The Reduced Density Matrix Method for Electronic Structure Calculations and the Role of Three-Index Representability Conditions

Zhengji Zhao (zz217nyu.edu)
Baastian Braams (braamsmathcs.emory.edu)
Mituhiro Fukuda (mituhirocims.nyu.edu)
Michael Overton (overtoncs.nyu.edu)
Jerome Percus (percuscims.nyu.edu)

Abstract: 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 work of R.~M.~Erdahl [Int.\ J.\ Quantum Chem.\ {\bf13}, 697--718 (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, SDCI, BD(T), CCSD(T) and full CI 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.

Keywords: Large Scale Semidefinite Programming

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Applications -- Science and Engineering (Basic Sciences Applications )

Category 3: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Technical Report, Courant Institute of Mathematical Sciences. Submitted to J. Chem. Phys.

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Entry Submitted: 10/16/2003
Entry Accepted: 10/16/2003
Entry Last Modified: 10/16/2003

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