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.

The reduced density matrix (RDM) method, which is a variational calculation based on the second-order reduced density matrix, is applied to the ground state energies and the dipole moments for 57 different states of atoms, molecules, and to the ground state energies and the elements of 2-RDM for the Hubbard model. We explore the well-known N-representability conditions (P, Q, and G) together with the more recent and much stronger T1 and T2(') conditions. T2(') condition was recently rederived and it implies T2 condition. Using these N-representability conditions, we can usually calculate correlation energies in percentage ranging from 100% to 101%, whose accuracy is similar to CCSD(T) and even better for high spin states or anion systems where CCSD(T) fails. Highly accurate calculations are carried out by handling equality constraints and/or developing multiple precision arithmetic in the semidefinite programming (SDP) solver. Results show that handling equality constraints correctly improves the accuracy from 0.1 to 0.6 mhartree. Additionally, improvements by replacing T2 condition with T2(') condition are typically of 0.1-0.5 mhartree. The newly developed multiple precision arithmetic version of SDP solver calculates extraordinary accurate energies for the one dimensional Hubbard model and Be atom. It gives at least 16 significant digits for energies, where double precision calculations gives only two to eight digits. It also provides physically meaningful results for the Hubbard model in the high correlation limit.

We present a novel linear scaling ab initio total energy electronic structure calculation method, which is simple to implement, easily to parallelize, and produces essentially the same results as the direct ab initio method, while it could be thousands of times faster. Using this method, we have studied the dipole moments of CdSe quantum dots, and found both significant bulk and surface contributions. We also found a strong geometry dependence for the bulk dipole contribution.

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We present a new linear scaling ab initio total energy electronic structure calculation method based on the divide-and-conquer strategy. This method is simple to implement, easily to parallelize, and produces very accurate results when compared with the direct ab initio method. The method has been tested using up to 8,000 processors, and has been used to calculate nanosystems up to 15,000 atoms.

Atomic scale surface structure plays an important role in describing many properties of materials, especially in the case of nanomaterials. One of the most effective techniques for surface structure determination is low-energy electron diffraction (LEED), which can be used in conjunction with optimization to fit simulated LEED intensities to experimental data. This optimization problem has a number of characteristics that make it challenging: it has many local minima, the optimization variables can be either continuous or categorical, the objective function can be discontinuous, there are no exact analytic derivatives (and no derivatives at all for categorical variables), and function evaluations are expensive. In this study, we show how to apply a particular class of optimization methods known as pattern search methods to address these challenges. These methods do not explicitly use derivatives, and are particularly appropriate when categorical variables are present, an important feature that has not been addressed in previous LEED studies. We have found that pattern search methods can produce excellent results, compared to previously used methods, both in terms of performance and locating optimal results.

With energy-efficient architectures, including accelerators and many-core processors, gaining traction, application developers face the challenge of optimizing their applications for multiple hardware features including many-core parallelism, wide processing vector-units and onchip high-bandwidth memory. In this paper, we discuss the development and utilization of a new application performance tool based on an extension of the classical roofline-model for simultaneously profiling multiple levels in the cache-memory hierarchy. This tool presents a powerful visual aid for the developer and can be used to frame the many-dimensional optimization problem in a tractable way. We show case studies of real scientific applications that have gained insights from the Integrated Roofline Model.