The density-to-pair-density map in density functional theory

Gero Friesecke
Technische Universtitat München

In my talk I discuss the relationship between two important collective variables in electronic structure theory, the single-electron density and the pair density. The traditional relationship, by integrating out degrees of freedom, is many-electron energy functional -> many-electron wavefunction -> pair density -> density. Abstract density functional theory establishes a somewhat converse path, which - in somewhat simplified form - reads density -> pair density -> exchange-correlation functional ->many-electron ground state energy.
In both paths, the weakest link is the first step: in the first path, one has to face a curse of dimension; and in the second path, one has to 'infer' a pair density model by combining physical principles with some kind of reference data. Pioneering contributions in the latter direction are due to Gunnarsson and Lundqvist (interpretation of the LDA via the pair density) and due to Burke, Perdew, Savin, and Wang (interpretation/construction of GGA's via the pair density).

In the talk, I will
(1) briefly recall the abstract construction by M.Levy of the density-to-pair-density map
(2) show how many common functionals including Dirac exchange arise from certain explicit approximations of this map, and
(3) present numerical computations of the exact map for one-parameter families of (1D) homogeneous and inhomogeneous model densities varying from 'concentrated' to 'dilute' (joint work with Huajie Chen (Beijing Normal University).
The pair densities are seen to develop remarkable multi-scale patterns which cross over from mean-field to strongly correlated behaviour and show strong dependence on the particle number. This highlights the difficulties in 'modelling' this map in a simple way, or - in ML language - in 'learning' such a map from a small dataset of supposedly representative pair densities.

Reference: H. Chen, G. F., Multiscale Model. Simul., 13(4), 1259ñ1289,

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