Linear-scaling self-consistent implementation of the van der Waals density functional

An efficient linear-scaling approach to the van der Waals density functional in electronic-structure calculations is demonstrated. The nonlocal correlation potential needed in self-consistent calculations is derived in a practical form. This enables also an efficient determination of the Hellmann-Feynman forces on atoms. The numerical implementation employs adaptive quadrature grids in real space resulting in a fast and an accurate evaluation of the functional and the potential. The approach is incorporated in the atomic orbital code SIESTA. The application of the method to the S22 set of noncovalently bonded molecules and comparison with the quantum chemistry data reveal an overall agreement but show that different exchange functionals should be used for different types of bonds. The van der Waals vdW interaction plays an important role in various processes in biochemistry, in surface science, and in physics of layered materials. Although the basic underlying physics of dispersion forces is understood, an accurate and efficient calculation of the vdW interaction is still challenging. The post-Hartree-Fock approaches, based on perturbation theory, are able to treat the vdW interaction. For instance, the coupled-cluster methods 1 especially coupled-cluster with single and double and perturbative triple excitations, CCSDT are very accurate and often used in benchmark calculations. Unfortunately, the high precision comes with enormous computational expenses, making the method fully applicable only for small complexes with less than a few dozen of atoms. The second-order Moller-Plesset perturbation theory MP2, although less expensive and less accurate, also contains the essential physics for describing the vdW interaction. Both of the two methods are not straightforward to apply to extended systems and the existing approaches to periodic systems, as those in Refs. 2 and 3, are fairly expensive and not in common use. Another class of methods for electronic-structure calculations is based on the density-functional theory DFT. The standard approximations to the exchange-correlation energy in DFT, such as the local-density approximation LDA and the generalized-gradient approximation GGA, are relatively cheap and easily applicable also to periodic systems. Unfortunately, these methods cannot describe the dispersion interaction properly. The problem has been addressed by Langreth, Lundqvist and co-workers, 4‐6 who constructed a nonlocal correlation functional aiming to preserve the essential mechanisms behind the vdW interaction. This approach has been successfully utilized for selected test systems and applications. 7‐10 However, most of these results are obtained either using an a posteriori energy correction scheme or a self-consistent calculation with a restricted capability of relaxing the structures. The reason for this is the computational complexity of the self-consistent approach of Ref. 11, which was employed in the earlier self-consistent calculations. In this Rapid Communication, we demonstrate our alternative highly efficient real-space approach. We derive the functional derivative of the vdW density functional vdWDF and obtain a representation, which enables a simpler evaluation of the nonlocal correlation potential and forces for self-consistent calculations with full atomic relaxation. Our implementation employs numerical quadratures on adaptive grids, which vary depending on the local properties of the electron density and are optimized to calculate the vdW-DF. We perform calculations on the widely used molecular benchmark set S22. 12 The results for its molecular complexes provide an extensive test for the reliability of the vdW-DF. Simple examples such as noble gas dimers are not suitable for these purposes as they do not provide a complete picture with respect to the variation in the nature of the bonding. Extended systems are either not discussed as a systematic discussion would require a versatile test set, which does not exist.