Strong electronic correlations in cerium oxides

One of my students, Elvis Shoko has just submitted his Ph.D thesis, entitled "A minimal model Hamiltonian for strong electron correlations in cerium and its oxides".
Well done, Elvis!

We welcome any feedback.

Mac vs. PC

I am due to buy a new laptop and so am considering switching from a PC to a Mac (probably a 13 inch Macbook Pro). Over the past couple of years some of my colleagues have made this switch and are happy with it. I welcome other views. John Fjaerestad sent me this link which appears to have lots of useful info and practical tips for physicists who make the switch to Mac's.

Something I have heard enthusiastic reviews about is Papers, the program for organising ones files and files of downloaded papers.

Give and take in organometallic complexes

Following up on my previous post on pi-back-bonding in organometallic complexes I came across this nice paper which considers a complex which has such strong interaction between the LUMO on the organic ligand and one of the d-orbitals on the metal (actually the metal plus some other ligands) that the bond is almost covalent.

Above I reproduce the nice figure from the online abstract since it summarises the paper so nicely and unfortunately is not in the paper.

Measuring Planck's constant by thermodynamics

An incredibly important and useful equation is the Sackur-Tetrode equation which gives the absolute entropy of an ideal gas and shows how

-Planck's constant enters statistical mechanics in a way that is measurable
-to resolve the Gibbs paradox

I did not appreciate just how significant the equation was until I taught a statistical mechanics course.

The latest APS News has a beautiful article by Richard Williams on the equation in the This Month in Physics History column.

Did you know Tetrode as only 17 when he derived the equation from Boltzmann's equation (see below)!

The limited role of quantum effects in proton transfer in enzymes

I am just finishing up a paper which gives a detailed analysis of the possible role of quantum effects in proton transfer in enzymes. Above I reproduce a key figure from the paper (if you left click on it, you can view a larger version)

This follows a series of posts:
the first discussed the claims that proton transfer in enzymes occurs predominantly by quantum tunneling below the transition state associated with the transition state.
the second reviewed beautiful experiments by Doll and Finke which were inconsistent with Klinman's hypothesis that enzymes have evolved to promote tunneling, and
the third post reviewed Quantum transition state theory and how instantons provide a natural description of quantum tunneling.

The Figure shows that a quantitative description of the Doll and Finke data is possible without instantons, i.e., the only quantum effects are those associated with fluctuations around the transition state.

John Cleese on reductionism

On the lighter side here is a video of John Cleese discussing genetic determinism, quantum physics, reductionism, God, coconut ice cream, and the usual silly stuff...

Why do molecules absorb and emit light?

Today at the fortnightly "cake meeting" (a condensed matter theory group meeting) I am giving a tutorial on optical transition dipole moments. This key quantity determines:

• the probability that a photon will be absorbed by a molecule which then makes a transition to an electronic excited state
• the polarisation of the light that is absorbed and emitted by the molecule
• the rate at which the excited state decays by emitting radiation
• the strength of the van der Waals interaction between pairs of the molecule
• FRET=Forster Resonant Energy Transfer, where an excited state can be transferred to a neighbouring molecule

Here are some of my notes which contain the key equations and ideas.
Some of the relations and key references are in this paper I co-authored on Transition dipole strength of melanin.

What determines these transition dipole moments? How do we make them large?
A key aspect is that they involve delocalisation of charge in both the ground and excited state.
This becomes apparent in this extract from Nitzan's book which shows how the transition dipole moment is proportional to the distance between the two centres over which the charge is delocalised and the Hamiltonian matrix element between orbitals localised on these two centres.

The direction of the transition dipole moment tends to be parallel to a vector joining the two centres.

Moving beyond structural biology

One of the key ideas in biophysics is that structure determines property determines function. On the PHYS3170 blog one of the students, Alex (aka. ack!) had a profound observation about chapter 11 of Nelson. Here is my paraphrase/version/extension of her point.

Today both biology teaching and research is driven by a paradigm: first determine the biomolecular structure, then deduce the relevant properties of the structure, and then explain the function of the biomolecule.

However, historically this is NOT how biology has operated, and Nelson illustrates this nicely when considering the case of molecular ion pumps and the mitochondria. One starts with a knowledge of the biological function, e.g., energy production and distribution, and one then deduces what physical property the system must have (e.g., the ability to maintain a non-equilibrium concentration gradient of ions), and one then makes a hypothesis about what kind of structure is necessary to have this property (e.g., an ion pump embedded in the cell membrane wall).

In the 21st century biology is moving away from a preoccupation with molecular biology to systems biology. Since this involves emergent properties the actual details of biomolecular structures are less important than the collective properties that they have as they interact with one another.

Not the ghost in the machine, but the machine in the membrane

Here are a few highlights from Chapter 11 of Nelson's Biological Physics. It is entitled, "Machines in Membranes."

Indirect physical arguments led to the hypothesis of the existence of the active ion pumps, long before the biochemical identity of these amazing molecular machines was known.
This is analogous to how the existence of a molecular carrier of genetic information (DNA) was proposed to exist long before its chemical or physical structure was known.

Biological question:
The internal and external chemical composition of cells is very different. Why doesn't osmotic flow burst (or shrink) the cell?

Physical idea:
This nonequilibrium, osmotically regulated state can be maintained by active ion pumps located in the cell walls.

The invention of batteries (i.e., voltaic cells) was stimulated by Volta's skepticism towards Galvani's claim that muscles could be a source of electricity.

The Nernst potential is the voltage that must be applied across a membrane to maintain a concentration gradient of a particular ion species across the membrane.
All animal cells have a sodium anomaly, i.e., the Nernst potential for sodium is much more positive than the actual membrane potential, i.e., the ion concentrations deviate significantly from equilibrium values.

This non-equilibrium is maintained by an ion current across the membrane which is proportional to the conductance per unit area of the membrane.

Ion pumps are embedded in cell membranes and hydrolyze ATP to obtain the free energy they use to pump sodium ions out of the cell.

Consider the industrial factory below. It generates, distributes, and utilizes energy.


Mitochondrion are like factories, systems of coupled machines. See the figure below.
They act as bus bars to generate, distribute, and utilize energy in a cell. The chemiosmotic mechanism proposes that ATP synthesis is indirectly coupled to respiration:

NADH + H+ 1/2 O2 -> NAD+ + H2O

Note that proton transfer plays a key role here.

Diamonds are a quantum nerds best friend

The past few years have seen significant advances in using NV (Nitrogen Vacancy) centres in diamond as qubits. A helpful review by Wrachtrup and Jelezko is here.
Recently, a group in Stuttgart reported produced a variety of two-particle and three-particle entangled states. They produced Bell states between the nuclear spins of two 13C nuclei associated with the NV centre. They also entangled two electron spin states with the two nuclear spin states to produced GHZ and W states, which have maximum three-particle entanglement.

The energy level structure is shown below.

It is interesting that the ground state of the NV centre is an electron spin triplet. Why?

Here is a possible simple argument to understand the essential physics behind the energy level structure and quantum numbers.
This is inspired by the figure below taken from a paper Jaime Merino, Ben Powell, and I wrote on a completely different topic, sodium cobaltate.

The NV centre has C3v symmetry and there are 4 electrons associated with it, one from each of the carbon atoms and one from the nitrogen atom.

A minimal model for the electronic structure is to take the three degenerate sp3 carbon orbitals and one nitrogen orbital at a lower energy, all directed towards the vacancy. A minimal Hamiltonian is a 3 site Hubbard model.
If t is positive and there are 4 electrons in the carbon orbitals (or t negative and there are 2 electrons) then the ground state is a spin triplet with spatial A symmetry. The blue lines represent spin triplet pairs.

The electronic state shown in the figure has a significant amount of entanglement. It is a resonating triplet bond state. But, this is not the entanglement that is created and manipulated in the experiment.

In passing I also mention that the singlet version of this figure was helpful to me for recent work Seth Olsen and I did in our recent papers on an effective Hamiltonian for flourescent protein chromophores and methine dyes.

I thank Professor Noel Hush for stimulating this post.

How do migratory birds navigate?

Do they have some magnetic "biochemical" compass?
Maybe. Nice reviews in PNAS and Physics Today discuss the status of this question.

Here a few of the demanding physical requirements for this hypothesis of "biochemical magnetoreception":

• There is biochemical reaction similar to that shown below.

• The relative yield of C depends of the direction of magnetic fields of the order of 50 microtesla.
• Note that the corresponding electron Zeeman energies are more than 6 orders of magnitude smaller than the thermal energy, k_BT!
• The relevant competing reactions must occur on the same timescale as the Larmor precession of the electron spins associated with the 50 microtesla field. (1 microsecond)
• The relevant biomolecules must be aligned predominantly in one direction so the relevant hyperfine tensor has a well defined direction relative to the terrestial magnetic field.

What is the source of random magnetic fields that could destroy the coherence of the spin quantum states?
Spin isotropic interactions such as those associated with the dielectric fluctuations or exchange interactions with the environment cannot produce such interactions.
What about decoherence due to random nuclear spins?
Experimental studies have quantified this for qubits based on quantum dots.

Donating electrons back to ....

What are the dominant orbitals and interactions in organometallic complexes?

Chapter 3 of the book, Molecular Orbitals of Transition Metal Complexes, by Yves Jean, [which has a forward by Roald Hoffmann] has a nice discussion of the pi orbitals in organic molecules with the d-electrons on the transition metal, pi back-bonding and the Dewar-Chatt-Duncanson model.


In the left figure above shows a pi orbital of ethylene with a sigma bond interaction with a d orbital. This leads to donation of electron charge from the ligand to the metal.

The right figure shows the overap of a pi* (anti-bonding) orbital with a metal d-orbital. This leads to back-bonding and charge transfer from the metal to the ligand. Experimental signatures of this interaction include:

• an increase in the length of the C-C bond in the ligand
• a softening of the ligand C-C stretch frequency
• substantial spectral weight associated with a metal-to-ligand-charge-transfer (MLCT) optical transition
• basicity of the ligand (i.e., its ability to receive protons) [remember acids like to give up protons] increases by orders of magnitude.
  

The second last point is nicely described in papers by Zwickel and Creutz and Reimers and Hush. They show that the transition dipole moment for the MLCT transtion is proportional to the hybridisation parameter (Huckel or hopping parameter) between the metal d and the ligand pi* orbital.

Ratcheting up my understanding

Something I have never quite understood has been discussions of ratchets which are driven by thermal fluctuations. Today in PHYS3170 we discussed these in the context of molecular motors. In Chapter 10 of Biological Physics, Nelson considers:

Biological question: How does a molecular motor convert chemical energy, a scalar quantity, into directed motion, a vector?

Physical idea: Mechanochemical cuoupling arises from a free energy landscape with a direction set by the geometry of the motor and its track. The motor executes a biased random walk on this landscape.

The figure below shows a protein (i.e., chain of amino acids) being irreversibly dragged to the right through a membrane.
Nelson considers the mechanical model below.
[We thought the S-ratchet and G-ratchet were from some profound nomenclature. But they are G&S = Gilbert and Sullivan!, who Nelson often uses for Socratic dialogues.]

This can be described by the potential energy curve below. On top of this random thermal motion (i.e, Brownian motion) occurs. One can see that this will lead to a nett motion to the right because the random thermal motion leads to small reversible displacements, except near the steps.
I did a web search for a simulation (e.g., a Java applet) of such a ratchet but could not find anything. Please let me know if you are aware of something like that.

Should experimentalists have to "explain" their data?

Sometimes when I talk to experimentalists about their latest results they say something like, "We have this really interesting data. But we can't explain it and so we are not going to publish it until we can explain it." No doubt this is sometimes what referees say.
But, I disagree. The most interesting experimental results have no clear explaination!
Furthermore, many of the "explainations" I read in experimental papers seem to be either naive, wrong, or highly speculative.

The point of non-radiative decay

The rate and efficiency of many photophysical processes is determined by conical intersections (points or more correctly seams) where two potential energy surfaces touch.
Much effort has been expended in the computational chemistry community in developing methods to simulate the dynamics of vibrational wave packets moving through such conical intersections. These methods have been used to simulate a number of important photophysical processes in condensed phases (e.g., the isomerisation of retinal in the protein rhodopsin).

I have often wondered what are the key energy, length, and time scales in this problem?

While in Toronto, Valentyn Prokhorenko, brought to my attention a really nice PRL that answers my questions. Unfortunately, this paper does not seem to have drawn the attention that it deserves.

The authors define a semi-classical expansion parameter g which is essentially the ratio of the particle deBroglie wavelength to the scattering length associated with the conical intersection. They estimate that for typical photochemical processes this parameter will be small, justifying the semi-classical approximation.


The probability of a wave packet to remain on the diabatic surface it starts on is shown to be
Pd = exp (-pi alpha^2/2)
where alpha is a dimensionless parameter proportional to the ratio of the impact parameter a of the wave packet and the scattering length rs.

A few questions I have are:

How do these results change with the shape of the conical intersection? e.g., if it is tilted?

How do these results relate to earlier work of Teller and Nikitin and more recently Malhado & Hynes?

How does decoherence change these results?

Theoretical chemistry at UW

Today I am giving a talk to the theory group in Chemistry at University of Washington.

I am looking forward to meeting Oleg Prezdho. I am currently learning a lot from reading a very nice review article he wrote with W.R. Duncan on modelling the ultrafast electron transfer (femtoseconds) from organic molecules to titanium dioxide. This is a key process in Gratzel solar cells.

A new class of chemical bonds

Shason Shaik and collaborators have a nice article in the latest Nature Chemistry, "charge-shift" bonding which is the essential component of the bonding of pairs of very electronegative identical atoms (e.g., flourine).

The essential physics of covalent bonding can be described by a two-site Hubbard model (with U>>t).

Can a negative U (or V>U) two-site Hubbard model describe the essential physics of charge-shift bonding?

The power of thermodynamics

Chapter 6 of Nelson's Biological Physics ends with a nice extract from a 2001 Science paper describing optical tweezer experiments which reversibly unfold single RNA molecules.

Figure b above shows how the amount of unfolded RNA as a function of applied force can be fit simply that predicted for a two-state equilibrium distribution (which has the same mathematical form as a Fermi-Dirac distribution!) The fit also gives a good estimate of the free energy of unfolding (about 80 k_B T at room temperature).

Nelson also gives simple arguments that give the probability distribution P(t) for the different waiting times t for folding and unfolding, P(t)=kexp(-kt) where k is the rate. The right panel of c. above shows the RNA length as a function of time for different applied forces. The resulting range of distribution times (note they are the order of seconds) are shown in panel d.

What is real new and cool here is observing all this for single molecules. Decades ago people found that the two-state model could describe the folding and unfolding of many biomolecules.



Here are the slides for a lecture I have given previously on how this two-state equilibria can quantitatively describe the unzipping of DNA and unfolding of proteins. The right panel figure above is just one example of a measurement the amount of unfolded molecule as function of temperature. It can be fit to the same two-state distribution.

As Nelson note, it is amazing and impressive that such incredibly simple expressions based on simple thermodynamic considerations can describe such complex systems and processes.

Informative section headings

As I posted before, I really like the way that Phil Nelson uses informative section headings in his book, Biological Physics: Energy, Information, and Life. More of us need to do this in papers we write. Specifically, we need to go beyond Introduction, Methods, Results, Discussion, and Conclusion!

Here are some headings from the chapter I am currently reading:

6.2.2 Entropy is a constant times the maximal value of disorder

6.4.1 Entropy increases spontaneously when a constraint is removed

6.5.1 The free energy of a subsystem reflects the competition between entropy and energy

6.5.2 Entropic forces can be expressed as derivatives of the free energy

6.5.3 Free energy transduction is most efficient when it proceeds in small, controplled steps

6.6.3 The minimum free energy principle also applies to microspcopic subsystems

6.6.4 The free energy determines the populations of complex two-state systems

Here is a challenge for you. How many of these can you prove without looking at the book?

A chemist, physicist, and mathematician.....

Since I am on holidays this week, all I have to offer is a joke I heard last week in Toronto. Apparently, David Tannor told it the week before at a Gordon Conference on Quantum Control. Here is my version:

A chemist, a physicist, and a pure mathematician all check into the same hotel. Unfortunately, a fire breaks out in each of their rooms in the middle of the night. The chemist wakes up rushes into the bathroom fills the bath with water and uses a wastepaper basket to rapidly bail water onto the fire. He quickly extinguishes the fire and goes back to bed.

The physicist wakes up sees the fire, looks at the thermometer on the wall, and does an order of magnitude estimate of the fire temperature and how fast it is spreading. He sits down at the desk and does a couple of quick calculations. He estimates it will require 3 liters of water to extinguish the fire. He measures out this water and pours it on the fire. He is very pleased that his estimate was correct. He goes back to sleep content that he has solved the problem.

The pure mathematician wakes up, sees the fire, and realises there is a problem to be solved. He sits at the desk in profound thought and then writes out some careful notes, ending with QED. He says to himself, "As I thought, there is a solution, but it is not necessarily unique." He goes back to bed content he has solved the problem.