Bastiaan Brams. Visiting Almut (Thursday/Friday).
Talk Friday, 2pm in Ker 228.
Title: "Electronic structure calculations via reduced density matrices
and semidefinite programming"
(Joint work with Shidong Jiang, Madhu Nayakkankuppam,
Michael L. Overton, Jerome K. Percus and Francois Oustry.)
Abstract:
The ground state variational problem for a many-electron system may be
formulated in terms of the one-body and two-body reduced density matrices
instead of the complete wavefunction [1-2]. A central issue is the
``representability problem'' of characterizing (through convex
inequalities) the class of functions that can arise as density matrix of a
many-electron system. The calculation of ground-state properties then
reduces to a linear optimization problem subject to the representability
constraints, of which the simplest ones are a finite set of linear
equalities and bounds on eigenvalues, as in semidefinite programming.
Significant work on this application was done in the 1970s [3], but
interest has waned since, partly because of the unsettled status of the
representability problem and partly because of the computational cost of
solving the associated optimization problem. Recent advances in the
application of interior-point and related methods to semidefinite programs
[4] have encouraged us to revisit the reduced density matrix approach to
electronic structure calculations. Our research follows two tracks. On
one hand we are using analytical and numerical approaches to develop new
representability conditions. Our new conditions extend fundamentally the
known "diagonal" conditions developed in the 1970s [5], which correspond
to the well-studied problem of characterizing the Cut Polytope [6]. While
a complete solution of this problem appears intractable, we hope that our
approach will bring the status of the complete representability problem to
the same level as has been achieved for the diagonal problem. On the
other hand, we are exploring numerically the strength of the known
representability conditions by calculations on model systems, in which we
compare the ground state energy found by the optimization approach with
the ground state energy found using full configuration interaction.
[1] A. J. Coleman: "Structure of fermion density matrices",
Rev. Mod. Phys. 35 (1963) 668--689.
[2] Claude Garrod and Jerome K. Percus: "Reduction of the N-particle
variational problem", J. Math. Phys. 5 (1964) 1756--1776.
[3] M. Rosina and C. Garrod "The variational calculation of reduced
density matrices", J. Comput. Phys. 18 (1975) 300--310.
[4] L. Vandenberghe and S. Boyd: "Semidefinite programming",
SIAM Rev. 38 (1996) 49--95.
[5] Walter B. McRae and Ernest R. Davidson: "Linear inequalities for
density matrices. II", J. Math. Phys. 15 (1972) 1527--1538.
[6] Michel Marie Deza and Monique Laurent: "Geometry of Cuts and Metrics",
Springer Verlag, Berlin, 1997.