Confusing bottom-up and top-down approaches to emergence

Due to emergence, reality is stratified. This is reflected in the existence of semi-autonomous scientific disciplines and subdisciplines. A major goal is to understand the relationship between different strata. For example, how is chemistry related to physics? How is genetics related to cell biology?

Before describing two alternative approaches —top-down and bottom-up —I need to point out that in different fields, these terms are used in opposite senses. That can be confusing!

In the latest version of my review article on emergence, I employ the same terminology traditionally used in condensed matter physics, chemistry, and biology. It is also consistent with the use of the term “downward causation” in philosophy. 

Top-down means going from long-distance scales to short-distance scales, i.e., going down in the diagrams shown in the figure above. In contrast, in the quantum field theory of elementary particles and fields (high-energy physics), “top-down” means the opposite, i.e., going from short to long distance length scales. This is because practitioners in that field tend to draw diagrams with high energies at the top and low energies at the bottom.

Bottom-up approaches aim to answer the question: how do properties observed at the macroscale emerge from the microscopic properties of the system? 

History suggests that this question may often be best addressed by identifying the relevant mesoscale at which modularity is observed and connecting the micro- to the meso- and connecting the meso- to the macro. For example, high-energy degrees of freedom can be "integrated out" to give an effective theory for the low-energy degrees of freedom.

Top-down approaches try to surmise something about the microscopic from the macroscopic. This has a long and fruitful history, albeit probably with many false starts that we may not hear about, unless we live through them or read history books. Kepler's snowflakes are an early example. Before people were completely convinced of the existence of atoms, the study of crystal facets and of Brownian motion provided hints of the atomic structure of matter. Planck deduced the existence of the quantum from the thermodynamics of black-body radiation, i.e. from macroscopic properties. Arguably, the first definitive determination of Avogadro's number was from Perrin's experiments on Brownian motion, which involved mesoscopic measurements. Comparing classical statistical mechanics to bulk thermodynamic properties gave hints of an underlying quantum structure to reality. The Sackur-Tetrode equation for the entropy of an ideal gas hinted at the quantisation of phase space. The Gibbs paradox hinted that fundamental particles are indistinguishable. The third law of thermodynamics hints at quantum degeneracy. Pauling’s proposal for the structure of ice was based on macroscopic measurements of its residual entropy. Pasteur deduced the chirality of molecules from observations of the facets in crystals of tartaric acid. Sometimes a “top-down” approach means one that focuses on the meso-scale and ignores microscopic details.

The top-down and bottom-up approaches should not be seen as exclusive or competitive, but rather complementary. Their relative priority or feasibility depends on the system of interest and the amount of information and techniques available to an investigator. Coleman has discussed the interplay of emergence and reductionism in condensed matter. In biology, Mayr advocated a “dual level of analysis” for organisms. In social science, Schelling discussed the interplay of the behaviour of individuals and the properties of social aggregates. In a classic study of complex organisations in business, understanding this interplay was termed differentiation and integration.

I thank Jeremy Schmit for requesting clarification of this terminology.

The quantum physics of life in red and green

Life is truly amazing!
Life is beautiful!
...and it involves quantum many-body physics...

There is a beautiful (short) review
Heme: From quantum spin crossover to oxygen manager of life 
Kasper Kepp

The article involves a plethora of topics that I have discussed before on this blog. I have included relevant links.

Kepp starts with the unique (chemically fine-tuned) properties of both iron and porphyrin that enable them to play a central role in two of the most important processes in life: respiration and photosynthesis. He has a beautiful paragraph (perhaps in the style of Roald Hoffmann):

Such ligand-field transitions of iron in porphyrin were familiar to our ancestors as the characteristic red color of blood that largely defines the human psychological and cultural connotations of the color representing courage, war, danger, and suffering. 

Incidentally, pi-pi* transitions within the porphyrin-derived chlorophylls are also responsible for the green color of plants, associated with nature, life and hope, so the reader may perhaps agree that porphyrin has had vast (but alas! rarely appreciated) cultural consequences.

The oxygen molecule is a spin triplet.
Iron(II) porphyrin is in a triplet spin state (S=1). The Fe(II) is a d6 configuration in a D_4h crystal field.
When they bind together the ground state is a spin-singlet.

There are two fundamental quantum chemistry questions that are discussed.

1. What is the electronic structure (many-body wave function) of the ground state for oxygen bound to heme?

2. What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

The first question has a long history. Like almost anything important and profound in quantum chemistry it goes back to Linus Pauling! In 1936 Pauling and Coryell argued that the ground state is

essentially a neutral O=O binding with two of its electrons to iron to produce a formally iron(II) if both the bonding electrons were confined to O2, corre- sponding to the non-bonding limit of neutral parts, but a formally iron(I) if the Fe–O bond were to be considered covalent. 

In 1960, McClure suggested a valence-bond formulation based on triplet–triplet coupling, which is appealing by the low promotion energies required to access these states, rather than the singlet states. In 1964, Weiss suggested, based on analogy to chemical reactions in aqueous solution, that the true ferrous hemeO2 adduct was mainly of the superoxo-iron(III) type caused by ‘‘electron transfer” from iron to O2. 

Goddard and Olafson suggested an ozone model of the adduct in 1975 which emphasized the four-electron three-center bond with maintained triplet state of dioxygen as in the McClure model with less electronic reorganization to explain the reversible binding. 

In 1977, Pauling maintained his original view again, the same year that Huynh, Case, and Karplus did a first attempt to bridge these views by performing early quantum chemical calculations that diplomatically emphasized the importance of both Weiss and Pauling resonance forms. 

However, interpretation depends on model language, orbital localization, and transformation between valence bond and orbital formalisms:  

In terms of molecular orbital theory, the wave function was a multi-configurational state dominated by the Pauling configuration; however, if one uses valence bond theory considerations, it can be interpreted as having large Weiss character. Thus, the multi-configurational state produced from CASPT2 is interpreted differently by different models. This partly explains why the trenches were so deeply dug during the exchange between Pauling, Goddard, McClure, and Weiss; all were right, and all were wrong. 

This is just another example of unnecessary conflicts about valence bond vs. molecular orbital (VB vs. MO). 

In terms of valence structures, the ground state was summarized by Shaik and Chen as having contributions from both Weiss, Pauling, and McClure forms, the first .. dominating. 

Ironically DFT ends up providing a useful language after all! 

The charge assignments to O2 are very dependent on calculation scheme, and both the orbitals, valence structures, and atomic charges that defined the Weiss-Pauling debate are non-observable. In contrast, the electron density is observable as are the geometries and spectroscopic data...

Molecular orbitals are not physical observables but calculational constructs. MO's don't exist.

In different words, one can take a many-body wave-function and make a linear unitary transformation of the molecular orbitals. The Slater determinants do not change. [The value of a determinant is invariant to a change of basis.]

Now. Question 2.
What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

Kepp talks about spin-orbit coupling and the fact that it is small for oxygen, motivating a discussion of a "broad crossing mechanism".  However, I am not sure this is relevant. I don't see the binding as necessarily spin forbidden. As the oxygen approaches the heme the two triplet states can mix to form a total spin singlet.
This is analogous to bringing two hydrogen atoms (each of which is spin 1/2) together to form a hydrogen molecule (which is spin zero). A multi-configurational wavefunction has no problem with this. But DFT-based approximations, which use a single determinant cannot describe this smooth crossover.

Other things of particular interest to me that are discussed in the review include the central role of back bonding and the success of the TPSSh functional in DFT calculations for organometallics.

Unfortunately, the review does not mention recent work by Weber et al, applying DMFT to the problem of oxygen binding to haemoglobin.

Spin ice in a nutshell

What is spin ice? What its definitive and experimental signatures?

A good place to start is the lucid discussion by Roderich Moessner and Art Ramirez in a 2006 article on Geometrical Frustration. They emphasise two organising principles: local constraints on neigbouring spins and the emergence of new entities such as gauge fields.

First, let's discuss the "ice" bit since this involves some beautiful chemistry, physics, statistical mechanics, and history. In the solid phase of water at atmospheric pressure (ice Ih) the water molecules form a hexagonal lattice, with the oxygen atoms located a the vertices of the lattice. The molecules interact with one another via hydrogen bonds.

Now the key point is that there are many different ways of orienting the water molecules (arranging the protons). The only constraint is that one has to have two protons covalently bonded to the oxygen and two protons on next-nearest neighbour water molecules hydrogen bonded to the oxygen. This is known as the ice rule. Suppose we assign an Ising spin variable (+1,-1)=(in, out)  = (covalent, Hbond) to each "bond" on the lattice. Then the ice rule is that on each tetrahedron the sum of the four "spins" must be zero.

How much degeneracy is there?
There are 2^4= 16 possible spin states on a tetrahedron. But, only six (a fraction of 3/8) satisfy the ice rule. To see this, put +1 on site one, then one must put +1 on one of the other three sites, and -1 on the other two. This gives 6 = 2 x 3 options.
If one neglects the interaction between vertices, the thermodynamic entropy per tetrahedron (water molecule) is

S = k ln (3/2)

Historical asides.
This "residual" entropy in ice was observed experimentally by William Giauque in the chemistry department at Berkeley in the 1930s.
Linus Pauling explained this in 1935, even arguing it as evidence for a specific crystal structure of ice.
Pauling's picture led to the ice-type models that are very important  (from a mathematical and conceptual point of view) in classical statistical mechanics as they are exactly soluble in two dimensions.
In 1956 Phil Anderson (who else!) noted that Pauling's problem was equivalent to that of Ising spins on a pyrochlore lattice.
It was not until four decades later than an experimental realisation was observed in a magnetic material. The experimental data is shown below.

But there is much more to spin ice. The local constraints lead naturally to an emergent gauge field (a pseudo-magnetic field), analogues of "magnetic monopoles", and unusual spin correlations (algebraic correlations without criticality). I now discuss the latter as they can be viewed as a "smoking gun" of spin ice.

The "magnetic field" B satisfies the constraint Div B =0. As a result the spin correlations have a dipolar form, i.e. they have a distance and directional dependence similar to the magnetic field associated with a magnetic dipole. This means the spin correlations fall off algebraically. This is in contrast to conventional magnets where spin correlations decay exponentially, except at a critical point. Furthermore, if one plots or measures the static spin structure factor S(q) one finds "pinch points" occur in high symmetry planes. The figure below shows an experimental measurement for Holonium Titanate, taken from here.

Should Hollywood make a Linus Pauling biopic?

The past few years has seen Hollywood make movies about famous scientists and mathematicians, including A Beautiful Mind (John Nash), The Theory of Everything (Stephen Hawking), The Imitation Game (Alan Turing), and now The Man Who Knew Infinity (Ramanujan).
The latter is to be released April 29 in the USA and May 5 in Australia.


Are there others?

This post is not about the important issue whether this is a good thing, particularly when you consider all the creative license taken, and whether the movies capture the science in an appropriate way.

First, what kind of scientist is an appropriate candidate for such a movie?
I think their life must have some significant components of romance, scandal, tragedy, and redemption. The list above does include substantial ingredients of most of these. Pure scientific heroism and brilliance just does not cut it.

Second, who might be some candidates from condensed matter physics or theoretical chemistry?

Feynman was one that came to mind, particularly because of the tragic death of his first wife and his involvement in the Challenger inquiry. However, I see that back in 1998 there was Infinity, starring Matthew Broderick and Patricia Arquette. It looks like it was box office flop. Has anyone seen it?
Interestingly, about 15 years ago, Alan Alda commissioned and was the lead actor in a play QED about Feynman. No signs of a movie.

So who might be other candidates? Greats like John Bardeen, Phil Anderson, Walter Kohn, are just too boring ....
But, what about Linus Pauling? There is the romance and partnership with his wife, amazing lecturing skills, the political activism and persecution, and the controversy of his views about vitamin C.

Perhaps William Shockley might make the cut, because of his role in starting Silicon Valley, conflict with everyone, and racist views, ...

Any other ideas?

Mrs. Pauling was right about two things

For Christmas (two years ago) my sister-in-law gave me a copy of
Ava Helen Pauling: Partner, Activist, Visionary by Mina Carson, a historian at Oregon State University, which is home to the Linus Pauling archives. I read it then but it has taken me a while to get around to writing this post.

Aside: There are many personal dimensions to this gift choice. My sister-in-law and her family live in Corvallis, and their younger daughter attends Linus Pauling Middle School. Of course, they knew about my great admiration of Pauling. But also, the author has been in a book club with my sister-in-law.

I enjoyed reading the book and it gave me a different perspective on Pauling's life. Although, some of the more intimate details in the book I would rather not have known about...

The author nicely highlights how Ava Helen was really the driving force behind Linus' political activism, which ultimately led to his second Nobel Prize (in Peace) for the Partial Nuclear Test Ban Treaty. It is interesting to wonder whether today she would have shared the prize with him.

But, here I want to focus on two things that really struck me from the book.

In the 1950s Mrs. Pauling advocated two positions that we (or at least most people) just take for granted today. Yet at the time, the Paulings were persecuted for their advocacy of these views, particularly by powerful political, governmental, and commercial interests.

1. Above ground nuclear testing and the associated radiation exposure is bad for peoples health.

2. Faculty at public universities should be allowed to hold and advocate any political views they choose.

The context of the second was the Loyalty Oath Controversy that ripped apart UC Berkeley from 1949-1951.

The Sommerfeld model is a Pauling point

A basic question that comes up in introductory solid state physics is:
Why does the Sommerfeld model for metals work so well?
It assumes that electrons are non-interacting fermions. Yet if you calculate the first order correction (in e^2 where e is the electronic charge) in the Coulomb energy you find it is comparable to the kinetic energy associated with the ground state.

Aside: the success of Sommerfeld is such a puzzle that Wigner mentioned it (for the wrong reasons in my view) at the end of his famous 1962 essay, The Unreasonable Effectiveness of Mathematics in the Physical Sciences.

The standard answer we give students is screening  plus Landau's Fermi liquid theory.
However, an interesting question is what happens if you try to actually do some sort of systematic many-body expansion with respect to the Coulomb interaction. Can you get the calculation to converge to experiment and see why Sommerfeld is good?

In Telluride last (northern) summer I heard a nice talk by Timothy Berkelbach that is relevant to this issue. The message I took away was that the Sommerfeld model is a Pauling point, i.e. by accident it gets the right answer for the wrong reasons.

The relevant paper has now appeared on the arXiv.

Spectral Functions of the Uniform Electron Gas via Coupled-Cluster Theory and Comparison to the GW and Related Approximations 
James McClain, Johannes Lischner, Thomas Watson, Devin A. Matthews, Enrico Ronca, Steven G. Louie, Timothy C. Berkelbach, Garnet Kin-Lic Chan

A key graph is below. It shows the quasi-particle dispersion relation for a uniform electron gas with r_s=4, the value relevant to sodium.

For comparison the "binding energy" [i.e. the k=0 energy] deduced from experiment is -2.6 eV, and the values for Sommerfeld and LDA are about -3.1 eV. Hartree-Fock gives -7.3 eV!

As you increase the "level of theory" [i.e. the sophistication of treatment] of electron correlations you go from Sommerfeld to HF to HF+GW to CCSD [Coupled Cluster Singles and Doubles].
Then one sees the answer at first gets worse and then improves and you almost get back to where you started!

Aside: This also illustrates how LDA is a Pauling point too!

A comparative appreciation of P.W. Anderson and Linus Pauling

Andrew Zangwill contacted me because he is working on scientific biography of Phil Anderson. I think this is overdue. I would argue that Phil is the greatest theoretical physicist of the second half of the twentieth century. I would argue this on similar grounds to why I think Linus Pauling was the greatest theoretical chemist of the first half of the twentieth century. Crucially, their scientific legacies have extended far beyond condensed matter physics and chemistry, respectively.

Specifically, Pauling did not just make essential contributions to our understanding of chemical bonding, x-ray crystallography, and quantum chemistry. His impact went far beyond chemistry. Francis Crick said Pauling was the "father of molecular biology." He proposed and elucidated alpha helices and beta sheets in proteins. Furthermore, he began the whole field of molecular medicine, by showing the molecular basis of a specific disease, sickle cell anemia.

Phil Anderson has made incredibly diverse and valuable contributions to condensed matter physics (anti-ferromagnetism, localisation, weak localisation, magnetic impurities in metals, Kondo problem, poor mans scaling, superfluid 3He, spin liquids, RVB theory of superconductivity... ).
I can think of three significant and profound influences of Phil beyond condensed matter physics.

Codifying and elucidating the concept of emergence (and the limitations of reductionism) in all of science, in More is Different in 1972.

Laying ground work for the Higgs boson in 1963 by connecting spontaneous gauge symmetry breaking and mass. 

Elucidating spin glasses in a way that was key to John Hopfield's development of a particular neural network and to the notion of a "rugged landscape", relevant in protein folding and evolution. Anderson described these connections nicely in two pages in Physics Today in 1990.

Are there other examples?

Who do you think is the greatest theoretical physicist of the second half of the twentieth century?
[n.b. If you are thinking Feynman, he did path integrals and QED before 1950].

Jacob's ladder is not the best Biblical metaphor for computational materials science

A more appropriate metaphor is Jacob wrestling with the angel (God).
This point was made in a nice talk that Mike Gillan gave last week at the NORDITA water meeting.

John Perdew has invoked the metaphor of Jacob's ladder to describe his "dreams of a final theory" and the quest for an "exact" exchange correlation functional for Density Functional Theory (DFT).
Perdew's metaphor was earlier reinvoked by Joost VandeVondele in his talk at the meeting.

This painting of Jacob's ladder is by Michael Willmann. In the Biblical account from Genesis 28

Jacob left Beersheba, and went toward Haran. He came to the place and stayed there that night, ....  And he dreamed, and behold, there was a ladder set up on the earth, and the top of it reached to heaven; and behold, the angels of God were ascending and descending on it! And behold, the Lord stood above it [or "beside him"] and said, "I am the Lord, the God of Abraham your father and the God of Isaac; the land on which you lie I will give to you and to your descendants; and your descendants shall be like the dust of the earth, and you shall spread abroad to the west and to the east and to the north and to the south; and by you and your descendants shall all the families of the earth bless themselves. Behold, I am with you and will keep you wherever you go, and will bring you back to this land; for I will not leave you until I have done that of which I have spoken to you." Then Jacob awoke from his sleep and said, "Surely the Lord is in this place; and I did not know it." And he was afraid, and said, "This is none other than the house of God, and this is the gate of heaven.

This was a dream.
The "ladder" of approximations is also a dream because it conveys the idea that with each rung of the ladder one is necessarily getting closer to the correct answer [heaven]. Things are not that simple. For example, Mike Gillan pointed out to me that the generalised gradient approximation (GGA) does worse than the local density approximation (LDA) for the surface energies of solids.   No doubt experts can provide other examples.

Similar issues arise in wave function based computational quantum chemistry. I think Pople first drew a diagram such as the one below (taken from this paper). The idea is that as one increases the size of basis set and the level of theory (i.e. treatment of electron correlation) one moves closer to reality (experiment).

Again, the problem is that the "convergence" to reality is not monotonic or uniform. This is reflected in the existence of Pauling points. Sometimes as one moves down or to the right one on the figure actually gets further away from the experimental value. This is discussed in detail for a specific example in a paper by Seth Olsen.

As Mike Gillan suggested a more appropriate Biblical metaphor than Jacob's ladder is the account in Genesis 32 of Jacob [whose name means deceiver] wrestling with an angel. Afterwards, he is renamed Israel [which means he who wrestles with God].

The painting is by Rembrandt (1659).

Computational materials science is a struggle. Jacob's ladder is a dream.

Pauling's last blackboard

Today I visited the Linus Pauling Archives at Oregon State University.
[I was actually on vacation in Corvallis visiting my sister-in-law but I just had to take a visit].

They have assembled a lot of fascinating material online, which is worth perusing. For example, how a funding agency convinced him to start working on proteins, and the details of correspondence about his (erroneous) ideas about quasi-crystals.

But, in the actual library there is a small display featuring Pauling's last blackboard, his desk, some molecular models, some calculators, his two Nobel Prize medals, and his signature beret.

Pauling is definitely one of my scientific heroes. He made multiple landmark contributions. Most of us would be happy to do just one of the things he is known for. He was truly the master of multi-disciplinarity. He brought quantum physics to chemistry, structural chemistry to biology, and molecular biology to medicine. But he had "clay feet", failing to see problems with his ideas about the triple helix for DNA, vitamin C and quasi-crystals.

Curious fact: I was allowed to touch the Nobel Prize medals but my brother-in-law asked if I could have my picture taken with Pauling's signature beret. The librarian said no!

Pauling on the role of quantum theory in chemistry

In an article The Nature of the Chemical Bond - 1992, Linus Pauling makes the following fascinating statement:

The concept of quantum mechanical resonance and the theorem that in quantum mechanics the actual structure of a system has a lower energy than any other structure have turned out to be especially important in chemistry. The energy could be calculated for an assumed wave function for a molecule. Any change that lowered the energy indicated some addition to the picture of the chemical bond. The polarization of bond orbitals and the partial ionic character of bonds were discovered in this way. The minimum-energy theorem led to the formulation of the electronegativity scale. Modern chemistry and molecular biology are the products of quantum mechanics. Chemistry has been changed by quantum mechanics even more than physics.

In 1936 in the Preface to The Nature of the Chemical Bond he also emphasized how quantum physics led to new chemical concepts.

The basics of electronegativity

Electronegativity is a simple and profound concept for understanding and giving a semi-quantitative description of broad classes of chemical bonds. It reflects the brilliant intuition of Linus Pauling who first introduced it in 1932. The bold idea is to assign a single number to each element of the periodic table; the relative value of the number between two elements then determines the magnitude of the polarity (charge distribution) in a chemical bond between two different elements.

I have been reviewing the concept because it is a key element to understanding hydrogen bonds, the recent IUPAC definition stating:

the hydrogen bond is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment X-H in which X is more electronegative than H, ...

Unlike earlier definitions it does not require that the acceptor be more electronegative than H, only that for X-H...Y-Z

the acceptor is an electron-rich region such as, but not limited to, a lone pair in Y or a pi-bonded pair in Y-Z. 

The Wikipedia page on electronegativity is a helpful introduction but I found section 6.4 in the classic Coulson's Valence extremely helpful.

There are several alternative definitions of electronegativity (Pauling, Mulliken, Alfred-Ronnow, Allen, ...). This highlights a few things:

   -like most intuitive chemical concepts they are not something that can be defined rigorously, without ambiguity, or in a reductionist manner (a point highlighted by Roald Hoffmann)

  -the concepts are useful for understanding semi-quantitative trends

  -these different definitions actually highlight the power of the concept because they show how a wide range of chemical and physical properties (bonding energies, dipole moments, charge distributions, ...) are correlated.
The graph below shows Pauling vs. Mulliken electronegativities

The "clearest" and most "precise" definition is that of Mulliken, where the electronegativity is the average of the ionisation energy and the electron affinity of the atom. This equals half of the ground state energy difference between the cation and the anion.
This means that if A and B have the same electronegativity that the ionic valence bond (VB) structures A+B- and A-B+ will have the same energy and so contribute equally to the full VB wave function, leading to no charge polarity.

To me the success of the concept also highlights the fact that predominantly chemical bonding is local.

Like most chemical concepts there are exceptions to their naive application. For example, carbon is less electronegative than oxygen, and so one might expect that in carbon monoxide (CO) there would be a net negative charge on the oxygen atom. However, the opposite is true.

Living and breathing quantum entanglement

Take a deep breath. You just created an entangled quantum state!

There is a really interesting paper on the arXiv

Quantum entanglement and Hund's rule are determinants to respiration
Cedric Weber, David D. O'Regan, Nicholas D. M. Hine, Peter B. Littlewood, Gabriel Kotliar, Mike C. Payne

They use DFT-DMFT (Density Functional Theory + Dynamical Mean-Field Theory) to study the binding of oxygen and carbon monoxide to iron-porphyrin (heme).

This is the process by which respiration occurs. Oxygen binds reversibly to the heme group in myoglobin. Unfortunately, CO does not bind reversibly and you die!

[Aside: Haemoglobin consists of four myoglobin molecules and they exhibit some interesting and important collective behaviour (allostery) first elucidated by Linus Pauling (who else!) from simple thermodynamic considerations.
This is nicely described in Thermal Physics by Kittel and Kroemer].

This new work shows that as the iron atom Hund's rule coupling J varies from 0 to 1 eV significant qualitative changes occur in the ground state. Specifically, it becomes a superposition of different iron spin (and valence) states.
Such behaviour cannot be captured by purely DFT-(Kohn-Sham) based calculations which are by assumption of single determinant character and so involve only one spin and charge state for the Fe atom.

As far as I am aware this is the first concrete application of DMFT to quantum chemistry. A few recent developments (e.g. this PRL from Columbia) were concerned with benchmark studies.

Earlier DFT studies show five different states within 15 kJ/mole [~0.15 eV] of one another. Hence, it is reasonable that small J value variations could change the character of the ground state.

The TOTAL spin of the ground state must be definite. If I recall correctly, experimentally it is found to be a singlet with oxygen bound.

The authors don't mention that back in 1979 Case, Huynh, and Karplus studied a Pariser-Parr-Pople (like an extended Hubbard) model for heme-O2 and heme-CO. They found that the ground state of the former was an equal mixture of Fe2+(S=0)O2(S=0) and Fe2+(S=1)O2(S=1).

In a Barley Peroxidase there is experimental evidence for a Quantum Mixed-Spin Heme State consisting of a superposition of S=5/2 and S=3/2.

Update. The published PRL version now references Case et al. There is also a PNAS paper that reports more results.

What is a Pauling point?

I first heard of the term "Pauling point" in a chemistry seminar that Fritz Schaefer gave at UQ several years ago. He asked the audience if anyone knew what it was. Only one person knew (and they were not from UQ! It was David Sholl who was on sabbatical). I think the term is originally due to Per Lowdin who here explains what it is:

At the Valadalen symposium in 1958, the author pointed out [Ref. 26, p. 23] that a characteristic feature of quantum chemistry was that even a fairly simple theory could sometimes give excellent agreement with experimental experience, but that this agreement may disappear whenever one tries to improve the theory. The point of excellent agreement was coined the "Pauling point" in honour of one of the great pioneers in our field who is also present here in Dubrovnik, not only because he could construct simple theories built on physical and chemical insight, but also because of his mastership in predicting figures which had not yet been measured. 

In the beginning of the 1930s one had constructed theories of chemical reactivity based on the properties of the valence electrons only to find that the good agreement disappeared when one included the inner shells leading to the concept of the "nightmare of the inner shells". In the MO-LCAO treatment of large molecules , one could get very good results without including the atomic overlap integrals, whereas in solid-state theory the inclusion may lead to the famous "non- orthogonality catastrophe". In the treatment of metal complexes, the original crystal- field theory for some reason seemed to give better agreement than the improved ligand- field theories. In the treatment of magnetic phenomena, the Hartree method seemed to give better results than the Hartree-Fock method, simply because the errors in treating parallel and antiparallel spins were better balanced in the former. Let me quickly add that my own doctoral thesis in 1948 treating the properties of ionic crystals by means of the independent-particle model is a typical example of a "Pauling point", where the good agreement with the experiments would disappear when one tries to include e.g. correlation in an unbalanced way. 

It goes without saying that, if one improves the theory more and more, the good agreement is expected to come back, but the simplicity of the theory is usually lost in this connection.

One should hence be somewhat suspicious, if a low-order perturbation theory seems to give excellent results - one may be at a "Pauling point".

The painful reality is that in quantum many-body theory we often do perturbation theory in dimensionless coupling constants that are of order one. This is not just in quantum chemistry. Another case is in spin-wave theory for quantum Heisenberg models where one expands in powers of 1/S where S is the total spin (usually S=1/2).


In lattice QCD (Quantum ChromoDynamics) one should worry about the size of the lattice a that one is using to approximate the space-time continuum. This PRL is one example of how a judicious choice of an effective Hamiltonian (action in field theory) [the O(a) technique] can give quite reasonable results for a relatively coarse lattice.


In quantum chemistry, one is often truncating other things such as the basis set for atomic orbitals or the size of the active space in CAS methods. One is always a long way from the asymptotic limit at which one expects to get the exact result. Yet there are many people who seem to assume that the bigger the basis set or the larger the active space the better. i.e., one is necessarily getting closer to the exact answer. The experience of "Pauling points" clearly shows this is not necessarily true and caution is in order.

Who is following who?

A landmark contribution of Linus Pauling was to show the intimate connection between molecular structure and the spatial arrangement of electronic wavefunctions. This is exemplified by the concept of orbital hybridisation. The figure above shows how by taking different linear combinations of s and p orbitals one can produce orbitals with different directionality.

However, as is often the case in quantum chemistry, it turns out not to be quite so simple. I was surprised to learn recently about the notion of imperfect orbital following. Specifically, the direction of the orbitals is not always the same as that of the nuclei, particularly, for non-equilibrium geometries. This can be seen for ammonia as it undergoes the umbrella inversion (the mode associated with the MASER). This phenomenon  of orbital following in ammonia was elucidated by Cohan and Coulson in 1956.  The figures below are from a JACS paper by Foster and Weinhold. [There is also a nice discussion in the book by Weinhold and Landis.]

In Figure 2 the angle associated with the direction of the orbital is plotted versus the HNH angle in the molecule. If the orbitals followed the nuclear geometry the solid line would lie on top of the dashed line. In the tetrahedral geometry, close to the equilibrium geometry, both angles are about 104 degrees, and the orbitals are approximately sp3. Halfway along the umbrella inversion reaction co-ordinate the molecule has D3 symmetry and both angles are 120 degrees. Roughly the orbitals consist of three sp2 orbitals and a lone pair p orbital.

What has quantum chemistry really achieved?

A small part only of the body of contributions of quantum mechanics to chemistry has been purely quantum-mechanical in character; only in a few cases, for example, have results of direct chemical interest been obtained by the accurate solution of the Schrodinger wave equation... The principal contribution of quantum mechanics to chemistry has been the suggestion of new ideas, such as the resonance of molecules among several electronic structures with an accompanying increase in stability.

Linus Pauling, Preface to The Nature of the Chemical Bond, First edition 1938

I wonder if this is still true today. Most computational chemists would bridle at that suggestion. But, I suspect that Pauling's point could still be argued today. Quantum mechanics has introduced important qualitative concepts such as potential energy surfaces, transition states, conical intersections, hybrid orbitals, ligand field theory, frontier orbitals, selection rules, .... These concepts are of far greater significance and success than the results of any detailed computations. Indeed the latter are largely of use to validate (and learn the boundaries of validity) of these concepts.
I welcome the views of readers.

The challenge of H-bonding

Hydrogen bonds are ubiquitous in biomolecules and a key to understanding their functionality. This was first appreciated by Pauling and exploited by Watson and Crick to decode the structure of DNA. It is also the origin of the unique and amazing properties of water. There is a helpful review article Hydrogen bonding in the solid state by Steiner. Here are a few things I learnt.

Energies vary by 2 orders of magnitude, 0.2-40 kcal/mol [10 meV to 2 eV]. This spans the energy range from van der Waals to covalent and ionic bonds. The amount of electrostatic, covalent, and dispersion character of the bond varies within this range.

For O-H ... O bonds the shift in frequency of the O-H bond correlates with the distance between the oxygen atoms. The O-H distance is correlated with the O..H distance.

All hydrogen bonds can be considered as incipient hydrogen transfer reactions.

Hydrogen bonds exhibit some unexplained isotope effects. Simple zero-point motion arguments suggest that deuterium substitution should lead to weaker bonds, as is observed in some cases. However, some bonds exhibit a negligible effect and others a negative effect.

Marrying Heitler-London and Pauling

The Linus Pauling archive at Oregon State University has lots of nice resources including original manuscripts, videos, quotes, and photos. Above is a photo of Heitler and London with Pauling's wife.

Bardeen International Airport?

I was intrigued and impressed that the Belgrade airport is called Belgrade Nikola Tesla Airport. This the only case I have encountered of an airport named after a scientist. They mostly seem to be named after politicians (e.g. JFK and Ronald Reagan) and rock stars (John Lennon in Liverpool). Here are a few proposals:

LAX - Linus Pauling
Urbana - John Bardeen
Newark - Phil Anderson
Adelaide - William Bragg

Any other ideas?

It is interesting reading the wikipedia page about Tesla, particularly the observation that he probably suffered from obessive-compulsive disorder. He is another example of where the dividing line between genius and mental illness is a fine one.

The origin of molecular medicine

The data in the Figure above changed the face of medical research.

It showed for the first time that disease can have a molecular basis.

It was another great scientific achievement and bold discovery of Linus Pauling.

The Figure is discussed in chapter 8 of Nelson's Biological Physics: Energy, Information, Life. He states:

In a historic discovery, Linus Pauling and coauthors showed in 1949 that the red blood cells of sickle-cell patients contained a defective form of hemoglobin. Today we know that the defect lies in parts of hemoglobin called the β-globin chains, which differ from normal β-globin by the substitution of a single amino acid, from glutamic acid to valine in position six. This tiny change (β-globin has 146 amino acids in all) is enough to create a sticky (hydrophobic) patch on the molecular surface. The mutant molecules clump together, forming a solid fiber of fourteen interwound helical strands inside the red cell and giving it the sickle shape for which the disease is named. The deformed red cells in turn get stuck in capillaries, then damaged, and destroyed by the body, with the net effect of creating pain and anemia.

In 1949, the sequence of β-globin was unknown. Nevertheless, Pauling and coauthors pinpointed the source of the disease in a single molecule. They reasoned that a slight chemical modification to hemoglobin could make a correspondingly small change in its titration curve, if the differing amino acids had different dissociation constants. 

A nice article  about the history is by William Eaton.

Marriage counseling for chemists

Roald Hoffmann, Philippe Hiberty, and Sason Shaik are probably my three favourite theoretical chemists. They write beautiful papers which focus on the quantum mechanical basis for chemical concepts and understanding rather than computation. Being a fly on the wall when they are all in one room for a scientific discussion would be fascinating. Well I don't have to dream. There is a really nice paper in Accounts of Chemical Research, A Conversation on VB vs. MO theory: a never ending rivalry. The three discuss and argue the relative merits and relationship between valence bond theory and molecular orbital theory. Here are a few choice quotes:

RH: A standard technique in marriage counseling (and I do think that MO and VB are a partnership) is to have the two parties stop and repeat, with an effort at understanding, what was said by the other partner. Can we try that? ......

PH: .......Pauling was smart enough to disguise all these “physical” elements of VB and packaged it as simple resonance theory. This was good for the 1930s, but now chemists have more theoretical savvy, and can digest these bits of physics, couldn’t they?

RH: Maybe. Some of them think theory is computation, and dignify that with the name of physical insight. The best physicists I’ve known - people like Ed Purcell - were after a quality of understanding that is ... almost chemical.

SS: ........ what tipped the balance in favor of MO may have been simply the computer implementation of MO-based theories. Chemists are a practical lot; they simply went where they could calculate.

RH: As they are doing now with the software available - it’s amazing what gains prominence just because there is a button in Gaussian to do it!