DOE ACTS COLLECTION Workshop Robust and High Performance Tools for Scientific Computing September 4-7, 2002

Calculate ground state energy of many-fermion systems by using Semidefinite Programming

The observables of a many-fermion system, and especially the ground state energy, can all be obtained from the first order and second order reduced density matrices of the system. Using these density matrices and a family of associated representability conditions one may obtain an approximation method for electronic structure theory that is in the mathematical form of Semidefinite Programming (SDP): minimize a linear matrix functional over a space of positive semidefinite matrices subject to linear constraints. This method(density matrix method with SDP) has great potential advantage over wave function methods when the particle number N is large. The wave function method gets ground state energy by calculate lowest eigen value of the Hamiltonian in configuration space. The dimension of the full configuration space increases exponentially with N, but in the density matrix method with SDP the dimension of the objective matrix increases only polynomially with N.

My research is focussed on application of Semidefinite Programming to many-fermion systems. This research includes two aspects: one is to establish a mature method to calculate ground state energy by using density matrix method with SDP;the other one is to find more representability conditions to make the optimization value given by SDP, which is lower bound of the ground state energy, closer to the correct ground state energy.

The main difficulty of the density matrix method with SDP comes from the fact that the number of linear constraints is huge when N is large. This requires a lot of memory and fast problem solver. Currently, I am using SeDuMi1.05, SDPT3.0, SDPA5.01, etc, as my problem solvers for relatively small model problems, but very soon I will use the DSDP4.5 package which can provide parallel calculation in PETSc environment.