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# Dynamics of Partial Differential Equations

## Volume 12 (2015)

### Number 2

### On the Hartree–Fock dynamics in wave-matrix picture

Pages: 157 – 176

DOI: http://dx.doi.org/10.4310/DPDE.2015.v12.n2.a4

#### Author

#### Abstract

We introduce the Hamiltonian dynamics with the Hartree–Fock energy in new *wave-matrix* picture. Roughly speaking, the wave matrix is defined as the square root of the density matrix.

The corresponding Hamiltonian equations are equivalent to an operator anticommutation equation. This wave-matrix picture essentially agrees with the density matrix formalism. Its main advantage is that it is Hamiltonian and allows an extension to infinite particle systems like crystals in contrast with the standard HF theory.

Our main result is the existence of the global “reduced” wave-matrix dynamics for finite-particle molecular systems, and the energy and charge conservation laws. For the proof we extend the techniques, based on Hardy’s and Sobolev’s inequalities, to the wave-matrix picture.

#### Keywords

Hartree–Fock equations, reduced Hartree–Fock equations, density matrix, Hamilton equation, wave matrix, trace, Hilbert–Schmidt operator, commutator, anticommutator, Hardy inequality, Sobolev inequality, energy, charge, local solution, global solution, a priori estimate

#### 2010 Mathematics Subject Classification

34L25, 35L10, 47A40, 81U05