Thursday, September 12, 2013
On persistent blogging, mental health, and a lesson from finite-time thermodynamics
Every once in a while, one finds in chemical physics a lesson for life. I have found one. The lesson comes from the field of finite-time thermodynamics. The lesson is simple:
Maximum power and minimum entropy production are incompatible under time constraint.
I found this reviewto be an excellent reference on the origin and foundations of this important physical principle.
It is clear to me that this principle is one of those that is always in your face -- so close, in fact that it is possible to miss it. Still, it is always there. Every time I am too rushed to put something back, it is there. Every time I get a reminder that, in the course of herding kids, I have failed to put my clothes in the hamper, it is there.
Indeed, it is all over this blog. Why did I start it? I have not contributed much, my last entry being in 2010 after the publication of my first sole-author paper, which I suppose was a momentous enough occasion. The analogy goes like this: blogging clarifies my own mental state, reducing its entropy. For some time now, I have been focussing on going hard and fast, and have not been concerned about the accumulating entropy.
All of a sudden, I am finding that my engine is wearing down. Maybe its time to think about entropy production. The last few years have included several significant knocks to the frame, including but not limited to having three kids in a span of less than four years.
Time to start blogging again?
Tuesday, April 13, 2010
Simplifying insight into the color of fluorescent protein chromophores
An article I recently wrote on the color of fluorescent protein chromophores is now officially in print and citable: Olsen. A Modified Resonance-Theoretic Framework for Structure−Property Relationships in a Halochromic Oxonol Dye. Journal of Chemical Theory and Computation (2010) vol. 6 (4) pp. 1089-1103.
In this paper, I show that the excitation energies of 29 dyes, formed by pairing different protonation states of a phenoxyl and imidazoloxyl group through a monomethine bridge, can be reproduced without loss of accuracy by a model which uses only two parameters per distinct terminal group. The set includes six distinct protonation states of the green fluorescent protein (GFP) chromophore.
The theoretical basis for the model goes back more than fifty years, and hinges on two key works by Brooker and by Platt. Both of these authors examined the color of methine cyanine dyes through a heuristic color theory based on the resonance of Lewis structures.
In the article, I make the case that the state space implicitly invoked in the resonance theory (specifically, a resonating pair of canonical Lewis structures) is actually not appropriate to describe the information content of the Brooker-Platt model. A straightforward interpretation of the Lewis structures as products of perfect bond-pairs over atomic orbitals contains too much information (relative to the information content implied by the Brooker Deviation Rule). Instead, I advocate a 3-orbital "fragment frontier orbital valence-bond" approach. The Lewis structure way of writing the resonance for the GFP chromophore motif would normally be:
Whereas I suggest a related but slightly different, less detailed ansatz, which averages over some of the structure on the rings themselves. The new ansatz uses three ring-localized orbitals:
When four electrons are distributed in these orbitals, six configurations arise:
There are a few arguments that lead to the 3-orbital model. Some of these are discussed in an upcoming Chem. Phys. Lett paper, which will shortly go to press. Another argument, that hasn't been published yet, is based purely on an information-theoretic "dimensional analysis" argument.
The argument goes as follows:
The classical resonance theory invokes two-dimensional state space, in the form of a resonating pair of canonical Lewis structures. In classical theories, Lewis structures are usually non-orthogonal (since the atomic orbitals in classical valence-bond theories are non-orthogonal).
There are multiple ways to orthogonalize two vectors. Many of these (e.g. Gram-Schmidt or Löwdin orthogonalization) map two non-orthogonal vectors onto two orthogonal vectors that span the same plane. However, there is another way to orthogonalize vectors, which maps the two non-orthogonal vectors into two orthogonal vectors in a THREE-dimensional space. This is analogous to Naimark's theorem in quantum information, which says you can always purify a mixed state by extending the Hilbert space with an appropriate "ancilla".
Applying the latter orthogonalization procedure to the canonical resonance theory suggests that the state space should be extended to a space spanned by three Lewis structures.
Now, note that there is much more information in a Lewis structure than is needed, because all of the bonds in the resonating pair change AT ONCE. This implies that each structure could be described by a contracted state with exactly ONE "perfect pair" worth of information (a perfect pair is a two-electron function, which is a superposition of a "covalent" and an "ionic" determinant).
This suggests that a more appropriate model space for the resonance theory would be spanned by three orthogonal pair states. This is isomorphic to a space spanned by a four-electron-in-three-orbital complete active space (CAS(4,3)).
The article I mention in the beginning shows that there is a self-consistent CAS(4,3) solution for the two lowest-lying electronic states of the green fluorescent protein chromophore, and explores the interesting consequences of this.
In this paper, I show that the excitation energies of 29 dyes, formed by pairing different protonation states of a phenoxyl and imidazoloxyl group through a monomethine bridge, can be reproduced without loss of accuracy by a model which uses only two parameters per distinct terminal group. The set includes six distinct protonation states of the green fluorescent protein (GFP) chromophore.
The theoretical basis for the model goes back more than fifty years, and hinges on two key works by Brooker and by Platt. Both of these authors examined the color of methine cyanine dyes through a heuristic color theory based on the resonance of Lewis structures.
In the article, I make the case that the state space implicitly invoked in the resonance theory (specifically, a resonating pair of canonical Lewis structures) is actually not appropriate to describe the information content of the Brooker-Platt model. A straightforward interpretation of the Lewis structures as products of perfect bond-pairs over atomic orbitals contains too much information (relative to the information content implied by the Brooker Deviation Rule). Instead, I advocate a 3-orbital "fragment frontier orbital valence-bond" approach. The Lewis structure way of writing the resonance for the GFP chromophore motif would normally be:
Whereas I suggest a related but slightly different, less detailed ansatz, which averages over some of the structure on the rings themselves. The new ansatz uses three ring-localized orbitals:
When four electrons are distributed in these orbitals, six configurations arise:
There are a few arguments that lead to the 3-orbital model. Some of these are discussed in an upcoming Chem. Phys. Lett paper, which will shortly go to press. Another argument, that hasn't been published yet, is based purely on an information-theoretic "dimensional analysis" argument.
The argument goes as follows:
The classical resonance theory invokes two-dimensional state space, in the form of a resonating pair of canonical Lewis structures. In classical theories, Lewis structures are usually non-orthogonal (since the atomic orbitals in classical valence-bond theories are non-orthogonal).
There are multiple ways to orthogonalize two vectors. Many of these (e.g. Gram-Schmidt or Löwdin orthogonalization) map two non-orthogonal vectors onto two orthogonal vectors that span the same plane. However, there is another way to orthogonalize vectors, which maps the two non-orthogonal vectors into two orthogonal vectors in a THREE-dimensional space. This is analogous to Naimark's theorem in quantum information, which says you can always purify a mixed state by extending the Hilbert space with an appropriate "ancilla".
Applying the latter orthogonalization procedure to the canonical resonance theory suggests that the state space should be extended to a space spanned by three Lewis structures.
Now, note that there is much more information in a Lewis structure than is needed, because all of the bonds in the resonating pair change AT ONCE. This implies that each structure could be described by a contracted state with exactly ONE "perfect pair" worth of information (a perfect pair is a two-electron function, which is a superposition of a "covalent" and an "ionic" determinant).
This suggests that a more appropriate model space for the resonance theory would be spanned by three orthogonal pair states. This is isomorphic to a space spanned by a four-electron-in-three-orbital complete active space (CAS(4,3)).
The article I mention in the beginning shows that there is a self-consistent CAS(4,3) solution for the two lowest-lying electronic states of the green fluorescent protein chromophore, and explores the interesting consequences of this.
Wednesday, January 13, 2010
Entropy and the Reaction Barrier
Proceeding with my goal to read the collected works of Raphael Levine, I recently came across a 1977 paper that he wrote with Noam Agmon, entitled, "Energy, Entropy and the Reaction Coordinate: Thermodynamic-Like Relations in Chemical Kinetics".
I was blown away.
The paper explains a series of empirical conjectures, collectively called "Free Energy Relationships (FERs)". FERs relate the (kinetic) rate of a reaction to its (thermodynamic) driving force. As such, they are thermodynamic-like rules for kinetic behavior. Note that this is strange! Thermodynamics is usually formulated for systems at equilibrium. These are unchanging, by definition, hence there is no rate!
What REALLY impressed me about this work was that they manage to explain FERs without any quantum mechanics at all! Instead, they use a very novel and simple statistical/information-theoretic model.
Agmon and Levine describe a model of a reaction where the progress of the reaction is parameterized by a number n between 0 and 1. This is a point in a 1-dimensional convex set of numbers (i.e. a closed line segment). Because the set is convex, there is a probability interpretation. They define an energy function that has a linear component, and introduce a barrier by adding a term that is just the Shannon entropy of mixing times a generalized "temperature". The entropy creates a barrier, and they proceed to show that this barrier follows the rules expected of LFERs as well as rules relating transition state position to exo/endoergicity (versions of "Hammond's Postulate").
Note that there is nothing quantum here, just probabilities! The convex set was just numbers, not density matrices.
The main idea that I took away is that FER's are more general than one might think (if one was trained as a "traditional" quantum chemist). They don't depend, for example, on details like how a transition state is defined for a molecule (or a series of molecules with similar shape).
I have been contrasting this in my head with another paper by Sason Shaik (another Israeli chemist of who I am a fan) and AC Reddy. It is not clear to me just yet how to bring the two pictures together, since the Shaik model of crossing VB surfaces intrinsically invokes two energies at the ends of the reaction coordinate (one for each surface). The Shaik/Reddy picture ultimately produces two potential surfaces with a diabatic interpretation. Also, it is clearly wavefunction-based. The Agmon/Levine model cannot be wavefunction based because the Hilbert space is not a convex set. It could be generalized to density matrices, but then it is not immediately clear how to discuss the "twin state" invoked by Shaik & Reddy. Since the states in the Shaik/Reddy picture are VB states and may be non-orthogonal, the definition of an appropriate mixing entropy is also not clear (see, for example, a paper by Dieks and van Dijk).
I was blown away.
The paper explains a series of empirical conjectures, collectively called "Free Energy Relationships (FERs)". FERs relate the (kinetic) rate of a reaction to its (thermodynamic) driving force. As such, they are thermodynamic-like rules for kinetic behavior. Note that this is strange! Thermodynamics is usually formulated for systems at equilibrium. These are unchanging, by definition, hence there is no rate!
What REALLY impressed me about this work was that they manage to explain FERs without any quantum mechanics at all! Instead, they use a very novel and simple statistical/information-theoretic model.
Agmon and Levine describe a model of a reaction where the progress of the reaction is parameterized by a number n between 0 and 1. This is a point in a 1-dimensional convex set of numbers (i.e. a closed line segment). Because the set is convex, there is a probability interpretation. They define an energy function that has a linear component, and introduce a barrier by adding a term that is just the Shannon entropy of mixing times a generalized "temperature". The entropy creates a barrier, and they proceed to show that this barrier follows the rules expected of LFERs as well as rules relating transition state position to exo/endoergicity (versions of "Hammond's Postulate").
Note that there is nothing quantum here, just probabilities! The convex set was just numbers, not density matrices.
The main idea that I took away is that FER's are more general than one might think (if one was trained as a "traditional" quantum chemist). They don't depend, for example, on details like how a transition state is defined for a molecule (or a series of molecules with similar shape).
I have been contrasting this in my head with another paper by Sason Shaik (another Israeli chemist of who I am a fan) and AC Reddy. It is not clear to me just yet how to bring the two pictures together, since the Shaik model of crossing VB surfaces intrinsically invokes two energies at the ends of the reaction coordinate (one for each surface). The Shaik/Reddy picture ultimately produces two potential surfaces with a diabatic interpretation. Also, it is clearly wavefunction-based. The Agmon/Levine model cannot be wavefunction based because the Hilbert space is not a convex set. It could be generalized to density matrices, but then it is not immediately clear how to discuss the "twin state" invoked by Shaik & Reddy. Since the states in the Shaik/Reddy picture are VB states and may be non-orthogonal, the definition of an appropriate mixing entropy is also not clear (see, for example, a paper by Dieks and van Dijk).
Y! Always Y!
Welcome to The Skeptycal Quantym Chemyst. I am your host, Seth Olsen.
This blog is to be a forum for me to float my increasingly heretical ideas about the state of modern theoretical chemistry and chemical physics.
On occasion, I will contrast these with the quite distinct field of molecular physics. One of my early rants will probably be about why these fields are NOT THE SAME.
I will be paying particular attention to the sub-field of molecular electronic structure, which somehow has managed to attract the traditional label of "Quantum Chemistry". I will not be limited to this, though, just as real quantum chemistry should not be limited to molecular electronic structure!
I was inspired to begin this blog by my colleague Prof. Ross H. McKenzie, who has demonstrated to me via his own blog "Condensed Concepts" (http://condensedconcepts.blogspot.com/) that this medium is actually a good way to foster scientific communication (without having to deal with annoying reviewers!). Ross' blog deals with some quantum chemical issues itself, as well as more general condensed matter physics.
Enjoy!
This blog is to be a forum for me to float my increasingly heretical ideas about the state of modern theoretical chemistry and chemical physics.
On occasion, I will contrast these with the quite distinct field of molecular physics. One of my early rants will probably be about why these fields are NOT THE SAME.
I will be paying particular attention to the sub-field of molecular electronic structure, which somehow has managed to attract the traditional label of "Quantum Chemistry". I will not be limited to this, though, just as real quantum chemistry should not be limited to molecular electronic structure!
I was inspired to begin this blog by my colleague Prof. Ross H. McKenzie, who has demonstrated to me via his own blog "Condensed Concepts" (http://condensedconcepts.blogspot.com/) that this medium is actually a good way to foster scientific communication (without having to deal with annoying reviewers!). Ross' blog deals with some quantum chemical issues itself, as well as more general condensed matter physics.
Enjoy!
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