Positivity 12 (2008), 653–666
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040653-14, published online April 5, 2008
Non-existence of a minimizer to the magnetic
Mattias Enstedt and Michael Melgaard
Abstract. In the presence of an external magnetic ﬁeld, we prove absence of
a ground state within the Hartree-Fock theory of atoms and molecules. The
result is established for a wide class of magnetic ﬁelds when the number of
electrons is greater than or equal to 2Z + K,whereZ is the total charge of
K nuclei. Positivity properties are instrumental in the proof of this bound for
the maximal ionization.
Mathematics Subject Classiﬁcation (2000). Primary 81V45; Secondary 35Q40,
Keywords. Magnetic Hartree-Fock equations, Ionization, Positivity.
In a recent paper  we proved existence of a solution in the form of a minimizer
for the nonlinear coupled Hartree-Fock equations of Quantum Chemistry in the
presence of an external magnetic ﬁeld described by a vector potential which is
supposed to be homogeneous of degree −1 at inﬁnity, roughly speaking. In the
opposite direction, we herein study absence of a minimizer. It turns out that much
weaker conditions on the magnetic ﬁeld are needed to establish nonexistence.
A molecule consisting of N electrons and K static nuclei with charges Z =
> 0, placed in an external magnetic ﬁeld B = ∇×A, A =
being the vector potential, is in quantum theory described
by the Hamiltonian
Expressed in Rydberg units.