Working Group 4 - Linear scaling and local orbitals

Pablo Ordejon, CSIC, Barcelona, Spain

Current Situation of the Area:

In the last decade, the use of local orbitals for ab-initio simulations in materials has experienced a tremendous growth in the solid state physics and materials science communities. One of the main reasons was the realisation that using locality concepts, the cost of electronic structure calculations can be reduced from the common N³ (N-cubed) scaling with the number of atoms, to a linear scaling (the so-called O(N)) methods; see Ref. [1]). After all, the bonding between atoms is a rather local affair, and there now exist ideas for how one can incorporate this fact in electronic structure calculations. This opens the possibility of performing, for the first time, realistic first-principles simulations in systems of up to several thousands of atoms (i.e., moving from the atomic scale to the nanoscale). This can obviously have a huge impact in many disciplines such as biology, nanotechnology or mineralogy, where the systems are very often too complicated for the standard techniques (see Ref. [2]).

The field is developing very fast in several directions. First, many theoretical advances are taking place, especially in the development of new types of localised basis sets that are able to exploit more efficiently the new localisation ideas. Besides, technical developments are being made to implement robust and efficient codes that use these techniques. Advance is also being made in the use of older, well established types of localised basis sets such as gaussians in a more efficient way, so as to achieve linear scaling.

Finally, the application of these programs to real problems is starting to have an impact on many different fields. The contribution of European research in this field has been very important: several of the members of this Working Group are world leaders, and have organised in Europe some of the most important workshops of this area in the last few years. We expect that this action to help us in maintaining a leading role in this strategic line.

Ongoing Projects in this Area:

An important part of the work being currently done in this area in Europe is the development of efficient simulation codes based on different types of local orbitals. Their degree of completeness at this point is quite different, but all of them will require intense effort in the next few years. The list is not comprehensive, but gives an idea of the current activity of the participants in this Working Group:

The SIESTA (P. Ordejon et al.) and PLATO (S. Kenny et al.) codes use basis sets of localised, finite range numerical atomic orbitals, which have the advantage of the ease of implementation of localisation ideas. The small number of orbitals per atom required to achieve a good accuracy provides an excellent performance, which allows the study of large systems: up to a few thousands of atoms in small computers, like Pentium PC's. However the accuracy attainable is not usually as high as with the plane wave codes that have the N³ scaling.
The CRYSTAL (R. Dovesi et al.) and GAPW (J. Hutter et al.) codes use gaussian basis sets. CRYSTAL has been used for a large number of years, and is not intended for linear scaling calculations, but is still being developed and used world-wide. GAPW is a recent development, with linear scaling in mind from the very beginning, and it shows great promise for efficient simulations in large systems, even for all-electrons calculations.
CONQUEST (D. Bowler et al.) is not based on atomic-like bases as the previous ones, but a real space grid technique, which makes the basis set essentially complete, providing a systematic accuracy comparable to that of Plane Waves. Being a real space technique, localisation is easily implemented. An efficient parallelisation enables CONQUEST to reach very large system sizes (up to tenths of thousands of atoms) in massive parallel computers. To be precise, this size has been reached in one demonstration calculation on a silicon crystal so that much development remains to be carried out.

Besides the methodological efforts, there is also intense work in the application of these newly developed methods to real problems in many disciplines, mainly by the groups developing the codes, but also by other users. For instance, SIESTA is being currently used to study problems as different and complex as the electronic conduction in DNA molecules (of obvious relevance for biology), the growth process of carbon nanotubes (of great importance in nanotechnology) and the melting and defect formation of SiO₂ (with important implications in mineralogy).

Goals for the Application Period (2003-2007):

The main goal of the Working Group is to make linear-scaling local-orbital methods a standard technique which can be used by non-specialist researchers in several disciplines. To achieve this, we need to achieve methodological and technical advances, as well as to improve the robustness and ease of use of the codes we are developing. Besides this general goal, we propose the following specific lines of research:

Integration or inter-relation of codes. The usage of parts (or at least technical advances) of one code into others would save a considerable amount of effort and redundancy.
The optimisation of basis sets can greatly benefit from code exchange: optimal, small atomic basis sets could be obtained from real-space grid calculations, which would considerably improve the transferability and accuracy of these bases.
Development of approximate, simplified methods like the Density Functional Tight Binding approach of G. Seifert and Th. Frauenheim. These have the advantage of being much more efficient than fully rigorous DFT techniques, at the cost of performing approximations.
The problem of metallic systems (which can not currently be described with O(N) techniques) must be tackled for these methods to be reliable for all kinds of systems. New ideas based on renormalisation group, wavelets, etc. need to be tested and implemented.
Besides solving the quantum-mechanical electronic equations in O(N) operations, a parallel effort needs to be devoted to the problem of dealing with the atomistic complexity of large systems. For instance, structural optimisations take an effort (in the sense of number of force evaluations) proportional to the number of atoms. Reducing this scaling will be essential for the study of complex systems as those in nanostructures or biomolecules.
Effort also needs to be devoted in the description of infrequent events (such as diffusion processes, or chemical reaction paths), which take place over time scales far beyond those accessible to standard Molecular Dynamics techniques.
Finally, we aim to extend the capabilities of current computational methods to systems out of equilibrium. For instance, electronic transport processes in nanostructures (in the basis of the current effort in moving from micro- to nano-electronics), involves the description of nanoscale systems in the presence of a potential drop and a current flowing through the device. This can not be tacked with the commonly used techniques, which assume thermodynamic equilibrium, and require new approaches.

Actions Proposed in this Working Group:

This Working Group should help coordinate much of the (large) effort that is being devoted in Europe in this subject. In this direction, we plan to carry out research visits between members, to coordinate the integration of codes and the distribution of technical advances among groups. Collaborations for the joint development of some of the lines of research described above will also be a main target.

The other main activity of the Working Group will be to reachout for new researchers entering the field. We will promote both the incorporation of developers (for improvement of existing codes or for the developement of new methods and programs) as well as users that can benefit from the existing codes in their research. For the first group, we plan to organize a number of workshops (as those organized in CECAM in 1998 and 2001) focused on the fundamental concepts and methodology of linear scaling and local orbital techniques. For the second group, we will organize hands-on courses on several of the existing codes, with the direct involvement of their developers. We will promote as much as possible the access of users to the existing codes, either as direct collaborations or via free access to the programs.

References:

[1] S. Goedecker, Linear Scaling Electronic Structure Methods, Rev. Mod. Phys. 71, 1085 (1999)

[2] P. Ordejon, Linear scaling ab-initio calculations in nanoscale materials with SIESTA. Phys. Stat. Sol (B) 217, 335 (2000)

MEMBERS OF THE WORKING GROUP [8 members]