20 research outputs found
Asymptotic behavior of the ground state energy of a Fermionic Fr\"ohlich multipolaron in the strong coupling limit
In this article, we investigate the asymptotic behavior of the ground state
energy of the Fr\"ohlich Hamiltonian for a Fermionic multipolaron in the
so-called strong coupling limit. We prove that it is given to leading order by
the ground state energy of the Pekar-Tomasevich functional with Fermionic
statistics, which is a much simpler model. Our main theorem is new because none
of the previous results on the strong coupling limit have taken into account
the Fermionic statistics and the spin of the electrons. A binding result for
Fr\"ohlich multipolarons is a corollary of our main theorem combined with the
binding result for multipolarons in the Pekar-Tomasevich model by the first
author and Griesemer in [AG14]. Our analysis strongly relies on the work of
Wellig [Well15] which in turn used and generalized methods developed by Lieb
and Thomas [LT97], Frank, Lieb, Seiringer and Thomas [FLST11] and Griesemer and
Wellig [GW13]. In order to take the Fermionic statistics into account, we
employ a localization method given by Lieb and Loss in [LL05]
Differentiability of the van der Waals interaction between two atoms
In this work we improve upon previous results on the expansion of the
interaction energy of two atoms. On the one hand we prove the van der
Waals-London's law, assuming that only one of the ground state eigenspaces of
the atoms is irreducible in an appropriate sense. On the other hand we prove
strict monotonicity of the interaction energy at large distances and, under
more restrictive assumptions, we provide the leading order of its first two
derivatives. The first derivative is interpreted as the force in Physics.
Moreover, the estimates of the first two derivatives provide a rigorous proof
of the monotonicity and concavity of the interaction energy at large distances
Asymptotic behavior of the ground state energy of a Fermionic Fröhlich multipolaron in the strong coupling limit
In this article, we investigate the asymptotic behavior of the ground state energy of the Fröhlich Hamiltonian for a Fermionic multipolaron in the so-called strong coupling limit. We prove that it is given to leading order by the ground state energy of the Pekar-Tomasevich functional with Fermionic statistics, which is a much simpler model. Our main theorem is new because none of the previous results on the strong coupling limit have taken into account the Fermionic statistics and the spin of the electrons. A binding result for Fröhlich multipolarons is a corollary of our main theorem combined with the binding result for multipolarons in the Pekar-Tomasevich model by [AG14]. Our analysis strongly relies on [Wel15] which in turn used and generalized methods developed in [LT97], [FLST11] and [GW13]. In order to take the Fermionic statistics into account, we employ a localization method given in [LL05]
Compactness of molecular reaction paths in quantum mechanics
We study isomerizations in quantum mechanics. We consider a neutral molecule composed of N quantum electrons and M classical nuclei and assume that the first eigenvalue of the corresponding N-particle Schr¨odinger operator possesses two local minima with respect to the locations of the nuclei. An isomerization is a mountain pass problem between these two local configurations, where one minimizes over all possible paths the highest value of the energy along the paths. Here we state a conjecture about the compactness of min-maxing sequences of such paths, which we then partly solve in the particular case of a molecule composed of two rigid sub-molecules that can move freely in space. More precisely, under appropriate assumptions on the multipoles of the two molecules, we are able to prove that the distance between them stays bounded during the whole chemical reaction. We obtain a critical point at the mountain pass level, which is called a transition state in chemistry. Our method requires to study the critical points and the Morse indices of the classical multipole interactions, as well as to improve existing results about the van der Waals force. This paper generalizes previous works by the second author in several directions
Differentiability of the van der Waals interaction between two atoms
In this work we improve upon previous results on the expansion of the interaction energy of two atoms. On the one hand we prove the van der Waals-London’s law, assuming that only one of the ground state eigenspaces of the atoms is irreducible in an appropriate sense. On the other hand we prove strict monotonicity of the interaction energy at large distances and, under more restrictive assumptions, we provide the leading order of its first two derivatives. The first derivative is interpreted as the force in Physics. Moreover, the estimates of the first two derivatives provide a rigorous proof of the monotonicity and concavity of the interaction energy at large distances
On boundedness of isomerization paths for non- and semirelativistic molecules
This article focuses on isomerizations of molecules, i.e. chemical reactions
during which a molecule is transformed into another one with the same atoms in
a different spatial configuration. We consider the special case in which the
system breaks into two submolecules whose internal geometry is solid during the
whole procedure. We prove, under some conditions, that the distance between the
two submolecules stays bounded during the entire reaction. To this end, we
provide an asymptotic expansion of the interaction energy between two
molecules, including multipolar interactions and the van der Waals attraction.
In addition to this static result, we proceed to a quasistatic analysis to
investigate the variation of the energy when the nuclei move. This paper
generalizes a recent work by M. Lewin and the first author in two directions.
The first one is that we relax the assumption that the ground state eigenspaces
of the submolecules have to fulfill. The second one is that we allow
semirelativistic kinetic energy as well