad metallic behaviour. The resistivity can monotonically increase with temperature to values which are much larger than the Mott-Ioffe-Regel limit, a value corresponding (in a simple Drude model) to a value where the mean-free path is smaller than a lattice constant.
Key questions are
  • Does the resistivity ever "saturate" at high temperatures?
  • If so, to what value is the maximum possible resistivity?
  • Could this be related to the large spectral weight spread out over a broad spectral range in the optical conductivity? (see the next figure below).
There is a nice summary of the issues in section VIIH of this RMP by Basov and Timusk.

Gunnarsson and collaborators have come up with a simple argument to address these questions [see this nice RMP colloquium].

At high enough temperatures there is no Drude peak in the optical conductivity. However, the latter must still satisfy the f-sum rule, which relates the total "low energy" spectral weight to the average kinetic energy, E_K. The spectral weight may be spread over the non-interacting band width W.
These observations can be summarised in the figure and equation below which provides an estimate for the dc conductivity  sigma(0) on the scale e^2/(h d) where d is the interlayer spacing.
A recent PRB by Bergeron, Hankevych, Kyung, and Tremblay calculates the optical conductivity for the Hubbard model at the level of a two-particle self-consistent approach, including the constraint of the f-sum rule. The curves below is for a doping p=0.2 , close to optimal,  and a temperature of T=0.2t.
Qualitatively, it seems consistent with the predictions of Gunnarsson et al.

Saturday, September 10, 2011

A unified description of hydrogen bonding

I have just finished a paper Unified description of hydrogen bonding by a two-state effective Hamiltonian. I welcome feedback. Many of the ideas in the paper have been discussed on this blog before over the past year. The paper sits right at the boundary of physics and chemistry and exemplifies the associated tensions. Some physicists may like it because of the simplicity and transparency. Some chemists may consider it glosses over and misses too many details. 
Practically all the ideas in the paper have been discussed in some form before. I see the key novelty as synthesis: providing a conceptual and semi-quantitative framework to understand a wide range of experimental and theoretical results. Consequently, I consider the paper may be the most significant one I have ever written...

I welcome feedback.
I will submit the paper to Physical Review Letters in about a week, after I have received more feedback.

Thursday, September 8, 2011

Deconstructing quantum nuclear effects in hydrogen bonding

Most molecules and solids exhibit simple isotope effects associated with vibrational modes. In the harmonic limit the vibrational frequency scales with the inverse square root of the mass. Hence, replacing hydrogen with deuterium (D) should reduce the frequency by a factor of 1.414 = sqrt(2). However, significant anharmonicity or non-adiabatic (violations of Born-Oppenheimer?) effects can lead to very different effects. 
An important case is that of hydrogen bonding. The graph below shows gamma = the ratio of the O-H stretch stretch frequency to the O-D frequency as a function of the O-H frequency. The latter is a measure of the H-bond strength and is directly correlated with the donor-acceptor distance. The graph is taken from this paper.
One sees that as the mode softens [as the H-bond gets stronger] the isotope effect gets smaller. Playing around with the lowest energy eigenvalues for a Morse potential one can reproduce this effect. However, the physics behind the upward turn for lower frequencies [which presumably are in the domain where the D-H potential no longer has an energy barrier] is not clear to me. This reflects non-trivial quantum nuclear effects, as discussed in this recent PNAS paper.

Wednesday, September 7, 2011

Scientific misnomers?

Who should get their name associated with a particular physical effect? Surely, they should have to be involved in the discovery or in understanding the actual effect. But, it seems this is sometimes not the case.
Also, if someone develops a theory for an observation, which later turns out to be the wrong theory, should their name remain associated with the effect?

Here are a few examples, up for discussion.

The Pauli paramagnetic limit for the upper critical field of a superconductor should be the Clogston-Chandrasekhar limit. Pauli worked out the Pauli paramagnetism which is involved in this effect, but Pauli never said anything about how that might relate to superconductivity.

A Fermi liquid should be a Landau fermion liquid.
The whole point is not Fermi statistics but the universality which Landau elucidated.

The Luttinger liquid should be a Haldane liquid.
Tomonaga and Luttinger introduced the Hamiltonian. Luttinger solved it incorrectly; Mattis and Lieb gave the correct solution. But, the important point is universality which is what Haldane elucidated, coining the phrase "Luttinger liquid".

Lebed magic angles in quasi-one-dimensional metals. Lebed made predictions about what would have happen for certain magnetic field directions. But, what is actually observed is quite different.

Yamaji angles for angle-dependent magnetoresistance oscillations in quasi-two-dimensional metals. Others (Kartsovnik, Kajita, ...) observed the effect experimentally. Yamaji's explanation is not the correct one because it involves quantised orbits, whereas the effect is semi-classical, as explained by Karstovnik and collaborators.

The Heisenberg limit in quantum measurements. It is certainly based on Heisenberg's uncertainty principle, but I am unaware of him discussing the limit referred to here.

Tuesday, September 6, 2011

Conference in a castle

On Saturday I am heading off to Germany to attend a workshop on Electronic structure of novel materials, organised by the Andersen group at the Max Planck Institute Stuttgart, and to be held in Ringberg Castle.

I will give a talk on Interlayer magnetoresistance as a probe of Fermi surface properties in strongly correlated metals. It will be an updated version of this earlier talk.

Deconstructing the Hall coefficient in strongly correlated metals

In basic solid state physics one learns that in metals sign of the Hall voltage is related to the sign of the charge carriers and the Hall coefficient is a measure of the density of the charge carriers, and is essentially temperature independent. However, in strongly correlated metals the situation is more complicated. Today I read a beautiful 1991 paper by N.P. Ong where he derives a simple geometrical interpretation of the transverse conductivity sigma_xy for a two-dimensional metal. It is proportional to the area swept out by the scattering length [Fermi velocity * scattering time] as one traverses the Fermi surface.

This result is based on a solution to the Boltzmann equation and so assumes a quasi-particle picture. Ong's result can be used to decipher the relative contributions from
  • the curvature of the Fermi surface
  • the Fermi surface area to circumference ratio
  • variation in the scattering rate at different points on the Fermi surface
The latter can be responsible for significant temperature dependence of the Hall coefficient and the Hall angle.
The figure above [taken from this review by Kontani] shows the temperature dependence of the Hall coefficient for cuprates at different dopings. The top and bottom are hole doped and electron doped respectively. The challenge for theory is to explain this strong doping and temperature dependence, which some claim goes beyond any Fermi liquid picture.

Monday, September 5, 2011

Towards research independence

I received a request:
"Can you talk about going from a research dependent (when your projects are fed to you) to someone able to write single author papers?" 
Here a few thoughts.
  • Get a good mentor. Ask them to help you with this.
  • Be aware of how committed the person paying your salary is to you making this transition.
  • Before and during any project (both those fed to you and ones you come up with) ask yourself, "Why am I doing this? Why is it important? What do I hope to discover? Is it realistic, particularly in the desired time frame? When and why might I abandon the project?"
  • Ask the same questions of your supervisor and/or collaborators.
  • Get in the habit of critically evaluating other peoples research (fellow students and postdocs, papers you read, talks you hear..). What is the basis for your positive and negative evaluations? 
  • Continually work on your writing skills.
Having said all that bear in mind that "independence" is not the be all and end all. It should be tempered with the following two points.

First, the ultimate goal is to do the best possible science. This usually involves collaboration. Different people have different strengths and weaknesses: choosing specific projects, putting together a collaboration, doing the actual calculations or experiments, writing code, making devices, writing the paper, getting the grants, promoting the work, connecting theory and experiment, ... 
A key is recognising your own strengths and weaknesses and working with people who complement you and appreciate your contribution.

Second, the importance of "independence" in career advancement varies significantly between different countries and institutions. Furthermore, I fear it is often evaluated in a fickle, highly subjective, and inconsistent manner. Hence, you may think you have become "independent" when you have published a single author paper; don't assume that others will see it that way. They may have completely different criteria or not even notice.

I welcome comments.

Friday, September 2, 2011

Effective Hamiltonians for transition metal compounds

Writing down the simplest possible effective Hamiltonian for a chemically and structurally complex material is a great challenge. This is particularly true of transition metal compounds because of the presence of multiple d-orbitals. Often crystal field effects split these orbitals and so one needs only consider a subset. Nevertheless, justify a selection and defining the relevant inter and intra-orbital interactions is difficult. Starting from the atomic limit one can consider the following Hamiltonian for a single atom [see this post about Hund's rule]
same orbital, and with opposite spins in the same orbital, maximizing S and then T.

This seems a lot simpler and transparent than forms I have seen with creation and annihilation operators. 
How does one determine U and J? Hotta has a helpful review where he discusses these in terms of Racah (Slater) parameters. He also discusses the overlap matrix elements associated with orbitals on neighbouring atoms. These are needed to determine a tight-binding band structure.

One should also be able to extract some of these parameters from related transition metal complexes studied by inorganic chemists.

Thursday, September 1, 2011

The role of theory in chemistry

I just read a nice article Theory in Chemistry by Roald Hoffmann. Although, written in 1974, I consider it just as poignant (or perhaps even more so) today. It should be read in full. But here are a few highlights to motivate you to read the 3 pages.
Is not the design and analysis of a beautiful experiment theory?
Given the human character, rationalisation poses no problem. Prediction is another matter. By and large, theory has not predicted much chemistry.
some exceptions.... I would call a credibility nexus - in which a group of experimentalists, otherwise skeptical of theory, suddenly found itself faced with the success of a simple theory. That set of specialists quickly became concerts, often zealots... 
the most important role of theory in chemistry is to provide a framework in which to think, to organize experimental knowledge
Hoffmann identifies six examples of "credibility nexus"
  • Huckel's rules for the stability of different conjugated organic molecules
  • Walsh's rules (1953) which showed how the geometry changes in excited states of polyatomic molecules could be derived from simple symmetry and bonding diagrams
  • Huckel theory descriptions of the spin distribution in anion radicals
  • crystal field theory for transition metal complexes which led to a renaissance of inorganic chemistry
  • critical phenomena
  • orbital symmetry conservation [for which Hoffmann later received the Nobel Prize]
I came across the article via a review article, "A Different Story of pi-Delocalization" by Shaik, Shurty, Davidovich, and Hiberty who claim that valence bond theory provides a  "credibility nexus".