Zhengji Zhao

The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions

The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used P, Q, and G conditions. The additional conditions (called T1 and T2 here) are implicit in the work of Erdahl [Int. J. Quantum Chem. 13, 697 (1978)] and extend the well-known three-index diagonal conditions also known as the Weinhold-Wilson inequalities. The resulting optimization problem is a semidefinite program, a convex optimization problem for which computational methods have greatly advanced during the past decade. Formulating the reduced density matrix computation using the standard dual formulation of semidefinite programming, as opposed to the primal one, results in substantial computational savings and makes it possible to study larger systems than was done previously. Calculations of the ground state energy and the dipole moment are reported for 47 different systems, in each case using an STO-6G basis set and comparing with Hartree-Fock, singly and doubly substituted configuration interaction, Brueckner doubles (with triples), coupled cluster singles and doubles with perturbational treatment of triples, and full configuration interaction calculations. It is found that the use of the T1 and T2 conditions gives a significant improvement over just the P, Q, and G conditions, and provides in all cases that we have studied more accurate results than the other mentioned approximations.

Large-scale semidefinite programs in electronic structure calculation

It has been a long-time dream in electronic structure theory in physical chemistry/chemical physics to compute ground state energies of atomic and molecular systems by employing a variational approach in which the two-body reduced density matrix (RDM) is the unknown variable. Realization of the RDM approach has benefited greatly from recent developments in semidefinite programming (SDP). We present the actual state of this new application of SDP as well as the formulation of these SDPs, which can be arbitrarily large. Numerical results using parallel computation on high performance computers are given. The RDM method has several advantages including robustness and provision of high accuracy compared to traditional electronic structure methods, although its computational time and memory consumption are still extremely large.

Simple Hamiltonians which exhibit drastic failures by variational determination of the two-particle reduced density matrix with some well known N-representability conditions

Calculations on small molecular systems indicate that the variational approach employing the two-particle reduced density matrix (2-RDM) as the basic unknown and applying the P, Q, G, T1, and T2 representability conditions provides an accuracy that is competitive with the best standard ab initio methods of quantum chemistry. However, in this paper we consider a simple class of Hamiltonians for which an exact ground state wave function can be written as a single Slater determinant and yet the same 2-RDM approach gives a drastically nonrepresentable result. This shows the need for stronger representability conditions than the mentioned ones.