Many biological and physical effects can be traced back to basic chemical processes. To develop a deeper understanding, it is crucial to simulate these fundamental processes efficiently and with good accuracy. The main challenge is to capture the underlying quantum mechanical structures properly. The forces governing the dynamics of molecules, their stable configurations and reaction probabilities are represented by potential energy surfaces (PES), which are hypersurfaces in a typically high-dimensional space. Unfortunately, the evaluation of PES is in general very time-consuming. Since simulations of a chemical process often require thousands or even millions of such force evaluations, it is essential to develop smart discretizations for both the dynamics of a molecule, and the underlying PES. It is a main goal of project B06 to to explore the interplay between simulation methods for molecular quantum dynamics and the discretization of PES. The focus lies on discretizations that preserve the relevant physical and geometric properties of the original system. B06 connects research topics ranging from abstract mathematical physics to the implementation of numerical algorithms for specific molecules. This facilitates new insights into both the theoretical concepts, and practical applications.

# B06

Potential Energy Surfaces

## Discretizing the Forces for Molecular Quantum Dynamics

**
Potential energy surfaces are at the origin of the analytic description of molecular quantum dynamics and they give insight in it. B06 explores the accessibility of these high-dimensional structures for simulations of chemical processes which are fundamental for our understanding of basic principles in nature.
**

**Group:**B. Dynamics-
**Principal Investigators:** **Universities:****Term:**2012 - 2016

#### Mission-

#### Scientific Details+

Potential energy surfaces (PES) are central for the Born-Oppenheimer approximation, which is the key approximation for quantum molecular dynamics. Defining the potential V for the nuclei's effective Schrödinger equation,

* i∂ _{t}ψ(t,q) = (- 1/2m Δ_{q} + V(q))ψ(t,q), ψ(0,q) = ψ_{0}(q)*

a PES is a Lipschitz continuous function mapping open subsets of *R ^{d}* to

*R*. Away from exceptional sets, the so-called crossing manifolds, a PES is an analytic function. We note that it is not only the hypersurface {

*(q, v) ∈ R*} but also the function V, which is typically referred to as PES. The dimension d is large (e.g., d ≥ 9 for molecules with five atoms) and the evaluation of V is expensive, since it involves the solution of electronic Schrödinger equations. In recent years, chemical physicists have successfully worked on discrete PES representations allowing the numerical simulation of nuclear Schrödinger equations with more than thirty degrees of freedom.

^{d+1}| v = V(q)The main goal of this project is the mathematical investigation of discrete PES representations with a focus on structure preservation. There are two natural lines of research associated with this goal, the one concerned with the analytic regime, the other concentrating on the singularities.

In regions, where the PES is an analytic function, the discrete representation should, for example, preserve the permutation invariance of like atoms and reflect the high regularity of its continuous counterpart. Therefore, our investigation will start with discrete representations built on permutationally invariant polynomials. We will pursue the numerical computation of Taylor coefficients in high dimensions, following the concept of contour integration with minimized condition number. Least squares fitting, which currently prevails the chemical literature, and the construction of appropriate weights will also be explored.

For fixed nucleonic configuration *q ∈ R*^{d}, the value of the potential *V(q)* is an eigenvalue of a self-adjoint linear operator, the electronic Schrödinger operator *H _{el}(q)* associated with the nucleonic configuration

*q*. PES singularities occur, if the multiplicity of

*V(q)*changes when varying

*q*, that is, if different PES intersect. In this case, the nucleiâ€™s effective dynamics are governed by a smooth matrix-valued potential whose eigenvalues are the relevant singular PESs. Our investigation will aim at the discrete representation of these effective potential matrices. Since generically the singular manifolds, where the different PESs intersect, are of low codimension, i.e. the codimension ranges between two and five, we will first draw from the expertise on discrete singularity in the Collaborative Research Center. The research on discrete PES representations in high dimensions will then be pursued emphasizing probabilistic methods.

#### Publications+

##### Papers

###### A new phase space density for quantum expectations

**Authors: **
J. Keller, C. Lasser, and
Ohsawa, T.

**Note: **24pp

**Date: **
2015

**Download: **
arXiv

###### An invariant class of Hermite type multivariate polynomials for the Wigner transform

**Authors: **
H. Dietert, J. Keller, and
Troppmann, S.

**Note: **22pp

**Date: **
2015

**Download: **
arXiv

###### Non-Hermitian propagation of Hagedorn wavepackets

**Authors: **
C. Lasser, R. Schubert, and
Troppmann, S.

**Note: **33pp

**Date: **
2015

**Download: **
arXiv

###### Quasi-classical description of molecular dynamics based on Egorov's theorem

**Authors: **
Keller, Johannes and
Lasser, Caroline

**Journal: **The Journal of Chemical Physics,
141(5)

**Date: **
Aug 2014

**DOI: **
10.1063/1.4891517

**Download: **
external
arXiv

###### Landau–Zener type surface hopping algorithms

**Authors: **
Belyaev, Andrey K. and
Lasser, Caroline and
Trigila, Giulio

**Journal: **The Journal of Chemical Physics,
140(22):-

**Date: **
Jun 2014

**DOI: **
10.1063/1.4882073

**Download: **
external
arXiv

###### Hagedorn Wavepackets in Time-Frequency and Phase Space

**Authors: **
Lasser, Caroline and
Troppmann, Stephanie

**Journal: **Journal of Fourier Analysis and Applications,
20(4):679-714

**Date: **
May 2014

**DOI: **
10.1007/s00041-014-9330-9

**Download: **
external
arXiv

###### Corrections to Wigner type phase space methods

**Authors: **
Gaim, Wolfgang and
Lasser, Caroline

**Journal: **Nonlinearity,
27(12):2951-2974

**Date: **
2014

**DOI: **
10.1088/0951-7715/27/12/2951

**Download: **
external
arXiv

###### Propagation of Quantum Expectations with Husimi Functions

**Authors: **
Keller, Johannes and
Lasser, Caroline

**Journal: **SIAM Journal on Applied Mathematics,
73(4):1557-1581

**Date: **
Jul 2013

**DOI: **
10.1137/120889186

**Download: **
external
arXiv

##### PhD thesis

###### Quantum Dynamics on Potential Energy Surfaces-Simpler States and Simpler Dynamics

**Author: **
Keller, J.

**Date: **
Oct 2015

**Download: **
internal