I am a mathematician, theoretical chemist, and computer scientist.

My research mainly concerns developing new methods for understanding the electronic structure of molecules. This involves three main things:

  • investigating the underlying mathematics of quantum chemistry;
  • developing algorithms using cutting-edge technology, including machine learning and GPUs;
  • applying the above to interesting molecular and biological systems.


Read summaries of the main areas of research I have been working on by clicking on any of the images below. A publication list and summary CV can be found lower down, and examples of my work can be found on my github page.

I am also a co-creator of the Argon educational molecular dynamics sandbox app.

Algorithm development and implementation for high-accuracy, fast quantum chemistry

Method development

Intermolecular interactions

Machine learning

Molecular dynamics

Statistical modelling

Basis sets

About Me

You can download a copy of my CV here. Please contact me for details of referees.

I have always been passionate about two things: mathematics and music. My love for these led to me spending a year at the Royal College of Music, followed by simultaneous undergraduate degrees in chemistry (at Sheffield) and maths (with the Open University). I have since worked and studied in Oxford, Bristol, and Southampton, before returning to do my PhD in Sheffield. My work is highly interdisciplinary - finding the links between these disciplines is fascinating, and lets me work with many different people. I have particularly enjoyed supervising several students and introducing them to the world of mathematical science.

Outside of work, you'll mostly find me playing guitar or piano, writing music, or with my friends debating which biscuit is best - all while wearing outrageously colourful clothes. Check out my profile on 500 Queer Scientists.


  1. libecpint: A C++ library for the efficient evaluation of integrals over effective core potentials, R. A. Shaw and J. G. Hill, J. Open Source Software, 2021, 6 (60), 3039
  2. Efficient enumeration of bosonic configurations with applications to the calculation of non-radiative rates, R. A. Shaw, A. Manian, I. Lyskov and S. P. Russo, J. Chem. Phys., 2021, 154 (8), 084102
  3. CHARMM-DYES: Parameterization of Fluorescent Dyes for Use with the CHARMM Force Field, R. A. Shaw, T. Johnston-Wood, B. Ambrose, T. D. Craggs and J. G. Hill, J. Chem. Theory Comput., 2020, 16 (12), 7817
  4. Computational investigations of dispersion interactions between small molecules and graphene-like flakes, T. J. Hughes, R. A. Shaw and S. P. Russo, J. Phys. Chem. A, 2020, 124 (26), 9552
  5. The completeness properties of Gaussian-type orbitals in quantum chemistry, R. A. Shaw, Int. J. Quantum Chem., 2020, 120 (17), e26264
  6. Psi4 1.4: Open-Source Software for High-Throughput Quantum Chemistry, D. Smith et al., J. Chem. Phys., 2020, 152 (18), 184108
  7. Below the FRET Limit, B. Ambrose, M. Willmott, T. Johnston-Wood, R. A. Shaw, J. G. Hill, and T. D. Craggs, Biophys. Journal, 2020, 118 (3), 615a
  8. A linear-scaling method for non-covalent interactions, R. A. Shaw and J. G. Hill, J. Chem. Theory Comput., 2019, 15 (10), 5352-5369
  9. A simple model for halogen bonds, R. A. Shaw and J. G. Hill, Inorganics, 2019, 7 (19)
  10. Midbond basis functions for weakly bound complexes, R. A. Shaw and J. G. Hill, Mol. Phys., 2018, 116 (11), 1460-1470
  11. Interplay between n→ π* interaction and hydrogen bond in an analgesic drug salicin, S. K. Singh, P. R. Joshi, R. A. Shaw, J. G. Hill and A. Das, PCCP, 2018, 20, 18361-18373
  12. Prescreening and efficiency in the evaluation of integrals over ab initio effective core potentials, R. A. Shaw and J. G. Hill, J. Chem. Phys., 2017, 147 (7), 074108
  13. Approaching the Hartree–Fock limit through the complementary auxiliary basis set singles correction and auxiliary basis sets, R. A. Shaw and J. G. Hill, J. Chem. Theory Comput., 2017, 13 (4), 1691-1698
  14. Halogen bonding with phosphine: Evidence for Mulliken inner complexes and the importance of relaxation energy, R. A. Shaw, J. G. Hill and A. C. Legon, J. Phys. Chem. A, 2016, 120 (42), 8461-8468


Click on the images below to view posters and talks I have presented in the course of my PhD.

Accelerating geometry optimisations with machine learning

Presented at the Helsinki Winter School in Theoretical Chemistry.

Integrals over effective core potentials

Presented at the Midlands Computational Chemistry Conference 2017 and TMCS Symposium 2017.

Accelerating high-accuracy electronic structure theory

Won first prize at the University of Sheffield Faculty of Science Showcase. Also presented at Midlands Computational Chemistry and RSC Theoretical Chemistry Group Conferences in 2018.

Analytical derivatives for basis function optimisation

Presented at the European Symposium on Computational Methods in Quantum Chemistry 2017.

Method development

Quantum chemistry involves solving a high-dimensional, non-linear partial differential equation for a system of fully interacting electrons and nuclei. This presents a massive challenge computationally, with the most accurate commonly used method - coupled-cluster - becoming 128 times as time consuming for each doubling of the system size. In my work, I have been developing and implementing a much more efficient, but just as accurate, approach specifically for non-covalent interactions.

As can be seen above, this allows much larger systems to be accurately treated. In particular, I am focusing on utilising modern technologies, such as graphical processing units, to allow for fully quantum calculations on condensed matter systems. For example, we are using the new method to look at the photoelectron spectra of solvated organic molecules, in collaboration with an experimental group.

Machine learning

As part of looking at ways to better handle the curse of dimensionality in quantum chemistry, I have been investigating using machine learning methods to aid in the solution of electronic structure problems. This has focused on three main strands:

  • optimising basis sets with predictable errors, using deep learning methods;
  • learning how to translate molecular graphs - skeleton diagrams of molecules - into near-optimal 3D geometries;
  • representing the electronic wavefunction directly as a neural network.

I am particularly interested in the last point, having proven that deep neural networks are in general mathematically mappable onto renormalisation group theories as used in statistical physics. This should allow for greater understanding of both the physical problem, and the underlying structure of the machine learning methods themselves.

Intermolecular interactions

Chemists like to neatly divide the world up in to molecules, and generally focus on the bonds within those molecules. However, the nominally weaker interactions between them explain phenomena as far-reaching as why asteroids form, to why ice is less dense than liquid water, and why geckos can stick to walls. Understanding the physics behind these interactions is very challenging, as they are weak enough to be difficult to measure and calculate, and usually happen in the condensed phases, making them hard to isolate.

Particularly prevalent and interesting are a type of interaction called a halogen bond. These are counter-intuitive, as they involve electrons interacting with halogens, which are generally considered to already be electron rich. Understanding the underlying mechanics of these bonds is important in the design of new drugs and materials, and has implications for many other similar interactions, such as chalcogen bonds.

Molecular dynamics

In collaboration with the group of Tim Craggs, I have been investigating the dynamics of fluorescent dyes attached to DNA (see above) and HIV-1 protease. Dr. Craggs is interested in how these dynamics affect a type of resonance energy transfer between the dyes, which he uses to experimentally measure the movement of biological systems. The insights gained could allow for development of better combination therapies for tackling HIV.

Statistical modelling

With rapid advances in computing, and increased ease of calculation, there is a huge amount of accurate chemical data available. When developing theories, it is necessary to make assumptions to make the problem soluble. The advantage of a statistical approach, therefore, is to look at this data in an unbiased manner, and reduce the dimensionality of the problem to glean information that may then be used going forward in choosing which assumptions are useful. This is heavily related to the machine learning and method development research above.

One particularly successful example is shown above, where a simple statistical model for halogen bonds outdoes considerably more expensive density-functional theory approaches. This is outlined in the paper "A simple model for halogen bonds".

Basis set development

One of the most interesting questions of quantum chemistry is how to represent wavefunctions, which are continuous, on discrete systems, i.e. computers. This is usually achieved with a set of mathematical functions, called a basis set. All computational methods currently used in quantum chemistry, and other branches of physics and maths, rely on these. Working in the Hill group, we specialise in developing compact, optimal basis sets for high accuracy applications. I am also interested in using novel approaches, such as machine learning, to streamline and improve this optimisation process. Finally, assumptions of completeness in the limit of an infinite number of basis functions are often made, and I have investigated the veracity of this for mean-field methods, such as Hartree-Fock. The extension to so-called correlated methods is an interesting problem in mathematical analysis.