Taxonomy of Empirical Valence Bond methods

The use of Empirical Valence Bond methods to describe chemical reactions in complex environments (e.g. solvents and proteins) was pioneered by Warshel. I found the useful table below in a Comment by Jan Florian, arguing that some "new" methods with new acronyms are actually misnomers. [Aside: this is the same issue as The Best Paper Title and Abstract Ever].

Table 1: Taxonomy of the EVB and Earlier Methods^a

year	1954	1980	1991	1996	1997	1998	2001
acronym	VBb	EVB	EVB	AVB	extendedEVB	MS-EVB	MC-MM
principalauthor	Coulson1	Warshel2-6	Miller12	McCammon11,19	Borgis15	Voth14	Truhlar10,13
no.ofVBstates	3	2−8	2	2−8	20	6−10	2
Hiic	Morse	Morse+MM	Morse+MM	Morse+MM	Morse+MM	Morse+MM	MM
Hijc	expfunction	constorexp function	expfunction	const	expfunction	generalfunction	generalfunction
analyticalforcesd	no	yes	no	yes	yes	yes	yes
solvente	no	inHii	no	inHii	inHii	inHiiandHij	no
studiedenergysurf	Hbonding	enzymecatal, solnreacns	double-well potentials	phospholipase catalysis	hydratedproton	hydratedproton	Htransfer
systsize(no.ofatoms)	3	2to104	n/a	2to104	400	400	3−13

Deconstructing Fermi liquid scattering

What is a factor of 2 to theorists?

There is an interesting preprint The normal state of URu$_2$Si$_2$: spectroscopic evidence for an anomalous Fermi liquid by Tom Timusk and collaborators.

From a measurement of the frequency dependency of the conductivity they aim to extract the frequency and temperature dependence of the scattering rate of the Fermi liquid quasi-particles. General considerations suggest it has the form:

where omega = frequency and T=temperature. What is the value of b?
The authors find b ~ 1 but suggest that Landau would have b=4.

A few comments:

1. A fuller discussion of the theoretical literature is in a review Quantum criticality in organic conductors? Fermi liquid versus non-Fermi-liquid behaviour by Martin Dressel.

2. Hewson's book on The Kondo Problem cites work on the Anderson single impurity model which gives b=1 [see equation 5.102].

3. The Kadowaki Woods ratio [as discussed here] will be proportional to b.

4. I believe that measurements of the frequency and temperature dependence of ultrasound (zero sound) attenuation in liquid 3He are consistent with b=4.

I thank Nigel Hussey (who is currently visiting UQ) for bringing the paper to my attention.

Spectroscopy of a strong Hydrogen bond

The paper Fundamental Excitations of the Shared Proton in the H5O2+ and H3O2- complexes contains the graph below. It shows the frequency dependence of the Infra-red absorption intensity of the complex H3O2-, which can be viewed as a water molecule hydrogen bonded to a hydroxide anion (OH)-.

The authors state:

The H₃O₂^-·Ar spectrum is presented in Figure 2B, and it is dominated by a very strong band at 697 cm^-1, far below the bands displayed by the cationic system. To put the intensity of the 697 cm^-1 feature in context, this band exhibits a transition moment that is approximately 1000 times larger than that associated with excitation of the free OH stretches in this complex. 

This is fascinating as it suggests to me something fundamentally new is happening. However, it is not clear to me what physics could give such a large effect  (n.b. a thousand fold increase in transition moment means a million fold increase in intensity!). A previous post considered the empirical correlation between hydrogen bonding energy and increase in intensity absorption. That will certainly not give an effect of this magnitude.

Some of the authors have a recent theory paper which compares to the above experimental results. However, it appears to me that it only discusses mode frequencies and not intensities. 

The Sum of it all

A characteristic of strongly correlated metals and of phase transitions (e.g. superconducting and the Mott metal-insulator transition) is that they lead to a redistribution of spectral weight (e.g. due to the opening of an energy gap in the excitation spectrum). The total spectral weight is often constrained by sum rules. However, it turns out that when comparing to experimental data one needs to be careful about the high energy cutoffs one uses in these sum rules.

An important sum rule is the Ferrel-Glover-Tinkham sum rule which related the total spectral weight in the optical conductivity in the normal state (NS) to that in the superconducting state (SC) and the superfluid density n_s

Note the integrals extend to infinite frequency.

The schematic figure below shows the frequency dependence of the real part of the optical conductivity in NS (upper curve) and the SC state (lower curve). The green shaded area is the area which collapses into a delta function peak at (energy) omega=0 with weight n_s, characteristic of the infinite conductivity in the superconducting state.  

In a lattice the sum rules become modified because of band effects and the mass has to be replaced by a band mass. Many-body effects can reduce the sum for the NS, by as much as a factor of ten. Roughly this sum scales with the average electronic kinetic energy. Furthermore, one has to worry about finite (energy) frequency cut-offs in the integrals.

Today I read a nice PRB by Maiti and Chubukov which performs a systematic study of how for different model self energies (in both SC and NS) the sums behave, particularly as a function of the cutoff used.
This paper is motivated by some analysis of experimental data for the cuprates which suggested "sum rule violation". This might be expected if there are no quasi-particle poles in the normal state or if superconductivity is extracting spectral weight from energy above 2Delta. [This violation is interpreted as that the kinetic energy decreases in SC, opposite to what happens in BCS, due to particle-hole mixing].

Maiti and Chubukov find that for various model cases the results are quite sensitive to the cutoff. Furthermore, a marginal Fermi liquid model study by Norman and Pepin which did produce a kinetic energy decrease in SC [and so attracted significant attention] turns out to be parameter dependent.

The Figure above is taken from a Nature Physics viewpoint by Basov and Chubukov. They use it to partially justify Homes law which relates the superfluid density to the product of Tc and the dc conductivity at T=Tc.

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.

A "handy" resource

I delighted to find yesterday that the 91st edition of the Handbook of Chemistry and Physics is online (to subscribers). I think looking at the data is great anti-dote to the hubris of theorists, particularly reductionists. For example, how many of the listed properties (or even trends therein) can we accurately calculate, whether it is melting temperatures of simple solids or electron affinities of organic molecules?

The fact that spectroscopic properties of diatomic molecules are parametrised in terms of the Morse potential shows how good that empirical potential is!

Now the meaning of "handbook" becomes even stranger. Many editions ago the book ceased to be something you could hold in your hand! Now, in some sense it isn't even a book!

Covalent character of hydrogen bonds

Originally, it was thought that hydrogen bonds were largely electrostatic. Consider the situation D-H..A (where D and A denote donor and acceptor atoms respectively). Then one can calculation the electrostatic potential associated with the charge density associated with a D- anion and the A atom and one finds a minimum somewhere near where the proton sits. However, it turns out (particularly for "short" D-A distances, i.e. strong bonds) that there can be significant covalent character to the D-H and H...A bonds. Three complementary ways to describe this are in terms of

1. a valence bond (VB) picture where there is a quantum superposition of D-H and H-A bonds (which are partially covalent and partially ionic)
2. a molecular orbital picture of 4 electrons in 3 orbitals
3. a donor-acceptor natural orbital picture

But is there any definitive experimental evidence for covalent character? Two are cited:

• Compton scattering from ice (I need to read the PRL and understand it) 
• J Coupling of the D and A nuclear spins observed via NMR (first observation described in this JACS paper).

For a physicist and non-expert on NMR it is hard to follow the latter. However, I suspect what is going on is the following.

Please correct me if I am wrong. 

There is a hyperfine coupling between the D nucleus (A nucleus) and an electron which is largely localised on the D atom (A atom). But this D electron forms a "bond" (i.e. its spin is partly entangled with) with another electron localised on the H atom. But this electron in the H atom orbital also forms a bond with the electron in the A orbital. The net effect is there is some entanglement (covalency) between the electrons in the D and A orbitals. This in turn leads to entanglement (i.e. a spin-spin interaction) between the D and A nuclear spins. 

In contrast, if everything was purely classical and electrostatic there would be no quantum mechanical phase coherence between electron spins associated with the D and A atoms.