Optimization Letters

, Volume 13, Issue 2, pp 429–437 | Cite as

A mathematical programming formulation for the Hartree–Fock problem on open-shell systems

  • Leo LibertiEmail author
  • Carlile Lavor
  • Nelson Maculan
Original Paper


The solutions of the time-independent Schrödinger equation provide a quantum description of the stationary state of electrons in atoms and molecules. The Hartree–Fock problem consists in expressing these solutions by means of finite dimensional approximations thereof. These are themselves linear combinations of an existing linearly independent set; best approximations are obtained when a certain energy function is minimized. In Lavor et al. (Europhys Lett 5(77):50006p1–50006p5, 2007) we proposed a new mathematical programming (MP) approach which enhanced the likelihood of attaining globally optimal approximations, limited to closed-shell atomic systems. In this paper, we discuss an extension to open-shell systems: this is nontrivial as it requires the expression of a rank constraint within an MP formulation. We achieve this by explicitly modelling eigenvalues and requiring them to be nonzero. Although our approach might not necessarily scale well, we show it works on two open-shell systems (lithium and boron).


Quantum chemistry Mathematical programming Linear independence Rank constraint 



The authors are thankful to Brazilian research agencies FAPERJ, FAPESP, CAPES and CNPq for financial support, and to M.A. Chaer Nascimento and T.M. Cardozo for useful discussions leading to the formulation of Eq. (4). The first author (LL) has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 764759 “MINOA”.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CNRS LIX, École PolytechniquePalaiseauFrance
  2. 2.Department of Applied Mathematics (IMECC-UNICAMP)University of CampinasCampinasBrazil
  3. 3.Universidade Federal do Rio de Janeiro, COPPE – UFRJRio de JaneiroBrazil

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