CECAM/Psi-k Workshop

Local orbitals and linear-scaling ab initio calculations

CECAM, Lyon, 3-7 September 2001

Abstracts for Thursday, 6 September 2001

A summary will be given of the current state of the CONQUEST linear-scaling code, which performs full first-principles calculations of total energy and forces, using DFT and the pseudopotential method. Examples will be given of its practical performance on systems of over 10,000 atoms. Then the close relation between the linear-scaling and embedding problems will be analysed. Practical examples will be presented, in which embedding calculations based on CONQUEST methods have been performed on a substitutional impurity and the vacancy in silicon, and the Si (001) reconstructed surface.

The localization of the density matrix or the electronic wave functions is the key concept behind linear scaling algorithms. The fact that these quantities are not localized for metals at zero temperature is what hinders their application to these systems. Localization is recovered in metals at high electronic temperatures, and therefore linear scaling algorithms can be used to describe metals when T is large enough [1]. However, the temperatures required to make linear scaling calculations feasible are very high, so much that the energetics and dynamics obtained are significantly different from those at zero temperature. Horsfield and Bratkovsky [2] suggested the possibility of using the approximation due to Gillan [3] to obtain an estimate of the T=0 energies and forces from the free energy at finite temperatures.

In this work, we revisit the problem, and show that the entropy term in the free energy shows localization properties that allow it to be computed in O(N) operations, for large enough temperatures. We implemented these ideas in the context of the Li-Nunes-Vanderbilt functional [4], and show that the errors associated with the approximation and truncation of the entropy and the extrapolation from finite to zero temperature are small, and comparable to those of the localization approximation for insulators. This makes it possible to study metallic systems at zero temperature with linear scaling, with errors similar to those in insulating systems.

1. J L Corkill and K-M Ho, Phys. Rev. B 54, 5340 (1996).
2. A P Horsfield and A M Bratkovsky, Phys. Rev. B 53, 15381 (1996).
3. M Gillan, J. Phys. Condens. Matter 1, 689 (1989).
4. X-P Li, R W Nunes and D Vanderbilt, Phys. Rev. B 47, 10891 (1993).

Use of the inverse Cholesky factor S^-L is a promising approach to the transformation between orthogonal and non-orthogonal representations defined by the overlap matrix S of Gaussian-Type Atomic Orbital (GTAO) basis functions. However, linear scaling methods for the computation of S^-L have not been forthcoming.

For diagonally dominant overlap matrices that fall off from the diagonal, S^-L is typically much more sparse than S^-1 and has matrix elements that fall off exponentially from the diagonal. Rate of this fall off is controlled by Cond(S).

In this talk I will introduce a sparse-blocked version of Benzi and Tuma's AINV [1] for directly computing the inverse Cholesky factor in O(N), avoiding an incomplete linear solve which would otherwise be necessary. The sparse-blocked AINV employs (1) atom-blocked DGEMMs for efficiency, (2) ordering with a space filling curve to achieve diagonal dominance, and (3) a-priori spatial thresholding (distance based truncation) to achieve an O(N) complexity for well conditioned matrices.

I will show that well constructed GTAO basis sets, in which primitive exponents are related by geometric progression (i.e. even tempered [2] or well tempered [3] sequences), yield well conditioned overlap matrices and an early approach to linear scaling computation of S^-L, while other more popular basis sets such as 6-31G have a much larger condition number and approach linear scaling slowly.

1. M Benzi, R Kouhia and M Tuma, paper presented at ECCOMAS 2000, Barcelona 11-14 September 2000. Available at http://www.hut.fi/~kouhia/papers.html
2. R D Bardo and K Ruedenberg, J. Chem. Phys. 59 5956 (1973).
3. S Huzinaga and M Klobukowski, Theochem 44 1 (1988).

Recently there has been renewed excitement in experimental studies [1-3] of nitrogen that have apparently found high-pressure non-molecular phases similar to those predicted long ago [4-6]. The experiments also show that there is a remarkable hysterisis that apparently was the reason the phases were not observed before in low temperature experiments [7] even though a transition was identified in high temperature shock wave experiments [8]. However, the structure of the new phase is not established experimentally. To study the metastability issues and search for possible new forms of nitrogen at high pressures, we are carrying out simulations using the SIESTA local orbital code. Results established so far are: agreement with previous LDA plane wave calculations; modifications of previous theoretical predictions due to inclusion of GGA functionals; general agreement with shock wave data and other recent high temperature simulations [9]; results for several potential molecular and non-molecular structures; quenching of a disordered non-molecular phase under conditions similar to experiments; and strong temperature dependence of the transition rates indicating the tendency for long-time-scale metastability below room temperature.

1. A F Goncharov, et al., Phys. Rev. Lett. 85, 1262 (2000).
2. M I Eremets, et al., Nature 411, 170(2001).
3. E Gregoryanz, et al., Phys. Rev. B 64, 052103 (2001).
4. A K McMahan and R LeSar, Phys. Rev. Lett. 54, 1929 (1985).
5. R M Martin and R J Needs, Phys. Rev. B 34, 5082 (1986).
6. C Mailhiot, L H Yang and A K McMahan, Phys. Rev. B 46, 14419 (1992).
7. H Olijnyk and A P Jephcoat, Phys. Rev. Lett. 83, 332 (1999).
8. W J Nellis, et al., Phys. Rev. Lett. 53, 1661 (1984).
9. J D Kress, et al., Phys. Rev. B 63, 024203 (2000).

We will present our effort in the development of linear scaling methods and their applications to biological systems. The divide-and-conquer method we developed in 1991 was the first linear scaling solution to electronic structure calculations. Its recent applications to the simulation of protein dynamics will be presented. We also will describe our variational linear scaling approach with nonothorgonal localized molecular orbitals, and its comparison with other localized orbital approaches. Our numerical calculations demonstrate the advantages of this method.

Selected References:

1. Weitao Yang, "Direct Calculation of Electron Density in Density-Functional Theory", Phys. Rev. Lett. 66, 1438 (1991).
2. Weitao Yang and Taisung Lee, "A Density-Matrix Divide-and-Conquer Approach for Electronic Structure Calculations of Large Molecules", J. Chem. Phys. 163, 5674(1995).
3. Jose M Perez-Jorda and Weitao Yang, "An Algorithm for 3D numerical integration that scale linearly with the size of the molecule", Chem. Phys. Lett. 241,469 (1995).
4. Jose M Perez-Jorda and Weitao Yang, "An Simple O( N log N) Algorithm for the Rapid Evaluation of Particle-ParticleInteractions", Chem. Phys. Lett. 247, 484 (1995).
5. Tai-Sung Lee, Darrin York and Weitao Yang, "Linear-Scaling Semiempirical Quantum Calculations for Macromolecules", J. Chem. Phys. 105, 2744 (1996).
6. Weitao Yang, "Absolute Energy Minimum Principles for Linear Scaling Quantum Mechanical Calculations", Phys. Rev. B 56, 9294-9297 (1997).
7. James P Lewis, Charles W Carter Jr, Jan Hermans, Wei Pan, Tai-Sung Lee and Weitao Yang, "Active Species for the Ground-State Complex of Cytidine Deaminase: A Linear-Scaling Quantum Mechanical Investigation", J. Am. Chem. Soc. 120, 5407-5410 (1998).
8. Weitao Yang and Jose M Perez-Jorda, "Linear Scaling Methods for Electronic Structure Calculations", 1496-1513, Encyclopedia of Computational Chemistry, edited by Paul v. R. Schleyer, John Wiley & Sons, New York(1998).
9. Shubin Liu, Jose M Perez-Jorda and Weitao Yang, "Nonothorgonal Localized Molecular Orbitals in Electronic Structure Theory", J. Chem. Phys. 112, 1634-1644 (2000).
10. Haiyan Liu, Marcus Elstner, Efthimios Kaxiras, Thomas Frauenheim, Jan Hermans and Weitao Yang, "Quantum Mechanics Simulation of Protein Dynamics on Long Time Scale Made Possible", Proteins: Structure, Function and Genetics 44 484-489 (2001).