We describe an implementation of Hedin's GW approximation for molecules and clusters, the complexity of which scales as O(N 3) with the number of atoms. Our method is guided by two strategies: (i) to respect the locality of the underlying electronic interactions and (ii) to avoid the singularities of Green's functions by manipulating, instead, their spectral functions using fast Fourier transform methods. To take into account the locality of the electronic interactions, we use a local basis of atomic orbitals and, also, a local basis in the space of their products. We further compress the screened Coulomb interaction into a space of lower dimensions for speed and to reduce memory requirements. The improved scaling of our method with respect to most of the published methodologies should facilitate GW calculations for large systems. Our implementation is intended as a step forward towards the goal of predicting, prior to their synthesis, the ionization energies and electron affinities of the large molecules that serve as constituents of organic semiconductors.
We thank Olivier Coulaud for useful advice on computing, and Isabelle Baraille and Ross Brown for discussions on chemistry, both in the context of the ANR project “NOSSI.” James Talman has kindly provided essential computer algorithms and codes, and we thank him, furthermore, for inspiring discussions and for correspondence. We are indebted to the organizers of the ETSF2010 meeting at Berlin for feedback and perspective on the ideas of this paper. Arno Schindlmayr, Xavier Blase, and Michael Rohlfing helped with extensive correspondence on various aspects of the GW method. D.S.P. and P.K. acknowledge financial support from the Consejo Superior de Investigaciones Científicas (CSIC), the Basque Departamento de Educación, UPV/EHU (Grant No. IT-366-07), the Spanish Ministerio de Ciencia e Innovación (Grant No. FIS2010-19609-C02-02), and the ETORTEK program funded by the Basque Departamento de Industria and the Diputación Foral de Guipuzcoa.
I. INTRODUCTION II. ELEMENTARY ASPECTS OF HEDIN'S GW APPROXIMATION III. TENSOR FORM OF HEDIN'S EQUATIONS IV. THE INSTANTANEOUS PART OF THE SELF-ENERGY V. USING SPECTRAL FUNCTIONS TO COMPUTE THE SELF-ENERGY A. The spectral function of a product of two correlators B. The second window technique VI. TESTING OUR IMPLEMENTATION OF GW ON A SMALL MOLECULE VII. COMPRESSION OF THE COULOMB INTERACTION A. Defining a subspace within the space of products B. Construction of the screened interaction from the action of the response function in the subspace C. The compression in the case of benzene VIII. MAINTAINING O(N 3) COMPLEXITY SCALING BY COMPRESSING/DECOMPRESSING A. A construction of the subspace response in O(N 3) operations B. A construction of the self-energy in O(N 3) operations IX. A SUMMARY OF THE COMPLETE ALGORITHM X. TESTS FOR DIFFERENT BASIS SET SIZES XI. TESTS FOR MOLECULES OF INTERMEDIATE SIZE XII. CONCLUSIONS AND OUTLOOK