Title: Semidefinite programming in electronic structure
 
Mituhiro Fukuda (New York University, Courant Institute of Mathematical Sciences), 

Zhengji Zhao (New York University, Department of Physics),
 
Bastiaan J. Braams (Emory University, Dept. of Mathematics and
Computer Science),

Michael L. Overton (New York University, Courant Institute of Mathematical Sciences), 
 
Jerome K. Percus (New York University, Courant Institute of Mathematical Sciences) 

The electronic structure problem we are investigating 
consists in determining the ground state energy of an
N-electron system subject to a given external potential.
It is one of the fundamental problems in quantum chemistry.

It was known since the 1960s that one could obtain a lower bound
for the ground state energy by solving a  semidefinite program,
however this approach faded away due to its great computational 
difficult.

Recently, with the advance of numerical semidefinite programming
computation, the electronic structure calculation through the
Reduced Density Matrix method has reinvigorated, and it
seems a very promising approach since it gives better accuracy
for the ground state energy than familiar approximations based 
on wavefunction for small molecules. At the same time, it is a very
challenging problem because the semidefinite programs which
arise here are huge-scale and they require about 7-8 digits
of accuracy for the objective function values.

One of the key issues to solve this problem is to choose
an appropriate formulation as a semidefinite program. 
Also since this problem is unusually large, it requires
parallel computation, careful memory management, check
points, etc. Several topics are under consideration now to solve 
larger problems like the conjugate gradient method, the
Bundle's method, the cutting plane method, and exploration 
of the problem structure.

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  MITUHIRO FUKUDA
  New York University
  Courant Institute of Mathematical Sciences
  mituhiro@cims.nyu.edu    Tel: +1-212-998-3120
  http://www.cims.nyu.edu/~mituhiro
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