Homotopy continuation methods for coupled-cluster theory in quantum chemistry

TL;DR

An overview of new interest in homotopy approaches to applied mathematics has emerged using both topological degree theory and algebraically oriented tools.

Abstract

Homotopy methods have proven to be a powerful tool for understanding the multitude of solutions provided by the coupled-cluster polynomial equations. This endeavor has been pioneered by quantum chemists that have undertaken both elaborate numerical as well as mathematical investigations. Recently, from the perspective of applied mathematics, new interest in these approaches has emerged using both topological degree theory and algebraically oriented tools. This article provides an overview of describing the latter development.

Figures (3)

• Figure 1: Sketch of possible homotopy paths. The solid line shows a path with no finite limit as $\lambda \to 0$, the dashed lines have the same limit, and the dotted-dashed line has a unique limit.
• Figure 2: Newton fractal of $p(z) = z^3 -1$. The white dots correspond to the roots $x_1 = 1+i \cdot 0$ and $x_{2,3} = -1/2\pm i \sqrt{3}/2$. The different colored regions, red, blue, and green, correspond to the basins of attraction of the roots $x_1$, $x_2$, and $x_3$, respectively.
• Figure 3: Visualization of the existence result of the Kowalski--Piecuch homotopy. Under certain assumptions there is a "tube" in amplitude space that connects a truncated CC solution to a FCC (FCI) solution. In principle, such trajectory allows us to select which solutions of a truncated CC calculation are "physical".