As well as the Schrödinger equation, the embedding method can be used to solve Maxwell’s equations, to calculate electromagnetic waves. There has been renewed interest in solving Maxwell’s equations over the last 20 or so years, and this has led to the development of photonics, plasmonics, and perhaps most important of all, the new science of metamaterials – all of these are about the manipulation of electromagnetic waves and photons, in various structures and situations. In the Schrödinger equation context, the main use of embedding has been to replace the semi-infinite substrate in surface calculations by the embedding potential, added on to the surface Hamiltonian. It can be used in an analogous way in Maxwell’s equations, to take care of the rest of space into which the electromagnetic waves can escape from some structure. But there’s another important use of electromagnetic embedding, and that is to to replace a dielectric object in a photonic structure, such as a metal sphere or cylinder. Why should we wish to do this? It’s because the electric field jumps across the surface of a dielectric, and this makes it more to difficult to calculate. If we replace these objects by embedding potentials, we only have to calculate the electric field outside, where it is likely to be smoothly varying and easier to calculate (by expanding it in plane waves, for example).
Having finished the chapter on the embedding potential/self-energy in tight-binding (actually, not quite), I’ve started the chapter on “Embedding Maxwell’s Equations”. The chapter on transport, which immediately follows the tight-binding chapter, I’ve put off until I have done some more background work and reading. The Maxwell chapter should be relatively straightforward to write, as this is the stuff which I’ve been writing papers on most recently. Beginning the chapter has reminded me of the difficulties I met with when I first started on Maxwell embedding, connected with the fact that one is dealing with the vector electric and magnetic fields, rather than the scalar wave-function of the Schrödinger equation. For a start, the embedding potential is replaced by an embedding tensor, a more complicated object. But more seriously, the solutions of Maxwell’s equations can become corrupted by approximate solutions of Laplace’s equation (the equation of electrostatics), which being approximate appear at finite frequency, mixed up with the solutions we are looking for! It still isn’t completely clear to me why these Laplace solutions crop up, but fortunately I found a way of dealing with them, pushing them down to zero frequency where they belong – if this was not possible, the embedding method wouldn’t have been much use for Maxwell’s equations. Needless to say, in the course of writing this chapter I have realised that there are more calculations I want to do, more figures I want to generate, and this all takes up time, good writing time!
I mentioned that I haven’t quite finished the chapter on embedding in tight-binding, and what remains to be done are sections on embedding in quantum chemical calculations. This includes work by Pisani and other quantum chemists from Turin, who had been working on embedding long before me. Their formalism is quite involved, and I still have to fully understand the rationale behind their method. I made some progress in understanding during a recent visit to my alma mater, Cambridge, where the excellent libraries provided the ideal working environment.
I’m writing this blog during a week’s holiday on the island of La Gomera, in the Canaries. What a beautiful place this is, and how lovely to have spring sunshine and temperatures – though it is very windy at the moment. I’ve been doing a lot of walking in the mountains of this tiny island, mountains formed by volcanic activity so they are very rough and rugged. Needless to say, in such terrain the walking is very hard. The flowers are one of the great delights, and the centre of the island is characterised by a sort of cloud forest, with tree heathers and laurels, – laurisilva. I’m staying in the main town on the island, San Sebastián; next week it’s back to another San Sebastián, the one in the Basque Country, to visit the physics institute there for a month of writing (and doing calculations).