Unfortunately, 0.5% error still isn’t enough to be useful to the working chemist. The energy in molecular bonds is just a tiny fraction of the total energy of a system, and correctly predicting whether a molecule is stable can often depend on just 0.001% of the total energy of a system, or about 0.2% of the remaining “correlation” energy. For instance, while the total energy of the electrons in a butadiene molecule is almost 100,000 kilocalories per mole, the difference in energy between different possible shapes of the molecule is just 1 kilocalorie per mole. That means that if you want to correctly predict butadiene’s natural shape, then the same level of precision is needed as measuring the width of a football field down to the millimeter.
With the advent of digital computing after World War II, scientists developed a whole menagerie of computational methods that went beyond this mean field description of electrons. While these methods come in a bewildering alphabet soup of abbreviations, they all generally fall somewhere on an axis that trades off accuracy with efficiency. At one extreme, there are methods that are essentially exact, but scale worse than exponentially with the number of electrons, making them impractical for all but the smallest molecules. At the other extreme are methods that scale linearly, but are not very accurate. These computational methods have had an enormous impact on the practice of chemistry – the 1998 Nobel Prize in chemistry was awarded to the originators of many of these algorithms.
Fermionic Neural Networks
Despite the breadth of existing computational quantum mechanical tools, we felt a new method was needed to address the problem of efficient representation. There’s a reason that the largest quantum chemical calculations only run into the tens of thousands of electrons for even the most approximate methods, while classical chemical calculation techniques like molecular dynamics can handle millions of atoms. The state of a classical system can be described easily – we just have to track the position and momentum of each particle. Representing the state of a quantum system is far more challenging. A probability has to be assigned to every possible configuration of electron positions. This is encoded in the wavefunction, which assigns a positive or negative number to every configuration of electrons, and the wavefunction squared gives the probability of finding the system in that configuration. The space of all possible configurations is enormous – if you tried to represent it as a grid with 100 points along each dimension, then the number of possible electron configurations for the silicon atom would be larger than the number of atoms in the universe!
This is exactly where we thought deep neural networks could help. In the last several years, there have been huge advances in representing complex, high-dimensional probability distributions with neural networks. We now know how to train these networks efficiently and scalably. We surmised that, given these networks have already proven their mettle at fitting high-dimensional functions in artificial intelligence problems, maybe they could be used to represent quantum wavefunctions as well. We were not the first people to think of this – researchers such as Giuseppe Carleo and Matthias Troyer and others have shown how modern deep learning could be used for solving idealised quantum problems. We wanted to use deep neural networks to tackle more realistic problems in chemistry and condensed matter physics, and that meant including electrons in our calculations.
There is just one wrinkle when dealing with electrons. Electrons must obey the Pauli exclusion principle, which means that they can’t be in the same space at the same time. This is because electrons are a type of particle known as fermions, which include the building blocks of most matter – protons, neutrons, quarks, neutrinos, etc. Their wavefunction must be antisymmetric – if you swap the position of two electrons, the wavefunction gets multiplied by -1. That means that if two electrons are on top of each other, the wavefunction (and the probability of that configuration) will be zero.
This meant we had to develop a new type of neural network that was antisymmetric with respect to its inputs, which we have dubbed the Fermionic Neural Network, or FermiNet. In most quantum chemistry methods, antisymmetry is introduced using a function called the determinant. The determinant of a matrix has the property that if you swap two rows, the output gets multiplied by -1, just like a wavefunction for fermions. So you can take a bunch of single-electron functions, evaluate them for every electron in your system, and pack all of the results into one matrix. The determinant of that matrix is then a properly antisymmetric wavefunction. The major limitation of this approach is that the resulting function – known as a Slater determinant – is not very general. Wavefunctions of real systems are usually far more complicated. The typical way to improve on this is to take a large linear combination of Slater determinants – sometimes millions or more – and add some simple corrections based on pairs of electrons. Even then, this may not be enough to accurately compute energies.