The problem of determining the ground state energy of a system of N electrons, known as the electronic structure problem, is one of the central challenges in quantum chemistry. Traditionally, this problem gives rise to large linear or nonlinear Hermitian eigenvalue problems. But the reduced density matrix (RDM) method allows to solve this problem much faster by using large-scale semidefinite programming ([1], [2]). Semidefinite programms (SDPs) are a class of convex optimization problems which can be summarized as maximization of a linear function on the intersection of a linear affine space and the convex cone of block-diagonal positive semidefinite matrices [3].The full set of sufficient representability conditions for the RDM is not known. A consequence is that the semidefinite program is a relaxation, and thus yields a lower bound of the ground state energy.

In the literature it has been pointed out that the semidefinite programming problems must be solved to high accuracy, typically 7 decimal digits for the optimal value. However, sometimes, owing to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be obtained. For example, numerical experiences show that for the semidefinite software packages SDPA and SDPARA the accuracy of the optimal value is very poor for some mole-cules such that the lower bound is not below but above the full CI ground state energy.

In our work we present a novel and time efficient algorithm for calculating rigorous lower bounds for the ground state energy of molecules employing the RDM approach; that is, the algorithm provides a guaranteed accuracy. The algorithm is based on the semidefinite programming method described in [4]. By carefully postprocessing the output of a semidefinite programming solver, all rounding errors due to floating point arithmetic are rigorously estimated. The ground state energy of molecules taken from Benchmark problems presented in [1] were calculated by the RDM approach in a semidefinite mathematical form, and then compared with our rigorous results. Additionally, our results were compared with coupled cluster (CCSD) and Hartree-Fock energies. With our algorithms an upper bound of the RDM SDP optimal value can also be computed.

References

[1] Mituhiro Fukuda, Bastiaan J. Braams, Maho Nakata, Michael L. Overton, Jerome K. Percus, Makoto Yamashita, and Zhengji Zhao. Large-scale semidefinite programs in electronic structure calculation. Math. Program., 109(2):553–580, 2007

[2] Zhengji Zhao, Bastiaan J. Braams, Mituhiro Fukuda, Michael L. Overton, and Jerome K. Percus. The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions. The Journal of Chemical Physics, 120(5):2095–2104, 2004.

[3] Aharon Ben-Tal, Arkadi Nemirowski. Lectures on Modern Convex Optimization.

SIAM, Philadelphia (2001)

[4] C. Jansson, D. Chaykin, and C. Keil. Rigorous Error Bounds for the Optimal Value in Semidefinite Programming. SIAM Journal on Numerical Analysis, 46(1):180–200, 2007.