# Research

A main interest in our group is the study of mathematical models of molecules and solids, based on a strong background in mathematics and a broad interdisciplinary perspective. We are interested in the full spectrum of scales, from electronic to atomistic to continuum.

#### People:

- Mi-Song Dupuy
- Gero Friesecke
- Heinz-Juergen Flad
- Soeren Behr
- Benedikt Graswald

The chemical behaviour of an atom or molecule relies crucially on its electronic structure. Unlike in many other areas of science, in electronic structure theory a unifying and very precise mathematical model is available, namely the (nonrelativistic, Born-Oppenheimer) time-independent Schrödinger equation, first written down by Schrödinger for the hydrogen atom in 1926, and by Dirac for arbitrary atoms and molecules in 1929. The catch is that the equation is extremely complicated.

One source of complexity is the high-dimensionality of the equation, leading to the so-called problem of exponential scaling: the Schrödinger equation for an atom or molecule with N electrons is a partial differential equation in 3N dimensions, so direct discretization of each coordinate direction into K gridpoints yields K3N gridpoints. Thus the Schrödinger equation for a single CO2 molecule (N=22) on a coarse ten point grid in each direction (K=10) already has a prohibitive 1066 degrees of freedom.

Second, understanding electronic structure is a tough multiscale problem: the electronic state of a particular system, and hence its chemical behaviour, is not governed by total energies (mathematically: Schrödinger eigenvalues), but by small energy differences between competing states. Even for single atoms, these differences can already be several orders of magnitude smaller than total energies. For instance the spectral gap between the Carbon atom ground state and the first excited state is only 0.1 percent of the total energy. But this tiny gap is of crucial chemical importance as the two states have different spin and angular momentum symmetry (3P respectively 1D). The angular momentum symmetry of the excited state is that of a transition metal like Scandium or Yttrium, which has entirely different chemical behaviour.

A particular interest of our group is the analysis, design and validation of reduced quantum chemical models which allow to understand and accurately quantify chemical properties of interest with a moderate number of degrees of freedom. One of our innovations is the use, to this end, of rigorous asymptotic analysis of complex models (such as the full Schrödinger equation) in appropriate scaling regimes.

Another interest is the understanding, and possibly improvement, of tensor network methods as underlying the QC-DMRG method. Our recent work is concerned with the largely unresolved but in examples extremely beneficial task of optimizing the network, and the connection between tensor networks (TNs) and deep neural networks (DNNs).

#### Publications

- Mi-Song Dupuy, Gero Friesecke, Inversion symmetry of singular values and a new orbital ordering method in tensor train approximations for quantum chemistry, https://arxiv.org/abs/2002.07459
- G.Friesecke, D.Voegler, Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces, 2017 https://arxiv.org/abs/1801.00341
- C.Cotar, G.Friesecke, C.Klueppelberg, Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional, 2017, to appear in Arch.Rat.Mech.Analysis https://arxiv.org/abs/1706.05676
- H.Chen, G. Friesecke, Pair densities in density functional theory, Multiscale Model. Simul., 13(4), 1259–1289, 2015 Article
- G.Friesecke, F.Henneke, K.Kunisch, Sparse control of quantum systems, http://arxiv.org/abs/1507.00768
- H.Chen, G. Friesecke, Ch.Mendl, Numerical Methods for a Kohn−Sham Density Functional Model Based on Optimal Transport, J. Chem. Theory Comput. 10, 4360-4368, 2014 Article
- C.Cotar, G.Friesecke, B.Pass, Infinite-body optimal transport with Coulomb cost, Calc. Var. PDE 54, no. 1, 717-742, 2015 Article Preprint
- G.Friesecke, Ch.Mendl, B.Pass, C.Cotar, C.Klüppelberg, N-density representability and the optimal transport limit of the Hohenberg-Kohn functional, J. Chem. Phys. 139, 164109, 2013 Article Preprint
- C.Cotar, G.Friesecke, C.Klüppelberg, Density functional theory and optimal transportation with Coulomb cost, accepted for publication in Comm. Pure Appl. Math., 2012 http://arxiv.org/abs/1104.0603
- Ch.Mendl, G.Friesecke, Efficient Algorithm for Asymptotics-Based CI and Electronic Structure of Transition Metal Atoms, J. Chem. Phys. 133, 184101, 2010 Article Preprint
- G.Friesecke, B.D.Goddard, Atomic structure via highly charged ions and their exact quantum states, Phys. Rev. A 81, 032516, 2010 Article Preprint
- G.Friesecke, B.D.Goddard, Asymptotics-based CI models for atoms: properties, exact solution of a minimal model for Li to Ne, and application to atomic spectra, Multiscale Model. Simul. Vol. 7, No. 4, pp. 1876-1897, 2009 Article Preprint
- G.Friesecke, B.D.Goddard, Explicit large nuclear charge limit of electronic ground states for Li, Be, B, C, N, O, F, Ne and basic aspects of the periodic table, SIAM J. Math. Analysis Vol. 41, No. 2, pp. 631-664, 2009 Article Preprint
- P.M.W.Gill, A.T.B.Gilbert, S.W.Taylor, G.Friesecke, M.Head-Gordon, Decay behaviour of least-squares coefficients in auxiliary basis expansions,
*J. Chem. Phys.*123, 061101, 2005 Article - G.Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions,
*Arch. Rat. Mech. Analysis*169, 35-71, 2003 Article - G.Friesecke, Pair correlations and exchange phenomena in the free electron gas,
*Commun. Math. Phys.*184, 143-171, 1997 Article

#### People:

- Annika Bach
- Marco Cicalese
- Gero Friesecke
- Gianluca Orlando

Many problems in analysis, geometry, physics, engineering, and economics can be cast into the form of minimizing a functional F(u) among a class of admissible functions u. Important early examples of such functionals minimized in nature are: time of travel of a light ray (Fermat's principle in optics, 1662), action of a trajectory of a mechanical system (Hamilton's principle, 1834), and energy of the electrostatic field outside a charged body (Dirichlet's principle, Dirichlet, Kelvin, Gauss, 1840s). Minimizers of the latter problem solve Laplace's equation Δu=0, linking the calculus of variations to the theory of partial differential equations. Many fascinating modern day minimization problems can be viewed as far-reaching nonlinear extensions of Dirichlet's principle, in that minimizers solve nonlinear partial differential equations. Another important contemporary research area are discrete problems, a paradigm being the travelling salesman problem of finding the shortest connection between a large number of cities.

Much of our own work is motivated by questions in continuum mechanics and atomistic mechanics. We are especially interested in

- the crystallization problem: under which conditions is it optimal for atoms to assemble into crystalline order?
- atomistic-to-continuum limits (including effective theories for thin films, shape models)
- 3D-to-2D reduction of nonlinear elasticity theory to membrane-, plate- and shell theories
- phase transitions and fracture in 3D elasticity.

A main mathematical tool in our work is Gamma convergence, introduced by De Giorgi and developed notably by Dal Maso, Braides and coworkers. It provides a powerful and rigorous mathematical framework to pass from a finer-scale (or higher-dimensional) variational principle to a coarser-scale (or lower-dimensional) effective variational principle.

#### Publications

- Y.Au Yeung, G.Friesecke, B.Schmidt, Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape, Calc. Var. PDE 44, 81-100, 2012 Article Preprint
- S.Capet, G.Friesecke, Minimum energy configurations of classical charges: Large N asymptotics, Appl. Math. Research Express, doi:10.1093/amrx/abp002, 2009 Article Preprint
- B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy,
*J. Math. Pures Appl.*88, 107-122, 2007 - B. Schmidt, Minimal energy configurations of strained multi-layers,
*Calc. Var. Partial Diff. Eq.*30, 477 - 497, 2007 - G.Friesecke, R.D.James, S.Müller, A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence,
*Arch. Rat. Mech. Analysis*180 No. 2, 183-236, 2006 Article Preprint - G.Friesecke, R.D.James, S.Müller, The Föppl-von Karman plate theory as a low energy Gamma limit of nonlinear elasticity,
*C. R. Acad. Sci. Paris*, 2004 - G.Friesecke, R.D.James, S.Müller, Rigorous derivation of nonlinear plate theory and geometric rigidity,
*C. R. Acad. Sci. Paris*Ser. I 334, 173-178, 2002 - G.Friesecke, R.D.James, S.Müller, A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity,
*Commun. Pure Appl. Math.*Vol LV, 1461-1506, 2002 Preprint - G.Friesecke, F.Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice,
*J. Nonl. Sci.*12 No. 5, 445-478, 2002 - G.Friesecke, R.D.James, A Scheme for the Passage from Atomic to Continuum Theory for Thin Films, Nanotubes and Nanorods,
*J. Mech. Phys. Solids*48, 1519-1540, 2000

#### People:

- Marco Cicalese
- Gero Friesecke
- Rufat Badal
- Gianluca Orlando

Mechanical examples of truly multiscale systems include bulk solids, polymers, biological membranes, proteins. Their large-scale mechanical response is governed by a small number of active "modes", but both these modes and the constitutive parameters driving their behaviour depend crucially on the details of the underlying microstructure at a large spectrum of length- and timescales (e.g., in case of a bulk alloy, positions and energy barriers of the individual atoms, crystallographic phases, grains, dislocations, and cracks). A key challenge, which in complex examples lies well beyond current mathematical understanding, is to relate or couple the relevant mathematical models at the different scales.

Our own research concerns carefully chosen model systems which exhibit central difficulties appearing in more complex systems. In particular, we study

- atomic-to-continuum limits in solid mechanics (including effective theories for thin films, crystallisation, shape prediction, fracture, elasticity)
- effective models for shape, interaction and long time evolution of coherent modes in lattice systems
- bridging of time scales in molecular dynamics simulations.

#### Recent workshop

Development and Analysis of Multiscale Methods

IMA Annual Program Year Workshop,

University of Minnesota, USA, November 3-7, 2008

**Organizers:**

Frank L. H. Brown, Chemistry and Biochemistry, University of California

Anne M. Chaka, National Institute of Standards and Technology

Gero Friesecke, Mathematics, TU Munich

Kurt Kremer, Max Planck Institute for Polymer Research, Mainz

Yousef Saad, Computer Science and Engineering, University of Minnesota

Gregory A. Voth, Theoretical and Physical Chemistry, University of Utah

#### Publications

- G. Friesecke, O. Junge, P. Koltai, Mean field approximation in conformation dynamics, Multiscale Model. Simul. Vol. 8 No. 1, 254-268, 2009 Article Preprint
- B. Schmidt, Qualitative properties of a continuum theory for thin films,
*Annales de l'I.H.P. - Analyse non linéaire*25, 43 - 75, 2008 - B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models,
*SIAM Mult. Model. Simul.*5, 664-694, 2006 - J. Giannoulis, A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials,
*Discrete Contin. Dyn. Syst.*Ser. B. 6, 493-523, 2006 - J. Giannoulis, A. Mielke, The nonlinear Schr"odinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities,
*Nonlinearity*17, 551-565, 2004 - G. Friesecke, R. D. James, A Scheme for the Passage from Atomic to Continuum Theory for Thin Films, Nanotubes and Nanorods,
*J. Mech. Phys. Solids*48, 1519-1540, 2000 - G. Friesecke, R. L. Pego, Solitary waves on FPU lattices I: qualitative properties, renormalization and continuum limit, with R. L. Pego,
*Nonlinearity*12, 1601-1627, 1999

#### People:

#### MD network

Gero Friesecke (in his role as part time Professor at University of Warwick, U.K.) is co-director of the EPSRC-funded UK network Mathematical challenges of molecular dynamics.

#### Projects

- Quantum-classical coupling, especially passage from quantum molecular dynamics to classical molecular dynamics in the presence of Coulomb singularities and eigenvalue crossings (Gero Friesecke, Jannis Giannoulis, Luigi Ambrosio (Pisa))
- Mean field approximation and sparse tensor product discretization of transfer operators and the computation of conformation changes in molecules (Oliver Junge, Peter Koltai, Tim Wartewig, Gero Friesecke)
- Objective molecular dynamics for Ethane (Richard D. James (Minneapolis), Gero Friesecke, Florian Rupp)

#### Publications

- L. Ambrosio, A. Figalli, G. Friesecke, J. Giannoulis, T. Paul, Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data, to appear in Comm. Pure Appl. Math., 2011 http://arxiv.org/abs/1006.5388
- L. Ambrosio, G. Friesecke, J. Giannoulis, Passage from quantum to classical molecular dynamics in the presence of Coulomb interactions, Communications in PDE 35, 1490-1515, 2010 Article Preprint
- G. Friesecke, O. Junge, P. Koltai, Mean field approximation in conformation dynamics, Multiscale Model. Simul. Vol. 8 No. 1, 254-268, 2009 Article Preprint

#### People:

- Gero Friesecke
- Alice Mikikits-Leitner

Our particular interest is in

- existence, stability and interaction of coherent modes in nonlinear lattices such as the Fermi-Pasta-Ulam chain
- solutions of integrable systems of algebro-geometric (quasi-periodic) or rational type

#### Publications

- G.Friesecke, A.Mikikits-Leitner, Cnoidal Waves on Fermi-Pasta-Ulam lattices, http://arxiv.org/abs/1208.2805
- J. Giannoulis, A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials,
*Discrete Contin. Dyn. Syst.*Ser. B. 6, 493-523, 2006 - J. Giannoulis, Young-measure solutions to a generalized Benjamin-Bona-Mahony equation,
*Math. Methods Appl. Sci.*28, 607-629, 2005 - G. Friesecke, R. L. Pego, Solitary waves on FPU lattices III: Howland-type Floquet theory, IV,
*Nonlinearity*17, 207-228, 2004 - G. Friesecke, R. L. Pego, Solitary waves on Fermi-Pasta-Ulam lattices IV: Proof of stability at low energy,
*Nonlinearity*17, 229-252, 2004 - J. Giannoulis, A. Mielke, The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities,
*Nonlinearity*17, 551-565, 2004 - G. Friesecke, K. Matthies, Geometric solitary waves in a 2D mass-spring lattice,
*Discrete and Continuous Dynamical Systems*B, 3, 105-114, 2003 - G. Friesecke, K. Matthies, Atomic-scale localization of high-energy solitary waves on lattices,
*Physica*D 171, 211-220, 2002 - G. Friesecke, R. L. Pego, Solitary waves on FPU lattices II: Linear implies nonlinear stability,
*Nonlinearity*15, 1343-1359, 2002 - G. Friesecke, R. L. Pego, Solitary waves on FPU lattices I: qualitative properties, renormalization and continuum limit, with R. L. Pego,
*Nonlinearity*12, 1601-1627, 1999 - G. Friesecke, J. A. D. Wattis, Existence theorem for solitary waves on lattices,
*Commun. Math. Phys.*161, 391-418, 1994

#### People:

- Gero Friesecke
- Daniela Vögler

Our particular interest is in

- multi-marginal optimal transport (for example with Coulomb cost as arising in the strong correlation limit of density functional theory)
- intrinsic sparsity and efficient computation of high-dimensional optimal plans (theory and algorithms)

#### Publications

- G. Friesecke, D. Matthes, B. Schmitzer, Barycenters for the Hellinger--Kantorovich distance over R^d, 2019 https://arxiv.org/abs/1910.14572 , to appear in SIAM J. Math. Analysis
- G.Friesecke, A simple counterexample to the Monge ansatz in multi-marginal optimal transport, convex geometry of the set of Kantorovich plans, and the Frenkel-Kontorova model, SIAM Journal on Mathematical Analysis 2019, Vol. 51, No. 6, pp. 4332-4355 Article Preprint
- G.Friesecke, D.Voegler, Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces, SIAM J. Math. Analysis Vol. 50 No. 4, 3996-4019, 2018 Preprint Article
- G.Friesecke, F.Henneke, K.Kunisch, Frequency-sparse optimal quantum control. Math. Control Relat. Fields 8, no. 1, 155–176, 2018 Preprint
- C.Cotar, G.Friesecke, C.Klueppelberg, Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional, Arch. Ration. Mech. Analysis 228, no. 3, 891–922, 2018 Preprint
- H.Chen, G. Friesecke, Pair densities in density functional theory, Multiscale Model. Simul., 13(4), 1259–1289, 2015 Article
- H.Chen, G. Friesecke, Ch.Mendl, Numerical Methods for a Kohn-Sham Density Functional Model Based on Optimal Transport, J. Chem. Theory Comput. 10, 4360-4368, 2014 Article
- C.Cotar, G.Friesecke, B.Pass, Infinite-body optimal transport with Coulomb cost, Calc. Var. PDE 54, no. 1, 717-742, 2015 Article Preprint
- G.Friesecke, Ch.Mendl, B.Pass, C.Cotar, C.Klüppelberg, N-density representability and the optimal transport limit of the Hohenberg-Kohn functional, J. Chem. Phys. 139, 164109, 2013 Article Preprint
- C.Cotar, G.Friesecke, C.Klüppelberg, Density functional theory and optimal transportation with Coulomb cost, Comm. Pure Appl. Math. 66, 548-599, 2013 Article Preprint (for improved presentation and literature discussion see the final Article version)