# Electronic structure of atoms and molecules

People: |
Gero Friesecke |

Heinz-Juergen Flad | |

Soeren Behr | |

Benjamin Graswald |

*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

*K*gridpoints. Thus the Schrödinger equation for a single Carbon atom (

^{3N}*N=6*) on a coarse ten point grid in each direction (

*K=10*) already has a prohibitive 10

^{18}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 (

^{3}P respectively

^{1}D). 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.

- Talk
^{}given at the occasion of the 60th birthday of Sir John Ball FRS. - Introductory tutorial
^{}given at a conference in Banff/Canada

## Publications

- 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^{}