## Introduction

Quest is a general purpose electronic structure code based on density functional theory. It uses pseudpotentials and a high quality local orbital basis of contracted Gaussian functions in a linear combination of atomic orbitals (LCAO) approach to solve the Kohn-Sham equations fully self-consistently. This page gives a (very) brief description of the formulation of the method.

## Background

The basic algorithm that gives Quest its computational efficiency is the decomposition of the density and associated potentials into a sum of reference atoms plus a residual on a regular (FFT) grid. This fundamental algorithm is based in the work of Peter Feibelman in the 1980’s, who never published the comprehensive description of his technique. Feibelman wrote a number of papers that obliquely described the method, including:

- “Efficient Solution of Poisson’s Equation in Linear Combination of Atomic Orbitals (LCAO) Electronic Structure Calculations”

P.J. Feibelman, J. Chem. Phys.**81**, 5864 (1984) - “Efficient Solution of Poisson’s Equation in Linear Combination of Atomic Orbitals (LCAO) Calculations of Crystal Electronic Structure”

P.J. Feibelman, Phys. Rev. B**33**, 719 (1986). - “Force and Total Energy Calculations for a Spatially Compact Adsorbate on an Extended, Metallic Crystal Surface”

P.J. Feibelman, Phys. Rev. B**35**, 2626 (1987). - “Pulay-type Formula for Surface Stress in a Local Density Functional, Linear Combination of Atomic Orbital Electronic Structure Calculation”

P.J. Feibelman, Phys. Rev. B**44**, 3916 (1991).

This decomposition leads to a solution of the Coulomb problem in O(N) time, where N is a characteristic size of the problem (e.g., number of atoms). This eliminates the computation of the Hamiltonian as the limiting factor in the electronic structure calculations, and reduces the scaling of the Hamiltonian construction to O(N).

The SeqQuest code is highly efficient and enables very large scale calculations using very modest resources. 100-300 atom problems are routinely performed with desktop machines and 1000-atom are possible on (large-memory) single-processor workstations. The asymptotic O(N) scaling is not only theoretically possible, but actually achieved in practice after a surprisingly small number of atoms. Until the definitive paper for Quest is written, the code is cited using the following:

- P.A. Schultz (unpublished), for a description of the method see: P.J. Feibelman, PRB
**35**, 2626 (1987).

Continuing the tradition of describing embellishments of the method, a discussion of how to rigorously solve the Coulomb problem in supercell calculations is described in:

- “Local electrostatic moments and periodic boundary conditions”

P.A. Schultz, Phys. Rev. B**60**, 1551 (1999). - “Charged local defects in extended systems”

P.A. Schultz, Phys. Rev. Lett.**84**, 1942 (2000). - “Theory of defect levels and the ‘band gap problem’ in silicon”

P.A. Schultz, Phys. Rev. Lett.**96**, 246401 (2006).

We explored replacing the standard dense eigensolver in Quest with a linear solver, and the following paper discusses the result:

- “Unconstrained and constrained minimization, localization, and the Grassmann manifold: Theory and application to electronic structure”

D. Raczkowski, C.Y. Fong, P.A. Schultz, R.A. Lippert, and E.B. Stechel, Phys. Rev.**B64**, 155203 (2001).

This linear solver is not currently in the production version of the code. In practice, it did not improve over standard dense solvers until huge systems beyond what is reasonable on single-processing workstations, and the implementation required demolishing large parts of the code. We are in the process of parallelizing the code, and we are exploring alternate linear solver methods that become viable with the new capability.

A couple of recent applications highlight some of the special capabilities of the code:

- “First-principles approach for the charge-transport characteristics of monolayer molecular electronic devices: Application to hexanedithiolate devices”

Y.-H. Kim, J. Tahir-Kheli, P.A. Schultz, and W.A. Goddard III, Phys. Rev. B**73**, 235419 (2006). - “Theory of defect levels and the ‘band gap problem’ in silicon”

P.A. Schultz, Phys. Rev. Lett.**96**, 246401 (2006). - “First principles site occupation and migration of hydrogen, helium, and oxygen in beta-phase erbium hydride”

R.R. Wixom, J.F. Browning, C.S. Snow, P.A. Schultz, and D.R. Jennison J. Appl. Phys.**103**, 123708 (2008). - “Simple intrinsic defects in gallium arsenide”

P.A. Schultz and O.A. von Lilienfeld, Modelling Simul. Mater. Sci. Eng.**17**, 084007 (2009).

A comprehensive list of papers using (and describing) the method can be found on the Quest bibliography page.