Symbolic Algebra in Quantum Chemistry

Anton Amann, Ulrich Müller‐Herold

Per‐Åke Malmqvist

Ichio Kikuchi, Akihito Kikuchi

In this article, we demonstrate the first-principles computation of quantum chemistry through symbolic computation, using computational algebraic geometry. We have developed a Python program that generates symbolic formulas of one- and two-electron integrals. The approximations of those integrals by multivariate polynomials yield the set of equations required by quantum chemistry. We solve these equations in hybrid ways where numeric and symbolic computations are intertwined. We discuss the possibility of the extension of our scheme to treat the models beyond the Born-Oppenheimer approximation.

Hinne Hettema

So Hirata

Today, state-of-the-art computational methods and programs for quantum theory for electron correlation may not be developed entirely manually. They are often constructed with the essential aid of symbolic algebra systems automating lengthy and error-prone mathematical derivation and computer implementation, inevitable in these developments. Recent progress in this new paradigm of chemical theory development—complete automation—is reviewed.

Jinmo Zhao

Computational many-body theories in quantum chemistry, condensed matter, and nuclear physics aim to provide sufficiently accurate description of and insights into the motion of many interacting particles. Due to their intrinsic complexity, the development of such theories generally involves very complex, tedious, and error-prone symbolic manipulations. Here a complete solution to automate the symbolics in many-body theory development is attempted. General data structures based on an existing computer algebra system are designed to specifically address the symbolic problems for which there is currently no satisfactory handling. Based on the data structures, algorithms are given to accomplish common symbolic manipulations and simplifications. Noncommutative algebraic systems, tensors with symmetry, and symbolic summations can all enjoy deep simplifications efficient enough for theories of very complex form. After the symbolic derivation, novel algorithms for automatic optimization of tensor contractions and their sums are devised, which can be used together with automatic code generation tools. In this way, the burden of symbolic tasks in theory development can be vastly reduced, with the potential to spare scientists more time and energy for the actual art and science of many-body theories.