Unfortunately, 0.5% error still isn’t enough to be useful to the working chemist. The energy in molecular bonds is only a tiny fraction of the overall energy of a system, and appropriately predicting whether a molecule is stable can often depend upon just 0.001% of the overall energy of a system, or about 0.2% of the remaining “correlation” energy.
As an example, while the overall energy of the electrons in a butadiene molecule is sort of 100,000 kilocalories per mole, the difference in energy between different possible shapes of the molecule is just 1 kilocalorie per mole. That implies that if you wish to appropriately predict butadiene’s natural shape, then the identical level of precision is required as measuring the width of a football field all the way down to the millimeter.
With the arrival of digital computing after World War II, scientists developed a big selection of computational methods that went beyond this mean field description of electrons. While these methods are available a jumble of abbreviations, all of them generally fall somewhere on an axis that trades off accuracy with efficiency. At one extreme are essentially exact methods that scale worse than exponentially with the variety of electrons, making them impractical for all however the smallest molecules. At the opposite extreme are methods that scale linearly, but are usually not very accurate. These computational methods have had an infinite impact on the practice of chemistry — the 1998 Nobel Prize in chemistry was awarded to the originators of a lot of these algorithms.
Fermionic neural networks
Despite the breadth of existing computational quantum mechanical tools, we felt a brand new method was needed to handle the issue of efficient representation. There’s a reason that the most important quantum chemical calculations only run into the tens of hundreds of electrons for even probably the most approximate methods, while classical chemical calculation techniques like molecular dynamics can handle thousands and thousands of atoms.
The state of a classical system might be described easily — we just should track the position and momentum of every particle. Representing the state of a quantum system is way more difficult. A probability needs to be assigned to each possible configuration of electron positions. That is encoded within the wavefunction, which assigns a positive or negative number to each configuration of electrons, and the wavefunction squared gives the probability of finding the system in that configuration.
The space of all possible configurations is gigantic — in the event you tried to represent it as a grid with 100 points along each dimension, then the variety of possible electron configurations for the silicon atom can be larger than the variety of atoms within the universe. This is precisely where we thought deep neural networks could help.
Within the last several years, there have been huge advances in representing complex, high-dimensional probability distributions with neural networks. We now know learn how to train these networks efficiently and scalably. We guessed that, given these networks have already proven their ability to suit high-dimensional functions in AI problems, possibly they might be used to represent quantum wavefunctions as well.
Researchers resembling Giuseppe Carleo, Matthias Troyer and others have shown how modern deep learning might be used for solving idealized quantum problems. We wanted to make use of deep neural networks to tackle more realistic problems in chemistry and condensed matter physics, and that meant including electrons in our calculations.
There’s only one wrinkle when coping with electrons. Electrons must obey the Pauli exclusion principle, which suggests that they’ll’t be in the identical space at the identical time. It is because electrons are a form of particle referred to as fermions, which include the constructing blocks of most matter: protons, neutrons, quarks, neutrinos, etc. Their wavefunction have to be antisymmetric. When you swap the position of two electrons, the wavefunction gets multiplied by -1. That implies that if two electrons are on top of one another, the wavefunction (and the probability of that configuration) will probably be zero.
This meant we needed to develop a brand new form of neural network that was antisymmetric with respect to its inputs, which we called FermiNet. In most quantum chemistry methods, antisymmetry is introduced using a function called the determinant. The determinant of a matrix has the property that in the event you swap two rows, the output gets multiplied by -1, similar to a wavefunction for fermions.
So, you possibly can take a bunch of single-electron functions, evaluate them for each electron in your system, and pack the entire results into one matrix. The determinant of that matrix is then a properly antisymmetric wavefunction. The key limitation of this approach is that the resulting function — referred to as a Slater determinant — just isn’t very general.
Wavefunctions of real systems are often much more complicated. The everyday strategy to improve on that is to take a big linear combination of Slater determinants — sometimes thousands and thousands or more — and add some easy corrections based on pairs of electrons. Even then, this may occasionally not be enough to accurately compute energies.
