An approximation strategy to compute accurate initial density matrices for repeated self-consistent field calculations at different geometries

ABSTRACT

Repeated computations on the same molecular system, but with different geometries, are often performed in quantum chemistry, for instance, in ab-initio molecular dynamics simulations or geometry optimisations. While many efficient strategies exist to provide a good guess for the self-consistent field procedure, little is known on how to efficiently exploit the abundance of information generated during the many computations. In this article, we present a strategy to provide an accurate initial guess for the density matrix, expanded in a set of localised basis functions, within the self-consistent field iterations for parametrised Hartree–Fock problems where the nuclear coordinates are changed along with a few user-specified collective variables, such as the molecule's normal modes. Our approach is based on an offline-stage where the Hartree–Fock eigenvalue problem is solved for some particular parameter values and an online-stage where the initial guess is computed very efficiently for any new parameter value. The method allows nonlinear approximations of density matrices, which belong to a non-linear manifold that is isomorphic to the Grassmann manifold, by mapping such a manifold onto the tangent space. Numerical tests on different amino acids show promising initial results.

GRAPHICAL ABSTRACT

KEYWORDS:

• Self-consistent field
• density guess
• ab-initio molecular dynamics
• geometry optimisation

Acknowledgments

This paper is dedicated to Prof. Jürgen Gauss in honour of his sixtieth birthday. F.L. would like to express his deepest gratitude to Prof. Gauss, not only for being an exceptional, dedicated mentor, who deeply cares for his pupils both from a scientific and a human point of view, but also for all the things done together, that range from going to the opera, to skiing, to enjoying a nice dinner and a good bottle of wine. Having the chance of working in Jürgen's group has made a big difference not only for my scientific growth, but also for my personal one.

This work has been created based on an interdisciplinary collaboration between theoretical chemistry and applied mathematics. The mathematics community had the chance to meet and interact with Prof. Gauss in interdisciplinary workshops, such as the Oberwolfach workshop on ‘Mathematical Methods in Quantum Chemistry’ held in March 2018 for example, and learnt to know him as a researcher who is interested in fundamental concepts and answers to scientific questions, which of course is a common interest with mathematics. In this regards, we think that this article reflects this particular facet of Prof. Gauss' research activities.

The roots of this project lie in a project given to students in the so-called CAMMP Week Pro (https://blog.rwth-aachen.de/cammp/angebot-fuer-studierende/) (where CAMMP stands for Computational And Mathematical Modelling Program) where students work during one week intensively in a team on a research-related problem. It was fascinating to see what students can achieve within one week and their work has definitively contributed to get the ball rolling in this project.

Finally, the authors acknowledge Eric Cancès and Yvon Maday for fruitful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the French ‘Investissements d'Avenir’ program, project Agence Nationale de la Recherche (ISITE-BFC) (contract ANR-15-IDEX-0003).