The Imminent Materials Science Revolution

Throughout most of the 20^th century, tremendous progress was made in the field of quantum chemistry. This field is defined as the use of quantum mechanics to derive the properties of atoms, molecules and solid state systems. Quantum chemistry breaks down a system into its component parts and respective energies and then solves the resulting Schrödinger equation.

Figure 1: How computational time scales with larger chemical systems. (Source: Wikipedia)

For example, a hydrogen atom consists of a proton and an electron, each moving through space with certain kinetic energies and with a potential energy between them. Solving the Hamiltonian associated with the Schrödinger equation, even for a system as simple as the hydrogen atom, requires approximations. For instance, when solving the Hamiltonian for a hydrogen atom, the gravitational potential between the proton and electron is neglected because of its small size compared to the electromagnetic potential between them. The proton is taken to be fixed in space (Born-Oppenheimer approximation) since its motion is slow when compared to the electron darting around. Relativistic effects are ignored in the hydrogen atom system because the electron’s speed is far from relativistic. Therefore, common sense approximations are used to simplify the Hamiltonian to the most important aspects.

Once these approximations have been made, the electron orbitals can be calculated, the Hamiltonian can be solved, and the wave functions and energies of the electron in the hydrogen atom can be obtained. This is typically done by splitting the problem into a radial part and an angular part, the solution of the latter turning out to be spherical harmonics, from which the s, p, d, f, etc. electron orbitals are derived. Finally, when the electron’s wave functions and energies are known, its spectrum, ionization energy and other important measurable properties can be derived. Therefore, through simple calculation we can predict how the hydrogen atom will behave. In theory, if quantum chemistry can solve the Schrödinger equation for any atom, molecule or solid state system, it can predict how the material will behave.

Of course, quantum chemists aren’t there yet. At least not for systems much larger than a few thousand atoms, and for even those systems that are smaller than that, the results aren’t very precise. The problem is that the Hamiltonians associated with these systems can get complicated very quickly. Even slightly heavier elements require more approximations. Take helium, for example: two protons and two neutrons in a nucleus orbited by two electrons. The protons and neutrons are treated as a single object, which is fine because the energetics within the nucleus are not significantly affecting the electron wave functions. The problem is a bit trickier when examining how the two electrons affect each other. In general, going from a two-body problem to a three-body problem gets messy. As these electrons are orbiting they experience a varying potential energy from each other. This makes their Hamiltonian much harder to solve.

Figure 2: Exponential vs. polynomial scaling of computational time for larger chemical systems. (Source: Wikipedia)

When dealing with smaller systems it makes sense to use perturbation theory, stopping at the order of correction makes sense for a certain calculation. This works well for capturing the many-body effects in atoms, but what happens when molecules consist of many atoms? For example, the simple water molecule contains three nuclei and 10 electrons. An octane molecule (C₈H₁₈) contains 26 nuclei and 66 electrons, all interacting with each other. It’s not hard to see that the problem can scale very quickly. Even with approximations such as reducing the number of dimensions used to describe core electrons and focusing on the valence electrons, using molecular orbitals that are described as a linear combination of the constituent atomic orbitals, and using the summation of Gaussians to approximate those constituent atomic orbitals (or core and valence electrons) because they are easier to integrate, the problem becomes too computationally expensive once molecules become complex. Fields can be used to further reduce dimensions in order to open the door to calculating larger systems, but when system size increases accuracy decreases. Figure 1 illustrates the issue.

So is the problem intractable? Quantum chemists have done what they can in terms of approximations for the past 50 years and changes were incremental at best. What hope is there for a future in which materials are designed on a computer? The answer lies in Figure 2.

Figure 2 again illustrates the exponential growth in computation time with respect to the number of dimensions involved (basis set is the general term for the dimensions used to approximate the system) for traditional computers (red line). The yellow line represents the computational time required for a quantum computer to do the same calculations, which scales polynomially for larger systems. At first glance the difference may not seem significant, but if the graph is extended the difference between the exponential growth and polynomial growth grows quickly. In other words, quantum computers could solve far more complicated molecular systems, and far more accurately.

The current accuracy of computational chemistry done on supercomputers with respect to solid state systems and larger biological systems is enough to get researchers in the ballpark of the desired properties. A quantum supercomputer could likely design a material for a particular purpose with predicable properties. Consider applications like custom superconductors, cancer treatments, stronger structural materials and lower thermal conductivity materials. The result would undoubtedly be specialized materials for every application/design. A revolution in materials that would make things lighter, stronger, smoother and more insulating, on demand. Nearly any property would be possible.

This revolution becomes possible when quantum computers scale to the size of current computers. Later this year Google will unveil the first quantum computer capable of solving certain problems that are beyond the abilities of ordinary computers, a feat known as quantum supremacy. From that point forward it’s just a matter of scaling. The first significant implications to quantum chemistry could start to materialize within about a decade, changing materials science forever.