The emergence of molecular structure from quantum theory

Most debates about the emergence of molecular structure centre around the issue of irreducibility. Specifically, can the existence of molecular structures be predicted from quantum theory without assuming their existence or invoking classical concepts?

Consider a molecule that contains Ne electrons and Nn atomic nuclei (ions). The full quantum-mechanical Hamiltonian for the system is 

where e is the electronic charge, rj is the position of the j-th electron, Zi  and Mi are the charge and mass, respectively, of the i’th ion with position co-ordinate Rj. This is the Hamiltonian that Laughlin and Pines dubbed “The Theory of Everything” because if the solution (i.e., eigenstates and eigenvalues of the Hamiltonian operator) could be found it would describe almost all of chemistry and materials science.

This Hamiltonian treats the electrons and nuclei on an equal footing. 

For isomers, the Hamiltonian is identical. However, as will be discussed in a later post, that does not preclude solutions to the Hamiltonian that can describe isomers. 

The Hamiltonian has global translational and rotational symmetry, where all the particles undergo the same rotation or translation. In contrast, molecular structures may have discrete rotational symmetries. However, this is not necessarily a problem, as an eigenstate of a quantum problem can transform according to a non-trivial irreducible representation of the symmetry. For example, except the s-orbitals all the orbitals of the hydrogen atom are spatially anisotropic.

The electrons are identical particles and so have permutation symmetry. They are fermions with spin-1/2 and so any eigenstate must be antisymmetric under the exchange of two electrons. The energies associated with this exchange are crucial to the formation of chemical bonds and the stability of molecular structures.

If two or more atoms in the molecule are identical, then any exact eigenstate must be consistent with permutation symmetry. If a nucleus is composed of an even (odd) number of nucleons, then it is a boson (fermion) with integer (half-integer) spin, and eigenstates must be symmetric (antisymmetric) under exchange of identical nuclei. However, the corresponding exchange energies are relatively small (because the quantum delocalisation of the nuclei is small) and consequently most practical calculations of the eigenstates do not make this requirement of the eigenstates. Nevertheless, if the electrons and nuclei are treated on equal footing, this should be done. Although this is challenging, it has been done recently, as discussed below. 

Full quantum solutions of the Hamiltonian

In most computational quantum chemistry, the Hamiltonian is solved in the Born-Oppenheimer approximation, which will be introduced and discussed later. This is a source of some confusion and contention in philosophical discussions about the emergence of molecular structure.

Due to advances in methodology and computational power over the past few decades, it has become possible in practise to solve the full quantum Hamiltonian for small molecules. There are three levels of complication associated with this: quantum nuclear motion, rotational symmetry, and some nuclei being identical particles. There are also two challenges: first, finding the eigenstates and second, deducing the molecular structure from the eigenstates.

To begin, I consider the simplest case and ignore the complications associated with rotational symmetry or identical nuclei. This provides some insight and undermines some objections in the philosophical literature.

The ground state eigenfunction can be written as

where r and R are 3Ne and 3Na -dimensional vectors, respectively. Note that this function will have a complicated structure as it will depend on the spin states of all the electrons, denoted by s.

A probability distribution (reduced density matrix) for the positions of the nuclei is given by

where the sum is over all the electron spin degrees of freedom.

For many molecules, but not all, this probability distribution will have a unique global maximum at the coordinates R_0. This set of coordinates defines the geometry of the molecular structure. The physics underlying the existence of well-defined maxima is that the mass of the nuclei is much larger than the mass of the electrons, and as a result, the zero-point motions of the nuclei are much smaller than the separation of the nuclei in the molecular structure.

Note that the nuclear probability distribution is regularly measured in scattering experiments (using X-rays, neutrons, or electrons), and its maxima are used to determine the structures of molecules and crystals. The Debye-Waller factor is a measure of the width of the probability distribution. At low temperatures, it is determined by quantum zero-point motion. In other words, it is well established experimentally that classical molecular structures are an approximation to a fluctuating quantum structure.

Not every molecule will have a probability distribution with a unique maximum. An example is ammonia. As discussed further below, it has two maxima; each represents an umbrella geometry, and they are related by an inversion symmetry. The ground state wavefunction of the whole system is a superposition of two quantum states, each being associated with one of the two umbrella geometries, and the electronic and nuclear degrees of freedom are entangled with one another.

A general quantum definition of molecular structure

Lang et al. have recently overcome the challenges mentioned above to determine molecular structure in a manner that treats the electrons and nuclei on an equal footing with regard to quantum theory. They have considered both rotational symmetry and nuclear permutation symmetry and given a general definition of molecular structure involving nuclear probability densities calculated from the full wavefunction. They have explicitly performed these calculations for D3+, (where D is deuterium). The result is that the molecule has the same triangular structure that is observed experimentally and calculated using the Born-Oppenheimer approximation. This work is significant because it explicitly shows that molecular structure can be predicted in practice, not just in principle, from quantum theory.

In a forthcoming post, I will discuss the Born-Oppenheimer approximation and some of the confusion associated with it.

Role of quantum nuclear motion in biomolecular systems

 Total I am giving a talk, "Effect of quantum nuclear motion on hydrogen bonds in complex molecular materials" at Light-matter Interactions from scratch: Theory and Experiments at the Border with Biology 

Here are the slides

The talk provides a concrete example of the tutorial on constructing simple model Hamiltonians for complex materials that I give before the talk. It relates to the bio theme of the meeting through work on isotopic fractionation in proteins and the recent paper below. It makes use of the simple model that I talk about.

Unusual Spectroscopic and Electric Field Sensitivity of Chromophores with Short Hydrogen Bonds: GFP and PYP as Model Systems

Chi-Yun Lin and Steven G. Boxer

Isotope effects in spin crossover materials

A range of isotope substitution experiments have been performed on spin-crossover materials. 

Just like for other systems such as superconductors their interpretation is subtle.

The first studies are reviewed in Section 2.3.5 of this review article.

Isotopic exchange was investigated for a tris(picolylamine)iron(II) system which exhibits a two-step spin transition. Results are shown in the figure below. Significant changes in the spin-state transition curve were observed only when the isotopic substitution (H/D and 14N/15N) was made for atoms directly involved in the hydrogen-bonding network that connects the spin-crossover molecules. For example, with C2H5OD/ND2 the crossover temperature was shifted to higher temperatures by about 15 K and the middle step was no longer present. 

I would not have expected such a large effect given the chemical complexity of these systems and that the H atoms are not immediately bonded to the iron atoms which undergo the spin-state transition.

I now mention two other studies. They are particularly helpful because they also measured how the enthalpy and entropy change associated with the spin-state transtion changed with isotopic substitution.

Weber et al. studied the iron(II) spin-crossover complex [FeL1(HIm)2] and the deuterium-substituted [FeL1(DIm)2] where Him is (not a man but) imidazole. Both exhibit a single-step transition with hysteresis. H/D exchange decreased both the transition temperature and the hysteresis width by a few K. Deuteration decreased the value of the enthalpy and entropy differences between the low spin and high spin states (determined from differential scanning calorimetry) by about twenty and ten percent, respectively. (See Table 2 in the paper). They estimated an interaction parameter J = 560 K, indicating strong intermolecular interactions, which they attributed to a hydrogen bond. They reference some earlier studies showing how the magnitude of the ligand field in a transition metal complex can be modified by hydrogen bonds involving the complex. 

Very recently, Jornet-Mollá et al. studied the iron(ii) salt [Fe(bpp)2](isonicNO)2·HisonicNO·5H2O, which with decreasing temperature undergoes a transition at 162 K. There is a width of about 5 K, associated with hysteresis. With deuteration, the transition temperature decreases to 155 K, the width increases to 7 K, and the enthalpy and entropy differences both increase by about fifteen percent. “Annealing the compound at lower temperatures results in a 100% LS phase that differs from the initial HS phase in the formation of a hydrogen bond (HB) between two water molecules (O4W and O5W) of crystallisation. Neutron crystallography experiments have also evidenced a proton displacement inside a short strong hydrogen bond (SSHB) between two isonicNO anions.” 

I am particularly interested in this because of previous work I have done on strong hydrogen bonds.

Again I am surprised at the magnitude of these effects because the zero-point energy associated with the relevant H atoms is only a small fraction of the total zero-point energy and the entropy contribution from vibrations.

I now start a preliminary discussion of how these experiments might be interpreted in terms of an Ising model picture, such as in a recent preprint. The Hamiltonian is

 where the pseudospin sigma=+1/-1 corresponds to high spin and low spin states.

The crossover temperature is independent of the Ising interactions J's and given by 

Thus, it is a property of the individual molecules and not directly dependent on the intermolecular interactions.

The presence of hysteresis and the difference between the transition temperature for increasing and decreasing temperature sweeps are determined by the magnitude of the Ising interactions. 

Our results in Appendix A of the preprint imply that there should be no dynamical isotope effects on the J’s, i.e., provided other parameters such as structural details and bond lengths do not change with isotope substitution.

This does not rule out changes in the crossover temperature. Both the enthalpy and entropy differences can change with isotope substitution (as is observed). The former due to changes in zero-point energies, and the latter due to changes in the vibrational contribution to the entropy change. 

Interpretation of isotope effects can be subtle

 Isotopic substitution has provided significant insights into molecular and solid-state physics. This involves the substitution of particular atoms in a compound by the same chemical element with a different nuclear mass (i.e. a nuclear isotope). An example is hydrogen/deuterium substitution which has shown the significant role that quantum nuclear motion can play in hydrogen bonding, particularly in strong hydrogen bonds. Of particular relevance to the discussion below is that isotopic substitution does not only change vibrational frequencies but can also change bond lengths. 

 A key piece of evidence on the road to the BCS theory of superconductivity in 1957 was the observation of an isotope effect. In 1950 a shift in the transition temperature of mercury was observed, suggesting that superconductivity resulted from electron-phonon interactions, as argued by Frohlich that same year. In particular, the magnitude of the shift was consistent with theoretical work by Herbert Frohlich. (Whether he predicted or postdicted the observed effect is a matter of debate, as discussed by Jorge Hirsch.) BCS theory gives that $\Delta T_c/T_c = - {1/over 2} \Delta M /M$, which arises from the fact that phonon frequencies scale with $1/\sqrt{M}$, consistent with the mercury experiments. 

However, in the 1960s there were many observations of “anomalous” isotope effects, particularly in transition metals, that were inconsistent with the prefactor in this equation. These anomalies were resolved by going beyond the BCS theory and allowing for strong-coupling effects. Following the discovery of cuprate superconductors in 1986, isotope effects were observed in some cuprates. However, the consensus now is that these observations do not support an electron-phonon mechanism for superconductivity but rather are due to structural changes due to the isotope substitution. For example, isotopic substitution changes the zero-point energy, and that can alter the unit cell volume and the hopping parameter t in a Hubbard model. 

This illustrates that there are subtleties in interpreting isotope experiments. This is because there are both structural and dynamical isotope effects. Changes in isotope can lead to changes in structure, such as bond lengths or lattice constants, and even in changes in crystal symmetry. These structural changes arise because the equilibrium structure of the system is that which minimises the total energy of the system. The contribution to this energy from the zero-point energy of the atomic vibrations changes with isotope substitution and with bond lengths. Dynamical effects are those that involve exchange of phonons such as in superconductivity. 

I am not sure how to sharpen this structural/dynamical or static/dynamical distinction. Or is it secondary and primary effects?

In the next post, I will discuss observations of isotope effects in spin-crossover materials and how that relates to recenttheoretical work with my collaborators.

Square ice on graphene?

As I have written many times before, water is fascinating, a rich source of diverse and unusual phenomena, and an unfortunate source of spurious research reports.
Polywater is the classic example of the latter.
I find the physics particularly interesting because of the interplay of hydrogen bonding and quantum nuclear effects such as zero-point motion and tunneling.

There is a fascinating paper
Polymorphism of Water in Two Dimensions
Tanglaw Roman and Axel Groß

The paper was stimulated by a Nature paper that claimed to experimentally observe square ice inside graphene nanocapillaries. Such a square structure is in contrast to the hexagonal structure found in regular three-dimensional ice.
Subsequent, theoretical calculations claimed to support this observation of square ice.
Here the authors use DFT-based methods to calculate the relative energies of a range of two-dimensional structures for free-standing sheets of water (both single layer and bilayers) and for sheets bounded by two layers of graphene.

The figure below summarises the authors results for free-standing layers showing how the relative stability of the different water structures depends on the area density of water molecules [which varies the length and strength of the hydrogen bonds].

On the science side, there are several interesting questions arise.
How much do the results depend on the choice of DFT functional used [RPBE with dispersion corrections]?
Would inclusion of the nuclear zero-point energy modify the relative stability of some of the structures, as it does for the water hexamer?
Quantum nuclear effects are particularly important when the hydrogen bond length [distance between oxygen atoms] is about 2.4 Angstroms. [I am not quite sure what area density this corresponds to for the different structures].

On the sociology side, this paper is another example of a distressingly common progression:
1. A paper in a luxury journal reports an exotic and exciting new result.
2. More papers appear, some supporting and some raising questions about the result.
3. A very careful analysis reported in a solid professional journal shows the original claim was largely wrong. This paper attracts few citations because the community has moved on to the latest exciting new "discovery" reported in a luxury journal.

I thank Tanglaw Roman for helpful discussions about his paper.

A spicy scientific scandal

I am often on the lookout for interesting molecules and solids which involve short hydrogen bonds, particularly biomolecules where this bond may play a key role in functionality. Such bonds are of interest from a physics point of view because then the quantum motion of the proton matters.
Consequently, the following paper (published in October 2016) caught my attention.

Proton Probability Distribution in the O···H···O Low-Barrier Hydrogen Bond: A Combined Solid-State NMR and Quantum Chemical Computational Study of Dibenzoylmethane and Curcumin Xianqi Kong, Andreas Brinkmann Victor Terskikh, Roderick E. Wasylishen, Guy M. Bernard∥, Zhuang Duan∥, Qichao Wu∥, and Gang Wu

The authors state their motivation.

Curcumin was selected in our study, in part because it is being touted as a wonder drug and is of intense interest to the pharmaceutical and medical community.31−33

This sounds quite exciting. Could low barrier hydrogen bonds be important in curing cancer?
Curcumin is a major ingredient of tumeric, the yellow spice, which features heavily in Asian cooking.
This got me Googling and it turns out the claims of a "wonder drug" are dubious.

Experimental studies of curcumin turn out to be particularly problematic, as explained in a blog post
Curcumin will waste your time by Derek Lowe. It is worth reading because it highlights the need for replication studies and publication of null results.

But it gets worse. References 31 and 32 have the same last author, Bharat Aggarwal, who it turns out has been the major proponent of the "wonder drug". In 2015 he "retired" from the University of Texas, following allegations of scientific fraud. By August 2106, eighteen published papers by him had been withdrawn.

To illustrate the problem of metrics, in 2016 Aggarwal had an h-index of 160, and in 2015, Thomson Reuters (ISI Web of Science) listed him among the World's Most Influential Scientific Minds.

I should stress that none of this invalidates the results of the hydrogen bonding paper that got me on this trail.

Tuning the electronic ground state of organic crystals by isotope substitution

One puzzle concerning organic charge transfer salts (such as those based on the BEDT-TTF molecule) is how the Mott metal-insulator transition can be tuned with substituting hydrogen with deuterium. I find it particularly puzzling because the relevant hydrogen bonds are weak and so one does not expect significant isotope effects.
Similar concerns are relevant to cases of isotopic polymorphism [where the actual crystal structure changes] in molecular crystals such as pyridine.

I recently came across a nice example that I do understand.

Hydrogen-Bond-Dynamics-Based Switching of Conductivity and Magnetism: A Phase Transition Caused by Deuterium and Electron Transfer in a Hydrogen-Bonded Purely Organic Conductor Crystal 
Akira Ueda, Shota Yamada, Takayuki Isono, Hiromichi Kamo, Akiko Nakao, Reiji Kumai, Hironori Nakao, Youichi Murakami, Kaoru Yamamoto, Yutaka Nishio, and Hatsumi Mori

The key to understanding how H/D substitution changes the electronic state is that there is a hydrogen bond between two of the organic molecules with an oxygen-oxygen distance of 2.45 A. As highlighted (and explained) in this paper, around this distance the geometric isotope effect is largest (the H bond length increases to almost 2.5 A), leading to a significant change in the energy barrier for proton transfer.

The figure below nicely shows, using DFT-based calculations and the measured crystal structures for both isotopes at two different temperatures, how the barrier changes, leading to a change in the charge state of the molecules.
The H and D isotopes are at the top and the bottom, respectively.

Hydrogen bonding at Berkeley

On Friday I am giving a talk in the Chemistry Department at Berkeley.
Here is the current version of the slides.

There is some interesting local background history I will briefly mention in the talk. One of the first people to document correlations between different properties (e.g. bond lengths and vibrational frequencies) of diverse classes of H-bond complexes was George Pimentel. 
Many correlations were summarised in a classic book, "The Hydrogen Bond" published in 1960.
He also promoted the idea of a 4-electron, 3 orbital bond which has similarities to the diabatic state picture I am promoting.
There is even a lecture theatre on campus named after him!

Hydrogen bonding talk at IIT-Kgp

Today I am giving a seminar, "Effect of quantum nuclear motion on hydrogen bonding" in the Chemistry Department at IIT Kharagpur. My host is Srabani Taraphder.

Here are the slides. The talk is mostly based on this paper.

A minor detail that matters in organic charge transfer salts

One helpful way to think about condensed matter is in terms of relative energy scales. This can help one decide what is important and what is not.
However, this does not always work, particularly in complex systems where new low energy scales can emerge.

For a long time there has been a "minor detail" about organic charge transfer salts based on the BEDT-TTF molecule that I have found rather annoying and puzzling.
It concerns the role of ethylene end groups on the molecule and their possible different conformations (eclipsed vs. staggered).

Why should the conformations matter?

I would think not. The overlap of the relevant electronic molecular orbitals which are largely centred on sulphur atoms are negligible as seen below in the HOMO (Highest Occupied Molecular Orbital) for a BEDT-TTF dimer.

The figures are taken from this paper by Edan Scriven and Ben Powell.

However, things are more subtle than I would have thought.

Here are some of the significant effects that result from these two different conformations. They have different energies and by thermal annealing in a crystal you can convert between them.
As a result disorder in a crystal can be controlled by varying the cooling rate.
In some materials there is even a glass transition around 80 Kelvin.

Examples of the dramatic effects of the disorder can be seen.

Resistance vs. temperature curve (see for example the figure below taken from here).

Suppression of the superconducting transition temperature.
This can be seen in the curves above.

Electrical noise experiments

Another dramatic effect of the ethylene groups that is much larger than most people expect is
Isotopic substitution of the hydrogen with deuterium in the ethylene groups can drive the Mott metal-insulator transition. 
This somehow arises from a geometrical isotope effect associated with hydrogen bonds between the ethylene groups and the anion.

It turns out that changing the conformation of the end group can have a significant effect on the parameters in the Hubbard model, that is the simplest possible effective Hamiltonian for these materials.
This is shown in this recent paper which estimates these parameters using DFT-based electronic structure calculations and Wannier orbitals to map onto a tight-binding model.

Influence of molecular conformations on the electronic structure of organic charge transfer salts Daniel Guterding, Roser Valentí, and Harald O. Jeschke .

In particular in going from Eclipsed (E) to Staggered (S) or visa versa is enough to cross the Mott insulator-metal phase boundary.
This provides a framework to understand the experimental puzzles discussed above.

One minor quibble. 
The authors estimate the Hubbard paper U (Coulomb interaction) for two holes on a BEDT-TTF dimer with a formula which is only valid in a particular limit.
The general formula for the energy of  two electrons on a two site Hubbard model is

where Um is the Hubbard interaction on a single dimer, V is the inter site Coulomb repulsion and t is the intersite hopping. The authors are assuming that Um - Vm is much larger than 4t which Scriven and Powell argue is not the case.
This will lead to quantitative changes but not change the main point that the conformational changes can produce a significant change in the Hubbard model parameters; particularly a large enough change to cross the Mott insulator-metal phase boundary.

Later I will write about the noise measurements (which I puzzled about before) which turn out to be a very sensitive probe of these two molecular conformations and their interconversion.

I thank Jens Muller for very helpful discussions about this work.

Another example of competing quantum effects in hydrogen bonds

Previously, I have highlighted how one of the organising principles for understanding quantum nuclear effects in hydrogen bonding is that of competing quantum effects.
This idea features in this talk and this recent review about water.

Basically, as the strength of the hydrogen bond in an X-H...Y systems increases, the zero point energy associated with the X-H stretch (bending) vibrational modes increases (decreases).
The effect manifests in a wide range of isotope effects where hydrogen is replaced with deuterium.
The relative magnitude of these competing effects changes with the bond strength, and so the sign of the isotope effects can be positive or negative.

This week I learned of another nice example of competing quantum effects in the paper.

Why Does Argon Bind to Deuterium? Isotope Effects and Structures of Ar·H 5O 2 + Complexes Laura R. McCunn, Joseph R. Roscioli, Ben M. Elliott, Mark A. Johnson, and Anne B. McCoy

The figure below shows the ground state geometry of the system before deuterium substitution.

When one H is replaced by a D it prefers to be one of the end H's not the central one, again due to zero point energy considerations. The paper answers the question as to where the Ar binds: to one of the end H's or the D? It turns out it is due to the D.
The H-bonding (or D-bonding) to the Ar lowers the stretch frequency and increases the bend frequency. It turns the zero point energy is lowered the most by D-bonding.

The conclusion nicely puts the work in a broader context.

in the case of deuterated water dimer, the deuterium-bound conformers of H2O · D2O or (HOD)2 have lower ZPEs than the H-bound conformers. 

Likewise, in the case of I-·DOH, for example, the D atom is preferentially in the bound position, whereas in F- · HOD, the H atom is in the bound position...  Cl- · DOH behaves like the I- complex. The OH-stretch frequency of the halide-bound OH bond in F- · H O is considerably lower than the OH-stretch frequency in water. This large difference in ZPE is the driving force for the H being in the shared position. This is analogous to the situation of the shared H in Zundel. In I-·H2O, the difference between the OH-stretch frequencies is small, and it is the dependence of the lower frequency in-plane and out-of- plane bends upon the location of the D atom that determines the energy ordering of the two isomers.

Review of nuclear quantum effects in water

Chemical Reviews just published an article

Nuclear Quantum Effects in Water and Aqueous Systems: Experiment, Theory, and Current Challenges 
Michele Ceriotti, Wei Fang, Peter G. Kusalik, Ross H. McKenzie, Angelos Michaelides, Miguel A. Morales, and Thomas E. Markland

(Trivia: 4 out of 7 authors have a surname beginning with M!)

One of the unifying themes in the review is that of competing quantum effects, illustrated above.

This article is a direct outcome of the NORDITA program, "Water - the most anomalous liquid" that I attended about 18 months ago.
Other reviews from the program will appear together in a special issue of the journal.
I must confess I was skeptical that we were going to be able to pull off these reviews, written by large teams of busy and opinionated individuals.
For ours, we are greatly in debt to Tom Markland for his perseverance and leadership.

We welcome any comments about the contents of the review.

Superconductivity, quantum hydrogen bonds, and rotten egg gas

I thought these three things had nothing to do with each other. Howevever, they do! Two enduring interests of mine, and commonly featured in this blog, are superconductivity and hydrogen bonding, particularly the role of quantum nuclear motion in the latter. The former is related to my beginnings as a condensed matter physicist and the latter more to my attempt, over the past decade, to become a chemical physicist. But, this post discusses how they can be intimately related.

About a year ago, a group reported superconductivity at 200 K, in a sulfur hydride compound under the incredibly high pressures above 100 Gigapascal (about 1000 kbar, which only a few years ago was not feasible). A nice commentary by Igor Mazin puts the discovery in context.

Aside: the starting compound H2S is commonly known as "rotten egg gas".

A longer paper is helpful and here I reproduce some of the abstract.

What superconducts in sulfur hydrides under pressure and why 
N. Bernstein, C. Stephen Hellberg, M. D. Johannes, I. I. Mazin, and M. J. Mehl

Intriguingly, superconductivity in the observed pressure and temperature range was predicted theoretically in a similar compound, H3S. Several important questions about this remarkable result, however, are left unanswered: 

(1) Does the stoichiometry of the superconducting compound differ from the nominal composition, and could it be the predicted H3S compound? 

(2) Is the physical origin of the anomalously high critical temperature related only to the high H phonon frequencies, or does strong electron-ion coupling play a role? 

We show that at experimentally relevant pressures H2S is unstable, decomposing into H3S and S, and that H3S has a record high Tc due to its covalent bonds driven metallic, which make this compound rather similar to MgB2, but unlike most other good conventional superconductors.

One thing that is striking is that there are few experiments (due to the high pressures) and the important role that theory (specifically, computations based on DFT-approximations) are playing. People are even debating differences of 20% in predictions of Tc!

The paper that particularly got my greatest interest was this one

Quantum Hydrogen-Bond Symmetrization and High-Temperature Superconductivity in Hydrogen Sulfide
Ion Errea, Matteo Calandra, Chris J. Pickard, Joseph Nelson, Richard J. Needs, Yinwei Li, Hanyu Liu, Yunwei Zhang, Yanming Ma, Francesco Mauri

It shows (again using DFT-based computations) that at the high pressures H3S undergoes a phase transition from a structure with a mixture of S-H covalent and S-H...S hydrogen bonds to a structure where the proton is delocalised between the two S atoms.

This structural phase transition is completely analogous to what happens in ice under pressure (and has a natural description in a simple model of H-bonding). Also there the quantum nuclear motion of the protons plays a significant role, leading to significant isotope effects (as observed in the superconductivity experiments).

Based on experience with these strong H-bonds, a couple of cautions are in order.

The S-H stretch vibrations (phonons) are highly anharmonic.

Computational results can vary significantly depending on what approximation (density functional or level of theory) is used.

Crystal structure transitions induced by isotopic substitution

At the level of the Born-Oppenheimer approximation replacing hydrogen with deuterium in a molecule or crystal should not change anything. The "chemical forces" responsible for all types of bonding, and encoded in a potential energy surface, remain the same. However, in reality changes can occur such as geometric isotope effects. This is because the zero-point energy associated with hydrogen bonds changes. The essential physics is described here.

In molecular crystals one can see not just small quantitative changes, such as changes in bond lengths of the order of a few hundredths of an Angstrom, but actual changes of the geometric arrangements of the molecules in the crystal. This "isotopic polymorphism" is nicely reviewed in a recent article by Klaus Merz and Anna Kupka.

A specific example is pyridine. The H and D polymorphs are shown below and taken from here. Note, the hydrogen bonds involved are relatively weak C-H...N bonds.

Why does this sensitivity to H/D matter in a broader context?

1. Understanding and calculating the relative stability of different possible crystal structures for organic molecular crystals represents a formidable theoretical challenge. This shows that one needs to have an accurate calculation of the relative zero-point energies of the competing structures, making the challenge even greater.

2. As I posted before, an intriguing and outstanding problem concerning superconducting organic charge transfer salts is how H/D substitution allows one to tune between Mott insulating and superconducting states.

3. Protons matter in molecular biology! Yet one cannot "see" them with X-ray crystallography. The alternative, which is increasing in viability and power, is neutron crystallography. However, this usually means replacing the hydrogens with deuterium. But, this means that the structure one determines is not necessarily the native structure. In many situations, the differences are probably small. However, in situations with short hydrogen bonds [see e.g. here] the difference could be significant.

4. The isotope effects can be particularly large in ferroelectric materials.

Quantum biology smells bad

I am skeptical of the grand and speculative claims of "quantum biology". 
There is a nice paper in PNAS which systematically considers the specific claim that smell is based on sensing the vibrational frequencies of particular molecules, and rebuts it from both theoretical and experimental points of view.

Implausibility of the vibrational theory of olfaction
Eric Block, Seogjoo Jang, Hiroaki Matsunami, Sivakumar Sekharan, Bérénice Dethier, Mehmed Z. Ertem, Sivaji Gundala, Yi Pan, Shengju Li, Zhen Li, Stephene N. Lodge, Mehmet Ozbil, Huihong Jiang, Sonia F. Penalba, Victor S. Batista, and Hanyi Zhuang.

I thank Suggy Jang for bringing the paper to my attention.

Telluride talk on competing quantum effects

Tomorrow I am giving a talk, "Competing quantum effects in hydrogen bonding: geometric isotope effects and isotope fractionation" at the meeting on Quantum effects in condensed phase systems

Here is the current version of the slides.

The talk is largely based on these two papers

Effect of quantum nuclear motion on hydrogen bonding

Isotopic fractionation in proteins as a measure of hydrogen bond length

Quantum nuclear effects in condensed phase chemistry

I am currently in Telluride for a meeting on Quantum effects in condensed phase systems. Two years ago I attended a similar meeting and in preparing it has been helpful to re-read several posts I wrote stimulated by that meeting.

In my first post, I listed possible quantum effects [zero-point motion, tunnelling, geometric phases, entanglement, ...] and pointed how generally one expects a condensed phase environment [protein, glass, solvent] for a molecular system will tend to reduce these quantum effects by decoherence.

I then asked two big questions.
Are there any instances where the environment can
A. enhance quantum effects?
B. lead to qualitatively new effects (e.g. associated with collective degrees of freedom) that are absent in the gas phase?

I clarified what I meant by a trivial vs. non-trivial enhancement of a quantum effect, from a physics point of view. An example of a "trivial" enhancement is where the environment changes the molecular geometry to enhance the effect. But I stressed that such an enhancement may be highly valuable from a chemistry or biochemistry point of view.

In a comment, Gautam Menon suggested that the Surface Enhanced Raman scattering was a nice example of a non-trivial enhancement. It is certainly spectacular, with enhancements as large as 10^11. However, I am not sure this is the type of quantum effect I am thinking of. The actual mechanism of the effect is still debated [see this paper] and I am not qualified to consider the relative merits of the alternative explanations, but it does look to me like it could be viewed as a semi-classical effect.

Tom Miller suggested to me that the solvation of single electrons and the associated polarons may be a suitable example of B.

I suggested that there were two important organising principles for describing and understanding quantum nuclear effects
1. Competing quantum effects
2. Rate processes can be dominated by rare quantum events.

I am looking forward to the meeting.

Calibrating a ruler for hydrogen bond lengths

I have just finished a paper with Bijyalaxmi Athokpam and Sai  Ramesh,
Isotopic fractionation in proteins as a measure of hydrogen bond length

If a deuterated molecule containing strong intramolecular hydrogen bonds is placed in a hydrogenated solvent it may preferentially exchange deuterium for hydrogen. This preference is due to the difference between the vibrational zero-point energy for hydrogen and deuterium.  It is found that the associated fractionation factor $\Phi$  is correlated with the strength of the intramolecular hydrogen bonds. This correlation has been used to determine the length of the H-bonds (donor-acceptor separation) in a diverse range of enzymes and has been argued to support the existence of short low-barrier H-bonds.

Starting with a potential energy surface based on a simple diabatic state model for H-bonds we calculate $\Phi$ as a function of the proton donor-acceptor distance $R$.  For numerical results, we use a parameterization of the model for symmetric O-H.... O bonds.  We consider the relative contributions of the O-H stretch vibration, O-H bend vibrations (both in plane and out of plane), tunnelling splitting effects at finite temperature, and the secondary geometric isotope effect. We
compare our total $\Phi$ as a function of $R$ with NMR experimental results for enzymes, and in particular with an empirical parametrisation $\Phi(R)$, used previously to determine bond lengths.

I welcome any comments or suggestions.

Future challenges with nuclear quantum effects in water

Last October I enjoyed attending a meeting, Water: the most anomalous liquid at NORDITA. One of the goals of the workshop was to produce a review article, co-authored by about a dozen working groups, each covering a specific aspect of water. I was in the group on "Nuclear quantum effects in water", led by Tom Markland. I was worried that this goal was a bit too ambitious. After all, I am into modest goals! However, it is all coming together, a great credit to the organisers. Our group is now finalising our "chapter". An important and difficult task is to write something concrete and useful about future challenges and directions.

Here I give a few of my own biased tentative thoughts. Comments and suggestions would be very welcome.

Over the past decade there have been several significant advances that are relevant to understanding nuclear quantum effects in water. It was only by writing this summary that I realised just how tangible and significant these advances are. I am not sure other fields I am familiar with have experienced comparable advances.

Experiment.
Deep inelastic neutron scattering reveals the momentum distribution of protons, and can be compared to path integral simulations, as described here. Furthermore, this has illuminated competing quantum effects, as described here.

Quantum chemistry.
New accurate intermolecular potential energy surfaces and force fields, such as MB-pol.

Computational.
Path integral simulations. Besides significant increases in computational power [Moore's law] making simulation of much larger systems and better "statistics" possible, there have been significant methodological advances, such as Ring Polymer Molecular Dynamics, and PIGLET.

New concepts and organising principles.
Competing quantum effects associated with the zero-point energy of O-H stretching and bending modes. The competition is particularly subtle in water, to the point that it can change the sign of isotope effects.
Dynamical properties such as proton transport being dominated by extremely rare events, associated with short hydrogen bonds.

Simple models.
The coarse-grained monatomic Water (mW) model captures many anomalies of classical water, showing their origin is in the tetrahedral bonding. A diabatic state model captures essential features of the potential energy surface of single hydrogen bonds, particularly the variation with the distance between oxygen atoms. The model does describes competing quantum effects.

These advances present some significant opportunities and challenges.

Experiment.
Resolving the ambiguity associated with interpreting the deep inelastic neutron scattering experiments. Going from the data to robust (i.e. non-controversial) spatial probability distributions for protons, particularly ones involving proton delocalisation would be nice.

Simulation.
The path integral simulations will only be as good at the potential energy surfaces that they use. For example, recent work shows how calculated isotope effects vary significantly with the DFT functional that is used. This is because the potential energy surface, particularly with respect to the proton transfer co-ordinate, is quite sensitive to the oxygen atom separation, and to the level of quantum chemical theory. This becomes particularly important for properties that are determined by rare events [i.e. thermal and quantum fluctuations to short hydrogen bonds].

Simple models.
Monatomic Water (mW) is completely classical. It would be nice to have a quantum generalisation that can describe how the water phase diagram changes with isotope (H/D substitution). Note there is already a problem because mW is so coarse-grained that it does not contain the O-H stretch. On the other hand, mW does describe the librational modes, and these do make a significant contribution to quantum nuclear effects in water, as described here.

I welcome suggestions and comments.

Quantum protons in enzymes

A number of proteins involve short strong hydrogen bonds [also known as low-barrier bonds] and there is considerable debate about how important or relevant they are for functionality. A particularly interesting enzyme is KetoSteroid Isomerase (KSI) which features such bonds. Its structure and mechanism has recently been elucidated by some beautiful experiments using mutants near the active site.

There is a nice paper
Quantum delocalization of protons in the hydrogen-bond network of an enzyme active site
Lu Wang, Stephen D. Fried, Steven G. Boxer, and Thomas E. Markland

This is a combined experimental and theoretical study of isotope substitution effects where the protons are replaced with deuterium. This allows one to probe the effects of the zero-point motion of the protons in hydrogen bonds. You can see zero-point energy with a pH meter.

The authors measure the change in the pKa [acidity] with H/D substitution of the different amino acid residues in the active site of KSI. Significantly, they find that for one of the KSI tyrosine's the pKa change is much larger than the change in water. Furthermore, they calculate this change using an ab initio path integral molecular dynamics simulation, obtaining a value in reasonable agreement with experiment.

The large isotope effect arises because of the significant quantum delocalisation of the protons in the H-bond network near the tyrosine's. This is illustrated in the figure below, showing the probability of finding a proton along the co-ordinate associated with proton transfer between the two different tyrosine's [when nu_16=0 the proton is equidistant between the Tyr16 and Tyr57 residues].

The simulation is a real tour de force. It uses a "force field" calculated "on the fly" from density functional theory with the B3LYP-D3 functional.

These simulations treat both the nuclear and electronic degrees of freedom quantum mechanically in the active-site QM region and also incorporate the fluctuations of the protein and solvent environment in the MM region. The simulations consisted of between 47 and 68 QM atoms and more than 52,000 MM atoms describing the rest of the protein and solvent. 

These simulations, which until recently would have been computationally prohibitive, were made possible by 

accelerating the path integral molecular dynamics convergence using a generalized Langevin equation, 

using new methods to accelerate the extraction of isotope effects, and 

exploiting graphical processing units (GPUs) to perform efficient electronic structure theory evaluations through an interface to the TeraChem code. 

Such a combination yielded almost three orders of magnitude speedup compared with existing AI-PIMD approaches.

Being able to perform such detailed stimulations will allow critical examination of controversial claims that short hydrogen bonds and proton tunnelling is a key ingredient in the functionality of specific enzymes.