Seminar on Numerical Analysis
# Grassmann extrapolation for Born-Oppenheimer Molecular Dynamics

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Aula Magna (Dipartimento di Matematica)
### Aula Magna

#### Dipartimento di Matematica

Description

Born-Oppenheimer Molecular Dynamics (BOMD) is a powerful, yet very demanding technique in computational quantum chemistry. Performing a BOMD simulation require, for each time step, to compute the energy and forces for a quantum mechanical system, which can also be embedded in a classical environment. When Density Functional Theory is used as a quantum mechanical model, the most time-consuming step for each energy/forces evaluation is the solution to the non-linear eigenvalue problem that stems from the discretization of the Kohn-Sham equations.

Limiting the number of iterations required to achieve a satisfactory convergence of the procedure is therefore paramount to reduce the cost - and therefore extend the applicability - of such simulations. The most important factor in reducing the number of iteration is starting from a good guess, usually in the form of a one-body reduced density matrix. In a BOMD simulation it is possible to extrapolate information from previous steps into a new guess, however, this is not straightforward due to the nonlinear constraints that the density matrix must satisfy.

In this contribution, we address this issue by performing the extrapolation on the tangent plane to the Grassmann manifold where the density is defined, by mapping the manifold to its tangent plane with a locally bijective map, which allows us to perform a linear extrapolation and then map the result back to the manifold, ensuring thus that all the

physical properties of the density matrix are correctly enforced.

Preliminary benchmark calculations show that the strategy is general and robust, and that even when using a naive linear extrapolation we obtain a strategy that outperforms the current state-of-the-art methods.

Streaming link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4