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Progress in Chemistry 2012, Vol. 24 Issue (06): 910-927 Previous Articles   Next Articles

• Special Issue of Quantum Chemistry •

The Band Gap Problem: the State of the Art of First-Principles Electronic Band Structure Theory

Jiang Hong   

  1. Beijing National Laboratory of Molecular Sciences, State Key Laboratory of Rare Earth Materials Chemistry and Applications, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
  • Received: Revised: Online: Published:
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Electronic band structure is one of the most fundamental properties of a material that plays a crucial role in many important applications, and its accurate description has been a long-standing challenge for the first-principles electronic structure theory. Kohn-Sham Density-functional theory (KS-DFT) within local density or generalized-gradient approximations (LDA/GGA), currently the “standard model” for first-principles computational materials science, suffers from the well-known band gap problem. A lot of efforts have been invested to improve the description of band gaps within the framework of Kohn-Sham DFT or its generalized formalisms. On the other hand, many-body perturbation theory (MBPT) based on Green's function G (GF) provides a different and conceptually more rigorous framework for electronic band structure. The central ingredient of the GF-based MBPT is the exchange-correlation self-energy ∑xc, which can be formally obtained by solving a set of complicated integro-differential equations, named Hedin's equations. The GW approximation, in which ∑xc is simply a product of G and the screened Coulomb interaction (W), is currently the most accurate first-principles approach to describe electronic band properties of extended systems. Compared to LDA/GGA, the computational efforts required for GW calculations are much heavier, so that its applications have been limited to relatively small systems. In this work, we review the basic principles, latest developments, and remaining challenges of first-principles electronic band structure theory from both DFT and GF-based MBPT perspectives. It is hoped that new ideas on further developments can be obtained by setting up the connection between the two different theoretical frameworks. Contents
1 Introduction:Electronic band structures and the band gap problem
1.1 Experimental measurements of electronic band structures
1.2 Theoretical treatments of electronic band structures and the band gap problem
2 Electronic band structures frommean-field approaches
2.1 Hartree theory
2.2 Hartree-Fock theory
3 Electronic band structures from density functional theory
3.1 Density functional theory and Kohn-Sham equations
3.2 The band gap problem and its origin
3.3 The optimized effective potential and related methods
3.4 Generalized Kohn-Sham methods
4 Electronic band structures from Green's function based many-body perturbation theory
4.1 Green's function
4.2 Self-energy and quasi-particle equations
4.3 Hedin's equations and GW approximation
4.4 The G0W0 approach and self-consistency
5 Concluding remarks
Key words: electronic band structure, the band gap problem, density-functional theory, generalized Kohn-Sham, Green&rsquo, s function, many-body perturbation theory, GW approximation

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