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Linear Scaling Electronic Structure Methods with Periodic Boundary Conditions

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 ( ie , 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.

Figure 1. General structure of our periodic Kohn-Sham DFT program.

The general structure of our program is shown in Fig.1 and the current scaling of the different components in Fig. 2.

Figure 2. Computational scaling of our current implementation of the program.

During the last 12 months of this research grant, we have continued to focus our attention on three fronts. These are (1) MP2 for periodic systems, (2) k-point integration in metallic systems, and (3) screened hybrid functionals for solids.

We have applied our atomic-orbital formulation of second-order Møller–Plesset (MP2) theory for periodic systems [3] to study van der waals interactions between infinite polyacetylene (PA) chains and their effect on band gaps. The results show that these are important interactions, not to be neglected. A paper on the subject is currently being written for publication.

In another interesting development, we have formulated and implemented a new method to evaluate self-consistently quasiparticle energies of periodic systems within the diagonal approximation for solving Dyson's equation.

Our algorithm for band connectivity resolution [4] has been applied to a number of metallic systems. As a spin-off of this project, we have also developed a KS-DFT method for studying solids at finite temperature. A manuscript on the subject has been submitted to the Journal of Chemical Physics.

A recently developed hybrid exchange-correlation functional developed with a screened Coulomb potential [5] has been applied to study the properties of solids. The results obtained for lattice parameters, bulk moduli, and band gaps of a number of solids, support the notion that this functional performs significantly better than most other existing hybrid functionals, at a fraction of their computational cost, and at a rather low overhead when compared to other recently developed meta-GGA functionals [6] . This manuscript is currently in press [7] .

References

[1] Linear Scaling Density Functional Calculations with Gaussian Orbitals , G. E. Scuseria, J. Phys. Chem. A 103 , 4782 (1999).

[2] Achieving linear scaling for the electronic quantum Coulomb problem , M. C. Strain, G. E. Scuseria and M. J. Frisch, Science 271 , 51 (1996).

[3] 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).

[4] Efficient algorithm for band connectivity resolution , O. V. Yazyev, K. N. Kudin, and G. E. Scuseria, Phys. Rev. B 65 , 205117 (2002).

[5] Hybrid functionals based on a screened Coulomb potential , J. Heyd and G. E. Scuseria, J. Chem. Phys. 118 , 8207 (2003).

[6] Climbing the density functional ladder: Non empirical meta-generalized gradient approximation designed for molecules and solids, J. Tao, J. P. Perdew, V. N. Staroverov and G. E. Scuseria, Phys. Rev. Lett. 91 , 146401 (2003).

[7] Assessment and validation of a screened Coulomb hybrid density functional, J. Heyd and G. E. Scuseria, J. Chem. Phys. in press.

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