When applying to graduate school, I found it frustrating to browse the websites of professors which touted their research interests in technical jargon but did nothing to inform me what they actually did on a daily basis. To remedy this, I will now try to unpack some key concepts I engage with every day in my research.
Molecular orbitals are the building blocks of molecular physics and quantum chemistry. Anyone who has taken high school chemistry has probably heard the word `orbital.' In nontechnical usage, this word refers to the fact that unlike the conventional illustration, electrons do not orbit around atomic nuclei, but in fact exhibit more complicated behavior. In simple physical terms, an orbital is simply a place in an atom or molecule where an electron can live, and specifying the `address' of the electron determines all of its physical properties, such as its energy and angular momentum.
As the laws of quantum mechanics were being discovered, physicists were able to figure out how to apply these laws to determine the orbitals of simple atoms like hydrogen with pen and paper mathematics. However, in more complicated atoms and molecules, determining the orbitals can be a very complicated mathematical task, even though we know the physics. For this reason, molecular physics has always been intertwined with computer science. The ability of computers to perform complicated mathematical calculations was incredibly beneficial to the task of determining orbitals, and molecular physicists and quantum chemists have been perfecting the process ever since.
Today, molecular orbitals are not really an area of active research. Their existence is assumed, as is the fact that their properties can be specified by a computer performing an appropriate algorithm, such as the Hartree-Fock method. Instead, researchers care about how electrons are transported between orbitals, and the interactions that they experience when occupying orbitals. The situation is analogous to that of an astrophysicist interested in studying the velocity of a particular planet as it orbits the Sun. The existence of all the planets, moons, and the Sun is assumed, and it is taken for granted that others have already calculated basic properties like the mass, period of rotation, and shape of these celestial bodies. With these basics specified, the researcher can perform more interesting calculations about the many-body system.
The basic system of interest in molecular physics consists of a set of known molecular orbitals and some electrons which can occupy them. Although the complexity of such a system grows exponentially with the number of orbitals and the number of electrons, a physicist can often understand the important features with a simple tool called a model Hamiltonian. To understand model Hamiltonians, it is first necessary to discuss undergraduate physics education and why it is the way it is. If you were interested in physics in high school, you probably discovered at some point that physicists know the four fundamental interactions that govern all physical processes: the electromagnetic, strong nuclear, weak nuclear, and gravitational interactions. Therefore one might expect the undergraduate physics curriculum to consist of courses such as `The Electromagnetic Interaction,' `The Strong Nuclear Interaction,' etc. While we do learn electromagnetism as undergraduates, the other three interactions are rarely discussed. Instead, the standard physics curriculum consists of classical mechanics, quantum mechanics, and statistical mechanics. These courses teach us how to determine the physical behavior of systems (of point particles, quantum mechanical wavefunctions, and large collections of particles/wavefunctions, respectively) for a variety of possible physical interactions. In contrast, courses like electromagnetism are restricted to a specific interaction that happens to be present in our physical universe. This restriction might seem slight given that we are always in our universe, but in fact our universe actually looks very different depending on one's location and length scale. At human length scales, gravity dominates, the nuclear forces are negligible, and electromagnetism has subtle yet important consequences, e.g. through friction. However, the `friction interaction' looks very different than the fundamental subatomic scale electromagntic interation as formulated in the theory of quantum electrodynamics. Since the fundamental interactions can look so different in different systems, it is better to teach students how physical systems behave without specifying the exact interaction.
This same philosophy is at the heart of research with model Hamiltonians. I will illustrate this using a simple example from undergraduate quantum mechanics. Let's say that our system is a stationary spin-1/2 particle, i.e. the operator Sz representing the
Returning to our simple example, in undergraduate quantum mechanics one learns that a stationary electron in a magnetic field applied along the z axis is described by the Zeeman Hamiltonian H = -μBgSzB/ℏ. This specifies that in our model Hamiltonian, Δ(B) = -μBgB/ℏ. In other words, it tells us two crucial things about how the electron spin interacts with the magnetic field: it is linear in the field strength B, and the constant of proportionality is μBg/ℏ. Here, the Bohr magneton μB and reduced Planck's constant ℏ are fundamental constants, but g, the so-called g-factor, is not. It turns out that the Zeeman hamiltonian is typically correct but the value of g depends on the system. Thus, the Zeeman Hamiltonian you learned can be thought of as a model Hamiltonian with the g-factor as the system specific parameter. Indeed, there is active research into realizing physical systems with fortuitous values of g, so-called g-factor engineering.
My own research applies model Hamiltonians to better understand molecular magnets. Two fundamentally important features of a molecular magnet are the preference of the molecular spin to align with a specific axis, the so-called uniaxial magnetic anisotropy D, and the preference of the molecular spin to align with other spins in the environment, the so-called exchange coupling J. As a result, an illustrative model Hamiltonian is given by H = Σi Di (Siz)2 + Σi,j Jij Si Sj where the indices i, j run over the different molecular spins. This paper is a good investigation of this model Hamiltonian by some of my colleagues. On a daily basis what I typically do is write code to solve such a model Hamiltonian and investigate scattering from such a model Hamiltonian. This necessitates a familiarity with the mathematics of second quantized quantum chemistry, as well as computational tools which are necessary for larger systems. Luckily, a lot of computational tools for quantum chemistry have been developed as a part of the PySCF package which has proved invaluable to me.