Finding the twin state for hydrogen bonding in malonaldehyde

I was quite excited when I saw the picture above when I recently visited Susanta Mahapatra.

One of the key predictions of the diabatic state picture of hydrogen bonding is that there should be an excited electronic state (a twin state) which is the "anti-bonding" combination of the two diabatic states associated with the ground state H-bond.
Recently, I posted about how this state is seen in quantum chemistry calculations for the Zundel cation.

The figure above is taken from
Optimal initiation of electronic excited state mediated intramolecular H-transfer in malonaldehyde by UV-laser pulses 
K. R. Nandipati, H. Singh, S. Nagaprasad Reddy, K. A. Kumar, S. Mahapatra

The figure hinted to me that for malonaldehyde the twin state is the S2 excited state, because of the valence bond pictures shown at the bottom of the figure and because the shape of the two potential energy curves is similar to that given by the diabatic state model.
Below I have plotted the curves for a donor-acceptor distance of R=2.5 Angstroms, comparable to that in malonaldehyde.

The vertical scale is such that D=120 kcal/mol, leading to an energy gap between the ground and excited state of about 4 eV, comparable to that in the top figure [which is in atomic units, where Energy = 1 Hartree = 27.2 eV].
Note that in the first figure, there is a gap on the vertical scale and the top and bottom part of the figures involve a different linear scale.
Hence, to make a more meaningful comparison I took the potential energy curves and replotted them on a linear scale using the polynomial fits given in the paper. The results are below.

There is reasonable agreement, but only at the semi-quantitative level.
[With regard to units the distances are in atomic units (Bohr radius = 0.5 Angstroms] and comparable].

The figure below shows how the calculated transition dipole moment between the ground state and the first excited state. I was surprised that it only varied by about five per cent with change in nuclear geometry.
However, Seth Olsen pointed out to me that this small variation reflects the Franck- Condon approximation [which is very important and robust in molecular spectroscopy].

I also calculated this with the diabatic state model, assuming that the dipole moment of diabatic states did not vary with geometry and the Mulliken-Hush diabaticity condition [that the dipole operator is diagonal in the diabatic state basis] held (see Nitzan for an extensive discussion).
[The units here are 1 atomic unit = 2.6 Debye].

The vertical scale here is the magnitude of the dipole moment in one of the diabatic states. This is also the approximate value of the dipole moment of the molecule at "high" temperatures above which there is no coherent tunnelling between the two isomers, i.e. the proton is localised on the left or right side of the value. The experimental value reported here is about 2.6 Debye.
This is consistent with the claim that the diabatic state model gets the essential physics of the electronic transition.

But, of course the real molecule is more complex. For example, between the S0 and S1 states there is a "dark" S1 state, characterised as a n to pi* transition where n is the lone pair orbital on the oxygen. The three states and their associated conical intersections are discussed in a nice paper by Joshua Coe and Todd Martinez.

I think a good way to more rigorously test/establish/disprove the diabatic state picture is to use one of the "unbiased" recipes to construct diabatic states, such as that discussed here, from high-level computational chemistry.

New coherent spectroscopies

Shaul Mukamel (UC Irvine) gave the last talk at conference, Coherent nonlinear optical spectroscopy of biological complexes: from nmr to x-rays. It contained several really new and stimulating ideas. A nice overview is in this recent Accounts of Chemical Research paper.

Heterodyne-detected four wave mixing produces a signal S(t1,t2,t3).
There are three time differences between the four pulses t1, t2, and t3.

Experiments will soon be done in UV and X-ray part of the spectrum.

Joke: Translating Othello into Yiddish with great improvement!

Two-dimensional correlation peaks S(omega1,t2,omega3) gives a direct handle on coupling of modes, and on diagonal fluctuations and noise from lineshapes.

Recent experiments on Helical J-aggregrates reveal the pattern of energy flow.

There are TWO alternative and equivalent theoretical descriptions of these experiments. Eigenstate vs. Quasiparticle (oscillator) description of nonlinear response of excitons.

Same effective Hamiltonian for excitons, and several other systems.
H = site energies + hopping +
two-particle interactions (K coupling is a measure of anharmonicity).

Schrodinger picture. Eigenstates.
Third-order measurements probe single and double excitons.
Feynman diagrams and associated rules greatly help visualisation of the essential physics and aid derivation of equations.

Alternative approach: quasi-particles. Write Heisenberg equations of motion.
If they are linear (because there are no interactions) there is no non-linear spectroscopy. Add bath and solve equations of motion. Response function looks very different from the eigenstate picture. How do we relate the two?

Bethe-Saltpeter equation.

Oscillator picture has advantage numerically since it scales with N, whereas the eigenstate picture scales with N^2. The latter also involves massive cancellations of terms which obscures the physics.

For the FMO complex 7 excitons, 21 double exciton states.

New idea for new experiments. 1.
Measure Double quantum coherence. S(t3,omega2,omega1) shows a picture of which single excitons contribute to the two-exciton wavefunction. This is sensitive to electron correlations.
Greg Scholes has done the first experiment. Photon echo technique is not sensitive to these correlations.

Derive Lindblad equation from a modified Redfield equation.
There are features in some of Flemings spectra which are evidence for 3 exciton correlations.

New idea for new experiments. 2.
Non-linear spectroscopy with entangled pairs of photons.
Possibility of selecting between pathways/diagrams. Different scaling with intensity. (seen in Boston experiments). Control entanglement time can be varied with femtosecond resolution.
Provides a new spectroscopic tool which will work at low intensities.

Quantum efficiency in photosynthetic systems

Josef Wachtveil (Frankfurt) gave a nice talk on quantum efficiency in photosynthetic systems. Here are my rough notes.

First a joke.
Quantum biology?
Moles tunnel into the classically forbidden region. But they make holes everywhere. They are delocalised!
[John Briggs asked if there was an entanglement measure of entropy per mole!].

Arrangement of electron carriers has a common structural motif in many proteins.

Optimisation principle. Change pigment and reduce quantum efficiency.

Electronic coupling between molecules is sensitive to orientation and relative direction of two molecules. Is it optimised?

There appears to be no functional role of the observed vibrational coherence associated with electron transfer.

Non-photochemical quenching (NPQ) in green plants. How does plant deal with excess photon flux? Triplet chlorophyl excitations can produce highly reactive singlet oxygen. Xanthophylll cycle induced by high light or change in pH.

"gear-shift" model as a mechanism for NPQ. Increase no. of conjugated bonds lowers excited state energy can act as an energy dump.

Quantum chemistry (DFT) calculations suggested that charge transfer is a potential quenching mechanism. Fleming group (Science 2005) tested this. Peak is at 1000 nm.

Artificial systems
Beyond vibrationally induced electron transfer in Gratzel cells and CdSe quantum dots coupled with methyl viogen. (Multi-excitons can be produced by a single photons).
Claims to see ballistic wave packet motion in CdSe nanocrystal and it is at 200 cm-1 which is LO phonon frequency.

Simulation and visualisation of quantum dynamics

ack, a student in PHYS3170 brought to the class attention the site for the WAVEPACKET simulation project. It has some really nice animations of quantum dynamics on potential energy surfaces and non-adiabatic dynamics near a conical intersection.

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.

Is optimal control quantum, semi-classical, or classical?

I had a great meeting with Paul Brumer today, where he answered many of the questions I posted previously about quantum control. Here are just a few of the things I learnt:

In a 2005 PRL, Paul and Hoki did a careful analysis of experiments on the optimum pulses for photo-isomerisation of the dye NK88 in methanol. They found that the optimum pulses corresponded to an incoherent pump-dump scenario and that quantum interference effects were absent.

In a J. Chem. Phys. papers in 2005 and 2006 Christopher, Shapiro, and Brumer considered the pulse sequences needed to optimise internal conversion of S2 to S2 in pyrazine by considering the full time evolution of an effective Hamiltonian for these two states taking into account 24 vibrational modes. They found "active control over internal conversion so as to almost completely suppress the process over time scales of ~50–100 fs [well in excess of the natural internal conversion times (~20 fs)] or to accelerate it to complete internal conversion in less than 5 fs". One thing I want to understand better is how this relates to a conical intersection picture for internal conversion.

A recent experimental paper in PNAS found quantum coherence did not play a significant role in the isomerisation of retinal in bacteriorhodopsin, presenting a different view from a 2006 Science paper from Miller's group in Toronto.

Trying to disentangle my incoherent thoughts

Here a few notes and comments on some of todays talks on the Conference on Quantum Information and Quantum Control in Toronto.

Andrew White (University of Queensland)
Quantum Chemistry on a Quantum Computer: First Steps and Prospects

He showed some nices pictures of potential energy surfaces. In passing I mention that John Polanyi, a long-time faculty member in Chemistry at U. Toronto who shared a Nobel Prize in 1986 for illuminating the significance of such surfaces for reaction kinetics.

Essentially the work described seems to be diagonalising a 2x2 matrix on a quantum computer (by the phase estimation algorithm). It was not clear from the talk how the matrix elements in this matrix were evaluated since they involve performing various real space integrals (i.e., matrix elements) of the real space Hamiltonian. Practical quantum chemists would say that evaluating such integrals is an essential part of a real calculation. Even disregarding this issue calculations with more realistic basis sets will require many more qubits. Hence, I wonder if a better direction for such simulations of quantum systems is to focus on simulating systems with small Hilbert spaces interacting with an environment. The simplest such model Hamiltonian would be the spin-boson model. Simulating the quantum dynamics of this on classical computer is a real challenge but in a quantum computer simulation one could have the significant advantage that an artiificial source of decoherence would have few cost overheads...

Shohini Ghose (Wilfrid Laurier University)
Entanglement and nonlocality in multiqubit pure states

This is based on a recent PRL.

Consider pairs of qubits in a pure state. Then the states are entangled if and only if they violate Bell-type inequalities. [If the state is mixed then Werner showed there exist states which are entangled but not violate Bell].

For 3 qubits one can uses the 3 particle tangle (introduced by Coffman, Kundu, and Wootters) to quantify entanglement and there is an inequality due to Svetlichny which is the 3-qubit generalisation of Bell-CSCH inequalities.

A few really striking aspects of the results presented
(they are "counter-intuitive" because they are different to what occurs in the 2 qubit case) :

• There exist tripartite entangled states that do not violate the Svetlichny inequality.
• The tangle is not a smooth function of the state coefficients.

Talking to Shohini afterwards she drew my attention to a paper by Cai et al. which introduces a measure for true 2N-particle entanglement. I am particularly interested in this because I want to quantify the amount of entanglement in the resonating valence bond state of benzene.

Chris Monroe (University of Maryland)
Quantum Networks with Ions, Phonons, and Photons

Chris showed how to simulate a 3 spin Ising model in a transverse field. The spin-spin interaction is mediated by phonons (i.e., relative motion of the ions). Given the state of ion trap technology extending this to as many as 10 spins. This should make possible some of the simulations that Gerard Milburn and I considered several years ago with a student John Paul Barjaktarevic and described here. These have the significant advantage that one does not have to worry about Trotter decomposition.

well its midnight in Toronto and 2pm in brisbane.... time to go back to bed and try and get over the jet lag....

Getting more out of quantum control

A problem is that one does not know what is the Hamiltonian of the system (this is not just true for a system as complex as a protein but even for a small organic molecule in the gas phase) is and so a priori one cannot predict what the optimal laser pulse sequence will be to steer the photochemical reaction. This might make the problem seem hopeless but in 1992 Judson and Rabitz proposed a very clever solution: to optimise the pulse sequence using a learning algorithm that iteratively improves the control scheme. This has now successfully been implemented by many experimental groups. A comprehensive review of what has been done in condensed phases is here.

Hence, the optimal (and anti-optimal) pulse sequence contains a significant amount of information about the system. I wonder is there a way to convert this into information about the Hamiltonian of the system? For example, details of the ground state and excited state potential energy surfaces. For example, for a reaction which passes through a conical intersection surely the optimal pulse sequence defines a wave packet at the Franck-Condon point on the excited state surface with a momentum that points in the direction towards the conical intersection, i.e., it tells us exactly which vibrational modes comprise the "reaction co-ordinate". The wave packet shape and speed may be optimised to minimise intersystem crossing at the conical intersection.

The figure below is taken from a paper by Hunt and Robb shows the relevant potential energy surfaces for photoisomerisation of a model cynanine dye.

Some questions about quantum control

I am struggling my way through trying to figure out what quantum control is really about and how quantum it is. On the plane to Toronto I read some of the book by Shapiro and Brumer. A few questions I have include:

When can quantum control be semi-classical?

Why is it possible to perform quantum control in condensed
phase systems at room temperature?

How much decoherence is needed to destroy it?

The basic idea of quantum control is to use multiple coherent laser pulses to create an initial state which is a coherent superposition of several quantum states. This state then evolves in a manner where the final products (branching ratios) depend on the relative phase and amplitude of the initial laser pulses.

One example is the Tannor-Rice scheme for the reaction
A+BC -> AB + C
which is shown in the figure below.

By the Franck-Condon principle absorption of a photon by the system in the electronic and vibrational ground state will produce a Gaussian vibrational wavepacket in the excited electronic state.
However, if one applies the relevant pulse sequence one can produce an excited state which has a vibrational wave packet with a net momemtum to the left or the right.

A similar issue arises with photo-isomerisation. Usually this occurs via a conical intersection between two potential energy surfaces.
One should be able to enhance the photo-isomerisation yield by producing an excited state in which the vibrational wave packet is a coherent state with a momemtum pointing towards the conical intersection.