Alumni Project
Linear Scaling Electronic Structure Methods with
Periodic Boundary Conditions
PI:
Gustavo E. Scuseria
Department of Chemistry, Rice University
Summary
We are developing electronic structure methods and linear scaling computational
programs for systems with periodic boundary conditions. Our efforts focus on density
functional, Hartree-Fock, and second-order perturbation theory (MP2) using Gaussian
orbitals. These tools will enable a variety of applications to polymers, surfaces,
and solids. Linear scaling algorithms are required to deal effectively with systems
containing a large number of atoms in the unit cell. Parallel algorithms for sparse
matrix multiplication are also a key ingredient.
In recent years, electronic structure methods have become an important tool in the
theoretical prediction of chemical properties and in aiding experiments in the
interpretation of new chemical phenomena. There is an extensive number of chemical
problems involving large systems with and without periodicity, where bond-breaking,
excitation energies, heats of reaction, and many other properties require a quantum
mechanical treatment for high-accuracy results. Examples of these types of systems
include technologically important conducting polymers and catalytic processes on
multiple surfaces.
Our research group has been active in the development of fast, linear-scaling
quantum chemistry methods for large scale applications. In previous work, we have
developed O(N) solutions for the methodological bottlenecks appearing in density
functional [1] (DFT) and Hartree-Fock (HF) theories. These developments include
the fast multipole method for linearizing the computational cost of the quantum
Coulomb problem [2], fast quadratures for the exchange-correlation potential, and
linear scaling alternatives to the diagonalization bottleneck. In this grant, these
methodological and computational tools are being adapted and expanded to deal with
periodicity (i.e., polymers, surfaces, and infinite solids).
The methods that we are developing will significantly enhance the current capabilities
of the scientific community to model and study periodic systems. This research impacts
many areas where quantum molecular modeling is routinely employed, including the chemical,
pharmaceutical, and defense industries. For these objectives to become reality, the
methodology needs to be first developed, followed by its computational implementation
on terascale machines.
The general structure of our program is shown in Fig.1 and the current scaling of
the different components in Fig. 2.
Figure 1. General structure of our periodic Kohn-Sham DFT program.
Figure 2. Computational scaling of our current implementation of the program.
During the first 18 months of this grant, we have focused our attention on three
fronts. These are (1) MP2 for periodic systems, (2) Brillouin zone integration in
metallic systems, and (3) alternatives to the Hamiltonian diagonalization.
(1) We have developed an atomic-orbital formulation of second-order Møller–Plesset
(MP2) theory for periodic systems. Notably, the inherent spatial decay properties of
the density matrix in the atomic orbital basis are exploited to reduce computational
cost and scaling. The multidimensional k-space integration is replaced by independent
Fourier transforms of weighted density matrices. The correlated amplitudes in the
atomic orbital (AO) basis are obtained in closed-form, compatible with a semi-direct
algorithm, thanks to the Laplace transform of the energy denominator. Similar to its
molecular counterpart, the Laplace quadrature can be accurately carried out by using
few integration points, 3 to7 depending on the application. This work has been published
in Atomic orbital Laplace-transformed second-order Møller–Plesset theory for
periodic systems, P. Y. Ayala, K. N. Kudin, and G. E. Scuseria, J. Chem. Phys.
115, 9698 (2001).
(2) An efficient algorithm for band connectivity (BC) resolution was developed. The
method uses only readily available band coefficients and the overlap matrix, and has
a low computational cost. The accuracy of the BC resolution is such that the method is
practical for meshes of k points typically used in systems with small unit cells. For
details see: Efficient algorithm for band connectivity resolution,
O. V. Yazyev, K. N. Kudin, and G. E. Scuseria, Phys. Rev. B 65, 205117 (2002).
(3) One interesting alternative to the Hamiltonian diagonalization step is to directly
work with the density matrix. We have proposed a powerful approach to purification of
the first-order density matrix based on minimizing the trace of a fourth-order polynomial,
representing the deviation from idempotency. Two variants of this strategy were discussed.
The first, based on a steepest descent minimization is robust and efficient, especially
when the trial density matrix is far from idempotency. The second, using a Newton–Raphson
technique, is quadratically convergent if the trial matrix is nearly idempotent. This
work has been published in Purification of the first-order density matrix using
steepest descent and Newton–Raphson methods, R. Pino and G. E. Scuseria,
Chem. Phys. Lett. 360, 117 (2002).
References
[1] Linear Scaling Density Functional Calculations with Gaussian Orbitals,
G. E. Scuseria, J. Phys. Chem. A 103, 4782-4790 (1999).
[2] Achieving linear scaling for the electronic quantum Coulomb problem,
M. C. Strain, G. E. Scuseria and M. J. Frisch, Science 271, 51-53 (1996).
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