Getting to the heart of quantum chemistry

For physicists like me struggling to understand quantum chemistry it often seems a plethora of acronyms, software, and approximations that appear to be detached from my knowledge of quantum many-body theory and chemical intuition. However, Seth Olsen recently recommended a review article by Michael Schmidt and Mark Gordon to me. I have started reading it and am fin

ding it understandable and helpful. I loved the introductory paragraph:

The essence of chemistry involves processes such as the formation and dissociation of chemical bonds, the excitation of an atom or molecule into a higher electronic state, and atomic or molecular ionization, in which electron pairs are separated. Although the end points of such processes (that is, the reactants and products) may frequently be reasonably well described using a simple wavefunction that corresponds to a single Lewis structure, this often cannot be said for the key species in between, such as transition structures, reactive intermediates, and excited electronic states. Such species often must be described with more complex wavefunctions in which several different arrangements of the electrons (electronic configurations) are taken into account.

[Aside: practically all chemical research concerning new materials for energy and information technologies is concerned with "key species in between". Density functional theory based methods work for the starting and ending species but generally fail miserably for the in between species.]

The key problem the article addresses is

The general form of a MCSCF [Multi-Configuration Self-Consistent Field] wavefunction iswhich is a linear combination of several configurations [referred to as configuration state functions (CSFs), Φ_K] [i.e. Slater determinants]. Each CSF differs in how the electrons are placed in the MOs [Molecular Orbitals], _i. The MOs are usually expanded in a basis of AOs [Atomic Orbitals], χ_μ. A MCSCF wavefunction is one in which both the configuration mixing coefficients A_K and the MO expansion coefficients C_μiare variationally optimized. Such a wavefunction is therefore distinct from a configuration interaction (CI) wavefunction, in which only the configuration mixing coefficients are variationally optimized.

When interactions do not seem to matter

At wednesday's meeting of the Quantum many-body theory reading group, we discussed the following points about the chapter, "Non-interacting electron gas".

To get more context on this chapter read, Chapter 2 of Ashcroft and Mermin, Solid State Physics. In particular, they show how the non-interacting fermion model of Sommerfeld can give a good semi-quantitative description of a wide range of properties of elemental metals such as heat capacity, magnetic susceptibility, and bulk compressibility.

Why is this success surprising? A simple estimate suggests that average potential energy due to the interactions of the electrons between each other is 1-10 times larger than the kinetic energy. Yet, the non-interacting fermion model ignores these electron-electron interactions.

So why does the theory work so well? For profound reasons embodied in Landau's Fermi liquid theory, the elementary excitations (quasi-particles) in a three-dimensional electron liquid (chapter 5) have a one-to-one correspondence to those of the non-interacting fermion model.

A key aspect to this is that the electron liquid has sufficiently high density that the Fermi energy is of the order of several eV (1 eV= 11,000 K)...

Consequently, most properties of elemental metals are determined by properties of the Fermi surface, i.e. only states near the Fermi energy.

All of this breaks down in one dimension (chapter 9) .

In two dimensions the existence of non-Fermi liquids is still controversial.

Next week we discuss chapter 2, "The Born-Oppenheimer Approximation."

An excellent (and depressing?) question

My colleague, Ben Powell, recently sent me the thought provoking email below. I think it raises an excellent question. I want to think about my answer more, before I post something. Although, it is a little depressing that I cannot immediately rattle off several things....

Others should feel free to post their thoughts, provided it does not involve self-promotion.

When I was walking home last night I started wondering about the following question (brought on by the fact I realised that it is now more than 10 years since I started my PhD).

What do we KNOW now about CMT [Condensed Matter Theory] that we didn't know 10 years ago?

I stress the T in CMT as I specifically want to rule things like "that the pnictides superconduct" as the seem like chance discoveries - however interesting they are, but really I want understanding about real materials, which is clearly rapped up in the border between theory & expt. I'd like things to be on a firm footing and generally agreed upon, but really I'm happy to be a little fast and lose with
that.

Advice for undergraduates

Next week classes for the year begin at Australian Universities. On sunday night I was at a dinner for new undergraduate students at Emmanuel College at UQ. The Principal, Stewart Gill, urged the students to look at a book What's wrong with University and how to make it work for you anyway, by a Canadian student Jeff Rybak.

I read some of the extracts on his website and found them quite helpful, partly because my daughter is starting university this year! She read some of it and we discussed it this morning.

Also, I think for faculty it is also important to look at such material to see things more from the students point of view, what they are struggling with, and how we can serve them better.

Reading group on quantum many-body theory

Today I am starting a weekly reading group on the basics of quantum many-body theory, mostly for postgraduate students. This is to partly fill the gap from the fact that we do not have graduate courses in Australia. We are going to read a chapter each week from the book, Advanced Solid State Physics, by Philip Phillips. I chose this for its clarity, brevity, and treatment of modern topics.

This initiative was inspired by a highly successful group that a colleague, Andrew Doherty, ran on quantum field theory last.

Previously, I was involved in groups that looked at Fulde's book, Electronic correlations in molecules and solids, and a review by Shaik and Hiberty on valence bond theory (now superseeded by their excellent book). I found both immensely valuable.

What is the value of such ventures?

I find understanding increases greatly when I read something and then talk about it. Furthermore, rather than getting stuck at points of derivations talking to someone else can remove the logjam.

The weekly meeting also provides accountability and discipline to keep reading.

Are you local or non-local?

I wrote in a previous post about the importance of listening to referees. I recently got back a referee report for this review on oxygen vacancies in cerium oxides (written with Elvis Shoko and Michael Smith). The report ended:

Finally, I cannot avoid suggesting the authors to have a look at: “A Conversation on VB vs MO Theory: A Never-Ending Rivalry? Roald Hoffmann, Sason Shaik, Philippe C. Hiberty. Accounts of Chemical Research 2003 36 (10), 750-756”. Perhaps, they will hear some familiar tones.

I read and enjoyed the paper [inspiring the post Marriage Counseling for Chemists] and am now trying to make concrete the connection with our work.

By a bond valence sum analysis of the structure around oxygen vacancies we consider the charge distribution arising from the two electrons left behind by removing an oxygen atom. We find rather subtle charge distributions; the two electrons do not simply localise on the two Ce ions next to the vacancy [the standard picture which is either assumed or claimed ot be supported by density functional theory based calculations]. Instead the two electrons can delocalise over the next nearest neighbours, but do not delocalise into the whole crystal.

At first the connection with the VB vs. MO debate was not clear but on reflection there may be some profound ones such as:

• Valence Bond (VB) theory tends to localise electrons too much. Molecular Orbital (MO) theory tends delocalise electrons too much.
• Our empirical valence bond sum approach is a very local picture and somehow capturing the same physics/chemistry as VB theory.
• LDA is close to MO theory (it is a band theory, i.e., a non-local picture) and tends to delocalise electrons too much. Many of the LDA, and LDA+U calculations on cerium oxides artificially force electrons to localise on cerium ions.

Can we make any more connection than the above?

Polite physicists do not discuss this at dinner parties

I had some interesting but brief discussions today with Andrew Briggs about Tony Leggett's perspective on the quantum measurement problem. (see his Viewpoint in Science in 2005). Personally, I find Leggett's perspective a little extreme. I just think we need to have a more nuanced view of what "reality" is.

He considers three different views on the interpretation of Quantum Mechanics (QM):

(a) QM is the complete truth about the physical world, at all levels, and describes an external reality.

(b) QM is the complete truth (in the sense that it will always give reliable predictions concerning the nature of experiments) but describes no external reality.

(c) QM is not the complete truth about the world; at some level between that of the atom and that of human consciousness, other non–quantum mechanical principles intervene.

...... Personally, if I could be sure that we will forever regard QM as the whole truth about the physical world, I think I should grit my teeth and plump for option (b).