Nonlinear PDE Days

** **

**Harry
Yserentant**

**Regularity
and approximability of electronic wave functions**

`The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces`

`in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, `

`three spatial dimensions for each electron. Approximating them is thus inordinately challenging, and it is `

`conventionally believed that a reduction to simplified models, such as those of the Hartree-Fock method or`

`density functional theory, is the only tenable approach. We indicate why this conventional wisdom need not `

`to be ironclad: the unexpectedly high regularity of the solutions, which increases with the number of electrons,`

`the decay behavior of their mixed derivatives, and their antisymmetry enforced by the Pauli principle contribute`

`properties that allow these functions to be approximated with an order of complexity which comes arbitrarily`

`close to that for a system of two electrons. It is even possible to reach almost the same complexity as in the`

`one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.`